Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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CONTENTS 26 CHAPTER 1 SPACE TIME AND NEWTONIAN PHYSICS PHYlOl PHYle or PHY215 So this discussion is important for everyone I suggest that you take the opportunity to re ect on this at a deeper level than you may have done before Newtom39zm physics is the stuff embodied in the work of Isaac Newton Now there were a lot of developments in the 200 years between Newton and Einstein but an important conceptual framework remained unchanged It is this frame work that we will refer to as Newtonian Physics and in this sense the term can be applied to all physics up until the development of Relativity by Einstein Reviewing this framework will also give us an opportunity to discuss how people came to believe in such a strange thing as the constancy of the speed of light and why you should believe it too note so far I have given you no reason to believe such an obviously ridiculous statement By the way you can basically think of Newtonian Physics as a precise formu lation of your intuitive understanding of physics based on your life experiences Granted as those of you who have taken PHYlOlle or 215 know there are many subtleties But still the basic rules of Newtonian physics ought to make sense7 in the sense of meshing with your intuition Hard question What does it mean for something to make sense 11 Coordinate Systems We7re going to be concerned with things like speed speed of light distance and time As a result coordinate systems will be very important How many of you have worked with coordinate systems Reminder A coord system is a way of labeling points Say on a line You need A zero A positive direction A scale of distance l l l Xl m X0 Xl m We7re going to stick with onedimensional motion most of the time Of course space is 3dimensional but 1 dimension is easier to draw and captures some of the most important properties In this course we7re interested in space and time 2For our purposes the most important part of this work was actually done by Galileo However we will also make use of the re nements added by Newton and the phrase Galilean Physics as re ned by Newton7 is just to long to use 11 COORDINATE SYSTEMS 27 15 t t 0 tls quotU ut these together to get a spacetime diagram7 t 1 5 t to r I X1m X0 X1m Note For this class as opposed to PHYZllPHYZlS7 t increases upward and 1 increases to the ght This is the standard convention in relativity7 Which we choose to adopt so that this course is compatible With all books Also note 0 The xaxis is the line t 0i 0 The t axis is the line t 0i Think of them this way It Will make your life easier lateri 12 REFERENCE FRAMES 29 The worldline of a car door t E LThe door 5 arms an event 0 L m 39T H u X1m X0 X1m That de nition was a simple thing now let s think deeply about it Given an event say the opening of a door how 0 we now w ere to draw it on a spacetime diagram say in your reference frame Suppose it happens in our 1D world 0 How can we nd out what time it happens One way is to give someone a clock and somehow arrange for them be present at the event They can tell you at what time it happened 0 How can we nd out where at what position it happens We could hold out a meter stick or imagine holding one out Our friend at the event in question can then read off how far away she is Note that what we have done here is to really de ne what we mean by the position and time of an event This type of de nition where we de ne something by telling how to measure it or by stating what a thing does is called an operational de nition They are very common in physics Food for thought Are there other kinds of precise de nitions How do they compare Now the speed of light thing is really weird So we want to be very careful in our thinking You see something is going to go terribly wrong and we want to be able to see exactly where it is Lets take a moment to think deeply about this and to act like mathematiciansi When mathematicians de ne a quantity they always stop and ask two questions 32 CHAPTER 1 SPACE TIME AND NEWTONIAN PHYSICS Yes they will The point is that clock 1 might actually pass through a as well as shown belowi Certainly the readings of and 7 should not depend on whether or not this happens5i So let us assume that it does If now 6 and 7 do not agree at A then a must disagree with at least one of these clocksi Suppose that it is clock 6 But assumption T states that clocks a and 6 must measure the same time interval between events B and Al Now they were synchronized at A and so read the same time there Their reading at A is just their reading at B plus the time interval so the readings at A must be the same as well Finally a proof We are beginning to make progress Since the time of any event is well de ned the difference between the times of any two events is well de ned Thus the statement that two events are at the same time in a given reference frame7 is wellde ned Erwin But might two events be at the same time in some reference frame but not in others Well here we go again Second Corollary to T Any two sets of synchronized ideal clocks measure the same time interval between any two events Proof Note from the rst corollary that the time of an event de ned with respect to a synchronized set of clocks is wellde ned no matter how many 5Such a statement follows from the idea that the reading on an ideal clock is independent of all external in uences 14 NEWTONIAN ADDITION OF VELOCITIES 33 clocks are in that synchronized set Thus we are free to add more clocks to a synchronized set as we like This will not change the times measured by that synchronized set in any way but will help us to construct our proof So consider any two events E1 and E2 Let us pick two clocks BX and 7X from set X that pass through these two events Let us now pick two clocks y and W from set Y that follow the same worldlines as BX and 7X If such clocks are not already in set Y then we can add them in Now BX and y were synchronized with some original clock ax from set X at some events B and Cl Let us also consider some clock ay from set Y having the same worldline as ax We have the following spacetime diagram Note that by assumption T clocks ax and ay measure the same time interval between B and Cl Thus sets X and Y measure the same time interval between B and Cl Similarly sets X and Y measure the same time intervals between B and E1 and between C and E2 Let TX A B be the time difference between any two events A and B as determined by set X and similarly for TyABi Now since we have both TXE1E2 TXE1B TXBC TX C E2 and TyE1 E2 TyE1 B TyB C TyC E2 and since we have just said that all of the entries on the right hand side are the same for both X and Y it follows that TX E1 E2 Ty E1E2i QED Note that it now also follows that the notions of time and position given above are wellde ned 14 Newtonian addition of velocities Lets go back and look at this speed of light business Remember the 99c example Why was it confusing Let VBA be the velocity of B as measured by A What relationship would you guess between VBA VCR and VGA Most likely your guess was VOA VCB VBA7 1 1 and this was the reason that the 99c example didn t make sense to you But do you know that this is the correct relationship Or why should you believe this 34 CHAPTER 1 SPACE TIME AND NEWTONIAN PHYSICS The answer still leaving the speed of light example clear as coal tar is because 11 follows from assumptions S and Ill Proof Let AB C be clocksi For simplicity suppose that all velocities are constant and that all three clocks pass through some one event and that they are synchronized therei VBA t VCB t timet V At C B WLOG take t0 here At time t the separation between A and C is VCAt but we see from the diagram that it is is also VCBt VBAti Canceling the t7s we have VOA VCB VBAA 1 2 QED Now our instructions about how to draw the diagram from the facts that our ideas about time and position are wellde ned came from assumptions T and S so the Newtonian formula for the addition of velocities is a logical consequence of T and S If this formula does not hold then at least one of T and S must be false Of course I have still not given you any real evidence to doubt 1 1 7 l have only heavily foreshadowed that this will come It is a good idea to start thinking now based on the observations we have just made about how completely any such evidence will make us restructure our notions of reality 0 Q Where have we used T o A In considering events at the same time 0 Q Where have we used S o A We used the fact that dBc is same as measured by anyone AB or C CHAPTER 1 SPACE TIME AND NEWTONIAN PHYSICS 44 CHAPTER 2 MAXWELL EampM AND THE ETHER As they grew to understand more and more physicists found it useful to describe these phenomena not in terms of the forces themselves but in terms of things called elds Here s the basic idea Instead of just saying that X and Y repell or that there is a force between them we break this down into steps 0 We say that X lls the space around it with an electric eld E7 0 Then it is this electric eld E that produces a force on Y Electn39c force on Y charge of YElectric eld at location of Y Fon Y QYE Note that changing the sign i of the charge changes the sign of the force The result is that a positive charge experiences a force in the direction of the eld while the force on a negative charge is opposite to the direction of the eld A T The arrows indicate the field V Red pos1t1ve charge moves left with the field Blue negative charge moves right against the field blue negative charge red positive charge Similarly a magnetic charge lls the space around it with a magnetic eld B that then exerts a force on other magnetic charges H Now7 you may think that elds have only made things more com plicated7 but in fact they are a very important concept as they al lowed people to describe phenomena which are not directly related to charges and forces For example the major discovery behind the creation of electric generators was Faraday s Law This Law says that a magnetic eld that changes in time pro duces an electric eld In a generator rotating a magnet causes the magnetic eld to be continually changing generating an electric eld The electric eld then pulls electrons and makes an electric current Conversely Maxwell discovered that an electn39c eld which changes in time produces a magnetic eld Maxwell codi ed both this observation and Faraday s law in a set of equations known as well Maxwellls equations Note the closed loop this makes If we make the electric eld change with time in the right way it produces a magnetic eld which changes with time This magnetic eld then produces an electric eld which changes with time which produces a magnetic eld which changes with time and so on This phenomenon is called an 50 CHAPTER 2 MAXWELL EampM AND THE ETHER a Light Ray hits side instead of reaching bottom Telescope moves through ether Must tilt telescope to see star This phenomenon had been measured using the fact that the earth rst moves in one direction around the sun and then six months later it moves in the opposite direction In fact someone else Fizeau had also measured the effect using telescopes lled with water The light moves more slowly through water than through the air so this should change the angle of aberration in a predictable way While the details of the results were quite confusing the fact that the e ect occurred at all seemed to verify that the earth did move through the ether and moreover that the earth did not drag very much of the ether along with it 222 Michelson Morely and their experiment Because of the confusion surrounding the details of Fizeau s results it seemed that the matter deserved further investigation Michelson and Morely thought that they might get a handle on things by measuring the velocity of the ether with respect to the earth in a different way Michelson and Morely used a device called an interferometer which looks like the picture belowi The idea is that they would shine light an electromagnetic wave down each arm of the interferometer where it would bounce off a mirror at the end and return The two beams are then recombined and viewed by the experimenters Both arms are the same length say L 22 THE ELUSIVE ETHER 51 T MiIror l l ra s are viewed l y Mirror a Aggie Light rays bounce off both mirrors What do the experimenters see Well if the earth was at rest in the ether the light would take the same amount of time to travel down each arm and return ow when the two beams left they were synchronized in phase meaning that wave crests and wave troughs start down each arm at the same timeF i Since each beam takes the same time to travel this means that wave crests emerge at the same time from each arm and similarly with wave troughsi Waves add together as shown below7 with two crests combining to make a big crest and two troughs combining to make a big troughi The result is therefore a a bright beam of light emerging from the device This is what the experimenters should see On the other hand if the earth is moving through the ether say to the right then the right mirror runs away from the light beam and it takes the light longer to go down the right arm than down the top armi On the way back though the time less to travel this arm because of the opposite effect A detailed calculation is required to see which effect is greater and to properly take into account that the top beam actually moves at an angle as shown below 6Michelson and Morely achieved this synchronization by just taking one light beam and splitting it into two pieces 7My apologies for the sharp corners but triangular waves are a lot easier to draw on a computer than sine waves 52 CHAPTER 2 MAXWELL EampM AND THE ETHER This one takes less time After doing this calculation one nds8 that the light beam in the right arm comes back faster than light beam in the top arm The two signals would no longer be in phase and the light would not be so bright In fact if the difference were great enough that a crest came back in one arm when a trough came back in the other then the waves would cancel out completely and they would see nothing at alll Michelson and Morely planned to use this effect to measure the speed of the earth with respect to the ether m m w asumdm However they saw no e ect whatsoever No matter which direction they pointed their device the light seemed to take the same time to travel down each arm Clearly they thought the earth just happens to be moving with the ether right now ilel bad timing So they waited six months until the earth was moving around the sun in the opposite direction expecting a relative velocity between earth and ether equal to twice the speed of the earth around the sun However they still found that the light took the same amount of time to travel down both arms of the interferometerl So what did they conclude They thought that maybe the ether is dragged along by the earth But then how would we explain the aberration effects Deeply confused Michelson and Morely decided to gather more data Despite the aberration effects they thought the earth must drag some ether along with it After all as we mentioned the details of the aberration experiments were a little weird so maybe the conclusion that the earth did not drag the ether was not really justi ednm If the earth did drag the ether along they thought there might be less of this effect up high like on a mountain top So they repeated their experiment at the top of a mountain Still they found no effect There then followed a long search trying to nd the ether but no luck Some people were still trying to nd an ether dragged along very ef ciently by everything7 in the 1920 s and 1930s They never had any luck 8See homework problem 272 CHAPTER 2 MAXWELL EampM AND THE ETHER 32 TIME AND POSITION TAKE II 57 To de ne position Build a framework of measuring rods and make sure that the zero mark always stays with the object that de nes the inertial reference frame Note that once we set it up this framework will move with the inertial observer without us having to apply any forces The measuring rods will move with the reference frame An observer at an event can read off the position in this reference frame of the event from the mark on the rod that passes through that event To de ne time Put an ideal clock at each mark on the framework of measuring rods above Keep the clocks there moving with the reference frame The clocks can be synchronized with a pulse of light emitted for example from t 0 A clock at z knows that when it receives the pulse it should read lIlC These notions are manifestly wellde ned We do not need to make the same kind of checks as before as to whether replacing one clock with another would lead to the same time measurements This is because the rules just given do not in fact allow us to use any other clocks but only the particular set of clocks which are bolted to our framework of measuring rods Whether other clocks yield the same values is still an interesting question but not one that affects whether the above notions of time and position of some event in a given reference frame are well de ne Signi cantly we use a different method here to synchronize clocks than we did in chapter 1 The new method based on a pulse of light is available now that we have assumption H which guarantees that it is an accurate way to synchronize clocks in an inertial frame This synchronization process is shown in the spacetime diagram below t1mc Note that the diagram is really hard to read if we use meters and seconds as units 58 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES t1 sec t 1 sec X1m X0 X1m Therefore it is convenient to use units of seconds and lightseconds 1L3 l secc 31 X 108m 3 X 10516721 This is the distance that light can travel in one second7 roughly 7 times around the equator Working in such units is often called choosing c 177 since light travels at lLsseci We will make this choice for the rest of the course7 so that light rays will always appear on our diagrams as lines at a 45 angle with respect to the vertical iiei slope ll t1 sec t lsec XlLS X0 XlLS 33 Simultaneity Our rst departure from Galileo and Newton The above rules allow us to construct spacetime diagrams in various reference frames An interesting question then becomes just how these diagrams are related Let us start with an important example Back in chapter 17 we went to some trouble to show that the notion that two events happen at the same time7 does not depend on which synchronized set of clocks we used to measure these times Now that we have thrown our T and S7 will this statement still be true Let us try to nd an operational de nition of whether two events occur simulta neously7 iiei7 at the same time in some reference frame We can of course read the clocks of our friends who are at those events and who are in our reference frame However7 we can also construct an operational de nition directly from postulate H about the speed of light Note that there is no problem in deter mining whether or not two things happen like a door closing and a recracker 33 SIM ULTANEITY 59 going off at the same event The question is merely whether two things that occur at different events take place simultaneouslyi Suppose that we are in an inertial frame and that we emit a ash of light from our worldlinei The light will travel outward both to the left and the right always moving at speed cl Suppose that some of this light is re ected back to us from event A on the left and from event B on the right The diagram below makes it clear that the two re ected pulses of light reach us at the same event if and only if A and B are simultaneous So if the re ected pulses do reach us at the same event we know that A and B are in fact simultaneous in our frame of reference X 0 Signals return at the same time Km 0 if and only if the events are simultaneous in my reference frame tus 1 sec t us0 Us Us But now suppose that some friend of ours is moving by a different inertial framei By our postulates this observer moves at constant speed with respect to us and in a straight line Let us assume that as shown below this observer passes through the event from which we emitted the pulses of light By postulate H these pulses of light also somehow both recede from our friend at speed cl In other words our friend might as well have emitted these pulses herself and can use them to determine which events are simultaneous in her framei Note however that pulses of light re ecting from A and B do not meet her worldline at the same eventll Thus events A and B cannot be simultaneous in our friends reference frameli 1Oh yes this is where the fun part begins H V Please bear with me CHAPTER 3 EINSTEIN AND INERTIAL FRAMES another inerital obs erver Signals do not return to them at the same time t usl sec t 70 OK7 so A and B are not simultaneous in our friends reference frame But7 which events are simultaneous in the second reference frame We can again use the light pulses to nd out All we need to do is to nd some event call it C on the left outwardmoving light ray such that pulses re ected from B and C intersect our friends worldline at the same event This is shown in the gure belowi The line connecting event7s B and C7 which are simultaneous in our friends reference frame7 is known as her line of simultaneityi7 It will be a line of constant time according to the moving reference frame friend s line of C simultaneity X t 1 X us 70 7 0 XI t 1 const A t const B tus 1 sec f C B t quoti0 tf 0 Us In the above figure7 the diagram on the left shows the situation as described in our reference frame while the diagram on the right is drawn in our friends frame of reference We have used postulate l to draw the second diagram from the rst Postulate l tells us that if two light rays intersect at some event say7 A in one inertial frame of reference7 that they must also intersect at A in any friend 5 line of s imultaneity 33 SIM ULTANEITY 61 other inertial framei Thus we need only mark events B and C which we know to be simultaneous in our friends frame on her diagram and draw in the light rays running as always along lines at 45 angles to the vertical in order to complete the gure Note that on either diagram the worldline If 0 makes the same angle with the light cone as the line of simultaneity tf consti That is the angles a and 6 above are equal You will in fact derive this in one of your homework problems M By the way we also can nd other pairs of events on our diagram that are simultaneous in our friends reference frame We do this by sending out signals from another observer in the moving frame say located 1m to the right of our friend For example the diagram below shows another event D that is simultaneous with C in our friends frame of reference X150 C Us In this way we can map out our friends entire line of simultaneity 7 the set of all events that are simultaneous with each other in her reference frame The result is that the line of simultaneity for the moving frame does indeed appear as a straight line on our spacetime diagrami This property will be very important in what is to come Before moving on let us get just a bit more practice and ask what set of events our friend the moving observer nds to be simultaneous with the origin the event where the her worldline crosses ours We can use light signals to nd this lines as well Let7s label that line tf 0 under the assumption that our friend chooses to set her watch to zero at the event where the worldlines crossi Drawing in a carefully chosen light box we arrive at the diagram belowi 62 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES X f 0 xu 0 Us 7 t I const A B t f 0 47 tus 1 sec friend s liny simultaneity a t us0 Note that we could also have used the rule noticed above that the worldline and any line of simultaneity make equal angles with the light cone As a nal comment note that while we know that the line of simultaneity drawn above tf const represents some constant time in the moving frame we do not yet know which time that is In particular we do not yet know whether it represents a time greater than one second or a time less than on second We were able to label the tf 0 line with an actual value only because we explicitly assumed that our friend would measure time from the event on that line where our worldlines crossedi We will explore the question of how to assign actual time values to other lines of 39 s ort yr Summary We have learned that events that are simultaneous in one inertial reference frame are not in fact simultaneous in a different inertial framei We used light signals to determine which events were simultaneous in which frame of reference 34 Relations between events in Spacetime It will take some time to absorb the implications of the last section but let us begin with an interesting observationi Looking back at the diagrams above note that a pair of events which is separated by pure space77 in one inertial frame ie is simultaneous in that frame is separated by both space and time in another Similarly a pair of events that is separated by pure time77 in one frame occurring at the same location in that frame is separated by both space and time in any other frame This may remind you a bit of our discussion of electric and magnetic elds where a eld that was purely magnetic in one frame involved both electric and magnetic parts in another frame In that case we decided that is was best to combine the two and to speak simply of a single 64 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES friend s line of s imultaneity Thus7 it is worthwhile to distinguish three classes of relationships that pairs of events can have These classes and some of their properties are described belowi Note that in describing these properties we limit ourselves to inertial reference frames that have a relative speed less than that of light case 1 A and B are outside each other s light conesi simultaneous frame frame in which A happens first In this case7 we say that they are spacelz39ke relatedr Note that the following things are true in this case a There is an inertial frame in which A and B are simultaneousi b There is are also frames in which event A happens rst and frames in which event B happens rst even more tilted than the simultaneous frame shown above However7 A and B remain outside of each others light cones in all inertial frames case 2 A and B are inside each others light cones in all inertial framesi 2We will justify this in a moment 34 RELATIONS BETWEEN EVENTS IN SPACETIME case 3 simultaneous frame frame in which A happens first In this case we say that they are timelike related Note that the fol lowing things are true in this case a There is an inertial observer who moves through both events and whose speed in the original frame is less than that of light b All inertial observers agree on which event A or B happened rstr c As a result we can meaningfully speak of say event A being to the past of event Br A and B are on each others light conesr In this case we say that they are lightlz39ke related Again all inertial observers agree on which event happened rst and we can meaningfully speak of one of them being to the past of the other Now why did we consider only inertial frames with relative speeds less than 6 Suppose for the moment that our busy friend the inertial observer could in fact travel at v gt 6 ie faster than light as shown below at left I have marked two events A and B that occur on her worldliner In our frame event A occurs rst However the two events are spacelike related Thus there is another inertial frame in which E occurs before A as shown below at right This means that there is some inertial observer the one whose frame is drawn at right who would see her traveling backwards in time 66 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES X 0 f X f 0 Xus 0 t f const g A tfconst friend s line of RB tf 0 simultaneity Worl ine moves faster than light This was too weird even for Einsteinl After all if she could turn around our fasterthan light friend could even carry a message from some observerls future into that observerls past This raises all of the famous what if you killed your grandparents7 scenarios from science ction fame The point is that in relativity travel faster than light is travel backwards in time For this reason let us simply ignore the possibility of such observers for awhile In fact we will assume that no information of any kind can be transmitted faster than c I promise that we will come back to this issue this later The proper place to deal with this turns out to be in chapter 5 X s t const 5 t0 Me 35 Time Dilation We are beginning to come to terms with simultaneity but as pointed out early we are still missing important information about how different inertial frames match up In particular we still do not know just what value of constant tf the 35 TIME DILATION 67 line marked friendls line of simultaneity77 below actually represents 0 tuslsec friend s line of C t E0 1quot 5 Us 5 imultaneity In other words we do not yet understand the rate at which some observers clock ticks in another observerls reference frame 351 Rods in the perpendicular direction To address this question we re going to introduce a second space dimension allowing observers to hold out meter sticks perpendicular to the direction that they move This however introduces a new question Suppose that you and l are both inertial observers but that we are moving toward each other We each hold out meter sticks to one side so that both sticks point in the direction perpendicular to the direction of motiongi My meter stick is by de nition one meter long in my frame of reference Your meter stick is by de nition one meter long in your frame of reference Now since time and space are going to have to get all screwed up we no longer have T and S do we have any idea how long your meter stick is in my frame of reference or how long mine is in your frame To gure this our let us rst note that the direction perpendicular to our relative motion is simpler than the direction of our relative motion itself For example two inertial observers actually do agree on which events are simultaneous in that direction To see this suppose that I am moving straight toward you from the front at some constant speed Suppose that you have two recrackers one placed one light second to your left and one placed one light second to your right Suppose that both explode at the same time in your frame of reference Does one of them explode earlier in mine 3Hmmmm do both observers agree on what is perpendicular to the direction of motion7 It will turn out that they do However it also turns out not to matter at this stage So really I could just say you hold your stick in the direction that is perpendicular to the direction of motion in your frame and I will do the same with my stick in my frame7 35 TIME DILATION 69 352 Light Clocks and Reference Frames The property just derived makes it convenient to use such meter sticks to build clocksi Recall that we have given up most of our beliefs about physics for the moment so that in particular we need to think about how to build a reliable clock The one thing that we have chosen to build our new framework upon is the constancy of the speed of light Therefore it makes sense to use light to build our clocksi We will do this by sending light signals out to the end of our meter stick and back For convenience let us assume that the meter stick is one lightsecond long This means that it will take the light one second to travel out to the end of the stick and then one second to come back A simple model of such a light clock would be a device in which we put mirrors on each end of the meter stick and let a short pulse of light bounce back and forth Each time the light returns to the rst mirror the clock goes tick7 and two seconds have passe Now suppose we look at our light clock from the side Lets say that the rod in the clock is oriented in the vertical direction The path taken by the light looks like this However what if we look at a light clock carried by our inertial friend who is moving by as at speed 1 and that the rod in her clock is also oriented verticallyi The relative motion is in the horizontal direction Since the light goes straight up and down in her reference frame the light pulse moves up and forward and then down and forward in our reference frame In other words it follows the path shown belowi This should be clear from thinking about the path you see a basketball follow if someone lifts the basketball above their head while they are walking past your ctuS 9 VI us The length of each side of the triangle is marked on the diagram abovei Here L is the length of her rod and tus is the time as measured by us that it takes 70 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES the light to move from one end of the stick to the other To compute two of the lengths we have used the fact that in our reference frame the light moves at speed c while our friend moves at spee The interesting question of course is just how long is this time tus We know that the light takes 1 second to travel between the tips of the rod as measured in our friends reference frame but what about in ours It turns out that we can calculate the answer by considering the length of the path traced out by the light pulse the hypotenuse of the triangle abovei Using the Pythagorean theorem the distance that we measure the light to travel is vtus2 L However we know that it covers this distance in a time tus at speed cl Therefore we have CQtis v2tis L2 31 we ti 7lt was lt17 Monti 32gt 7 L Thus we measure a time tus 7 CW between when the llght leaves one mirror and when it hits the next This is in contrast to the time thend Lc lsecond measured by our friend between these same to events We can therefore conclude that measure is related to the time tfreind measured by our friend by through tfreind 3 3 t Between any two events where our friends clock ticks the time tus that we Finally we have learned how to label another line on our diagram above Xf0 1 tf llVZCZ Xu570 tT1 sec T 1v2c2 N 74 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES Event A xp 0 Xpend L t X0 V ts Lv ll Now the student must nd that the professor takes a time Lv to traverse the length of the students measuring rodi Let us refer to the event marked in magenta where the moving professor arrives at the right end of the students measuring rod as event A77 Since this event has its Lv we can use our knowledge of time dilation 33 to conclude that the professor assigns a time tp Lv 171162 to this event Our goal is to determine the length of the students measuring rod in the pro fessor7s frame of reference That is we wish to know what position zpend the professor assigns to the rightmost end of the students rod when this end crosses the professors line of simultaneity tp i To nd this out note that from the professor s perspective it is the students rod that moves past him at speed vi Recall that we determined above that the professor nds that it takes the rod a time tp Lv 1 7 1162 to pass by Thus the students rod must have a length LP L1 7 1162 in the professor s frame of reference The professor7s rod of course will similarly be shortened in the student s frame of reference So we see that distance measurements also depend on the observers frame of reference Note however that given any inertial object there is a special inertial frame in which the object is at rest The length of an object in its own rest frame is known as its proper length The length of the object in any other inertial frame will be shorter than the objects proper length We can summarize what we have learned by stating An object of proper length L moving through an inertial frame at speed 1 has length LxT 7 0262 as measured in that inertial framei There is an important subtlety that we should explore Note that the above statement refers to an object However we can also talk about proper distance between two events When two events are spacelike related there is a special 76 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES t ground 20 t ground 10 t ground 0 T T r r ia ia Tunnel Tunnel E tr Exit n n x n 0 x 100m B F ground ground a 1 C O k 11 t X 0 l 00 KT T Suppose that one robber sits at the entrance to the tunnel and that one sits at the exit When the train nears they can blow up the entrance just after event 1 and they can blow up the exit just before event 2 Note that in between these two events the robbers nd the train to be completely inside the tunnel Now what does the train think about all this How are these events described in its frame of reference Note that the train nds event 2 to occur long before event ll So can the train escape Letls think about what the train would need to do to escape At event 2 the exit to the tunnel is blocked and from the trains perspective the debris blocking the exit is rushing toward the train at half the speed of light The only way the train could escape would be to turn around and back out of the tunnel Recall that the train nds that the entrance is still open at the time of event 2 Of course both the front and back of the train must turn around How does the back of the train know that it should do this It could nd out via a phone call from an engineer at the front to an engineer at the back of the train or it could be via a shock wave that travels through the metal of the train as the front of the train throws on its brakes and reverses its engines The point is though that some signal must pass from event 2 to the back of the train possibly relayed along the way by something at the front of the train Sticking to our assumption that signals can only be sent at speed c or slower the earliest possible time that the back of the train could discover the exit explosion is at the event marked D on the diagram Note that at event D the back of the train does nd itself inside the tunnel and also nds that event 1 has already occurred The entrance is closed and the train cannot escape 78 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES 3 8 Homework Problems 31 Use your knowledge of geometry andor trigonometry to show that the angle a between the worldline of an inertial observer and the lightcone drawn in an inertial frame using units in which light rays travel at 45 degrees is the same as the angle 6 between that observer7s line of si multaneity and the lightconei Hint Many people nd it easier to use trigonometry to solve this problem than to use geometryi X0 S t const green line of C t p0 simultaneity Suppose that you and your friend are inertial observers that is that both reference frames are inertiali Suppose that two events A and B are simultaneous in your own reference frame Draw two spacetime diagrams in your reference frame For the rst arrange the relative velocity of you and your friend so that event A occurs before event B in your friends reference frame For the second arrange it so that event B occurs rst in your friends frame In both cases include one of your friends lines of simultaneity on the diagrami Draw a spacetime diagram in an inertial reference frame a Mark any two spacelike related events on your diagram and label them both Si b Mark any two lightlike related events on your diagram and label them both Li c Mark any two timelike related events on your diagram and label them both Ti You and your friend are again inertial observers and this time your relative velocity is lei Also suppose that both of your watches read t 0 at the 2 CHAPTER 3 EINSTEIN AND INERTIAL FRAMES 41 MINKOWSKIAN GEOMETRY 89 41 Minkowskian Geometry Minkowski was a mathematician and he is usually credited with emphasizing the fact that time and space are part ofthe same spacetime whole in relativityi It was he who emphasized spacetime diagrams together with the fact that this spacetime has a special kind of geometry It is this geometry which is the underlying structure and the new logic of relativity Understanding this geometry will provide both insight and useful technical tools For this reason7 we now pursue what at rst sight will seem like a technical aside in which we rst recall how the familiar Euclidean geometry is relates quantities in different coordinate systems We can then build an analogous technology in which Minkowskian geometry relates different inertial frames 411 Invariants Distance vs the Interval Recall that a fundamental part of familiar Euclidean geometry is the Pythagore an theoremi One way to express this result is to say that distance2 A12 AyQ7 41 where distance is the distance between two points and AI Ay are respectively the differences between the z coordinates and between the y coordinates of these points Here the notation A12 means AI2 and not the change in 12 Note that this relation holds in either of the two coordinate systems drawn belowi y1 y2 AX 90 CHAPTER 4 MINKOWSKIAN GEOMETRY Now recall that l have made analogies between changing reference frames and rotationsi Note that when I perform a rotation distances do not change and ifl compare coordinate systems with one rotated relative to the other I nd A1 Ay A13 Ag Letls think about an analogous issue involving changing inertial framesi Consid er for example two inertial observersi Suppose that our friend ies by at speed v For simplicity let us both choose the event where our worldlines intersect to be t 0 Let us now consider the event on his worldline where his clock ticks tf Ti Note that our friend assigns this event position If 0 since she passes through it What coordinates do we assign Our knowledge of time dilation tells us that we assign a longer time t Tdl 7 1126 For position recall that at tus 0 our friend was at the same place that we are mus 0 Therefore after moving at a speed 1 for a time tus Tdl 7 0262 our friend is at zus T Udl 7 11262 Xs0 Xvt US US ts1 sec 1 tusi l 1v2c2 t 1 sec US t 0 US 0 us Now we7d like to examine a Pythagoreanlike relationi But of course we can t just mix I and t in an algebraic expression since they have different units But we have seen that z and ct mix well Thinking of the marked event where our friends clock ticks is 12 ct2 the same in both reference frames Clearly no since both of these terms are larger in our reference frame than in our friends mus gt 0 and ctus gt ctfl Just for fun let s calculate something similar but slightly different Let7s com pare I 7 ctf2 and 125 7 ctus 2 I know you don7t actually want to read through lines of algebra here so please stop reading and do this calculation yourself 92 CHAPTER 4 MINKOWSKIAN GEOMETRY spacelike separation Similarly if the events are spacelike separated there is an inertial frame in which the two are simultaneous 7 that is in which 0 The distance between two events measured in such a reference frame is called the proper distance d Much as above d A12 7 62At2 S Ari Note that this seems to go the opposite way77 from the length contraction effect we derived in section 36 That is because here we consider the proper distance between two particular events In contrast in measuring the length of an object different observers do NOT use the same pair of events to determine lengthi Do you remember our previous discussion of this issue lightlike separation Two events that are along the same light ray satisfy AI icAti It follows that they are separated by zero interval in all reference frames One can say that they are separated by both zero proper time and zero proper distancei 412 Curved lines and accelerated objects Thinking of things in terms of proper time and proper distance makes it easier to deal with say accelerated objectsi Suppose we want to compute for example the amount of time experienced by a clock that is not in an inertial framei Perhaps it quickly changes from one inertial frame to another shown in the blue worldline marked B belowi This blue worldline B is similar in nature to the worldline of the muon in part b of problem 3 x4 x0 x4 Note that the time experienced by the blue clock between events a and b is equal to the proper time between these events since on that segment it could be in an inertial framei Surely the time measured by an ideal clock between a and b cannot depend on what it was doing before a or on what it does after bl Similarly the time experienced by the blue clock between events b and c should be the same as that experienced by a truly inertial clock moving between these events ie the proper time between these events Thus we can nd the total proper time experienced by the clock by adding the proper time between 42 THE TWIN PARADOX 93 a and b to the proper time between b and We also refer to this as the total proper time along the clocks worldlinei A red observer R is also shown above moving between events 074 and 0 4 Let A751 and A751 be the proper time experienced by the red and the blue observer between times t 74 and t 4 that is between events a and d Note that A75472 gt A7 72 and similarly for the other time intervals Thus we see that the proper time along the broken line is less than the proper time along the straight line Since proper time ie the interval is analogous to distance in Euclidean ge ometry we also talk about the total proper time along a curved worldline in much the same way that we talk about the length of a curved line in space We obtain this total proper time much as we did for the blue worldline above by adding up the proper times associated with each short piece of the curve This is just the usual calculus trick in which we approximate a curved line by a sequence of lines made entirely from straight line segments One simply replaces any A1 or At denoting a difference between two points with dz or dt which denotes the difference between two in nitesimally close points The rationale here of course is that if you look at a small enough in nitesimal piece of a curve then that piece actually looks like a straight line segment Thus we have d7 dtxT 7 0262 lt dt or AT de fxl 7v2c2dt lt At 4 dt 0 dTdt1 vzc2 t4 x4 x0 x4 Again we see that a straight inertial line in spacetime has the longest proper time between two events In other words in Minkowskian geometry the longest line between two events is a straight line 42 The Twin paradox Thatls enough technical stuff for the moment We re now going to use the language and results from the previous section to discuss a relativity classic The twin paradoxi77 Using the notions of proper time and proper distance turns out to simplify the discussion signi cantly compared to what we would have had to go through before section 41 Lets think about two identical twins who for obscure historical reasons are named Alphonse and Gaston Alphonse is in an inertial reference frame oating in space somewhere near our solar systemi Gaston on the other hand will travel to the nearest star Alpha Centauri and back at i8ci Alpha Centauri is 94 CHAPTER 4 MINKOWSKIAN GEOMETRY more or less at rest relative to our solar system and is four light years away During the trip Alphonse nds Gaston to be aging slowly because he is traveling at i8ci On the other hand Gaston nds Alphonse to be aging slowly because relative to Gaston Alphonse is traveling at 8c During the trip out there is no blatant contradiction since we have seen that the twins will not agree on which event birthday on Gaston7s worldline they should compare with which event birthday on Alphonse s worldline in order to decide who is older But who is older when they meet again and Alphonse returns to earth 3G Alphonse s t i o n A1553 yr A 5 1 A1 3 yr Al ha Centauri x 4 The above diagram shows the trip in a spacetime diagram in Alphonse7s frame of reference Let7s work out the proper time experienced by each observer For Alphonse AI 0 How about At Well the amount of time that passes is long enough for Gaston to travel 8 lightyears there and back at i8ci That is At SlyTBc lOyTi So the proper time An experienced by Alphonse is ten years On the other hand we see that on the rst half of his trip Gaston travels 4 light years in 5 years according to Alphonse7s frame i hus aston experiences a proper time of 52 7 42 Byearsi The same occurs on the trip back So the total proper time experienced by Gaston is ATG 6yea39rsi ls Gaston really younger then when they get back together Couldnlt we draw the same picture in Gaston s frame of reference and reach the opposite conclu sion NO we cannot The reason is that Gaston7s frame of reference is not an inertial framel Gaston does not always move in a straight line at constant speed with respect to Alphonse In order to turn around and come back Gaston must experience some force which makes him noninertial Most importantly Gaston knows this When say his rocket engine res he will feel the force acting on him and he will know that he is no longer in an inertial reference frame M The point here is not that the process cannot be described in Gaston s frame of reference Gaston experiences what he experiences so there must be such a description The point is however that so for we have not worked out the rules to understand frames of reference that are not inertial Therefore we can not simply blindly apply the time dilationlength contraction rules for inertial frames to Gaston7s frame of reference Thus we should not expect our results so far to directly explain what is happening from Gaston7s point of view 42 THE TWIN PARADOX 95 But you might say Gaston is almost always in an inertial reference frame He is in one inertial frame on the trip out and he is in another inertial frame on the trip back What happens if we just put these two frames of reference together Lets do this but we must do it carefully since we are now treading new ground First we should draw in Gaston s lines of simultaneity on Alphonse s spacetime diagram above His lines of simultaneity will match2 one inertial frame for the trip out and a different frame for the trip back Then I will use those lines of simultaneity to help me draw a diagram in Gastonls notquite inertial frame of reference much as we have done in the past in going from one inertial frame to anot erl Since Gaston is in a different inertial reference frame on the way out as on the way back I will have to draw two sets of lines of simultaneity and each set will have a different slopel Now two lines with different slopes must intersect a G a s l o 7 n G7 6 yrs39 Alphonse E G7 3 yrs t 3 yrs G E t A 5 Alpha 1 Centauri A 0 xA 0 XA 1 Here I have marked several interesting events on the diagram and l have also labeled the lines of simultaneity with Gaston s proper time at the events where he crosses those lines Note that there are two lines of simultaneity marked tg Syearsll l have marked one of these 3 which is just before77 Gaston turns around and l have marked one 34r which is just after77 Gaston turns around lfl simply knit together Gastonls lines of simultaneity and copy the events from the diagram above I get the following diagram in Gastonls frame of reference By the way it is safe to use the standard length contraction result to nd that in the inertial frame of Gaston on his trip out and in the inertial frame of Gaston on his way back the distance between Alphonse and Alpha Centauri is 4Lyrd1 7 452 125Lyrl 2Strictly speaking we have de ned lines of simultaneity only for observers who remain inertial for all time However for an observer following a segment 0 an inertial worldline it is natural to introduce lines of simultaneity which match the lines of simultaneity in the corresponding fully inertial frame 96 CHAPTER 4 MINKOWSKIAN GEOMETRY x 125L x 0 G yr G D tG6yr C B G73yr A G70 There are a couple of weird things here For example what happened to event E In fact what happened to all of the events between B and C By the way how old is Alphonse at event B In Gaston7s frame of reference which is inertial before tg 3 so we can safely calculate things that are con ned to this region of time Alphonse has traveled 125Ly7x in three years So Alphonse must expe rience aproper time of 432 7 12502 x9 714425 48125 95yea39rsi Similarly Alphonse experiences 95yea39rs between events C and D This mean s that there are 10 7 185 325yea39rs of Alphonse7s life missing from the diagram Oooops It turns out that part of our problem is the sharp corner in Gaston7s worldlinei The corner means that Gaston s acceleration is in nite there since he changes velocity in zero time Let7s smooth it out a little and see what happens Suppose that Gaston still turns around quickly but not so quickly that we cannot see this process on the diagram If the turnaround is short this should not change any of our proper times very much proper time is a continuous function of the curvelll so Gaston will still experience roughly 6 years over the whole trip and roughly 3 years over half Lets say that he begins to slow down and therefore ceases to be inertial after 29 years so that after 3 years he is momentarily at rest with respect to Alphonse Then his acceleration begins to send him back home A tenth of a year later 31 years into the trip he reaches 8 c his rockets shut off and he coasts home as an inertial observer We have already worked out what is going on during the periods where Gaston is inertial But what about during the acceleration Note that at each instant Gaston is in fact at rest in some inertial frame 7 it is just that he keeps changing from one inertial frame to another One way to draw a spacetime diagram for Gaston is try to use at each time the inertial frame with respect to which he is at rest This means that we would use the inertial frames to draw in more of Gaston7s lines of simultaneity on Alphonse s diagram at which point we can again copy things to Gaston s diagram A line that is particularly easy to draw is Gaston7s tg 3yea39r line This is because at tg Byears Gaston is momentarily at rest relative to Alphonse This means that Gaston and Alphonse share a line of simultaneityl Of course they label it differently For Alphonse it it tA Byea39rsi For Gaston it is 42 THE TWIN PARADOX 97 tg 3years On that line Alphonse and Gaston have a common frame of reference and their measurements agree G a s t o 7 n G7 6yrs39 Alphonse D G7 31 yx C t 29 yrs G girlJ light nun G7 0 E t 3 yrs A 5 G B Alpha Centauri A 0 XA0 XA4 Note that we nally have a line of simultaneity for Gaston that passes through event Elll So event E really does belong on Gaston s tg 3year line after all By the way just for fun77 I have added to our diagram an light ray moving to the left from the origin We are almost ready to copy the events onto Gaston s diagram But to properly place event R we must gure out just where it is in Gaston s frame In other words how far away is it from Gaston along the line tg 3years Recall that along that particular line of simultaneity Gaston and Alphonse measure things in the same way Therefore they agree that along that line event E is four light years away from Alphonse Placing event E onto Gaston7s diagram connecting the dots to get Alphonse7s worldline we nd x 4 L yr xJ125Lyr xG0 D tG6yr E C r if G73yr nld IlLlIl My 39 A G70 There is something interesting about Alphonse7s worldline between B and E It is almost horizontal and has speed much greater than one lightyear per year What is happening Notice that l have also drawn in the gold light ray from above and that it too 98 CHAPTER 4 MINKOWSKIAN GEOMETRY moves at more than one lightyear per year in this frame We see that Alphonse is in fact moving more slowly than the light ray which is good However we also see that the speed of a light ray is not in general equal to c in an accelerated reference frame In fact it is not event constant since the gold light ray appears bent on Gaston s diagrami Thus it is only in inertial frames that light moves at 3 X 108 meters per second This is one reason to avoid drawing diagrams in noninertial frames whenever you can Actually though things are even worse than they may seem at rst glancewi Suppose for example that Alphonse has a friend Zelda who in an inertial observer at rest with respect to Alphonse but located four light years on the other side of Alpha Centauri We can then draw the following diagram in Alphonse7s frame of reference G Once again we simply can use Gaston7s lines of simultaneity to mark the events TUVWXYZ in Zeldals life on Gaston7s diagrami In doing so however we nd that some of Zeldals events appear on TWO of Gaston s lines of simultaneity 7 a magenta one from before the turnaround and a green one from after the turnaround In fact many of them like event W appear on three lines of simultaneity as they are caught by a third during the turnaround when Gaston7s line of simultaneity sweeps downward from the magenta t 29 to the green t 31 as indicated by the big blue arrowl Marking all of these events on Gaston s diagram taking the time to rst calcu late the corresponding positions yields something like this 42 THE TWINPARADOX 99 o z 3 Y 39X W XG 125 Lyr x 4L x 125L r x 0 x 7245L G yr J V G yr DIG6yr V U E C IIY V 7 7 B LG3y X A X tG0 V U 0 T The events T U V W at the very bottom and W X Y Z are not drawn to scale but they indicate that Zeldals worldline is reproduced in that region of the diagram in a more or less normal fashioni Let us quickly run though Gaston7s description of Zeldals life Zelda merrily experiences events T U V W X and Y Then Zelda is described as moving backwards in time77 through events Y X W V and Ur During most of this period she is also described as moving faster than one lightyear per year After Gaston s tg Silyea39r line Zelda is again described as moving forward in time at a speed of 4 lightyears per 5 years experiencing events V W X Y for the third time and nally experiencing event Z The moral here is that noninertial reference frames are all screwed up77 Ob servers in such reference frames are likely to describe the world in a very funny way To gure out what happens to them it is certainly best to work in an inertial frame of reference and use it to carefully construct the noninertial s pacetime diagrami M By the way there is also the issue of what Gaston would see if he watched Alphonse and Zelda through a telescope This has to do with the sequence in which light rays reach him and with the rate at which they reach himi This is also interesting to explore but I will leave it for the homework see problem 4 43 MORE ON MINKOWSKIAN GEOMETRY 101 1 sec proper time to the future There are similar hyperbolae representing the events one second of proper time in the past7 and the events one lightsecond of proper distance to the left and right We should also note in passing that the light light rays form the some what degenerate hyperbolae of zero proper time and zero proper distance 1 sec proper time to the future Us of origin 1 ls proper distance 1 ls proper distance to the right of origin of origin t 0 us 1 sec proper time to the past of origin 432 Changing Reference Frames Note that on the diagram above I have drawn in the worldline and a line of simultaneity for a second inertial observer moving at half the speed of light relative to the first How would the curves of constant proper time and prop er distance look if we re drew the diagram in this new inertial frame Stop reading and think about this for a minute Because the separation of two events in proper time and proper distance is invariant iiei7 independent of reference frame7 these curves must look exactly the same in the new frame hat is7 any event which is one second of proper time to the future of some event A say7 the origin in the diagram above in one inertial frame is also one second of proper time to the future of that event in any other inertial frame and therefore must lie on the same hyperbola 12 7 c2 71366 The same thing holds for the other proper time and proper distance 102 CHAPTER 4 MINKOWSKIAN GEOMETRY hyperbolaei 1 sec proper time to the future Us of origin 1 ls proper distance of origin 1 sec proper time to the past of origin We see that changing the inertial reference frame simply slides events along a given hyperbola of constant time or constant distance7 but does not move events from one hyperbola to another Remember our Euclidean geometry analogue from last time The above obser vation is exactly analogous to what happens when we rotate an object4i he points of the object move along circles of constant radius from the axis7 but do not hop from circle to circ er 4Actually it is analogous to what happens when we rotate but the object stays in place This is known as the different between an active7 and a passive7 rotation It seemed to me however that the main idea would be easier to digest if I did not make a big deal out of this subtlety 43 MORE ON MINKOWSKIAN GEOMETRY 103 By the way the transformation that changes reference frames is called a boostf Think of an object being boosted up to a higher level or strength speed or think of a booster stage on a rocket So what I mean is that boosts are analogous to rotationsi77 433 Hyperbolae again In order to extract the most from our diagrams let s hit the analogy with circles one last time lfl draw an arbitrary straight line through the center of a circle it always intersects the circle a given distance from the center What happens if I draw an arbitrary straight line through the origin of our hyperbolae 104 CHAPTER 4 MINKOWSKIAN GEOMETRY If it is a timelike line it could represent the worldline of some inertial observeri Suppose that the observers clock reads zero at the origin Then the worldline intersects the future AT lsec hyperbola at the event where that observers clock reads one second Similarly since a spacelike line is the line of simultaneity of some inertial ob server It intersects the d 1L3 curve at what that observer measures to be a distance of 1L3 from the origin What we have seen is that these hyperbolae encode the Minkowskian geometry of spacetime The hyperbolae of proper time and proper distance which are different manifestations of the same interval are the right way to think about how events are related in spacetime and make things much simpler than trying to think about time and space separatelyi 434 Boost Parameters and Hyperbolic Trigonometry I keep claiming that this Minkowskian geometry will simplify things So you might rightfully ask what exactly can we do with this new way of looking at things Lets go back and look at how velocities combine in relativityi This is the question of why don7t velocities just add77 Or ifl am going at 12 c relative to Alice and Charlie is going at 12 c relative to me how fast is Charlie going relative to Alice Deriving the answer is part of your homework in problem 3 10 As you have already seen in Einstein7s book the formula looks like vABCvBcC i 44 1 39UAB39UBCC2 vAcc It is interesting to remark here that this odd effect was actually observed experi mentally by Fizeau in the 1850s He managed to get an effect big enough to see by looking at light moving through a moving uid say a stream of water The point is that when it is moving through water light does not in fact travel at speed c Instead it travels relative to the water at a speed cn where n is around 43 MORE ON MINKOWSKIAN GEOMETRY 105 15 Thus it is still moving at a good fraction of the speed of light77 Anyway if the water is also owing say toward us at a fast rate then the speed of the light toward us is given by the above expression in which the velocities do not just add together This is just what Fizeau found5 though he had no idea why it should be true Now the above formula looks like a mess Why in the world should the com position of two velocities be such an awful thing As with many questions the answer is that the awfulness is not in the composition rule itself but in the lter the notion of velocity through which we view it We will now see that when this lter is removed and we view it in terms native to Minkowskian geometry the result is quite simple indeedi Recall the analogy between boosts and rotationsi How do we describe rotations We use an angle 9 Recall that rotations mix I and y through the sine and cosine functionsi 12 rsint9 yg rcost9i 45 Now one of the basic facts associated with the relation of sine and cosine to circles is the relation sin2 9 cos2 9 1 46 Similarly there are other natural mathematical functions called hyperbolic sine sinh and hyperbolic cosine cosh that satisfy coshg t9 7 sinhg t9 1 47 so that they are related to hyperbolaei These functions can be de ned in terms of the exponential function e 9 79 eie sinh t9 5Recall that Fizeau7s experiments were one of the motivations for Michelson and Morely us we now understand the results not only of their experiment but also of the experiments that prompted their work 106 CHAPTER 4 MINKOWSKIAN GEOMETRY 9 79 cosht9 if 48 2 You can do the algebra to check for yourself that these satisfy relation 47 above By the way although you may not recognize this form these functions are actually very close to the usual sine and cosine functionsi lntroducing i 5 one can write sine and cosine as 61976719 6 sm 399 2 399 z 71 cost9 i 49 Thus the two sets of functions differ only by factors of i which as you can imagine are related to the minus sign that appears in the formula for the squared interv Now consider any event A on the hyperbola that is a proper time 739 to the future of the origin Due to the relation 47 we can write the coordinates tz of this event as t Tcosht z c739sinht9i 4 10 t0 x crsinhe X 0 On the diagram above I have drawn the worldline of an inertial observer that passes through both the origin and event A Note that the parameter 9 gives some notion of how different the two inertial frames that of the moving observer and that of the stationary observer actually are For 9 0 event A is at z 0 and the two frames are the same while for large 9 event A is far up the hyperbola and the two frames are very different We can parameterize the points that are a proper distance d from the origin in a similar way though we need to flip z and t7 43 MORE ON MINKOWSKIAN GEOMETRY 107 t dcsinht z dcosht9i If we choose the same value of 9 then we do in fact just interchange z and t ipping things about the light cone77 Note that this will take the worldline of the above inertial observer into the corresponding line of simultaneityi In other words a given worldline and the corresponding line of simultaneity have the same hyperbolic angler7 coshe t0 i d x dsmhe Again we see that 9 is really a measure of the separation of the two reference frames In this context we also refer to 9 as the boost parameter relating the two frames The boost parameter is another way to encode the information present in the relative velocity and in particular it is a very natural way to do so from the viewpoint of Minkowskian geometry In what way is the relative velocity v of the reference frames related to the boost parameter 9 Let us again consider the inertial observer passing from the origin through event A This observer moves at spee z c739sinht9 sinht9 v 7 t Tcosht iCCOShO Ctanhgl and we have the desired relationi Here we have introduced the hyperbolic tangent function in direct analogy to the more familiar tangent function of trigonometryi Note that we may also write this function as 9 79 e 7 e tanht i 69 6 9 The hyperbolic tangent function may seem a little weird but we can get a better feel for it by drawing a graph like the one belowi The vertical axis is tanht9 and the horizontal axis is 9 110 CHAPTER 4 MINKOWSKIAN GEOMETRY 441 Aberration in Relativity Recall the basic setup of the aberration experiments Starlight hits the earth from the side but the earth is moving forward77 so this somehow means that astronomers can t point their telescopes straight toward the star if they actually want to see it This is shown in the diagram belowl iX Light Ray hits side instead of reaching bottom Telescope moves through ether To reanalyze the situation using our new understanding of relativity we will have to deal the fact that the star light comes in from the side while the earth travels forward relative to the star Thus we will need to use a spacetime diagram having three dimensions 7 two space and one time One often calls such diagrams 21 dimensional77 These are harder to draw than the 11 dimensional diagrams that we have been using so far but are really not so much different After all we have already talked a little bit about the fact that under a boost things behave reasonably simply in the direction perpendicular to the action of the boost neither simultaneity nor lengths are affected in that direction We7ll try to draw 21 dimensional spacetime diagrams using our standard con ventions all light rays move at 45 degrees to the verticals Thus a light cone looks like this Must tilt telescope to see star 44 21 DIMENSIONS ABERRATION 111 LIGHT CONE We can also draw an observer and their plane of simultaneity lane of Simultaneity In the direction of the boost this plane of simultaneity acts just like the lines of simultaneity that we haVe been drawing However in the direction perpendicular to the boost direction the boosted plane of simultaneity is not tilted This is the statement that simultaneity is not a ected in this direction 112 CHAPTER 4 MINKOWSKIAN GEOMETRY Naw let s put ms all cogsma I also mm m display Lhe mnvmg ubsava Idea of ughb and leftquot so I have dxawn Lhe plane ufevaqm that Lhe mnvmg ubsava nds a beshajghb ha ha mg m m the left and nut at all m mm um behmd he Hae Lhe ubsava 1s mnvmg muss ms ma m ha ughb and 1am axe more as la mm and out of Lhe ya a Muvmg onsexm 39s Lemmqu Plane mm m me mnvm of course we would us ca kmw haw ths all lacks whsm re g ubsava s xefezence flame One chmg um we kmw s that may xay Bunyan I Lhe mnvmg ast mm ms plane mummng mass 115m xays Thus mass m 115m xays do not navel skughb mm an muve mmewhab m Lhe Wkwudsv anemonw ms 1s haw Lhe abaxauun e ms xe exmce fume of aux sun m we mnvmg xe exmce fume a out of Lhe page but Instead a s dacnbed m xasuwcy 311717086 mu m Lhe Sta bang viewed than we assays 1s ram flame mm mm m we g mmmgquot xefezaqce flame of ms euhh ms 115m my appeazs m be muvmg a m backwade quot Thus askummas must mm Lhax Hampa a m fmwaxd mmm A meowmmn GEOMETRY L lght Rays dawn aquotd a cause he nu we my n 5 mnvauaw m dxspmd mm a a he Thae 5 one no my mung emuand m an d m d Aida9411mm m m 5M dlxnmswns 5m d m talk dam m we a no my my part 5 m m dmgam and m draw M b wdapmm donyd mum Mame an wdx t bdve dug g mn p ndm ad dms d dapm mama Mk Wanda hm my 1mm added dim mm 119w xvi mm ammndysmmkaemdmnmdm 7 mm t d m m ongmx n quotW m drawthe mnspondmg Mamm ms m m mavmg mamas 44 21 DIMENSIONS ABERRATION 115 perspective As we saw before a given light ray from one reference frame is still some light ray in the new reference frame Therefore the effect of the boost on the light cone can be described by simply moving the various dots to appropriate new locations on the circle For example the light rays that originally traveled straight into and out of the page now fall a bit behind7 the moving observeri So they are now moved a bit toward the back Front back left and right now refer to the new reference frame Front Right Light Circle in the Second Frame Note that most of the dots have fallen toward our current observer7s back side 7 the side which represents in the current reference frame the direction of motion of the rst observerl Suppose then that the rst observer were actually say a star like the sun In it s own rest frame a star shines more or less equally brightly in all directions 7 in other words it emits the same number of rays of light in all directions So if we drew those rays as dots on a corresponding lightcircle in the stars frame of reference they would all be equally spread out as in the rst light circle we drew abovei M What we see therefore is that in another reference frame with respect to which the star is moving the light rays do not radiate symmetrically from the star Instead most of the light rays come out in one particular direction In particular they to come out in the direction that the star is moving Thus in this reference frame the light emitted by the star is bright in the direction of motion and dim in the opposite direction and the star shines like a beacon in the direction it is moving For this reason this is known as the headlight effecti By the way this effect is seen all the time in high energy particle accelerators and has important applications in materials science and medicine Charged particles whizzing around the accelerator emit radiation in all directions as described in their own rest frame However in the frame of reference of the laboratory the radiation comes out in a tightly focussed beam in the direction of the particles7 motion This means that the radiation can be directed very precisely at materials to be studied or tumors to be destroyed 116 CHAPTER 4 MINKOWSKIAN GEOMETRY 443 Multiple boosts in 21 dimensions Cool eh But wait We7re not done yet I want you to look back at the above two circles of light rays and notice that there is a certain symmetry about the direction of motion So suppose you are given a circle of light rays marked with dots which show as above the direction of motion of light rays in your reference frame Suppose also that these light rays were emitted by a star or by any other source that emits equally in all directions in its rest frame Then you can tell which direction the star is moving relative to you by identifying the symmetry axis in the circle There must always be such a symmetry axis The result of the boost was to make the dots ow as shown below Back Front Right green font and back dots are on the symmetry axis and so do not move at all The yellow and So just for fun let s take the case above and consider another observer who is moving not in the forwardbackward direction but instead is moving in the direction that is leftright relative to the moving observer above To nd out what the dots looks like in the new frame of references we just rotate the ow shown above by 90 degrees as shown below Left symmetry axis Right and apply it to the dots in the second frame The results looks something like this 44 21 DIMENSIONS ABERRATION 117 Back Front Symmetry axis Right The new symmetry axis is shown above Thus with respect to the original observer this new observer is not moving not along a line straight to the right Instead the new observer is moving somewhat in the forward direction as well But waitnn something else interesting is going on herein the light rays don7t line up right Note that if we copied the above symmetry axis onto the light circle in the original frame it would sit exactly on top of rays 4 and 8 However in the gure above the symmetry axis sits halfway between 1 and 8 and 3 and 4 This is the equivalent of having rst rotated the light circle in the original frame by 116 of a revolution before performing a boost along the new symmetry axis The new observer differs from the original one not just by a boost but by a rotation as we 1 In fact by considering two further boost transformations as above one acting only backward and then one acting to the right one can obtain the following circle of light rays which are again evenly distributed around the circle You should work through this for yourself pushing the dots around the circle with care 6 Right Thus by a series of boosts one can arrive at a frame of reference which while it is not moving with respect to the original fame is in fact rotated with respect to the original framei By applying only boost transformations we have managed to turn our observer by 45 degrees in space This just goes to show again that time and space are completely mixed together in relativity and that boost 122 CHAPTER 4 MINKOWSKIAN GEOMETRY 5 N THE UNIFORMLY ACCELERATIN G WORLDLINE 125 Friend 2 In fact to do this properly we should switch friends and reference frames fast enough so that we are always using a reference frame in which the rocket is moving only in nitesimally slowlyi Then the relativistic effects will be of zero size In other words we wish to borrow techniques from calculus and take the limit in which we switch reference frames continually always using the momentarily comoving inertial framei So welll have lots of fun with calculus in this chapter Anyway the thing that we want to be constant in uniform acceleration is called the proper acceleration Of course it can change along the rocket s worldline depending on how fast the rocket decides to burn fuel so we should talk about the proper acceleration at some event on the rocket s worldlinei7 To nd the proper acceleration a at event E rst consider an inertial reference frame in which the rocket is at rest at event El xU x0 0 10 Al E the rocket is at rest in this frame Momentarin Crymoving rame The proper acceleration aE at event E is just the acceleration of the rocket at event E as computed in this momentarily comoving reference frame Thus we have aE d UEth 5 1 51 THE UNIFORMLY ACCELERATING WORLDLINE 129 A simple way to work this out is to rst consider the familiar case where we are inertial observers and to think about other inertial observers moving relative to our reference frame We have seen that we can label events on their worldlines using proper time 739 and boost parameter 9 as well as the Cartesian coordinates z and ti 139 t0 xcrsinh6 A X 0 The relation between zt and 739 9 is encoded in hyperbolic trigonometry z c739sinht9 t Tcosht9i 57 For the case of inertial observers the boost parameter 9 is constant so we have the relations dz E cs1nh 9 58a dt E 7 cosh 9 58b This last relation 58b is just the time dilation formula rewritten in terms of the boost parameter 9 One can also check independently that cosht9 4 17v2c2 We would now like to apply equation 58b to the case of a uniformly acceler ating worldlinei This is justi ed since as we have discussed the clock on the rocket ship could be replaced by an inertial clock in the momentarily comoving inertial frame which would measure the same time interval Using the chain rule again we nd that d9 7 d9 d7 7 a l E 7 7 cosht Multiplying both sides by cosh Odt we get 5 9 cosh 949 Edt 510 C 132 CHAPTER 5 ACCELERATIN G REFERENCE FRAMES r r r 52 Exploring the uniformly accelerated refer ence frame We have now found that a uniformly accelerating observer with proper acceler ation 1 follows a worldline that remains a constant proper distance 6201 away from some event Just which event this is depends on where and when the observer began to accelerate For simplicity let us consider the case where this special event is the origin Let us now look more closely at the geometry of the situation 521 Horizons and Siumtaneity The diagram below shows the uniformly accelerating worldline together with a few important light rays Future Acceleration Horizon Signals from 39 39 39 end ofthe world The Light rays from thi 5 event never catch up with the rocket The rocke annever send signals to this region Past Accelerat10n H0r1zon Note the existence of the light ray marked future acceleration horizoni77 It marks the boundary of the region of spacetime from which the uniformly accel erated observer can receive signals since such signals cannot travel faster than cl This is an interesting phenomenon in and of itself merely by undergoing uniform acceleration the rocket ship has cut itself off from communication with a large part of the spacetime In general the term horizon7 is used whenever an object is cut off in this way On the diagram above there is a light ray marked past acceleration horizon77 which is the boundary of the region of spacetime to which the uniformly accelerated observer can send signalsi 52 THE UNIFORMLY ACCELERATED FRAME 133 When considering inertial observers we found it very useful to know how to draw their lines of simultaneity and their lines of constant positioni Presumably we will learn equally interesting things from working this out for the uniformly accelerating rocketi But what notion of simultaneity should the rocket use Let us de ne the rocket7s lines of simultaneity to be those of the associated momentarily co moving inertial framesi It turns out that these are easy to drawi Let us simply pick any event A on the uniformly accelerated worldline as shown belowi l have also marked with a Z the event from which the worldline maintains a constant proper distancei Dgt Recall that a boost transformation simply slides the events along the hyperbolai This means that we can nd an inertial frame in which the above picture looks like this In the new frame of reference the rocket is at rest at event A Therefore the rocket7s line of simultaneity through A is a horizontal line Note that this line passes through event Z This makes the line of simultaneity easy to draw on the original diagrami What we have just seen is that Given a uniformly accelerating observer there is an event Z from which it maintains proper distance The observer7s line of simultaneity through any event A on her worldline is the line that connects event A to event Z 134 CHAPTER 5 ACCELERATIN G REFERENCE FRAMES i i i Thus the diagram below shows the rocket s lines of simultaneityi Let me quickly make one comment here on the passage of time Suppose that events 72 71 above are separated by the same sized boost as events 710 events 01 and events 1 2 From the relation 9 aTc it follows that each such pair of events is also separated by the same interval of proper time along the worldlinei M But now on to the more interesting features of the diagram above Note that the acceleration horizons divide the spacetime into four regions In the right most region the lines of simultaneity look more or less normal However in the top and bottom regions there are no lines of simultaneity at all The rocket s lines of simultaneity simply do not penetrate into these regions Finally in the leftmost region things again look more or less normal except that the labels on the lines of simultaneity seem to go the wrong way moving backward in timer7 And of course all of the lines of simultaneity pass through event Z where the horizons crossi These strangesounding features of the diagram should remind you of the weird effects we found associated with Gaston s acceleration in our discussion of the twin paradox in section 42 As with Gaston one is tempted to ask How can the rocket see things running backward in time in the leftmost region77 In fact the rocket does not see or even know about anything in this region As we mentioned above no signal of any kind from any event in this region can ever catch up to the rocket As a result this phenomenon of nding things to run backwards in time is a pure mathematical artifact and is not directly related to anything that observers on the rocket actually noticei 522 Friends on a Rope In the last section we uncovered some odd effects associated with the the ac celeration horizonsi In particular we found that there was a region in which the lines of simultaneity seemed to run backward However we also found that the rocket could neither signal this region nor receive a signal from it As a result the fact that the lines of simultaneity run backward here is purely a mathematical artifacti 52 THE UNIFORMLY ACCELERATED FRAME 135 Despite our discussion above you might wonder if that funny part of the rocket s reference frame might somehow still be meaningful It turns out to be productive to get another perspective on this so let s think a bit about how we might actually construct a reference frame for the rocket Suppose for example that I sit in the nose the front of the rocket I would probably like to use our usual trick of asking some of my friends or the students in class to sit at a constant distance from me in either direction I would then try to have them observe nearby events and tell me which ones happen Where We would like to know what happens to the ones that lie below the horizon Let us begin by asking the question what worldlines do these fellow observers follow Let7s seen Consider a friend who remains a constant distance A below us as measured by us that is as measured in the momentarily comoving frame of ref erence This means that this distance is measured along our line of simultaneityi But look at what this means on the diagram below Upper Observer Recall that a distance measured in some inertial frame between two events on a given a line of simultaneity associated with that same inertial frame is in fact the proper distance between those events Thus on each line of simultaneity the proper distance between us and our friend is Al But along each of these lines the proper distance between us and event Z is 0162 Thus along each of these lines the proper distance between our friend and event Z is 162 7 Ar In other words the proper distance between our friend and event Z is again a constant and our friends worldline must also be a hyperbolal Note however that the proper distance between our friend and event Z is less than the proper distance 6201 between us and event Z This means that our 52 THE UNIFORMLY ACCELERATED FRAME 137 Front of Rocket Back of Rocket Note that the front and back of the rocket do in fact have the same lines of simul taneity so that they agree on which events happen at the same time77 But do they agree on how much time passes between events that are not simultaneous Since they agree about lines of simultaneity it must be that along any such line both ends of the rocket have the same speed 1 and the same boost parameter 9 However because the proper acceleration of the back is greater than that of the front the relation 9 1762 then tells us that more proper time 739 passes at the front of the rocket than at the back In other words there is more proper time between the events AF BF below than between events A3 B3 138 CHAPTER 5 ACCELERATIN G REFERENCE FRAMES i i i Rocket M Here it is important to note that since they use the same lines of simultaneity both ends of the rocket agree that the front top clock runs fasterl Thus this effect is of a somewhat different nature than the time dilation associated with inertial observersi This of course is because all accelerated observers are not equivalent 7 some are more accelerated than others By the way we could have read off the fact that Arpmm is bigger than ATBaCk directly from our diagram without doing any calculations This way of doing things is useful for certain similar homework problems To see this note that between the two lines t ito of simultaneity for the inertial framell drawn below the back of the rocket is moving faster relative to the inertial frame in which the diagram is drawn than is the front of the rocket You can see this from the fact that the front and back have the same line of simultaneity and therefore the same speed at events BF BB and at events AF ABi This means that the speed of the back at BB is greater than that of the front at DF and that the speed of the back at A3 is greater than that of the front at Cpl 53 HOMEWORK PROBLEMS 139 DF Front of Rocket tt 0 CF Back of Rocket Thus relative to the inertial frame in which the diagram is drawn the back of the rocket experiences more time dilation in the interval ito to and it s clock runs more slowly Thus the proper time along the backs worldline between events AB and BB is less than the proper time along the fronts worldline between events CF and DR M We now combine this with the fact that the proper time along the fronts worldline between AF and BF is even greater than that between CF and DR Thus we see that the front clock records much more proper time between AF and BF than does the back clock between A3 and B3 53 Homework Problems 51 Suppose that you are in a small rocket and that you make the following trip You start in a rocket in our solar system at rest with respect to the Sun You then point your rocket toward the center of the galaxy and accelerate uniformly for ten years of your irer proper time with a proper acceleration of lg lOmSQr Then you decelerate uniformly for ten years of proper time at lg so that you are again at rest relative to the sun a Show that lg is very close to llight 7 yeaTyeaTQr Use this value for g in the problems belowr b Draw a spacetime diagram showing your worldline and that of the Sun in the Suns reference frame c As measured in the Suns reference frame how far have you traveled CHAPTER 5 ACCELERATIN G REFERENCE FRAMES 148 CHAPTER 6 DYNAMICS ENERGY AND The third law of Newtonian Physics When two objects A and B exert forces FA on B and F3 on A on each other these forces have the same size but act in opposite directions To understand why this is a problem letls think about the gravitational forces between the Sun and the Earthi Earth Sun Recall that Newton said that the force between the earth and the sun is given by an inverse square law F GM MWquot where d is the distance between themi In particular the force between the earth and sun decreases if they move farther apart Let7s draw a spacetime diagram showing the two objects moving aparti At some time t1 when they are close together there is some strong force F1 acting on each object Then later when they are farther apart there is some weaker force F2 acting on each object However what happens if we consider this diagram in a moving reference frame I have drawn in a line of simultaneity the dashed line for a different reference frame above and we can see that it passes through one point marked F1 and one point marked F2 This shows that Newton7s third law as stated above cannot possibly hold1 in all reference frames So Newton7s third law has to go But of course Newton7s third law is not completely wrong 7 it worked very well for several hundred years So as with the law of composition of velocities and Newton7s second law we may expect that it is an approximation to some other more correct law with this approximation being valid only for velocities that are very small compared to cl It turns out that this was not such a shock to Einstein as there had been a bit of trouble with Newton7s third law even before relativity itself was understood Again the culprit was electromagnetismi 1You might wonder if you could somehow save the third law by having the concept of force depend on which inertial frame you use to describe the system Then in the moving frame the forces would not be F1 and F2 However the forces in the moving frame must still somehow be determined by F1 and F2 Thus if F1 and F2 do not agree neither can the forces in the moving frame 152 CHAPTER 6 DYNAMICS ENERGY AND 631 Lasers in a box Anyway suppose that we start with a box having a powerful laser5 at one end When the laser res a pulse of light the light is near the left end and pressure from this light pushes the box to the left The box moves to the left while the pulse is traveling to the right Then when the pulse hits the far wall its pressure stops the motion of the box The light itself is absorbed by the wall and disappears Before After Now momentum conservation says that the total momentum is always zero Nevertheless the entire box seems to have moved a bit to the left With a large enough battery to power the laser we could repeat this experiment many times and make the box end up very far to the left of where it started Or perhaps we do not even need a large battery we can imagine recycling the energy used the laser If we could catch the energy at the right end and then put it back in the battery we would only need a battery tiny enough for a single pulse By simply recycling the energy many times we could still move the box very far to the left This is what really worried our friend Mr Einstein 632 Center of Mass The moving laser box worried him because of something called the center of mass Here s the idea lmagine yourself in a canoe on a lake You stand at one end of the canoe and then walk forward However while you walk forward the canoe will slide backward a bit A massive canoe slides only a little bit but course lasers did not exist when Einstein was working on this He just used a regular light source but if lasers had been around that7s what he would have used in his example 63 ON TO RELATIVITY 153 a light canoe will slide a lot It turns out that in nonrelativistic physics the position technically known as the center of mass does not move This follows from Newton7s third law and momentum conservation To under stand the point suppose that in the above experiment we throw rocks from left to right instead of ring the laser beam While most of the box would shift a bit to the left due to the recoil with each rock thrown the rock in ight would travel quite a bit to the right In this case a sort of average location of all of the things in the box including the rock does not move Suppose now that we want to recycle the rock taking it back to the left to be thrown again We might for example try to throw it back But this would make the rest of the box shift back to the right just where it was before It turns out that any other method of moving the rock back to the left side has the same effect To make a long story short since the average position cannot change a box can never move itself more than one boxlength in any direction and this can only be done by piling everything inside the box on one side In fact when there are no forces from outside the box the center of mass of the stuff in the box does not accelerate at all In general it is the center of mass that responds directly to outside forces 633 Mass vs Energy So what s going on with our box Letls look at the experiment more carefully Before After After the experiment it is clear that the box has moved and in fact that every single atoms in the ox has slid to the left So the center of mass seems to have moved But Einstein asked might something else have changed during the experiment which we need to take into account Is the box after the experiment really identical to the one before the experiment began The answer is not quite77 Before the experiment the battery that powers the laser is fully charged After the experiment the battery is not fully charged What happened to the associated energy It traveled across the box as a pulse 154 CHAPTER 6 DYNAMICS ENERGY AND of light It was then absorbed by the right wall causing the wall to become hot The net result is that energy has been transported from one end of the box where it was battery energy to the other where it became heat Before Battery fully charged After Battery not fully charged HOt W811 So Einstein said perhaps we should think about something like the center of energy as opposed to the center of mass But of course the mass must also contribute to the center of energy so is mass a form of energy Anyway the relevant question here is Suppose we want to calculate the center of massenergy Just how much mass is a given amount of energy worth77 Or said another way how much energy is a given amount of mass worth Well from Maxwellls equations Einstein could gure out the energy transport ed He could also gure out the pressure exerted on the box so that he new how far all of the atoms would slider Assuming that the center of massenergy did not move this allowed him to gure out how much energy the mass of the box was in fact worth The computation is a bit complicated so we won7t do it hereF r However the result is that an object of mass m which is at rest is worth the energy E mc2 63 Note that since 62 9 X 1015721232 is a big number a small mass is worth a lot of energy Or a reasonable amount7 of energy is in fact worth very little mass This is why the contribution of the energy to the center of massenergy7 had not been noticed in preEinstein experiments Let7s look at a few We buy electricity in kilowatthours7 kWh 7 roughly the amount of energy it takes to run a house for an hour The mass equivalent of l kilowatthour is 7 1kWh 7 1kWh 3600366 1000W 7 3 6 X 106 m 2 c2 hm 1kW 9X1015 4 X10 10kgr 64 C In other words not much 6Perhaps some senior physics major would like to take it up as a course project 64 MORE ON MASS ENERGY AND MOMENTUM 159 an arrow Now an arrow that you draw on a spacetime diagram can point in a timelike direction as much as in a spacelike direction Furthermore an arrow that points in a purely spatial7 direction as seen in one frame of reference points in a direction that is not purely spatial as seen in another frame So spacetime vectors have time parts components as well as space parts A displacement in spacetime involves cAt as much as a A1 The interval is actually something that computes the size of a given spacetime vector For a displacement it is A12 7 cAt Together the momentum and the energy form a single spacetime vector The momentum is already a vector in space so it forms the space part of this vector It turns out that the energy forms the time part of this vector So the size of the energymomentum vector is given by a formula much like the one above for displacements This means that the rest mass m0 is basically a measure of the size of the energymomentum vector Furthermore we see that this size does not depend on the frame of reference and so does not depend on how fast the object is moving However for a rapidly moving object both the time part the energy and the space part the momentum are large 7 it s just that the Minkowskian notion of the size of a vector involves a minus sign and these two parts largely cancel against each other 643 How about an example As with many topics a concrete example is useful to understand certain details of what is going on In this case I would like to illustrate the point that while energy and momentum are both conserved mass is not conserved Let7s consider a simple system called positroniumf This consists of an electron negatively charged and a positron positively charged which orbit each other due to their electrical attraction This system is nice because both particles have exactly the same rest mass which we can call me e for electron This gives things a nice symmetry 160 CHAPTER 6 DYNAMICS ENERGY AND Electron g Positron A snapshot of the orbiting particles is shown above Let us suppose that they are orbiting at speed 465 as measured in an inertial frame where the particles just go round and round each other and do not y away At the time shown the electron is going straight upward and the positron is going straight downwardi Each particle has a momentum 4 me mec 612 W 17v2C2 3 and an energy 771862 5 2 e 6 13 xli UQCZ 3mc Now what is the energy and momentum of the positronium system as a whole Well the momenta are of the same size but they are in opposite directions So they cancel out and the total momentum is zero However the energies are both positive so they add together We nd E ppositronium 07 10 2 Eposmmum gmeci 614 So what is the rest mass of the positronium system 100 E2 7 p262 Vmic 615 So the rest mass of the positronium system is given by dividing the right hand side by 64 The result is 7218 which is signi cantly greater than the rest mass of the electron plus the rest mass of the positron i Similarly two massless particles can in fact combine to make an object with a nite nonzero massi For example placing photons in a box adds to the mass of the box Welll talk more about massless particles and photons in particular belowi 111 have cheated heavily here A real positronian system must of course have less rest mass than would the particles separately The difference is due to the electrical potential energy which is negative due to the attractive force between the particles 1 have ignored this here but it is the negative potential energy which makes real positronian hold together More properly one would say that the calculation here gives the rest mass of a system made by tying the two particles together with strong light string and then spinning things up 168 CHAPTER 6 DYNAMICS ENERGY AND 67 Homework Problems Note Energy can be measured in various units7 like Joules J or kiloWatt hours kVV hrsi7 this is the unit that Niagara Mohawk uses on my electric bill You can use any unit that you like You may nd the following relations between the various units usefu i lkgm2s2 lJouleJ lWatt 7 secondWs 31 6 gtlt lO F kW 7 ins 61 How much energy would it take to accelerate you up to 96 62 I pay Niagara Mohawk 0107 per kW hri How much would it cost me to accelerate you up to 90 63 Consider a box containing two photons traveling in opposite directions 9 If the box has a rest mass m0 and each photon has an energy E07 what is the rest mass of the combined boxplus photons system Hint How much energy and momentum does each of the three objects have 64 In particle accelerators7 one can collide an electron with a positron and sometimes they turn into a protonanti proton pair The rest mass of an electron or a positron is 911 X lO glkgi The rest mass of a proton or an antiproton is 1673 gtlt 10 27kgi Suppose that the protonantiproton pair is created at rest and that the electron and positron had equal speed in opposite directions How fast must the electron and positron have been moving for this reaction to be allowed by conservation of energy Give the answer both in terms of speed 1 and boost parameter i 65 Here7s a good calculation if you know a little physicsi It has to do with how your TV and computer monitor wor Particle physicists often use a unit of energy called the electronVolt eVl This amount of energy that an electron picks up when it accelerates across a potential of one Volt14l Since the charge on an electron is 1 6 X 10 19 Coulombs7 one electronVolt is 1 6 X 10 191 a If an object at rest has a total energy of leV7 what is its mass b The mass of an electron is 911 X lO glkgi What is the energy in eV of an electron at rest 14You can look at a PHY212 or 216 physics book for a de nition of the Volt but you won7t actually need to know it for this problem 170 CHAPTER 6 DYNAMICS ENERGY AND 172 CHAPTER 7 RELATIVITY AND THE GRAVITATIONAL FIELD c Gravity d Friction e One object pushing anotheri f Pressure and so onnm Now the rst two of these forces are described by Maxwell7s equations As we have discussed Maxwellls equations t well with and even led to relativity Unlike Newton7s laws Maxwell7s equations are fully compatible with relativity and require no modi cations at all Thus we may set these forces aside as complete7 and move on to the others Letls skip ahead to the last three forcesi These all have to do in the end with atoms pushing and pulling on each ot er ln Einstein7s time such things we believed1 to be governed by the electric forces between atoms So it was thought that this was also properly described by Maxwell7s equations and would t well with relativity You may have noticed that this leaves one force gravity as the odd one out Einstein wondered how hard can it be to make gravity consistent with relativ 71 The Gravitational Field Letls begin by revisiting the prerelativistic understanding of gravity Perhaps we will get lucky and nd that it too requires no modi cation 711 Newtonian Gravity vs relativity Newton7s understanding of gravity was as follows Newton s Universal Law of Gravity Any two objects of masses m1 and 7212 exert gravitational7 forces on each other of magnitude mlmZ FG d2 7 1 F1 F2 1This belief was not entirely correct A large part of such forces7 comes from an e ect that is not in fact described as a force7 today This e ect is known as the Pauli exclusion principle7 and states that no two electrons can occupy the same quantuln state7 basically that they cannot be stacked on top of each other Today we recognize this e ect as coming from the fundamental quantum nature of the electron Protons and other fermions7 behave similarly while photons and other bosons7 do not Quantum mechanics is a different kettle of sh altogether but in the end it does t well with special relativity N N THE GRAVITATIONAL FIELD 173 directed toward each other where G 667 X 10 11NmQkg2 is called Newton s Gravitational Constanti77 G is a kind of intrinsic measure of how strong the gravitational force is It turns out that this rule is not compatible with special relativity In particular having learned relativity we now believe that it should not be possible to send messages faster than the speed of light However Newton7s rule above would allow us to do so using gravity The point is that Newton said that the force depends on the separation between the objects at this instant Example The earth is about eight lightminutes from the sun This means that at the speed of light a message would take eight minutes to travel from the sun to the earth However suppose that unbeknownst to us some aliens are about to move the sun Then based on our understanding of relativity we would expect it to take eight minutes for us to nd out But Newton would have expected us to nd out instantly because the force on the earth would shift changing the tides and other things Force before T Force after 0 712 The importance of the eld Now it is important to understand how Maxwell7s equations get around this sort of problemi That is to say what if the Sun were a positive electric charge the earth were a big negative electric charge and they were held together by an Electro Magnetic eld We said that Maxwell7s equations are consistent with relativity 7 so how what would they tell us happens when the aliens move the sun The point is that the positive charge does not act directly on the negative charge Instead the positive charge sets up an electric eld which tells the negative charge how to move l 7 39 When the positive charge is moved the electric eld around it must change but it turns out that the eld does not change everywhere at the same time 2Note that there is also an issue of simultaneity here Which events on the two separated worldlines should one compare to compute the distance Which notion of this instant7 would one use to pick out these events 174 CHAPTER 7 RELATIVITY AND THE GRAVITATIONAL FIELD lnstead7 the movement of the charge modi es the eld only where the charge actually is This makes a ripple7 in the eld which then moves outward at the speed of light In the gure below7 the black circle is centered on the original position of the charge and is of a size ct7 where t is the time since the movement begani Thus7 the basic way that Maxwellls equations get around the problem of instant reaction is by having a eld that will carry the message to the other charge or7 say7 to the planet at a nite speedi Oh7 and remember that having a eld that could carry momentum was also what allowed Maxwellls equations to t with momentum conservation in relativityi What we see is that the eld concept is the essential link that allows us to understand electric and magnetic forces in relativityi Something like this must happen for gravity as well Let7s try to introduce a gravitational eld by breaking Newton7s law of gravity up into two parts The idea will again be than an object should produce a gravitational eld 9 in the spacetime around it7 and that this gravitational eld should then tell the other objects how to move through spacetimei Any information about the object causing the gravity should not reach the other objects directly7 but should only be communicated through the eld Old F fmiyzc New Fm ml mlg7 mgG d i g 72 Some observations 1 should mention that these notes will address our new topic General Relativity from a somewhat different point of view than your readings do I do not mean 72 SOME OBSERVATIONS 177 light light K 6 6 light light Let us suppose that gravity does not effect light and consider the following process 5 e light w Energy gt E l e e atrest E02mc2 E1 ONeJr with E 1 gt E 0 2 They fall 4 Shine E 1gt E 0 UP e e N 3 Makes light w Energy E1gt E0 1 First we start with an electron mass m and a positron also mass m at rest Thus we have a total energy of E0 27216 i 2 Now these particles fall a bit in a gravitational eld They speed up and gain energy We have a new larger energy E1 gt Eoi 3 Suppose that these two particles now interact and turn into some light By conservation of energy this light has the same energy E1 gt Eoi 4 Let us take this light and shine it upwards back to where the particles started This is not hard to do 7 one simply puts enough rnirrors around the region where the light is created Since we have assumed that gravity does not affect the light it must still have an energy E1 gt Eoi 5 Finally let us suppose that this light interacts with itself to make an electron a positron again By energy conservation these particles must have an energy of E1 gt Eel 72 SOME OBSERVATIONS 179 Bottom Top Butmi isn7t each wave crest supposed to move at the same speed c in a vacuum It looks like the speed of light gets faster and faster as time passes Perhaps we have done something wrong By the way do you remember any time before when we saw light doing weird stu Hmmmmim something is de nitely funny in the diagram above Nothing is really changing with time so each crest should really act the same as the one before and move at the same speed at least when the wave is at the same place Let7s choose to draw this speed as a 450 line as usual In that case our diagram must look like the one belowi However we know both from our argument above and from Pound and Rebke s experiment that the time between the wave crests is larger at the top So what looks like the same separation must actually represent a greater proper time at the top Bottom Top gt 25ec lsec C L constant distance This may seem very odd Should we believe that time passes at a faster rate 73 THE EQ UIVALENCE PRINCIPLE 181 Note how this ts with our observation about clocks higher up running faster than clocks lower down We said that this exactly matches the results for an accelerating rocket with a 9 As a result things that accelerate relative to the lab will behave like things that accelerate relative to the rocket In particular it is the freely falling frame that accelerates downward at 9 relative to the lab while it is the inertial frame that accelerates downward at 9 relative to the rocket Thus clocks in a freely falling frame act like those in an inertial frame and it is in the freely falling frame that clocks with no relative motion in fact run at the same ratell Similarly a lab on the earth and a lab in a rocket with its engine on and say accelerating at lOmSQ are very similar They have the following features in common Rocket Lab 1 Lab 2 l Clocks farther up run faster in both cases and by the same amount 2 If the nongravitational force on an object is zero the object falls relative to the lab at a certain acceleration that does not depend on what the object is 3 If you are standing in such a lab you feel exactly the same in both cases Einstein7s guess insight was that in fact Under local measurements a gravitational eld is completely equivalent to an acceleration This statement is known as The Equivalence Principle In particular gravity has NO local effects in a freely falling reference frame This ideas turns out to be useful even in answering nonrelativistic problems For example what happens when I drop a hammer held horizontally Does the heavy end hit rst or does the light end Try using the equivalence principle idea to predict the answer So then what would be the best way to draw a spacetime diagram for a tower sitting on the earth By best what I mean is in what frame of reference do we most understand what is going on The answer of course is the frame that acts like an inertial frame In this case this is the freely falling reference frame We have learned that in such a reference frame we can ignore gravity completely 182 CHAPTER 7 RELATIVITY AND THE GRAVITATIONAL FIELD Now how much sense does the above picture really make Let7s make this easy and suppose that the earth were really big it turns out that in this case the earth7s gravitational eld would actually be constant and would not become weaker as we go up Does this mesh with the diagram above Not really We said that the diagram above is effectively in an inertial frame However in this case we know that if the distance between the bottom and top of the tower does not change then the bottom must accelerate at a faster rate than the top does But we just said that we want to consider a constant gravitational eld So what s up Side note No it does not help to point out that the real earth7s gravitation al eld is not constant The point here is that the earth7s gravitational eld changes in a way that has nothing to do with the relationship a 2l from the accelerated rocket 732 How Local Well we do have a way out of this We realized before that the idea of freely falling frames being like inertial frames was not universally true After all freely falling objects on opposite side of the earth do accelerate towards each other In contrast any two inertial objects experience zero relative acceleration However we did say that inertial and freely falling frames are the same locallyf Lets take a minute to re ne that statement How local is local Well this is much like the question of when is a velocity small compared to the speed of light77 What we found before was that New tonian physics held true in the limit of small velocities In the same way our statement that inertial frames and freelyfalling frames are similar is supposed to be true in the sense of a limit This comparison becomes more and more valid the smaller a region of spacetime we use to compare the two You should take a moment to reflect on this However after doing so you might still point out that we still have a right to see just how accurate this comparison is In other words we will need to know just which things agree in the above limit and we will need to understand better just what this limit is To nd out let s consider a tiny box of spacetime from our diagram above 73 THE EQ UIVALENCE PRINCIPLE 183 This acceleration was matched to g For simplicity consider a square box of height 6 and width CE This square should contain the event at which we matched the gravitational eld 9 to the acceleration of the rocket How wrong would be be about for example the nal position of the top of the rocket if we thought that the freely falling reference ame was really inertial Well the answer is governed by the difference in acceleration between the top and bottom of the rocket in the inertial case this after all is the source of the discrepancy Recall that the acceleration a is a function of the position I Let7s use a little calculus to remember that for a small separation 61 CE the difference is approximately 6a oce 73 So how does this affect the position For short times small 6 and therefore small relative velocity we can forget about relativity and just use the old I at2 in our case AI 6a 62 This tells us that we make a mistake in the position of the top of the rocket by an amount 1 2 AI 5 a 666 74 Now lengths are kind of messy by which I mean that if the rocket was twice as long we would get twice this answer because 61 would have been 266 So a better measure of our error is the fractional error AI 1 2 a e 75 61 2 In fact the details are not important here The important part is just the dependence on 6 So let me write this as I 2 76 1006 184 CHAPTER 7 RELATIVITY AND THE GRAVITATIONAL FIELD For historical reasons3 the factor of proportionality is known as Ri Thus A j Re For the physicists in the audience note that R has dimensions of length 2 and so is distinctly not a radius 0 To summarize what we have found is that locally a freely falling reference frame is almost the same as an inertial frame If we think about a freely falling reference frame as being exactly like an inertial frame then we make a small error in computing things The fractional error is proportional to 62 where e is the size of the spacetime region needed to make the measurement 74 Going beyond locality Einstein Chapters XXXXH In section 731 we talked about the fact that locally a freely falling frame in a gravitational eld acts like an inertial frame does in the absence of gravity However we saw that freely falling frames and inertial frames are not exactly the same if they are compared over any bit of spacetime of nite size No matter how small of a region of spacetime we consider we always make some error if we interpret a freely falling frame as an inertial framei So since any real experiment requires a nite piece of spacetime how can our local principle be useful in practice The answer lies in the fact that we were able to quantify the error that we make by pretending that a freely falling frame is an inertial framei We found that if we consider a bit of spacetime of size 6 then the fractional error in position or equivalently time measurements is proportional to 62 This acceleration was matched to g AI 2 E 0 e i 77 3It comes from the mathematician Riemann about whom we will say more in the next chapter 74 GOING BEYOND LOCALITY 185 Let me pause here to say that the conceptual setup with which we have sur rounded equation 77 is much like what we nd in calculusi ln calculus we learned that locally any curve was essentially the same as a straight line Of course over a region of finite size curves are generally not straight lines How ever the error we make by pretending a curve is straight over a small finite region is smalli Calculus is the art of carefully controlling this error to build up curves out of lots of tiny pieces of straight lines Similarly the main idea of general relativity is to build up a gravitational eld out of lots of tiny pieces of inertial frames Suppose for example that we wish to compare clocks at the top and bottom of a tall toweri We begin by breaking up this tower into a larger number of short towers each of size All H 51321 If the tower is tall enough the gravitational field may not be the same at the top and bottom 7 the top might be enough higher up that the gravitational field is measurably weakeri So in general each little tower 012 will have a different value of the gravitational field 9 goglggmii lfl is the distance of any given tower from the bottom we might describe this by a function gli 741 A Tiny Tower Let7s compare the rates at which clocks run at the top and bottom of one of these tiny towersi We will try to do this by using the fact that a freely falling frame is much like an inertial framei Of course we will have to keep track of the error we make by doing this Recall that in any accelerating rocket the front and back actually do agree about simultaneityi As a result all of our clocks in the towers will also agree about simultaneityi Thus we can summarize all of the interesting information in a rate function7 pl which tells us how fast the clock at position l runs compared to the clock at position zero pa 78 We wish to consider a gravitational field that does not change with time so that p is indeed a function only ofl and not of ti So let us model our tiny tower as a rigid rocket accelerating through an inertial frame A spacetime diagram drawn in the inertial frame is shown belowi 186 CHAPTER 7 RELATIVITY AND THE GRAVITATIONAL FIELD Now the tiny tower had some acceleration 9 relative to freely falling framesi Let us suppose that we match this to the proper acceleration a of the back of the rocket In this case the back of the rocket will follow a worldline that remains a constant proper distance d 6201 from some xed eventi Note that the top of the rocket remains a constant distance d Al from this event As a result the top of the rocket has a proper acceleration amp r As we have learned this means that the clocks at the top and bottom run at different rates A7301 7 abottom 7 d Al 7 Al A7720an 7 atop 7 d 71 d 79 In terms of our rate function this is just 0 N W Ax Ax 1 710 W W pa Thus we have AP am 711 pl 7 d c2 Now how much of an error would we make if we use this expression for our tiny tower in the gravitational eld Well the above is in fact a fractional change in a time measurement So the error must be of size Al So for our tower case we have Ap 7 glAl pa c for some number kl Here we have replaced a with 9 since we matched the acceleration 91 of our tower relative to freely falling frames to the proper acceleration a MAI 7 12 Chapter 8 General Relativity and Curved Spacetime Read Einstein ch 2329 Appendices 35 In chapter 7 we saw that we could use the equivalence principle to calculate the effects of a gravitational eld over a nite distance by carefully patching together local inertial frames If we are very very careful we can calculate the effects of any gravitational eld in this way However this approach turns out to be a real messi Consider for example the case where the gravitational eld changes with time Then it is not enough just to patch together local inertial frames at different positions One must make a quilt of them at different places as well as at different times 3 31 32 33 ll 22 23 f l ll f 3 X As you might guess this process becomes even more complicated if we consider all 31 dimensions One then nds that clocks at different locations in the gravitational eld may not agree about simultaneity even if the gravitational 197 81 A RETURN TO GEOMETRY 199 Suppose on the other hand that the rock is released to the person s sidei Then Newton would say that both person and rock accelerate toward the center of the earth However this is not in quite the same direction for the person as for the rock I So again there is a relative accelerationi This time however the person nds the rock to accelerate toward her So she would draw a spacetime diagram for this experiment as follows The issue is that we would like to think of the freely falling worldlines as inertial worldlinesi That is we would like to think of them as being straight lines in spacetimei7 However we see that we are forced to draw them on a spacetime diagram as curvedi Now we can straighten out any one of them by using the reference frame of an observer moving along that worldlinei However this makes the other freely falling worldlines appear curvedi How are we to understand this 200 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME 811 Straight Lines in Curved Space Eventually Einstein found a useful analogy with something that at rst sight appears quite different 7 a curved surface The idea is captured by the question What is a straight line on a curved surface77 To avoid language games mathematicians made up a new word for this idea geodesici A geodesic can be thought of as the straightest possible line on a curved surface77 More precisely we can de ne a geodesic as a line of minimal distance 7 the shortest line between two points The idea is that we can de ne a straight line to be the shortest line between two points Actually there is another de nition of geodesic that is even better but requires more mathematical machinery to state preciselyi lntuitively it just captures the idea that the geodesic is straightil It tells us that a geodesic is the path on a curved surface that would be traveled for example by an ant or a person walking on the surface who always walks straight ahead and does not turn to the right or left As an example suppose you stand on the equator of the earth face north and then walk forwardi Where do you go If you walk far enough over the ocean etc you will eventually arrive at the north pole The path that you have followed is a geodesic on the sphere Note that this is true no matter where you start on the equator So suppose there are in fact two people walking from the equator to the north pole Alice and Bob As you can see Alice and Bob end up moving toward each other So if we drew a diagram of this process using Alicels frame of reference so that her own path is straight it would look like this Alice Bab 2Technically a geodesic is a line of locally minimal distance meaning that the line is shorter than any nearby ne 81 A RETURN TO GEOMETRY 201 By the way the above picture is not supposed to be a spacetime diagram It is simply supposed to be a map of part of the two dimensional earth7s surface on which both paths have been drawn This particular map is drawn in such a way that Alice s path appears as a straight line As you probably know from looking at maps of the earth7s surface no at map will be an accurate description globally over the whole earth There will always be some distortion somewhere However a at map is perfectly ne locally say in a region the size of the city of Syracuse if we ignore the hil s Now does this look or sound at all familiar What if we think about a similar experiment involving Alice and Bob walking on a funnelshaped surface Alice BO In this case they begin to drift apart as they walk so that Alice s map would look like this Alice Bab So we see that straight lines geodesics on a curved surface act much like freely falling worldlines in a gravitational eld It is useful to think through this analogy at one more level Consider two people standing on the surface of the earth We know that these two people remain the same distance apart as time passes Why do they do so Because the earth itself holds them apart and prevents gravity from bringing them together The earth exerts a force on each person keeping them from falling freely Now what is the analogy in terms of Alice and Bobls walk across the sphere or the funnel Suppose that Alice and Bob do not simply walk independently but that they are actually connected by a stiff bar This bar will force them to always remain the same distance apart as they walk toward the north pole The point is that in doing so Alice and Bob will be unable to follow their natural geodesic paths As a result Alice and Bob will each feel some push or pull from the bar that keeps them a constant distance apart This is much like our 202 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME two people standing on the earth who each feel the earth pushing on their feet to hold them in place 812 Curved Surfaces are Locally Flat Note that straight lines geodesics on a curved surface act much like freely falling worldlines in a gravitational eld In particular exactly the same prob lems arise in trying to draw a at map of a curved surface as in trying to represent a freely falling frame as an inertial frame A quick overview of the errors made in trying to draw a at map of a curved surface are shown below Naive straight line Bob s geodesic eventually curves away Locally geodesics remain paralell We see that something like the equivalence principle holds for curved surfaces at maps are very accurate in small regions but not over large ones In fact we know that we can in fact build up a curved surface from a bunch of at onesi One example of this happens in an atlas An atlas of the earth contains many at maps of small areas of the earth7s surface the size of states sayi Each map is quite accurate and together they describe the round earth even though a single at map could not possibly describe the earth accurately Computer graphics people do much the same thing all of the time They draw little at surfaces and stick them together to make a curved surface 81 A RETURN TO GEOMETRY 203 This is much like the usual calculus trick of building up a curved line from little pieces of straight lines In the present context with more than one dimension this process has the technical name of differential geometry77 813 From curved space to curved spacetime The point is that this process of building a curved surface from at ones is just exactly what we want to do with gravityl We want to build up the gravitational eld out of little pieces of at inertial frames Thus we might say that gravity is the curvature of spacetime This gives us the new language that Einstein was looking for 1 Global lnertial Frames ltgt Minkowskian Geometry ltgt Flat Spacetime We can draw it on our at paper or chalk board and geodesics behave like straight lines 2 Worldlines of Freely Falling Observers ltgt Straight lines in Spacetime 3 Gravity ltgt The Curvature of Spacetime Similarly we might refer to the relation between a worldline and a line of simul taneity as the two lines being at a right angle in spacetime It is often nice to use the more technical term orthogonal for this relationship By the way the examples spheres funnels etc that we have discussed so far are all curved spaces A curved spacetime is much the same concept Howev er we can t really put a curved spacetime in our 3 D Euclidean space This is because the geometry of spacetime is fundamentally Minkowskian and not Euclidean Remember the minus sign in the interval Anyway what we can do is to once again think about a spacetime diagram for 21 Minkowski space 7 time will run straight up and the two space directions x and y will run to the sides Light rays will move at 45 degree angles to the vertical taxis as usuali With this understanding we can draw a 11 curved spacetime inside this 21 spacetime diagrami An example is shown below 3As opposed to a right angle on a spacetime diagram drawn in a given frame 82 MORE ON CURVED SPACE 205 turns out to create only a few The point is that curvature is fundamentally associated with twodimensional surfaces Roughly speaking the curvature of a fourdimensional spacetime labelle by x y z t can be described in terms of It curvature yt curvature etc associated with twodimensional bits of the spacetime However this is relativity in which space and time act pretty much the same So if there is It yt and 2t curvature there should also be my yz and 12 curvaturel This means that the curvature can show up even if we consider only straight lines in space determined for example by stretching out a string in addition to the effects on the motion of objects that we have already discussed For example if we draw a picture showing spacelike straight lines spacelike geodesics it might look like this Y Two geodesics So curved space is as much a part of gravity as is curved spacetime This is nice as curved spaces are easier to visualize Let us now take a moment to explore these in more depth and build some intuition about curvature in gener Curved spaces have a number of fun properties Some of my favorites are O 27rR The circumference of a circle is typically not 2 times its radius Let us take an example the equator is a circle on a sphere What is its center We are only supposed to consider the twodimensional surface of the sphere itself as the third dimension was just a crutch to let us visualize the curved twodimensional surface So this question is really what point on the sphere is equidistant from all points on the equator7 In fact there are two answers the north pole and the south pole Either may be called the center of the sphere Now how does the distance around the equator compare to the distance measured along the sphere from the north pole to the equator The arc running from the north pole to the equator goes 14 of the way around the sphere This is the radius of the equator in the relevant sense Of course the equator goes once around the sphere Thus its circumference is exactly four times its radius A WRQZ The area of a circle is typically not 7r times the square of its radius Again the equator on the sphere makes a good example With the radius 206 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME de ned as above the area of this circle is much less than NR llll let you work out the math for yourself 0 2angles 1800 The angles in a triangle do not in general add up to 1800 An example on a sphere is shown below ocBylt1807 J ocB7gt1807 gtQ J30 D Woo Squares do not Close A polygon with four sides of equal length and four right angles aka a square in general does not close J 96 90 Vectors arrows parallel transported around Closed curves are rotated This one is a bit more complicated to explain Unfortunately to describe 82 MORE ON CURVED SPACE 207 this property as precisely as the ones above would require the introduc tion of more complicated mathematics Nevertheless the discussion below should provide you with both the avor of the idea and an operational way to go about checking this property In a at space like the 3D space that most people think we live in until they learn about relativityim we now what it means to draw an arrow and then to pick up this arrow and carry it around without turning it The arrow can be carried around so that it always remains parallel to its original direction Now on a curved surface this is not possible Suppose for example that we want to try to carry an arrow around a triangular path on the sphere much like the one that we discussed a few examples back For concreteness let s suppose that we start on the equator with the arrow also pointing along the equator as shown below We now wish to carry this vector to the north pole keeping it always pointing in the same direction as much as we can Well if we walk along the path shown we are going in a straight line and never turningi So since we start with the arrow pointing to our left we should keep the arrow pointing to our left at all times This is certainly what we would do in a at space When we get to the north pole the arrow looks like this 208 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME Now we want to turn and walk toward the equator along a different side of the triangle We turn say to the right but we are trying to keep the arrow always pointing in the same direction So the arrow should not turn with us As a result it points straight behind us We carry it down to the equator so that it points straight behind us at every step Finally we wish to bring the arrow back to where it started We see that the arrow has rotated 900 relative to the original direction All of these features will present in any space say a surface of sirnultaneity in a curved spacetirnei Now since we identify the gravitational eld with the curvature of spacetirne then the above features must also be encoded in the gravitational eld But there is a lot of information in these features In partic ular there are independent curvatures in the my yz and 12 planes that control say the ratio of circumference to radius of circles in these various planesi 83 GRAVITY AND THE METRIC 209 Q But wait doesn7t this seem to mean that the full spacetime curvature gravita tional eld contains a lot more information than just specifying an acceleration g at each point After all acceleration is related to how thing behave in time but we have just realized that at least parts of the spacetime curvature are asso ciated only with space How are we to deal with this For the answer proceed on to the next section below 83 Gravity and the Metric Einstein XXIIIXXVH Let7s recall where we are A while back we discovered the equivalence principle that locally a gravitational eld is equivalent to an acceleration in special rela tivity Another way of stating this is to say that locally a freely falling frame is equivalent to an inertial frame in special relativity We noticed the parallel be tween this principle and the underlying ideas being calculus that locally every curve is a straight line What we found in the current chapter is that this parallel with calculus is actually very direct A global inertial frame describes a flat spacetime 7 one in which for example geodesics follow straight lines and do not accelerate relative to one another A general spacetime with a gravitational eld can be thought of as being curved Just as a general curved line can be thought of as being made up of tiny bits of straight lines a general curved spacetime can be thought of as being made of of tiny bits of at spacetime 7 the local inertial frames of the equivalence principle This gives a powerful geometric picture of a gravitational eld It is nothing else than a curvature of spacetime itself 210 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME Our task for this section is to learn how to describe this in a useful way For example we noticed above that this new understanding of gravity means that the gravitational eld contains more information than just giving an accelera tion at various points in spacetimei The acceleration is related to curvature in spacetime associated with a time direction say in the It plane but there are also parts of the gravitational eld associated with the purely spatial my 12 and yz planesi Letls begin by thinking back to the at spacetime case special relativity What was the object which encoded the at Minkowskian geometry It was the in terval interval2 762 At2 AI This is a special case of something known as a metric which we will explore further in the rest of this section 831 Building Intuition in at space To understand fully what information is contained in the interval it is perhaps even better to think rst about at space for which the analogous quantity is the distance As between two points A82 A12 AyQi Much of the important information in geometry is not the distance between two points per se but the closely related concept of length For example one of the properties of at space is that the length of the circumference of a circle is equal to 2 times the length of its radiusi Now in at space distance is most directly related to length for straight lines the distance between two points is the length of the straight line connecting themi To link this to the length of a curve we need only recall that locally every curve is a straight line In particular what we need to do is to approximate any curve by a set of tiny in nitesimal straight lines Because we wish to consider the limit in which these straight lines are of zero size let us denote the length of one such line by dsi The relation of Pythagoras then tells us that d32 dIQ dy2 for that straight line where dz and dy are the in nitesimal changes in the z and y coordinates between the two ends of the in nitesimal line segment To nd the length of a curve we need only add up these lengths over all of the straight line segments In the language of calculus we need only perform the integral Length d3 dIV dz2 dyQi 81 You may not be used to seeing integrals written in a form like the one above Let me just pause for a moment to note that this can be written in a more familiar form by say taking out a factor of dz from the square root We have 83 GRAVITY AND THE METRIC 211 2 Length Aim dz 1 i 82 So if the curve is given as a function y yz the above formula does indeed allow you to calculate the length of the curve M Now what does this all really mean What is the take home7 lesson from this discussion The point is that the length of every curve is governed by the formu a d82 dz2 dyQi 83 Thus this formula encodes lots of geometric information such that the fact that the circumference of a circle is 2 times its radius As a result 83 will be false on a curved surface like a sphere A formula of the form d32 stuff is known as a metric as it tells us how to measure things in particular it tells us how to measure lengths What we are saying is that this formula will take a different form on a curved surface and will not match with 83 832 On to Angles What other geometric information is there aside from lengths Here you might consider the examples we talked about last time during class that at spaces are characterized by having 1800 in every triangle and by squares behaving nicely So one would also like to know about angles Now the important question is Is information about angles also contained in the metric77 It turns out that it is You might suspect that this is true on the basis of trigonometry which relates angles to ratios of distances Of course trigonom etry is based on at space but recall that any space is locally at and notice that an angle is something that happens at a point and so is intrinsically a local notion To see just how angular information is encoded in the metric let s look at an example The standard Cartesian metric on at space d32 dz2 dy2 is based on an orthogonal7 coordinate system 7 one in which the constant 1 lines intersect the constant y lines at right angles What if we wish to express the metric in terms of z and say some other coordinate 2 which is not orthogonal to z 83 GRAVITY AND THE METRIC 213 coordinates with a tiny spacing so that i is very short or coordinates with a huge spacing so that i is large What the metric tells us directly are the dot products of these vectors 0 i 9117 i gm 3239 39 3239 Hwy 87 Anyway this object gag is called the metric or the metric tensor for the space It te ls us how to measure all lengths and angles The corresponding object for a spacetime will tell us how to measure all proper lengths proper times angles etc It will be much the same except that it will have a time part with 9 negative4 instead of positive as did the at Minkowski spacer Rather than write out the entire expression 86 all of the time especially when working in say four dimensions rather than just two physicists use a condensed notation called the Einstein summation conventionli To see how this works let us rst relabel our coordinatesi Instead of using I and y let7s use 1112 with 11 z and 12 yr Then we have 2 2 d32 Z Zgagdzo dz gagdzo dz i 8 8 a1 1 It is in the last equality that we have used the Einstein summation convention 7 instead of writing out the summation signs the convention is that we implicitly sum over any repeated indexi 834 A rst example To get a better feel for how the metric works lets look at the metric for a at plane in polar coordinates 7 9 It is useful to think about this in terms of the unit vectors 729 4More technically 9 should have one negative and three positive eigenvalues at each point in spacetime 214 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME From the picture above we see that these two vectors are perpendicular T 0 Normally we measure the radius in terms of length so that T has length one and T T l The same is not true for 9 one radian of angle at large T corresponds to a much longer arc than does one radian of angle at small 7 In fact one radian of angle corresponds to an arc of length 7 The result is that 9 has length T and 3 7 2 So for theta measured in radians and running from 0 to 2 the metric turns out to be d32 dT2 7 2d62 89 Now let s look at a circle located at some constant value of 7 r const To nd the circumference of the circle we need to compute the length of a curve along the circle Now along the circle T does not change so we have T 0 Recall that dT is just the in nitesimal version of AT So we have d3 Tdt Thus the length is 27r 27r C d3 Tdt 27T7 810 0 0 Letls check something that may seem trivial What is the Tadz39us of this ciTcle The radius R is the length of the curve that runs from the origin out to the circle along a line of constant 9 Along this line we have d9 0 So along this curve we have d3 dT The line runs from T 0 to T T so we have R dT7 811 0 So we do indeed have C 27rR Note that while the result R T may seem obvious it is true only because we used an T coordinate which was marked off in terms of radial distance In general this may not be the case There are times when it is convenient to use a radial coordinate which directly measures something other than distance from the origin and in such cases it is very important to remember to calculate the actual Radiusl the distance from the origin to the circle using the metric 220 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME region As a result we will be describing the gravitational eld of an object the earth a star etci only in the region outside of the object This would describe the gravitational eld well above the earth7s surface but not down in the interior For this case the Einstein equations were solved by a young German mathemati cian named Schwarzschildi There is an interesting story here as Schwarzschild solved these equations during his spare time while he was in the trenches ghting on the German side in World War If I believe the story is the Schwarzschild got his calculations published but by the time this happened he had been killed in the war Because of the spherical symmetry it was simplest for Schwarzschild to use what are called spherical coordinates Tt9 as opposed to Cartesian Coordinates z y Here 7 tells us how far out we are and 9 45 are latitude and longitude coordinates on the sphere at any value of Ti r10 r20 Schwarzschild found that for any spherically symmetric spacetime and outside of the matter the metric takes the form 2 Rs 2 dTQ 2 2 2 2 d3 7 7 17 dt 1 R 7quot d9 s1n Odqb 817 T 7 s r Here the parameter R5 depends on the total mass of the matter insider In particular it turns out that R5 QMGcz The last part of the metric T2dt92 sin2 9d 2 is just the metric on a standard sphere of radius Ti This part follows just from the spherical symmetry itselfi Recall that 9 is a latitude coordinate and 45 is a longitude coordinatei The factor of sin2 9 encodes the fact that circles at constant 9 ie with d9 0 are smaller near the poles t9 07r than at the equator 222 CHAPTER 8 GENERAL RELATIVITY AND CURVED SPACETIME Actually there is an interesting story about Mercury and its orbit Astronomers had been tracking the motion of the planets for hundreds of years Ever since Newton they had been comparing these motions to what Newton7s law of grav ity predictedl The agreement was incredible In the early 1800s they had found small dis crepancies 30 seconds of arc in 10 years in the motion of Uranus For awhile people thought that Newton7s law of gravity might not be exactly right How ever someone then had the idea that maybe there were other objects out there whose gravity affected Uranusi They used Newton7s law of gravity to predict the existence of new planets Neptune and later Plutoi They could even tell astronomers where to look for Neptune within about a degree of angle on the sky However there was one discrepancy with Newton7s laws that the astronomers could not explain This was the precession of Mercuryls orbit The point is that if there were nothing else around Newton7s law of gravity would say that Mercury would move in a perfect ellipse around the sun retracing its path over and over and over Mercury Of course there are small tugs on Mercury by the other planets that modify this behavior However the astronomers new how to account for these effects Their results seemed to say that even if the other planets and such were not around Mercury would do a sort of spiral dance around the sun following a path that looks more like this Ellipse does not close M er cury Here I have drawn the ellipse itself as rotating aikiai precessing7 about the sun After all known effects had been taken into account astronomers found that Mercury s orbit precessed by an extra 45 seconds of are per century This is certainly not very much but the astronomers already understood all of the other planets to a much higher accuracy So what was going wrong with Mercury Most astronomers thought that it must be due to some sort of gas or dust 85 EXPERIMENTAL VERIFICATION OF GR 223 surrounding the Sun a big solar atmosphere7 that was somehow affecting Mercury s orbiti However Einstein knew that his new theory of gravity would predict a preces sion of Mercuryls orbit for two reasons First he predicted a slightly stronger gravitational eld since the energy in the gravitational eld itself acts as a source of gravity Second in Einstein7s theory space itself is curved and this effect will also make the ellipse precessi The number that Einstein calculated from his theory was 43 seconds of arc per centuryi That is his prediction agreed with the experimental data to better than 1 Clearly Einstein was thrilled This was big news However it would have been even bigger news if Einstein had predicted this result before it had been measured Physicists are always skeptical of just explaining known effectsi After all maybe the scientist intentionally or not fudged the numbers or the theory to get the desired result So physicists tend not to really believe a theory until it predicts something new that is then veri ed by experiments This is the same sort of idea as in double blind medical trials where even the researchers don7t know what effect they want a given pill to have on a patient 852 The Bending of Starlight Luckily Einstein had an idea for such an effect and now had enough con dence in his theory to push it through The point is that as we have discussed light will fall in a gravitational eld For example a laser beam red horizontally across the classroom will be closer to the ground on the side where it hits the far wall it was when it left the laser Similarly a ray of light that goes skimming past a massive object like the sun will fall a bit toward the sun The net effect is that this light ray is bent Suppose that the ray of light comes from a start What this means in the end is that when the Sun is close to the line connecting us with the star the star appears to be in a slightly different place than when the Sun is not close to that light rayi For a light ray that just skims the surface of the Sun the effect is about 875 seconds of arc star appears to be here earth actual star 39 39 actual path falls toward the Sun Sun However this is not the end of the story It turns out that there is also another effect which causes the ray to bend This is due to the effect of the curvature of space on the light rayi This effect turns out to be exactly the same size as the rst effect and with the same sign As a result Einstein predicted a total 92 ON BLACK HOLES 237 This gives us a natural guess for what is going on near the Schwarzschild radiusi In fact let us recall that any curved spacetime is locally ati So if our frame work holds together at the Schwarzschild radius we should be able to match the region near 7 R5 to some part of Minkowski spacer Perhaps we should match it to the part of Minkowski space near an acceleration horizon Let us guess that this is correct and then proceed to check our answer We will check our answer using the equivalence principle The point is that an accelerating coordinate system in at spacetime contains an apparent gravita tion e ere is some nontrivial proper acceleration a that is required to remain static at each position Furthermore this proper acceleration is not the same at all locations but instead becomes in nitely large as one approaches the horizon What we want to do is to compare this apparent gravitational eld the proper acceleration 13 where s is the proper distance from the horizon near the acceleration horizon with the corresponding proper acceleration 13 required to remain static a small proper distance 8 away from the Schwarzschild radiusi If the two turn out to be the same then this will mean that static observers have identical experiences in both cases But the experiences of static observers are related to the experiences of freely falling observersi Thus if we then consider freely falling observers in both cases they will also describe both situations in the same way It will then follow that physics near the event horizon is identical to physics near an acceleration horizon 7 something that we understand well from special relativity Recall that in at spacetime the proper acceleration required to maintain a constant proper distance 8 from the acceleration horizon eg from event Z is given see section 513 y a 623 911 240 CHAPTER 9 BLACK HOLES Future Horizon Future Interlor r R S Right Left 7 Exterior EXtSrlOf r 9 00 Past Interior Past Horizon 925 A summary of where we are Let us review our discussion so fari We realized that so long as we were outside the matter that is causing the gravitational eld any spherically symmetric aikiai round7 gravitational eld is described by the Schwarzschild metrici This metric has a special place at T R5 the Schwarzschild Radius7 Any object which is smaller than its Schwarzschild Radius will be surrounded by an event horizon and we call such an object a black hole If we look far away from the black hole at T gt R5 then the gravitational eld is much like what Newton would have predicted for an object of that mass There is of course a little gravitational time dilation and a little curvature4 but not much Indeed the Schwarzschild metric describes the gravitational eld not only of a black hole but of the earth the Sun the moon and any other round objecti However for those more familiar objects the surface of the object is at T gtgt Rsi For example on the surface of the Sun TRs N 5 X 105 So far from a black hole objects can orbit just like planets orbit the Sun By the way remember that orbiting objects are freely falling 7 they do not require rocket engines or other forces to keep them in orbit However suppose that we look closer in to the horizon What happens then In your recent homework you saw that something interesting happens to orbit ing objects when they orbit at T 31352 There an orbiting object experiences no proper time 739 0 This means that the mint at this Tadz39us is a light like path In other words a ray of light will orbit the black hole in a circle at 4dCdR is a bit less than 27L 92 ON BLACK HOLES 241 T SRSZi For this reason this region is known as the photon spherei7 This makes for some very interesting visual effects if you would imagine traveling to the photon spherei A few years ago NASA funded a guy to make some nice computer generated movies showing how this would look He hasn7t included is the effect of gravity on the color of light but otherwise what he has done gives a very good impression of what you would see You can nd his movies5 at httpantwrpigsfcinasalgovhtmltestrjnbhtihtmli This is not to say that light cannot escape from the photon sphere The point is that if the light is moving straight sideways around the black hole then the black holes gravity is strong enough to keep the light from moving farther away However if the light were directed straight outward at the photon sphere it would indeed move outward and would eventually escapei And what about closer in at T lt SRSQ Any circular orbit closer in is spacelike and represents an object moving faster than the speed of light So given our usual assumptions about physics nothing can orbit the black hole closer than T SEQ2i Any freely falling object that moves inward past the photon sphere will continue to move to smaller and smaller values of Ti However if it ceases to be freely falling by colliding with something or turning on a rocket engine then it can still return to larger values of Ti Now suppose that we examine even smaller T and still have not run into the surface of an object that is generating the gravitational eld If we make it all the way to T R5 without hitting the surface of the object we nd a horizon and we call the object a black hole Recall that even though it is at a constant value of T the hoTizon contains the onldlines of outwaTd diTected light Tays To see what this means imagine an expanding sphere of light like one of the ones produced by a recracker at the horizon Although it is moving outward at the speed of light which is in nite boost parameter the sphere does not get any bigger The curvature of spacetime is such that the area of the spheres of light do not increase A spacetime diagram looks like this 5The movies are in MPEG format so if you don7t already have an MPEG player then you may need to install one on your computer to see them 242 CHAPTER 9 BLACK HOLES Here I have used arrows to indicate the direction in which the time coordinate t increases on this diagrami Not only do light rays directed along the horizon remain at T R5 any light ray at the horizon which is directed a little bit sideways and not perfectly straight outward cannot even stay at T R5 but must move to smaller T The diagram below illustrates this by showing the horizon as a surface made up of light rays If we look at a light cone emitted from a point on this surface only the light ray that is moving in the same direction as the rays on the horizon can stay in the surface The other light rays all fall behind the surface and end up inside the black hole at T lt R5 Similarly any object of nonzero mass requires an in nite acceleration directed straight outward to remain at the horizon With any nite acceleration the object falls to smaller values of Ti At any value of T less than 5 no object can ever escape from the black hole This is clear from the above spacetime 93 BEYOND THE HORIZON 243 diagram since to move from the future interior to say the right exterior the object would have to cross the light ray at T R5 which is not possible 93 Beyond the Horizon Of course the question that everyone would like to answer is What the heck is going on inside the black hole77 To understand this we will turn again to the Schwarzschild metrici In this section we will explore the issue in quite a bit of detail and obtain several useful perspectives 931 The interior diagram To make things simple let s suppose that all motion takes place in the Tt plane This means that d9 dab 0 and we can ignore those parts of the metric The relevant pieces are just d82 717 gs0d 916 dT2 l 7 RsT Let7s think for a moment about a line of constant T with dT 0 For such a line d32 71 7 RsTdt2i The interesting thing is that for T lt R5 this is positive Thus for T lt R5 a line of constant T is spacelike You will therefore not be surprised to nd that near the horizon the lines of constant T are just like the hyperbolae that are a constant proper time from where the two horizons meet rRS rgtRS rRS The coordinate t increases along these lines in the direction indicated by the arrows This means that the t direction is actually spacelike inside the black hole The point here is not that something screwy is going on with time inside a black hole Instead it is merely that using the Schwarzschild metric in the 93 BEYOND THE HORIZON 245 in the exterior since the symmetry means that they are all the same Two such slices are shown belowi Note that they extend into both the right exterior7 with which we are familiar and the left exterior a region about which we have so far said littlei constant t slice another constantt slice r RS lgnoring say the 9 direction and drawing a picture of 7 and 45 at the equator t9 7r2 any constant t slice looks like this This is the origin of the famous idea that black holes can connect our universe right exterior to other universes left exterior or perhaps to some distant region of our own universei If this idea bothers you donlt worry too much as we will discuss later the other end of the tunnel is not really present for the black holes commonly found in nature Note that the left exterior looks just like the right exterior and represents another region outside7 the black hole connected to the rst by a tunneli This tunnel is called a wormhole7 or EinsteinRosen bridgei7 So what are these spheres inside the black hole that I said shrink with time They are the throat7 of the wormholei Gravity makes the throat shrink and 246 CHAPTER 9 BLACK HOLES begin to pinch of Does the throat ever pinch off completely That is does it collapse to T 0 in a nite proper time We can nd out from the metric Lets see what happens to a freely falling observer who falls from where the horizons cross at T R5 to T 0 where the spheres are of zero size and the throat has collapsed Our question is whether the proper time measured along such a worldline is nite Consider an observer that starts moving straight up the diagram as indicated by the dashed line in the gure below We rst need to gure out what the full worldline of the freely falling observer will be Suppose worldline of free faller starts rRS Will the freely falling worldline curve to the left or to the right Recall that since t is the space direction inside the black hole this is just the question of whether it will move to larger t or smaller ti What do you think will happen Well our diagram is exactly the same on the right as on the left so there seems to be a symmetTyi In fact you can check that the Schwarzschild metric is unchanged if we replace t by it So both directions must behave identically If any calculation found that the worldline bends to the left then there would be an equally valid calculation showing that the worldline bends to the right As a result the freely falling worldline will not bend in either direction and will remain at a constant value of ti So we can compute the proper time by using a worldline with dt 0 For such a worldline the metric yields dT2 T 277 2 2 d7 7 d3 RSTil R5 deT i 917 lntegrating we have 0 T Tde7 R577 918 93 BEYOND THE HORIZON 247 It is not important to compute this answer exactlyi What is important is to notice that the answer is nite We can see this from the fact that near T m R5 the integral is much like near I 0 This latter integral integrates to J and is nite at z 0 Also near T 0 the integral is much like REde which clearly gives a nite resulti Thus our observer measures a nite proper time between T R5 and T 0 and the throat does collapse to zero size in nite timer 932 The Singularity This means that we should draw the line T 0 as one of the hyperbolae on our digrami It is clearly going to be a rather singular line7 to paraphrase Sherlock Holmes again and we will mark it as special by using a jagged line As you can see this line is spacelike and so represents a certain time We call this line the singularity Note that this means that the singularity of a black hole is not a place at a l The singularity is most properly thought of as being a very special time at which the entire interior of the black hole squashes itself and everything in it to zero sizei Note that since it cuts all of the way across the future light cone of any events in the interior such as event A below there is no way for any object in the interior to avoid the singularity rRS r R 5 By the way this is a good place to comment on what would happen to you if you tried to go from the right exterior to the left exterior through the wormholei Note that once you leave the right exterior you are in the future interior regioni From here there is no way to get to the left exterior without moving faster than light Instead you will encounter the singularity What this means is that the wormhole pinches off so quickly that even a light ray cannot pass through it from one side to the other It turns out that this behavior is typical of wormholesi Lets get a little bit more information about the singularity by studying the motion of two freely falling objects As we have seen some particularly simple 248 CHAPTER 9 BLACK HOLES geodesics inside the black hole are given by lines of constant ti 1 have drawn two of these at t1 and t2 on the diagram belowi rR S One question that we can answer quickly is how far apart these lines are at each T say measured along the line T const That is What is the proper length of the curve at constant T from t t1 to t t277 Along such a curve dT 0 and we have d32 RsTil t So 8 t1 7t2 R57 7 1 As T A 0 the separation becomes in nite Since a freely falling object reaches T 0 in nite proper time this means that any two such geodesics move in nitely far apart in a nite proper time It follows that the relative acceleration aka the gravitational tidal force diverges at the singularity This means that the spacetime curvature also becomes in nite Said differently it would take an in nite proper acceleration acting on the objects to make them follow non geodesic paths that remain a nite distance apart Physically this means that it requires an in nite force to keep any object from being ripped to shreds near the black hole singularity 933 Beyond the Singularity Another favorite question is what happens beyond after the singularity77 The answer is not at all clear The point is that just as Newtonian physics is not valid at large velocities and as special relativity is valid only for very weak spacetime curvatures we similarly expect General Relativity to be an incomplete description of physics in the realm where curvatures become truly enormousi This means that all we can really say is that a region of spacetime forms where the theory we are using General Relativity can no longer be counted on to correctly predict what happens The main reason to expect that General Relativity is incomplete comes from another part of physics called quantum mechanics Quantum mechanical effect s should become important when the spacetime becomes very highly curvedi Roughly speaking you can see this from the fact that when the curvature is strong local inertial frames are valid only over very tiny regions and from the 93 BEYOND THE HORIZON 249 fact the quantum mechanics is always important in understanding how very small things work Unfortunately no one yet understands just how quantum mechanics and gravity work together We say that we are searching for a theory of quantum gravity77 It is a very active area of research that has led to a number of ideas but as yet has no de nitive answersi This is in fact the area of my own research Just to give an idea of the range of possible answers to what happens at a black hole singularity it may be that the idea of spacetime simply ceases to be meaningful there As a result the concept of time itself may also cease to be meaningful and there may simply be no way to properly ask a question like What happens after the black hole singularity77 Many apparently paradoxical questions in physics are in fact disposed of in just this way as in the question which is really longer the train or the tunnel In any case one expects that the region near a black hole singularity will be a very strange place where the laws of physics act in entirely unfamiliar waysi 934 The rest of the diagram and dynamical holes There still remains one region of the diagram the past interior about which we have said littlei Recall that the Schwarzschild metric is time symmetric under t A it As a result the diagram should have a topbottom symmetry and the past interior should be much like the future interiori This part of the spacetime is often called a white hole7 as there is no way that any object can remain inside everything must pass outward into one of the exterior regions through one of the horizonsl rRS rRS rgtRS rgtRS 1quotRs rR S As we mentioned brie y with regard to the second exterior the past interior does not really exist for the common black holes found in nature Lets talk about how this works So far we have been studying the pure Schwarzschild solution As we have discussed it is only a valid solution in the region in which no matter is present Of course a little bit of matter will not change the picture muchi However if the matter is an important part of the story for example 250 CHAPTER 9 BLACK HOLES if it is matter that causes the black hole to form in the rst place then the modi cations will be more important Let us notice that in fact the hole whether white or black in the above space time diagram has existed since in nitely far in the past If the Schwarzschild solution is to be used exactly the hole including the wormhole must have been created at the beginning of the universe We expect that most black holes were not created with the beginning of the universe but instead formed later when too much matter came too close together Recall that a black hole must form when for example too much thin gas gets clumped togetheri Once the gas gets into a small enough region smaller than its Schwarzschild radius we have seen that a horizon forms and the gas must shrink to a smaller size No nite force and in some sense not even in nite force can prevent the gas from shrinking Now outside of the gas the Schwarzschild solution should be valid So let me draw a worldline on our Schwarzschild spacetime diagram that represents the outside edge of the ball of gas This breaks the diagram into two pieces an outside that correctly describes physics outside the gas and an inside that has no direct physical relevance and must be replaced by something that depends on the details of the matter r R S This part of the Eternal Black Hole is not relevant to a black OUtSlde Edge Of the matter hole that forms from the couapse Of maner39 Here the Schwarzschild solution correctly describes the spacetime r Rs r R S We see that the second exterior7 and the past interior7 are in the part of the diagram with no direct relevance to relevance to black holes that form from collapsing matter A careful study of the Einstein equations shows that inside the matter the spacetime looks pretty normal A complete spacetime diagram including both then region inside the matter and the region outside would look like this 93 BEYOND THE HORIZON 251 Schwarzschild here Not Schwarzschild r R along Horizon from here on out here S Outside Edge of the matter r 0 Center of the matter 935 Visualizing black hole spacetimes We have now had a fairly thorough discussion about Schwarzschild black holes include the outside the horizon the inside and the extra regions77 second exterior and past interiori One of the things that we emphasized was that the spacetime at the horizon of a black hole is locally at just like everywhere else in the spacetime Also the curvature at the horizon depends on the mass of the black hole The result is that if the black hole is large enough the spacetime at the horizon is less curved than it is here on the surface of the earth and a person could happily fall through the horizon without any discomfort yet It is useful to provide another perspective on the various issues that we have discussed The idea is to draw a few pictures that I hope will be illustrative The point is that the black hole horizon is an effect caused by the curvature of spacetime and the way that our brains are most used to thinking about curved spaces is to visualize them inside of a larger at space For example we typically draw a curved twodimensional sphere as sitting inside a at t ree dimensional space Now the Tt plane of the black hole that we have been discussing and drawing on our spacetime diagrams forms a curved twodimensional spacetimei It turns out that this twodimensional spacetime can also be drawn as a curved surface inside of a at threedimensional spacetz39mei These are the pictures that we will draw and explore in this section To get an idea of how this works let me rst do something very simple I will draw a flat twodimensional spacetime inside of a at threedimensional spacetimei As usual time runs up the diagram and we use units such that light rays move along lines at 450 angles to the vertical Note that any worldline of a zsz gamma 9 BLACK HOLES hshnay mumbemmmmw mymmeznswxmmmw Ame 5m ban whaethe hm me am an Mm 6345 m 5 New um wewem the xdaa m 5w yum apmnre um 12png me 27D m pbueolonrblaakhal dravmasamlrv summsld aQDMWE n lack like ms 3 3 BEYOND m5 HORIZON 253 new x We am m m 4m ulmnsbm 7 so um yum m mule um mm as quotmum my 7 5 mm m quot2 mm mm magma m hum mm m 4mm Wang v 0m memsmsm 1m mam and at 12png me ruin 2mm mg m mm Impunam um to name 5 um we 4 m um 1 ememm 0139th mm an m M and 7 12mm mum manned 2 5mm A am My 39m mpst wuzldlm a They We as spot m 1 km xvs m m bb4k hole 5mm 1e madmi um ya law 1m m 25A gamma 9 BLACK HOLES me meme shaved yum ebwe emebieekieiemneeedeie on me any New We W2 May we not me e M WWW mm We whee 97 Time 1e we humans ofthebbak Me Anmlia dung we we see m age dusxams is me symmatry we dmss neswa eme 70 Mime meg me mesmebeeewsym metry e we 115a an In M Mae me Meek we we symmen39y me ems m e spasme dmm We eee 4e see m ms Mme w mae me Wide mesme wee Ame w Mme M Tidal effects in 94 Stretching and Squishin General Relativity e e w enemy in smeei Mammy whae gm by new gig me 5 dung sq mg a mob We Meek hm kpmblms mm e ee em mew has ebeemm away was um gmy Gauss vi a mg ubsmmswmmate elem to We owe m to new mesmeaug w vwmgwuzldhnsmybaidw e w bewaeweyx pmblmizidzlaskediww mame meeeeee me His Eel me m The w 268 CHAPTER 9 BLACK HOLES into a disk as shown in the picture7 belowi This is why astronomers often talk about accretion disks7 around black holes and neutron stars Now an important point is that a lot of energy is released when matter falls toward a black hole Why does this happen Well as an object falls its speed relative to static observers becomes very large When many such of matter bump into each other at high these speeds the result is a lot of very hot matter This is where those Xrays come from that I mentioned awhile back The matter is hot enough that Xrays are emitted as thermal radiationi By the way it is worth talking a little bit about just how we can calculate the extra kinetic energy7 produced when objects fall toward black holes or neutron stars To do so we will run in reverse a discussion we had long ago about light falling in a gravitational eld Do you recall how we rst argued that there must be something like a gravi tational time dilation effect It was from the observation that a photon going upward through a gravitational eld must loose energy and therefore decrease in frequency Well let s now think about a photon that falls down into a grav itational eld from far away to a radius Ti Recall that clocks at 7 run slower than clocks far away by a factor of x l 7 RsTi Since the lower clocks run more slowly from the viewpoint of these clocks the electric eld of the photon seems to be oscillating very quickly So this must mean that the frequency of the photon measured by a static clock at 7 is higher by a factor of 141 7 RsT than when the frequency is measured by a clock far away Since the energy of a photon is proportional to its frequency the energy of the photon has increased by 1x1 RsTi Now in our earlier discussion of the effects of gravity on light we noted that the energy in light could be turned into any other kind of energy and could then be turned back into light We used this to argue that the effects of gravity on light must be the same as on any other kind of energy So consider an object of mass m which begins at rest far away from the black hole It contains an energy me So by the time the object falls to a radius 7 its energy measured locally must have increased by the same factor was would the energy of a photon to E mc2lMl 7 RSTi What this means is that if the object gets 7This picture was created by Jillian Bornak a past PHY312 student 272 CHAPTER 9 BLACK HOLES Positive energy escapes as rRS 962 Penrose Diagrams or How to put in nity in a box77 There are a few comments left to make about black holes and this will re quire one further technical tooli The tool is yet another kind of spacetime diagram called a Penrose diagraml and it will be useful both for discussing more complicated kinds of black holes and for discussing cosmology in chapter 10 Actually it is not all that technicali The point is that as we have seen it is often useful to compare what an observer very far from the black hole sees to what one sees close to the black hole We say that an observer very far from the black hole is at in nity77 Comparing in nity with nite positions is even more important for more complicated sorts of black holes that we have not yet discussed However it is difficult to draw in nity on our diagrams since in nity is after all in nitely far away How can we draw a diagram of an in nite spacetime on a nite piece of paper Think back to the Escher picture of the Lobachevskian space By squishingl the space Escher managed to draw the in nitely large Lobachevskian space inside a nite circle If you go back and try to count the number of sh that appear along on a geodesic crossing the entire space it turns out to be in nitei Its just that most of the sh are drawn incredibly smalli Escher achieved this trick by letting the scale vary across his map of the space In particular at the edge an in nite amount of Lobachevskian space is crammed into a very tiny amount of Escher s map In some sense this means that his picture becomes in nitely bad at the edge but nevertheless we were able to obtain useful information from it We want to do much the same thing for our spacetimesi However for our case there is one catch As usual we will want all light rays to travel along lines at 45 degrees to the vertical This will allow us to continue to read useful information from the diagrami This idea was rst put forward by Sir Roger Penroselo so that the resulting pictures are often called Penrose Diagrams77 They are 10Penrose is a mathematician and physicist who is famous for a number of things 96 BLACK HOLE ODDS AND ENDS 273 also called conformal diagrams77 7 conformal is a technical word related to the rescaling of size Letls think about how we could draw a Penrose diagram of Minkowski space For simplicity let s consider our favorite case of 11 dimensional Minkowski spacer Would you like to guess what the diagram should look like As a rst guess we might try a square or rectangle However this guess has a problem associated with the picture belowi To see the point consider any light ray moving to the right in 11 Minkowski space and also consider any light ray moving to the left Any two such light rays are guaranteed to meet at some event The same is in fact true of any pair of leftward and rightward moving objects since in 1 space dimension there is no room for two objects to pass each otherl Le and right moving objects always collide when space has only one dimension However if the Penrose diagram for a spacetime is a square then there are in fact leftward and rightward moving light rays that never meetl Some examples are shown on the diagram belowi These light rays do not meet So the rectangular Penrose diagram does not represent Minkowski spacer What other choices do we have A circle turns out to have the same problemi After a little thought one nds that the only thing which behaves differently is a diamond That is to say that in nity or at least most of it is best associated not which a place or a time but with a set of light raysl 1n 31 dimensions we can as usual decide to draw just the Tt coordinatesi In this case the Penrose diagram for 31 Minkowski space is drawn as a halfdiamond 274 CHAPTER 9 BLACK HOLES 963 Penrose Diagrams for Black holes Using the same scheme we can draw a diagram that shows the entire spacetime for the eternal Schwarzschild black hole Remember that the distances are no longer represented accuratelyl As a result some lines that used to be straight get bentul For example the constant 7 curves that we drew as hyperbolae before appear somewhat different on the Penrose diagraml However all light rays still travel along straight 45 degree lines The result is r 0 r0 As you might guess I did not introduce Penrose diagrams just to draw a new diagram for the Schwarzschild black hole It turns out though that Schwarzschild black holes are not the only kind of black holes that can exist Recall that the Schwarzschild metric was correct only outside of all of the matter7 which means anything other than gravitational elds and only if the matter was spherically symmetric round7l Another interesting case to study occurs when we add a little bit of electric charge to a black hole In this case the charge creates an electric eld which will ll all of space This electric eld carries energy and so is a form of matterf Since we can never get out beyond all of this electric eld the Schwarzschild metric by itself is never quite valid in this spacetimel lnstead the spacetime is described by a related metric called the ReissnerNor39dster RN metric The Penrose diagram for this metric is shown below 11 his is the same effect that one nds on at maps of the earth where lines that are really straight geodesics appear curved on the map 96 BLACK HOLE ODDS AND ENDS 275 JVAVAVAVAVAVAVAF H H 0 Actually this is not the entire spacetimeim the dots in the diagram above indi cate that this pattern repeats in nitely both to the future and to the past This diagram has many interesting differences when compared to the Schwarzschild diagrami One is that the singularity in the RN metric is timelike instead of be ing spacelikei Another is that instead of there being only two exterior regions there are now in nitely many The most interesting thing about this diagram is that there does exist a timelike worldline describing an observer that travels more slowly than light that starts in one external region falls into the black hole and then comes back out through a past horizon7 into another external regioni Actually is possible to consider the successive external regions as just multiple copies of the same external regioni 276 CHAPTER 9 BLACK HOLES In this case the worldline we are discussing takes the observer back into the same universe but in such a way that they emerge to the past of when the entered the black holel ww However it turns out that there is an important difference between the Schwarzschild metric and the RN metric The Schwarzschild metric is stable This means that while the Schwarzschild metric describes only an eternal black hole in a space time by itself without for example any rocket ships near by carrying observers who study the black hole the actual metric which would include rocket ships falling scientists and students and so on can be shown to be very close to the Schwarzschild metricl This is why we can use the Schwarzschild metric itself to discuss what happens to objects that fall into the black hole It turns out though that the RN metric does not have this property The exterior is stable but the interior is not This happens because of an effect illustrated on the diagram belowl Suppose that some energy say a light wave falls into the black hole From the external viewpoint this is a wave with a long wavelength and therefore represents a small amount of energy The two light rays drawn below are in fact in nitely far apart from the outside perspective illustrating that the wave has a long wavelength when it is far away 96 BLACK HOLE ODDS AND ENDS 277 lt lt T g lt lt lt gt ltgt Inner horizon gt gt lt gt lt gt lt t 00 39 llL39llI 39d separation V JAAAAAAA However7 inside the black hole7 we can see that the description is different Now the two light rays have a nite separation This means that that near the light ray marked inner horizon77 what was a long wavelength light ray outside is now of very short wavelength7 and so very high energy In fact7 the energy created by any small disturbance will become in nite at the inner horizoni77 It will come as no surprise that this in nite energy causes a large change in the spacetime The result is that dropping even a small pebble into an RN black hole creates a big enough effect at the inner horizon to radically change the Penrose diagrami The Penrose diagram for the actual spacetime containing an RN black hole together with even a small disturbance looks like this 278 CHAPTER 9 BLACK HOLES Some of the researchers who originally worked this out have put together a nice readable website that you might enjoy It is located at httpwww theorieiphysikiunizhich drozinsidei Actually I have to admit that no one believes that real black holes in nature will have a signi cant electric charge The point is that a black hole with a signi cant say positive charge will attract other negative charges which fall in so that the nal object has zero total c argei However real black holes do have one property that turns out to make them quite different from Schwarzschild black holes they are spinning and typically do so quite rapidlyi Spinning black holes are not round but become somewhat disk shaped as do all other spinning objects As a result they are not described by the Schwarzschild metric The spacetime that describes a rotating black hole is called the Kerr metrici There is also of course a generalization that allows both spin and charge and which is called the KerrNewman metrici It turns out that the Penrose diagram for a rotating black hole is much the same as that of an RN black hole but with the technical complication that rotating black holes are not roundi One nds the same story about an unstable inner horizon in that context as well with much the same resolution I would prefer not to go into a discussion of the details of the Kerr metric because of the technical complications involved but it is good to know that things basically work just the same as for the RN metric abovei 964 Some Cool Stuff Other Relativity links In case you havenlt already discovered them the SU Rel ativity Group the group that does research in Relativity maintains a page of Relativity Links at httpphysicsisyrieduresearchrelativityRELATlVlTYihtmli The ones under Visualizing Relativity7 httpphysicsisyrieduresearchrelativityRELATlVlTYihtmlVisualizing Rel ativity can be a lot of fun 282 CHAPTER 9 BLACK HOLES Sketch a worldline on this diagram describing you falling into the black hole as described above b Describe what people on the rocket see as they watch you fall toward the black hole c Describe what you feel and see as you fall in Is there a difference between falling into a large7 black hole and a small7 black hole If so explain what it is Hintz Have you read section 94 on tidal forces yeti For each of the problems below the term black hole7 refers to the round Schwarzschild black hole that we have been studying in class 5 Use your knowledge of static observers in a gravitational eld to answer the following questions a Is it possible for a rocket to remain static ie to remain at constant 7 t9 at the photon sphere b If you were placed in such a rocket would you remain alive Hint ow heavy would you feel If the answer depends on the black hole describe roughly for which black holes you would survive and for which you would not 6 Remember that near the horizon of a black hole a good picture of space time looks like 286 CHAPTER 10 COSMOLOGY Well the stars are not in fact evenly sprinkled We now know that they are clumped together in galaxies And even the galaxies are clumped together a bit However if one takes a suf ciently rough average then it is basically true that the clusters of galaxies are evenly distributed We say that the universe is homogeneous Homogeneous is just a technical word which means that every place in the universe is the same 1011 Homogeneity and Isotropy In fact there is another idea that goes along with every place being essentially the same This is the idea that the universe is the same in every direction The technical word is that the universe is isotropic To give you an idea of what this means I have drawn below a picture of a universe that is homogeneous but is not isotropic 7 the galaxies are farther apart in the vertical direction than in the horizontal direction In contrast a universe that is both homogeneous and isotropic must look roughly like this 1012 That technical point about Newtonian Gravity in Homogeneous Space By the way we can use the picture above to point out that technical problem I mentioned with Newtonian Gravity in in nite space I will probably skip this part in class but it is here for your edi cation The point is that to compute the gravitational eld at some point in space we need to add up the contributions from all of the in nitely many galaxies This is an in nite sum When you discussed such things in your calculus class you 101 THE COPERNICAN PRINCIPLE AND RELATIVITY 287 learned that some in nite sums converge and some do not Actually this sum is one of those interesting inbetween cases where the sum converges if you set it up right but it does not converge absolutely What happens in this case is that you can get different answers depending on the order in which you add up the contributions from the various objects To see how this works recall that all directions in this universe are essentially the same Thus there is a rotational symmetry and the gravitational eld must be pointing either toward or away from the center Now it turns out that New tonian gravity has a property that is much like Gauss7 law in electromagnetism In the case of spherical symmetry the gravitational eld on a given sphere de pends only on the total charge inside the sphere This makes it clear that on any given sphere there must be some gravitational eld since there is certainly matter inside But what if the sphere is very small Then there is essentially no matter inside so the gravitational eld will vanish So at the center the gravitational eld must vanish but at other places it does not But now we recall that there is no center This universe is homogeneous mean ing that every place is the same So if the gravitational eld vanishes at one point it must also vanish at every other point This is what physicists call a problem However Einstein7s theory turns out not to have this problem In large part this is because Einstein7s conception of a gravitational eld is very d ifferent from Newton7s In particular Einstein7s conception of the gravitational eld is local while Newton7s is not 1013 Homogeneous Spaces Now in general relativity we have to worry about the curvature or shape of space So we might ask what shapes are compatible with the idea that space must be homogeneous and isotropic77 It turns out that there are exactly three answers 1 A threedimensional sphere what the mathematicians call 53 This can be thought of as the set of points that satisfy 1 1 I 13 R2 in fourdimensional Euclidean space 2 Flat three dimensional space 290 CHAPTER 10 COSMOLOGY t This is just a fake model system to better understand what a is I make no claims that this represents any reasonable solution of Einstein7s equations N evertheless let s think about what happens to a freely falling object in this universe that begins at rest7 meaning that it has zero initial velocity in the reference frame used in equation 101 If it has no initial velocity then we can draw a spacetime diagram showing the rst part of its worldline as a straight vertical line X Now when a shrinks to zero what happens to the worldline Will it bend to the right or to the left Well we assumed that the Universe is isotropic right So the universe is the same in all directions This means that there is a symmetry between right and left and there is nothing to make it prefer one over the other So it does not bend at all but just runs straight up the diagrami In other words an object that begins at z 0 with zero initial velocity will always remain at z 0 Of course since the space is homogeneous all places in the space are the same and any object that begins at any I 10 with zero initial velocity will always remain at z 10 From this perspective it does not look like much is happening However consider two such objects one at 11 and one at 12 The metric d32 contains a factor of the scale a So the actual proper distance between these two points is proportional to a Suppose that the distance between 11 and 12 is L when a 1 at t 0 Then later when the scale has shrunk to a lt l the new distance between this points is only aLi In other words the two objects have come closer together Clearly what each object sees is another object that moves toward it The reason that things at rst appeared not to move is that we chose a funny sort of coordinate system if you like you can think of this as a funny reference frame though it is nothing like an inertial reference frame in special relativity The funny coordinate system simply moves along with the freely falling objects 7 cosmologists call it the co moving7 coordinate systemi It is also worth pointing out what happens if we have lots of such freely falling objects each remaining at a different value of 1 In this case each object sees 292 CHAPTER 10 COSMOLOGY const X 0 Shown here in the reference frame of observer 0 that observer appears to be the center of the expansion However we know that if we change reference frames the result will e 1 0 1 2 3 W t const In this new reference frame now another observer appears to be the centerf These discussions in at spacetime illustrate three important points The rst is that although the universe is isotropic spherically symmetric there is no special centerf Note that the above diagrams even have a sort of big bang7 where everything comes together but that it does not occur any more where one observer is than where any other observer is The second important point that the above diagram illustrates is that the surface that is constant t in our comoving cosmological coordinates does not represent the natural notion of simultaneity for any of the comoving observersi The homogeneity7 of the universe is a result of using a special frame of reference in which the t const surfaces are hyperbolaei As a result the universe is not in 104 OBSERVATIONS AND MEASUREMENTS 297 In the above form this equation can be readily solved to determine the behavior of the universe for the three cases k 710 li We don7t need to go into the details here but let me draw a graph that gives the idea of how a changes with t in each case at proper time Note that for k 1 the universe expands and then recontracts whereas for k 071 it expands forever In the case k 0 the Hubble constant goes to zero at very late times but for k 71 the Hubble constant asymptotes to a constant positive value at late timesi Note that at early times the three curves all look much the same Roughly speaking our universe is just now at the stage where the three curves are be ginning to separate This means that the past history of the universe is more or less independent of the value of kl 104 Observations and Measurements So which is the case for our universe How can we tell Well one way to gure this out is to try to measure how fast the universe was expanding at various times in the distant past This is actually not as hard as you might think you see it is very easy to look far backward in time All we have to do is to look at things that are very far away Since the light from such objects takes such a very long time to reach us this is effectively looking far back in time 1041 Runaway Universe The natural thing to do is to try to enlarge on what Hubble did If we could gure out how fast the really distant galaxies are moving away from us this will tell us what the Hubble constant was like long ago when the light now reaching us from those galaxies was emittedi The redshift of a distant galaxy is a sort of average of the Hubble constant over the time during which the signal was in 104 OBSERVATIONS AND MEASUREMENTS 299 at proper time The line for A gt 0 is more or less independent of the constant kl So should we believe this The data in support of an accelerating universe has held up well for three years nowi However there is a long history of problems with observations of this sort There are often subtleties in understanding the data that are not apparent at rst sight as the various effects can be much more complicated than one might naively expecti Physicists say that there could be signi cant systematic errors7 in the technique All this is to say that when you measure something new it is always best to have at least two independent ways to nd the answer Then if they agree this is a good con rmation that both methods are accurate 1042 Once upon a time in a universe long long ago It turns out that one way to get an 39 A t t 39 constant is tied up with the story of the very early history of the universe This is of course an interesting story in and of itself Let7s read the story backwards Here we are in the present day with the galaxies spread wide apart and speeding away from each other Clearly the galaxies used to be closer together As indicated by the curves in our graphs the early history of the universe is basically independent of the value of A or i So imagine the universe as a movie that we now play backwards The galaxies now appear to move toward each other They collide and get tangled up with each other At some point there is no space left between the galaxies and they all get scrambled up together 7 the universe is just a mess of stars Then the universe shrinks some more so that the stars all begin to collide There is no space left between the stars and the universe is lled with hot matter squeezing tighter and tighter The story here is much like it is near the singularity of a black hole even though squeezing the matter increases 302 CHAPTER 10 COSMOLOGY roughly galaxystyle clumps today This is an interesting fact by itself Galaxies do not require special seeds to start up They are the natural consequence of gravity amplifying teeny tiny variations in density in an expanding universe Well that s the rough story anyway Making all of this work in detail is a little more complicated and the details do depend on the values of A k and so on As a result if one can measure the CMB with precision this becomes an independent measurement of the various cosmological parameters The data from COBE con rmed the whole general picture and put some constraints on A The results were consistent with the supernova observations but by itself COBE was not enough to measure A accurately A number of recent balloon based CMB experiments have improved the situation somewhat and in the next few years two more satellite experiments MAP and PLANCK will measure the CMB in great detail Astrophysicists are eagerly awaiting the results 1043 A cosmological Problem7 Actually the extreme homogeneity of the CMB raises another issue how could the universe have ever been so homogeneous For example when we point our radio dish at one direction in the sky we measure a microwave signal at 2 Kelvin coming to us from ten billion lightyears away Now when we point our radio dish in the opposite direction we measure a microwave signal at the same temperature to within one part in one hundred thousand coming at us from ten billion lightyears away in the opposite direction Now how did those two points so far apart know that they should be at exactly the same temperature Ahl You might say Didn t the universe used to be a lot smaller so that those two points were a lot closer together77 This is true but it turns out not to help The point is that all of the models we have been discussing have a singularity where the universe shrinks to zero size at very early times An important fact is that this singularity is spacelike as in the black hole The associated Penrose diagram looks something like this In nite Future Qw Singularity Here I have drawn the Penrose diagram including a cosmological constant but the part describing the big bang singularity is the same in any case since as we have discussed A is not important when the universe is small The fact that the singularity is spacelike means that no two points on the sin gularity can send light signals to each other even though they are zero distance apart Thus it takes a nite time for any two places to be able to signal 104 OBSERVATIONS AND MEASUREMENTS 303 each other and tell each other at what temperature they should be In fact we can see that if the two points begin far enough apart then they will never be able to communicate with each other though they might both send a light or microwave signal to a third observer in the middle The light rays that tell us what part of the singularity a given event has access to form what is called the particle horizon7 of that event and the issue we have been discussing of which places could possibly have been in thermal equilibrium with which other places is called the horizon problemi7 There are two basic ways out of this but it would be disingenuous to claim that either is understood at more than the most vague of levels One is to simply suppose that there is something about the big bang itself that makes things incredibly homogeneous even outside of the particle horizonsi The other is to suppose that for some reason the earliest evolution of the universe happened in a different way than we drew on our graph above and which somehow removes the particle horizonsi The favorite idea of this second sort is called in ation Basically the idea is that for some reason there was in fact a truly huge cosmological constant in the very earliest universe 7 suf ciently large to affect the dynamics Let us again think of running a movie of the universe in reverse In the forward direction the cosmological constant makes the universe accelerate So running it backward it acts as a cosmic brake and slows things down The result is that the universe would then be older than we would otherwise have thought giving the particle horizons a chance to grow suf ciently large to solve the horizon problemi The resulting Penrose diagram looks something like this Infinite Future Singularity The regions we see at decoupling now have past light cones that overlap quite a bit So they have access to much of the same information from the singularity 2More precisely they will be unable to interact and so reach thermal equilibrium until some late ime PHY312 lecture 9 Simon Catterall Recap j Saw that free fall frames serve as good approximations to inertial frames Indeed effects of gravity can be almost eliminated by jumping to such a FFF Relies on equality of gravitational and inertial masses Equivalent to old observation of Galileo that all bodies fall equally fast under gravity independent of their mass composition etc Einstein realized importance of this elevated it to a principle J PHY312 lecture 9 p 2 Principle of equivalence rThere are no local experiments that can distinguish free fall I in a gravitational field from uniform motion in the absence of a gravitational field Perhaps gravity is not a property of any body but a property of spacetime itself What about the caveat local L J PHY312 lecture 9 p 3 General frames of reference 7 7 0 Since FFF play a special role and these are accelerating one is motivated to formulate laws of physics so that they look same in any FOR not just inertial o This is why the resultant theory is called General Relativity GR as opposeed to special relativity which describes only inertial frames Have seen that such theory will necessarily involve gravity L J PHY312 lecture 9 p 4 Newtonian gravity j Newtonian Gravity Universal attractive force of gravity acts between all bodies G G F GN 21m2 GN IS Newton s constant GN 667 x 10 11m3kg1s2 mG is the gravitational mass Analogous to the electric charge in electromstatics specifies the strength of the coupling between the particle and the gravitational electric field Gravity is however a much weaker force then electromagnetism For example the ratio of gravitational to electric forces between two electrons FGFE N 10 22 Succesful prediction of planetary orbits tides etc J PHY312 lecture 9 p 5 What s wrong V 7 Requires an instantaneous gravitational force in conflict with special relativity which states that no physical disturbance or interaction can propagate with a speed greater than the speed of light L J PHY312 lecture 9 p 6 L Gravity vs acceleration W What do I mean by locally Think about the elevator free falling above surface of Earth Release two ball bearings 20m apart at each side What happens Initially they will remain 20m apart and stationary in falling frame But after 8 secs 315 m fall they will be 1mm closer together I Why the balls fall towards center of Earth on slowly converging trajectories This would not happen for balls released in an accelerating elevetor without gravity J PHY312 lecture 9 p 7 Can distinguish gravity from uniform acceleration Tidal gravity j Real gravitational fields vary in space and their effects cannot be eliminated by jumping to an accelerated FOR In the context of the moon s gravitational pull on Earth these effects result in tides the moon pus more strongly on the side of Earth nearest to it and the oceans flow under the influence of this difference of forces Hence real gravity is tidal gravity can detect a difference because of tidal effects But ifl do the 2 ball bearing expt over shorter times or with a smaller initial separation the effect is smaller and eventually undetectable thus over local regions of spacetime we cannot distinguish gravity from pure J acceleration More on tidal effects 7 7 o The magnitude of these tidal gravitational effects depends on the strength of the gravitational field and hence the mass of the gravitating body 0 As we saw they are also proportional to the size of the spacetime region Hence they can always be made smaller than the resolution of my measuring apparatus by going to small regions 0 Bodies falling toward the center of a spherically symmetric object like the Earth or a blackhole are stretched along their line of motion and squashed transverse to it L J PHY312 lecture 9 p 9 Consequences light bending V 7 o Light is bent by a gravitational field Imagine firing a beam of photons from the frame of an accelerating elevator they will follow a curved path since can think of them as particles of small mass m Ec2 But by the principle of equivalence the same thing should hold in a gravitational field 0 Light grazing the Sun should be bent by an angle GMS A N qb R562 See in experiments L J PHY312 lecture 9 p 10 More V 7 o Clocks are slowed in a gravitational field lmagine placing a clock in an accelerating FOR Time dilation small time interval says i 2 dTdt1 vt2C2 Taa2 But distance gone is As aT2 so find Integrating t 30 C L But use POE to swap a for 9 J PHY312 lecture 9 p 11 continuing V 7 o Time measured on a Clock at rest in a gravitational field should run more slowly compared to one at infinity For Earth subsitute radius R for As 1 GMi C2 R L J PHY312 lecture 9 p 12 r o Consider photon emitted out from a spherical Gravitational redshift j gravitational field with energy E Has mass Ec2 and hence a potential energy GMEc2R Needs to lose this amount of kinetic energy to escape Lowers its measured energy at infinity for photons also have E hf Lower frequency longer wavelength redshift AE Af GM 1 E f N c2 R Consistent with clock slowing thinking of oscillations of wave as like a clock J PHY312 lecture 9 p 13 r 9 Einstein argued that all observers whether inertial or General relativity j not should be capable of discovering the correct laws of physics Thus he proposed the following as a logical completion of special relativity Principle of General Relativity All observers are equivalent Einstein used the principle of equivalence this principle of general relativity together with the requirements that his theory be the simplest possible extension of Newton s ideas and reduce to the same predictions for small velocities etc to derive his theory of GR From our discussions so far it is clear that such a theory has the ability to describe not just accelerated frames but gravity J PHY312 lecture 9 p 14 Relativity and Cosmology lecture 18 Recap lecture 17 j Discussed radial motion of massive and massless bodies in Schwarschild spacetime Used several frames global far away frame local shell observer and freely falling observer Locally laws of SR apply light travels at speed c massive bodies with o lt c acceleration energy follows Newton for small 2 7 gt 7 5 Globally things different For BH event horizon singularity What about more general motion orbits J Relativity and Cosmology p 2 More constants of motion If In analysing radial motion key was to find a conserved 7 quantity the energyatinfinity o For orbital motion we need to find another such conserved quantity relativistic generalization of angular momentum o This when combined with metric will allow us to solve the motion L J Relativity and Cosmology p 3 Angular momentum a la Newton j For linear motion we know that in absence of external forces acceleration is zero and hence momentum p mo is constant If switch on a force F this is not true However if force is purely radial Fa y z F7 there is still a conserved 2d6 quantity angular momentum L mm mr Why Take cross product of F 771 with 7 F x 7 0 Hence 771W 0 Thus mt X 7 constant In polar COOl dS v 7 J Relativity and Cosmology p 4 Example 1 9 For simple circular motion with radius a L ma2w where w must be constant if radius a is constant J Relativity and Cosmology p 5 Schwarzshild metric If Again principle of extremal ageing geodesic motion 7 allows us to find angular momentum L m y AT 0 Differs only from classical expression by using proper time as expected in denominator L J Relativity and Cosmology p 6 Derivation V 7 o Consider a motion comprising two sections A and B delineated by three events with coordinates p 7 Ar 0 gt 7 gb the A section with mean r coordinate m and elapsed proper time TA 7 gb gt 7 Ar 1 the B section with mean r coordinate 7 3 and elapsed proper time T3 o The final angle lt1 is considered fixed and only the intermediate angle gb will be varied L J Relativity and Cosmology p 7 continued 9 Setting 3 7 O and using metric we find 9 T2EZT ID Cb ATA B 7quotB Thus we see that 73 constant and can be identified with a conserved angular momentum per unit mass for motion in a spherically symmetric spacetime Notice similarity to derivation of energy L J Relativity and Cosmology p 8 r 9 Notice that we now have two conserved quantities Remarks j energy E mc2Arj and angular momentum L m dT39 The fundamental reason why these two quantities do not change with time is related to the fact that the components of the Schwarzschild metric do not depend explicitly on either time t or angle gb This independence of the metric on certain coordinates is referred to as a symmetry isometry It is intimately connected to the existence of conservation laws J Relativity and Cosmology p 9 General motion 7 7 o Starting from some initial position 7 gb and the constants L and E we can imagine computing changes in t and gb using the equations 2 1 2GMC27 L Agb 2mm 7 9 Using the form of the Schwarzschild metric leads to an expression for the change in rcoordinate Emc22 1 2S2 1 5 m1 Relativity and Cosmology p 10 Ari L Comments j Choose small AT use equations to go from owUAn A Use these as new 7 gb and do again Keep iterating General path 7 T qbT Computer simulation Setting 7quot rrg AT Arc7 5 6 Emc2 l Lmcrs find 1 5 Awi u ng JAH Relativity and Cosmology p 11 PHY312 lecture 14 Simon Catterall Summary so far j POGR and POE guided Einstein to propose a radical new way of thinking about tidal gravity as curved spacetime Curvature related to distribution of energymomentum specified in field equations of GR Test particles follow geodesics closest thing to straight lines on such a space For small velocitiescurvature reduce to Newtonion picture Early successes gravitational red shift perihelion of Mercury bending of light by Sun used approx solutions J PHY312 lecture 14 p 2 Exact solutions j If Exact solutions rare and hard to find Numerical solutions challenging eg binary black hole collision project needs supercomputer level effort 9 However in some very simple cases possible eg Schwarzschild solution First exact solution to GR Describes spacetime outside spherically symmetric static mass distribution 1915 o Applies to Sun Earth neutron star black hole L J PHY312 lecture 14 p 3 Schwarzschild metric V 7 o In 21 polar coordinates 7 6 z direction suppressed metric is A52 A7 C2At2 1 A 2 2A62 Apr 7 7 9 Where A7 1 2G1 0 Unique spherically symmetric time independent solution 0 Yields flat Minkowski space for 7 gt 00 andor M gt O 3 7 and t are far away coordinates L J PHY312 lecture 14 p 4 Physical interpretation j Consider two events in 7 t 6 frame with A6 At O Spacelike separation As Arshe This is the spatial distance an observer at r coordinate 7 would measure if he dropped a plumb line radially inward a small distance Physical Analogous to value of proper time measured by a clock in comoving FOR in SR timelike interval Such an observer called a shell observer and this distance is a shell distance Notice to stay at a fixed r he must be accelerating outward Not in a FFF J PHY312 lecture 14 p 5 Examples 7 7 9 What is the distance between the two shells with r coordinate 7 1 and r2 if 9 7 1 695 9806771 and 7 2 69598116771 for Sun With mass 2 x logocg 7 1 4km and r2 5km for solar mass black hole Use G 67 x 10 11 mgkgs2 9 Should see that A is quite differentin 2 cases 9 Large deviations direct measure of spacetime curvature L J PHY312 lecture 14 p 6 Time coordinate j Now consider 2 events at same r coordinate and angle 6 What is proper time measured between these events AT ArAt Proper time lt time measured by far away observer t This is slowing of clocks in a gravitational field now done exactly It is another manifestation of spacetime curvature Consider light propagating outward Number of wavecrests is fixed Frequency hence changes as time between crests changes Gravitational redshift J PHY312 lecture 14 p 7 739 tshell time measured by stationary observer at 7 Examples again What is gravitational redshift 57f A 12 for light emitted from surface of Sun and from 7 4km from solar mass black hole Note Metric varies continuously in spacetime All these Ar At s etc should be differentials infinitesmally small or local J PHY312 lecture 14 p 8 Event horizon j Notice something odd The time measured by a shell observer vanishes if A O or 7 7 5 2GMc2 Schwarschild radius Depends only on mass M Infinite redshift at 7 7 5 all light emitted from this point is shifted to infinitely long wavelength as it propagates out to infinite 7 Time passes infinitely slowly relative to a far away observer at this point like traveling at v c in SR Radial velocity lt cAr for light goes to zero there J PHY312 lecture 14 p 9 More j Notice something else for 7 lt r5 role of time and space coordinates is interchanged the singularity 7 O is in the future of any test particle in this region Thus the event horizon marks a boundary in spacetime Particles outside this may escape to infinity Those within it even light are trapped and will eventually reach the singularity Caveat Schwarzschild only applies to exterior of mass distribution Thus bodies must be very dense for r5 gt physicalradius Only then does body have event honzon What is 7 5 for Sun J PHY312 lecture 14 p 10 Embedding diagrams j Can draw a picture which allows this stretching of space to be visualized Fix farway time t The spatial curvature can be visualized by embedding the surface in a flat 3D space This extra dimension is an aid to visualization only Resulting picture is called an embedding diagram The profile of the surface is given by function 147 Clear now why the distance between two points at different r coordinate is bigger than than the mere difference in 7 The event horizon corresponds to the point where the slope of the profile is infinite this representation does not work for points inside the event horizon Helps make it clear that once inside the event horizon J escape is impossible More on event horizon V 7 o Have seen that radial velocity of light goes to zero as 7 gt 7 5 It then reverses Light falls towards the singlarity 7 0 why called singularity because curvature R gt 00 there Spacetime is so curved that the light cones of SR tip over and even light cannot escape 0 Event horizon divides spacetime into 2 causally disconnected regions nothing from inside the event horizon can influence what happens outside L J PHY312 lecture 14 p 12 L3 More on FOR j Notice that an observer close to 7 5 will not notice anything odd velocity of a radially moving light beam will be cjust as per normal Heshe will see no change as the light ray crosses the event horizon no violent redshifting to himher etc Locally all will be well It is only globally as seen from the far away FOR 7 23 that it is obvious a threshold has been crossed Only one physical motion Many FOR of reference can be used to view it eg shell FFF far away global frame They agree only on invariant intervals not things like distance times velocities etc Different systems may be betterworse for figuring out different things eg presence of event horizon J PHY312 lecture 14 p 13 V 7 Relativity and Cosmology PHY312 Simon Catterall Whats the course about V 7 0 Modern Physics stands on 2 pillars Quantum Mechanics and Relativity All current theories of fundamental physics assume that these theories are correct and almost all efforts at improving our understanding of nongravitational physics assume them too Theory of relativity has 2 parts special and general theories L J Relativity and Cosmology PHY312 p 2 Special Relativity Inseparable with name of Einstein 1905 j Radically new way of describing and understanding motion Relates observations of motion from special frames of reference inertial frames From rather modest assumptions provides a radical new view of world challenges ideas about time and space In most situations provides a accurate description of the world Mathematics straightforward J Relativity and Cosmology PHY312 p 3 General Relativity V 7 Generalizes the special theory to all frames of reference 1916 9 Provides a new theory of gravity more accurate than Newton Needed to understand cosmology black holes etc 0 Mathematics more involved Curved spacetime Will try to avoid most of this Focus on motion in simplest spacetimes L J Relativity and Cosmology PHY312 p 4 Cosmology j For a long time this was barely physics A mathematical theory of the Universe based on solutions to GR OK not quite fair Hubble s law Big Bang nucleosynthesis Now it has become a bustling experimental subject with lots of new precision data arriving every year This data is radically altering our picture of the structure and evolution of the Universe Dark matter dark energy cosmic acceleration J Relativity and Cosmology PHY312 p 5 To do list If Lot of time on special theory Key to all that follows 7 Most useful 9 Much less time on general theory Enough to talk about motion in black hole spacetimes Schwarzschild solution 9 A very little on cosmology Big Bang picture Successes and failures New ideas L J Relativity and Cosmology PHY312 p 6 bbbb b Mechanics of class j Here Tuesdays and Thursdays 111220 Lecture Work examples in groups 1 homework per week On Thursday Back week later Course page will have homeworks lectures project suggestions announcements Final grade based on homeworks midsemester and final exams and a final project Need PHY211212 and calculus J Relativity and Cosmology PHY312 p 7 Relativity in Newton s mechanics j If Philosophiae Naturalis Principia Mathematica 1686 I do not know what I may appear to the world but to myselfl seem to have been only like a boy playing on the seashore and diverting myself in now and then nding a smoother pebble or a prettier shell than ordinary whilst the great ocean of truth lay all undiscovered before me 9 Newton s laws of motion 1 Every body remains in a state of rest or uniform motion unless acted on by a force 2 Fmd 3 Two bodies in interaction exert equal and opposite forces on one another L J Relativity and Cosmology PHY312 p 8 What these laws tell us W This set of laws can be used to describe and predict the motion of material bodies Revolutionary I no precedent in history of science Still in wide use today eg cars bridges planes washing machines Only when you ask detailed questions about the structure of atoms about particles traveling at very high speeds or the behavior of exotic objects like black holes or neutron stars do you need anything else What does it tell us implicitly J Relativity and Cosmology PHY312 p 9 Frames of reference j Newton s first law seems benign but is perhaps the most important It implicitly says that I can find a frame of reference FOR in which certain types of motion look very simple A frame of reference is just some system for measuring distances between objects and time intervals eg fill space with set of Cartesian axes and place a series of clocks at each point Suppose I find such a frame then I can immediately find many others by taking my original frame of reference and moving it at constant speed in some direction J Relativity and Cosmology PHY312 p 10 Inertial frames j Such frames are called inertial frames Newton s first law really says that such frames exist and motions looks very simple when seen from such a frame In fact all such frames are equally good for describing some physical situation One can use accelerating frames rotating frames and indeed this is what GR allowsneeds but motions are necessarily more complicated when viewed from such frames J Relativity and Cosmology PHY312 p 11 Events j Consider some physical occurrence at some place and some time as measured in particular FOR Eg cartesian grid plus time 51325 Call this an event Everything that happens in the Universe is a collection of such events eg As a ball moves it traces out a succession of positions at certain times a series of events Called the worldline of the ball J Relativity and Cosmology PHY312 p 12 r o Notice I cannot really make the statement the ball is Everything is relative j at rest I can only say the ball is at rest relative to me But is it absolutely at rest No I am standing on a spinning planet which is orbiting the Sun which is orbiting the center of Milky Way which has some motion relative to distant stars etc etc Newton realized this As far as we know there is no fundamental FOR forthe Universe to which all motion can be refered And even ifthere was how could we find out what it was Therefore it is necessary to couch a mechanics purely in terms of relative motion J Relativity and Cosmology PHY312 p 13 Things to do If What is your speed relative to the center of the Earth j What is your speed relative to the Sun 9 How big are these speeds relative to the speed of light 3 X 108 ms L J Relativity and Cosmology PHY312 p 14 Relating different inertial FOR 78 7 1 513 513 112 2 t t eems obvious Galilean transformation Right Newton assumes absolute time Absolute true and mathematical time of itself and from its own nature flows equany without relation to anything external Einstein was forced to change this to make theory compatible with experiments done at end of 19th century But he chose to keep the Principle of Relativity L J Relativity and Cosmology PHY312 p 15 Principle of Newtonian Relativity V 7 All inertial FOR are equally good for discovering the laws of mechanics and for predicting motion L J Relativity and Cosmology PHY312 p 16 PHY312 lecture 2 Simon Catterall Review W All physical phenomena observed from some frame of reference xy 223 Life is simple ifthis frame is inertial Observe uniform motion of object in absence of forces Newtonian mechanics obeys a Relativity Principle that is all such inertial FOR are equally good Deduce same fundamental laws of Physics from any such frame Newton assumed absolute time t t F ma takes same form in all such FOR J PHY312 lecture 2 p 2 Crisis j Late 19th century new laws were discovered governing the phenomena of electricity and magnetism In particular light seen as an electromagnetic wave with a fixed speed c 2 30 x 108 ms But how can a law of physics contain a fundamental constant with the dimensions of velocity Seems to imply existence of a preferred FOR Violates relativity principle Can show that EM equations do not have same form when xy 21 gt x y z t for two FOR with relative velocity 2 using Galilean transformations J PHY312 lecture 2 p 3 Resolution j OK throw out relativity principle Assume preferred reference frame the ether Velocity of light is measured relative to ether But experients designed to measure speed of Earth relative to ether all returned null result MichelsonMorley 1897 Ether dragging Einstein used this as cornerstone of a new theory special relativity which aimed to preserve the relativity principle by throwing out the Newtonian way of relating distance and time between 2 inertial FOR J PHY312 lecture 2 p 4 Einstein s idea j Relativity principle is central Assume laws of electricity and magnetism correct Forced to conclude that all inertial observers must agree on the numerical value of the speed of light This is in conflict with the Newtonian way of computing relative velocity A light beam receeds at the same speed independent of how fast I run after it Not like any other wave motion J PHY312 lecture 2 p 5 A thought experiment j Consider rocket moving along xaxis of some inertial FOR at speed 2 Imagine it sends out a light beam in ydirection which bounces off the rocket side and returns to the starting point Emission and reabsorption are 2 events If rocket has diameter h and light travels at speed c it will return in time 25R 2hc according to rocket FOR J PHY312 lecture 2 p 6 More rockets j How do they look from original FOR Simple geometry shows total distance gone is now 2yh2 2222 ift is time separation in original FOR and If assume it moves at speed 2 still takes a time t 2 h2 0220 to return as measured by the original FOR Algebra gives Not the same ll 25 gt 25R J PHY312 lecture 2 p 7 Time is not absolute V 7 Time measured in stationary frame longer than moving frame Time dilation 0 Notice also spatial separations AacRO AxvAt Observers who are in uniform relative motion ascribe different spatial and temporal differences to the same 2 physical events Is there anything they agree on L J PHY312 lecture 2 p 8 Spacetime interval Easy to see that the quantity A52 C2At2 A5132 invariant same for both sets of observers Notice that for c gt 00 just yields absolute time Called the square of the spacetime interval dimensions of length Generalizes the notion of notion ofdistance in ordinary space Thus familiar with the idea that the coordinates of the endpoints of ruler are not invariant under rotations or Galilean transformations But length of ruler is This is the analog 4 PHY312 lecture 2 p 9 Things to do j Suppose you open the door to a room event 1 and walk at a constant speed of 1 ms to a chair 10 m away and sit down event 2 How much time elapses in the frame of the room between the two events What is the spatial distance between the two events as seen from the room What is the distance in spacetime associated with these two events What is the spatial separation between the two events as measured in a frame which moves with you How much time is recorded on your watch How would things change if you could walk at 35 speed of light I PHY312 lecture 2 p 10 What follows j Also allows us to define proper time between the two events 739 Asc Time interval measured by Clock traveling with the rocket For the proper time to be a real number the velocity must be less than cl Maximum velocity Length of a worldline in spacetime is proportional to the proper time measured by a Clock that travels along that path What is shape of this spacetime path for a free particle What is the length of the path for a photon J PHY312 lecture 2 p 11 Some philosophy 7 7 o Why should I believe such wierdness s It agrees extremely well with experiment o It follows logically from the relativity principle which is really forced upon us by the impossibility of knowing how to establish a preferred FOR Einstein merely extends Newton s ideas to all physical laws notjust mechanics 1 It reduces to Newtonian mechanics as 00 gt O o While it violates intuitive feelings about how the world works there is no reason to trust these in regimes far from our commonsense experience This is similar to quantum mechanics L J PHY312 lecture 2 p 12 C onclusmns If Notice once more that in Special Relativity there is no 7 absolute observer independent notion of space and time separately only a funny fusion of the two the thing we have been calling spacetime Hendeforth space by itself and time by itself are doomed to fade away into mere shadows and only a union of the two will preserve an independent reality Herman Minkowski 18641909 L J PHY312 lecture 2 p 13 PHY312 lecture 7 Simon Catterall Review 7 7 energymomentum vector 79 ECppypz 0 Contains relavistic energy momentum Invariant length is rest mass 0 Conserved L J PHY312 lecture 7 p 2 Example j A photon moving with energy E collides with a stationary atom with rest mass m The photon is absorbed and the recoils Work out formulae for The mass of the atom after collision The momentum of the atom after the collision How fast it is traveling after collision as viewed from the original FOR at which it was at rest A typical visible light photon carries approx 1 x 10 18J of energy while a hydrogen atom has mass approx 10 26 kg What is the velocity of the recoiling H atom J PHY312 lecture 7 p 3 Solution 0 Conservation of energy and momentum E 77102 Ef p EC Pf Total mass M2c2 Eiac2 p 2Em m2c2 To find velocity equate Mm Ec Find 2 4 C 1 mC2E Putting E 10 18m 10 26 c 3 x 108 find g 10 8 ie few ms L J PHY312 lecture 7 p 4 Photon rockets j Perfect rocket engine combines matter and antimatter to create photons directed backwards Initial mass M After all fuel is burnt moves with speed 21 and has mass fM where f lt 1 What is v as function of f Conservation of 79 needed J PHY312 lecture 7 p 5 Solution 7 7 0 Consider before fuel burning and after all fuel burnt Apply conservation of momentum and energy M62 Eradnyc2 EradC p where p2 nyc2 M2f2c2 from invariant length of final rocket EP vector Find f2 2f710 L J PHY312 lecture 7 p 6 Consequences r1 fv v2 1 What f fOPy 10 What fOPy 100 Recap causality j Said 2 events that are spacelike separated cannot be causally related used argument that no physical signal can propagate faster than speed of light Here give another more direct argument Consider 2 events A 0 0 and B 51325 and assume A causes B In some FOR 25 gt O B happens later Question can one jump to another FOR with coords ast where t lt 0 Le in such frame B occurs earlier Lorentz transformation implies U 323 a lt O C Compatible with s2 22252 5132 Since oc lt 1 we are J safe for timelike intervals only PHY312 lecture 7 p 8 Inertial frames again W In practice it is easy to find a inertial FOR in which electric magnetic nuclear forces are all negligible But what about gravity One option go to interstellar space away from all masses etc There is another way In presence of gravity jump to a FOR which is falling freely under gravity Quick thought experiment Imagine an observer in a freely falling elevator He throws a coin What does he observe He will see coin move at constant velocity l Thus freely falling frames FFF serve as almost inertial frames even presence of gravity PHY312 lecture 7 p 9 Newtonian analysis r Denote coordinates of coin relative to Earth by ME coordinates of freely falling elevator frame relative to Earth by xFE and coordinates of coin relative to elevator by SUCF J we have CECF CECE CEFE 3 But from Newton s 2nd law d2a I FE G d2a I CE G m0 dt2 ng Provided m mG I 2 J d CECF 039 dt2 PHY312 lecture 7 p 10 Conclusions j Thus while motion of coin would follow parabola in Earth frame it is uniform in FFF Requires inertial massgravitationa mass It took Einstein to understand the significance of this Can use the laws of special relativity within such FFF The effects of gravity can be almost eliminated within such a frame J PHY312 lecture 7 p 11 Accelerating elevator j Consider now a rocket in empty space Imagine accelerating the rocket Throw the coin again What will you see Relative to the rocket the trajectory of the coin will be curved d2CF dt2 This looks like equation of motion for coin near Earth s surface where acceleration due to gravity is a Thus gravity can be mimicked by accelerating frames of reference J PHY312 lecture 7 p 12 Principle of equivalence rThere are no local experiments that can distinguish free fall 7 in a gravitational field from uniform motion in the absence of a gravitational field Requires equality of gravitational and inertial mass Follows from the equal gravitational acceleration of all bodies independent of their mass Galileo What about the caveat local L J PHY312 lecture 7 p 13 PHY312 lecture 10 Simon Catterall Recap j FFF serve as good approxs to inertial frames in presence of gravity Principle of equivalence POE Using POE learn that light is bent clocks slow and light is redshifted in gravitational field Characterized by dimensionless number 6 Principle of general relativity all observers in any FOR should be able to discover the correct laws of physics Real gravity is tidal gravity J PHY312 lecture 10 p 2 More on tidal gravity j Detect real gravity by watching tidal motion of test particles over finite region of spacetime from FFF Motion does not depend on mass composition electric charge etc etc Seen that measurements of time suffer distortion in accelerating frames clock slowing Similarly one can argue that length measurements are also deformed by gravityacceleration Worldlines of unaccelerated particles are now bent Is there someway to think of tidal motion as resulting from free motion in a deformed or curved spacetime What do I mean by this J PHY312 lecture 10 p 3 Deforming space by acceleration V 7 o Consider disk rotating at high speed relative to inertial frame An observer at rest would measure is diameter to be d and circumference C using a ruler at rest in inertial frame Finds Cd 7r as usual for flat Euclidean space 0 Observer on disk would measure a distance less than 0 since he would see ruler as contracted by SR along its length Thus for him Cd lt 7r How can this be 9 Imagine drawing a circle on the surface of a sphere a curved space You will find that Cd lt 7r in such a case L J PHY312 lecture 10 p 4 Deforming space 11 But disk is accelerating 7 Thus if one uses accelerating FOR one can expect to see effects which can be interpreted as resulting from an underlying curved geometry But by POE same effects should be expected for a disk at rest but sitting in a gravitational field Thus gravity can be pictured as associated with curved spacetime J PHY312 lecture 10 p 5 Gravity as curved spacetime W In accelerating frames clocks slow spatial geometry can become curved By POE expect that effects of gravity can be similarly associated to an underlying curved curved spacetime Perhaps possible to think of tidal motions of test particles in a gravitational field as really free motion in a background curved spacetime Einsteins guess It is correct J PHY312 lecture 10 p 6 Motion on a sphere j Consider two 2d observers confined to surface of sphere Set them off a small distance apart at the equator Let them move freely along lines of longitude You will see that they start to converge and eventually meet at the North Pole just like fortidal gravitational motions Indeed ifthey didn t know they lived on a curved space they might infer that a myesterious force gravity was drawing them together Reality is that they are moving along straight lines in an underlying curved space J PHY312 lecture 10 p 7 r o Take a hint from Newton Mass causes Newtonian What causes curving j gravity Presumably some relavistic generaliation of mass is responsible for spacetime curvature energymomentum Nice picture Force of gravity is just curvature of spacetime But clearly this curving is independent of the coordinates one uses on the spacetime I can use a variety of different coordinates systems to describe the surface of the sphere Thus physical laws should not depend on the coordinate system used to describe the underlying space or equivalently all coordinate systems frames of reference should be equally good choices for describing the curvature of spacetime This is just J the principle of general relativity again PHY312 lecture 10 p 8 PHY312 lecture 6 Simon Catterall r Review W Vectors in spacetime sets of4 numbers at am ay az which transform like spacetime coordinates under LT Length squared a a a a2 same for all inertial observers lmportant example energymomentum vector 79 Eopwpymz All laws of physics written in terms of such vectors ensures satisfy relativity principle J PHY312 lecture 6 p 2 Energymomentum vector j Special relativity unites time with space but also energy with momentum and mass with energy I Ec mocy Hence E gt 00 as v gt 2 Physical reason why cannot go faster than speed of light p mom Momentum also infinite as v gt 2 Notice relativistic momentum exceeds Newtonian expression since y 2 1 What is mass Energy measured in FOR in which particle is at rest J PHY312 lecture 6 p 3 Examples 7 7 o Eg A ball of mass 1 kg is travelling in the xdirection with speed 1 ms According to Einstein what is its energy and what is its momentum What values would Newton ascribe to the ball Now consider a proton of mass 167 x 10 27 kg moving at 09c in the xdirection What is its relativistic energy and momentum What would be its energy and momentum according to Newton L J PHY312 lecture 6 p 4 r o The Fermilab Tevatron is the largest US accelerator for More examples j doing high energy physics In it protons and antiprotons are accelerated to speeds close to the speed of light and made to collide The protons in the beam have energies of 900 Gev 1 Gev is 16 x 10 10 Joules The proton rest mass is 167 x 10 27 kgs What is the ratio of the speed of the particles to the speed of light take 2 30 x 108ms The protons and antiprotons are accelerated around a circlar ring of circumference approx 4 miles How long in the rest frame of the accelerator does it take a protonantiproton to go once around the ring How long does one of the protons think it takes J PHY312 lecture 6 p 5 Many particles j For system of particles get the total energymomentum vector by adding up the energies and momenta for each Ptot E ZC7Zp i What is mass of system Length of total energy momentum vector In general this is not the sum of all rest masses ie the total length of a vector is not in general the sum of a single component of each vector This is the price one pays for unifying mass energy with momentum One loses the notion of absolute mass that all observers agree on PHY312 lecture 6 p 6 Examples 7 7 0 Consider 2 particles both of rest mass m one at rest the other moving with total energy 4mc2 What is 1 Total energy of the system 2 Total momentum of the system 3 Total mass of the system 9 ans 577102 V15mc 10m L J PHY312 lecture 6 p 7 More r 9 Photons are massless particles Hence 771202 O E2C2 p2 ie E pc 0 Consider two photons initially travelling in same direction with energies E 30 and E 1c 1 What is their total energy 2 Hence what is their total momentum 3 What is the mass of the 2 photons 4 What happens when photons headed in opposite directions 9 ans 4c 40 120 L J PHY312 lecture 6 p 8 Dioral j Mass has different interpretation in relativity than Newtonian theory Mass becomes rest mass length of particles energymomentum vector Total mass of system is just length of total energymomentum vector of system In rest frame of particle proportional to energy via E 771062 Energy and momentum are united into more fundamental object energymomentum vector 79 J PHY312 lecture 6 p 9 Conservation of 73 V 7 9 In Newtonian mechanics E and p conserved separately In relativity simply have conservation of 7 Valid in any inertial FOR o Length of 79 conserved conservation of mass also I L J PHY312 lecture 6 p 10 Example j A photon moving with energy E collides with a stationary atom with rest mass m The photon is absorbed and the recoils Work out formulae for The mass of the atom after collision The momentum of the atom after the collision How fast it is traveling after collision as viewed from the original FOR at which it was at rest A typical visible light photon carries approx 1 x 10 18J of energy while a hydrogen atom has mass approx 10 26 kg What is the velocity of the recoiling H atom J PHY312 lecture 6 p 11 Solution 0 Conservation of energy and momentum E 77102 Ef p EC Pf Total mass M2c2 Eiac2 p 2Em m2c2 To find velocity equate Mm Ec Find 2 4 C 1 mC2E Putting E 10 18m 10 26 c 3 x 108 find g 10 8 ie few ms L J PHY312 lecture 6 p 12 Relativity and Cosmology lecture 20 r o Discussed effective potential for motion of massive Recap lecture 19 j bodies in black hole Schwarzschild spacetime 1 0 0 No stable orbits for small L Capture by singularity inevitable for large enough E Bound orbits possible for smaller 6 Emc2 and large enough 2Lmcr5 2 M Generically correspond to precessing ellipse Stableunstable circular orbits possible by tuning E with L J Relativity and Cosmology p 2 Today r p Example escape from a black hole 9 Massless particle orbits 9 More on the Newtonian limit L J Relativity and Cosmology p 3 Escape 9 Satellite fired at 900 to radial direction by an observer at distance 7 0 573 from black hole The measured local speed is 22 Angular momentum and energy are given by Lm ToVshellvsheu l 2Lmcr5 575 Emc2 Ar012y8h ll 1033 0 So has enough energy to escape to infinity if launched radially but it has a nonzero L so that is not the case Need to look at effective potential J Relativity and Cosmology p 4 Escape II V 7 o Construct effective potential Find circular orbits Timetable 334 rstable 2972 9 Height of effective potential at 7 rummble is Vmax 109 9 Thus 6 1033 lt Vmax and satellite escapes to infinity L J Relativity and Cosmology p 5 Photon motion j For massive bodies used extremal ageing to find constants of motion and from there solved for the motion For light AT 0 always tricky Consider massive particle and carefully take limit m gt 0 First replace drdT by drdt in radial equation 2 j gt2ltj gt2 with d7 E AltrgtltEmc2gt J Relativity and Cosmology p 6 Continued If Find W 1d7 2 2 3 771 022 E2 gltggt lt1 M 1 mm ltEgt ltEgtT2 l 1 7 57 Cdt E 7 2 Take limit m gt 0 Edi 7 2 C 2 l i rsr 1 1 7 57 L m Rescaling etc 7 7 o Notice that LcE is constant for massless photons Impact parameter b cLE 9 Defining 7 Wig2 t 225073 and b 2b7 5 find dr i1 1 1 56 12 W 7quot 7quot and L J Relativity and Cosmology p 8 Photon effective potential r o RHS of radial equation possesses no constant like 6 Hard to think of an effective potential formulation 0 However m has one dtshell drshell2 1 27b2r2 dtshell or 1 drshell 2 1 V 52 dtshell 52 r where W 1 207 j J Relativity and Cosmology p 9 Orbits etc j V7 has a maximum at 7 3273 Horizontal lines show values of 1b2 Light with impact parameter I such that 1b2 is bigger than peak will be captured by black hole Conversely light with 12 less than peak will scatter off black hole bend but leave Notice possibility of unstable circular orbit for light with 1b2 peak 26719 May go to infinity or be captured subsequen y J Relativity and Cosmology p 10 Realtivity and Cosmology lecture 17 Recap lecture 16 F Energy EGRmC2 1 2GM At AT 327 9 Radial speed Ar 2GM 12GM At C 327 327 9 Shell observer sees just 1 2 M C Atshell 027 L J Realtivity and Cosmology p 2 Energy measured by shell observer 7 7 2GM 1 C27 Eshell where E mc2 is energy measured by far away observer As 7 gt R5 energy available to local observer becomes infinite L J Realtivity and Cosmology p 3 Time to crunch If Once pass event horizon object will reach 7 0 in finite j proper time Ar 2GM 12 AT C 327 Integrate to find tota proper time T0 o What is T0 for solar mass black hole L J Realtivity and Cosmology p 4 More examples W What if we throw the object with some initial speed into the black hole Conservation of energy leads to dt 7 1 T s7 where y corresponds to the initial value of Emc2 for an object in radial motion inwards Substitute into this the expression for d7 from the Schwarzsohild metric and we find 2 dr 1 E C1 T s7 1 1 7 57 J We see that these limiting speeds remain the Another variation r Dropping from a finite rcoordinate Answer clt1 mm l Cljffof and a similar expression for the shell velocity 0 Notice that the shell speed approaches the speed of light at the horizon independent of the initial r coordinate 7 0 ll L J Realtivity and Cosmology p 6 r o Consider some shell observer at rcoordinate 7 0 corrections to Newton j Differentiate the above result with respect to tsheu and reexpress the expression as a function only of shell quantities to derive an expression for the locally 2 defined graVItational acceleration in GR W shell 2 d T sheii GM 2 2 disheii 7 0 NIH 1 2GMr0 Newton approximates GR for plunging object for small Ushell and GMT J Realtivity and Cosmology p 7 Continued dt dtshell I 9 From Em 1 7 57 CCll subsitute Ell j m7 Use the expression you know for Use also the hell expression d72 dtghell drgheuc2 to find Hence show that for small enough UshellC and rSr that 1 2 GM Em N1 Evshell T L J Realtivity and Cosmology p 8 PHY312 lecture 13 Simon Catterall Summary so far j Tidal gravity caused by geodesic motion on curved spacetime Curved spacetime described by metric matrix which varies from point to point If spacetime flat possible to choose FFF where 911 17922 I 933 I 944 I 1 Laws of SR hold Minkowski space In general curvature R nonzero So such frames exist only locally Geometrical picture automatically contains POGR POE J PHY312 lecture 13 p 2 Field equations j Need equation that determines how curvature R is determined by massenergymomentum density T Schematically R constant T where constant G4 For small We and small curvature reduces to Newtonian gravity What about geodesic motion J PHY312 lecture 13 p 3 r 0 Conservation of energymomentum implies that Geodesic motion j AIL A T constant 0 Or d2 0 in inertial FOR d72 0 Consider motion in some arbitrary FOR y ya Z d7 j 8yj d7 ML ij y dT2 j 87f dT2 8yj8yk d7 d7 J PHY312 lecture 13 p 4 Finally rThus free motion looks in funny FOR looks like j w z EM O dT2 jk d7 d7 o By the principle of equivalence this equation should also hold for geodesic motion in a curved spacetime since inertial effects are equivalent to gravitational The difference between the gravitational situation and the flat space free motion is that there will not exist any global coordinate system in which all the F8 vanish L J PHY312 lecture 13 p 5 Newtonian limit again 1 Limit of small vC expect only F711 to be important Equation of motion looks like d2xquot 1 2 C Corresponds to Newton s second law if F39il force Cgl g 3 9111CQ2 J PHY312 lecture 13 p 6 Conclusions 7 7 Thus Einstein s field equations and the geodesic rule for motion in a curved spacetime indeed contain Newton s law of gravity plus his famous second law of motion in the limit when 21 ltlt 3 and gravity is weak L J PHY312 lecture 13 p 7 Approx solutions r o Einstein was able to show that GR contained Newton s 7 theory plus small corrections 9 Was able to compute these corrections and correctly reproduced 0 Bending of light verified by Eddington in solar eclipse of 1919 3 Slowing of clocks Perihelion precession of Mercury Discrepancy 42 seconds of arc per century L J PHY312 lecture 13 p 8 Schwarzschild solution j If Exact solutions rare and hard to find Numerical solutions challenging eg binary black hole collision project needs supercomputer level effort 9 However in some very simple cases possible eg Schwarzschild solution First exact solution to GR Describes spacetime outside spherically symmetry static mass distribution 1915 o Applies to Sun Earth neutron star black hole L J PHY312 lecture 13 p 9 b Metric j First lets write spacetime of 21 SR in polar coordinates 7 6 A52 C2At2 A5132 A342 becomes A52 C2At2 A72 7 2A62 Note spacetime is still flat but metric is now nontrivial in polar coordinates 27w is distance round spherical object at r coordinate 7 Switch on gravity How will this change Spherical symmetry and time independence suggests we try J PHY312 lecture 13 p 10 A52 Ar c2t2 B7 A7 2 7 2A62 Solution o Plug into field equations Find solution provided 7 1 B7 1A7 Am 1 WM C27 3 Dimensions o Time it is far away time Measured by clock at infinity L J PHY312 lecture 13 p 11 Comments j Einstein wrote to Karl Schwarzschild I had not expected that the exact solution to the problem could be formulated Your analytic treatment seems to me splendid This is valid for timeike separated events For spacelike events multiply by minus 1 This is nonrotating uncharged spherically symmetric structure The solution for a spinning black hole was only published in 1963 almost 50 years later Holds all info on the external spacetime Note 7 gt oo corrections go to one flat spacetime Also for M gt O correct flat spacetime again J PHY312 lecture 13 p 12 PHY312 lecture 15 Simon Catterall Summary of lecture14 j Exterior spacetime to spherically symmetric time independent nonspinning gravitational source is given by Schwarzschild metric 1 A52 A7 C2At2 A7 2 7 2A62 7 A with An 1 7 5 2GMc2 Schwarzschild radius Global coordinate system 7 and t far away coordinate and time Visualize at fixed 25 by embedding in 3D space gt surface Profile is just A7 J PHY312 lecture 15 p 2 Consequences V 7 o Controls gravitational redshift stretching of space 9 If rmdius lt 7 5 spacetime possesses an event horizon L J PHY312 lecture 15 p 3 Event horizon j A 0 at event horizon Time passes infinitely slowly relative to far away observer Photons emitted from event horizon are redshifted to infinite wavelength and fail to escape to infinity A cA gtOasr gtr5 For 7 lt 7 5 all matter including light must fall to the singularity 7 0 Time and space coordinates flip Thus event horizon divides spacetime into 2 causally disconnected regions stuff inside the event horizon can never influence what happens outside J PHY312 lecture 15 p 4 r 9 Notice that an observer close to 7 5 will not notice What happens at 7 m j anything odd velocity of a radially moving light beam will be cjust as per normal Arshell c for light Atsheu In fact the metric in shell coordinates is flat l SR applies Heshe will see no change as the light ray crosses the event horizon no violent redshifting etc For a freely falling observer gravity does not even exist so that there cannot be anything that happens at 7 7 5 provided that tidal effects are small J PHY312 lecture 15 p 5 Different frames If Thus while the global coordinates mt tells us 7 correctly that something irreversible has happened on crossing 7 5 it will not seem so to an observer in either a FFF or a shell frame close to the horizon L J PHY312 lecture 15 p 6 FOR in GR j Only one physical motion through spacetime Many FOR of reference can be used to view it eg shell FFF far away global frame They agree only on invariant intervals not things like distance times velocities etc Different systems may be betterworse for figuring out different things eg presence of event horizon Notice that she and FFF are physical frames where observers can make local measurements Global frame different No real global observer but does give usefu picture of entire trajectory of test particle in spacetime J PHY312 lecture 15 p 7 Radial stretching done right j Said that strictly should treat Ar At etc as infinitessimal But so far taken them finite OK provided Ar small compared to the characteristic scale of the curvature 7 5 Calculation of Sun stretching fine But the solar mass black hole Should really use an integral Hence T2 dr 1 7 57 12 J Can be done exactly or numerically Lecture 8 Curved Spaces Lets think about the twodimensional sphere It is perhaps the simplest example of a nonEuclidean geometry One coordinate system we could try to use to discuss the surface of the sphere is the 3D cartesian system x y The sphere is just given by the equation x2 y2 22 R2 Notice that the intrinsically two dimensional surface is given by three coordinates and a equation which expresses the fact that 3 say is given in terms of x and y Better to use spherical polar coordinates 73055 Now the equation is just 7 R and we can just use 2 coordinates 0 qb to specify a point on the surface To specify the geometry of the surface we really need to talk about the distances between 2 such neighboring points 0 qb and 0 A0 qb Aqb If I approximate the surface by a set of points joined together by straight lines it should be clear that I can reconstruct the surface to high accuracy by knowing all the distances between points As I add more and more points the discrete surface I am constructing will approach the target smooth surface 7 in this case the 2sphere Simple considerations show that this distance As is given by A32 R2 A02 sing 0M2 Notice that I can rewrite this in a suggestive form A82 911A12 922A3722 where x1 0 x2 55 and R2 R2 sin2 0 911 922 The objects 911 and 922 are components of an object called the metric Knowledge of the metric at each point on the surface tells us how to construct invariant distances in the neighborhood of that point It speci es the surface uniquely If we had a flat space the components would be constants think about ordinary 2d flat space where 911 922 1 Thus a nonconstant met ric in some coordinate system is a necessary but not sufficient condition for the space to be curved If we have the metric we can compute the length of nite paths on surface Consider the path 55 constant varying 0 This is a path going running from north to south poles lines of longi tude The total length is R13 10 7rR as expected for 12 the circumference of a circle radius R What about the set of paths 0 constant varying gb Length is Rsin fg 7rd 27rR sin 0 These are the lines of constant latitude At this point we can ask the simple question 7 what are the curves with the shortest length between any two points on the sphere i the answer as any pilot for a commercial airliner knows are the segments of great circles circles defined by the intersection of a plane through the 2 points and the center of the sphere with the sphere itself The lines of longitude are examples of such geodesic curves They are the curves on a curved surface which locally look like straight lines but which yield the sortest distance between two points in the space It seems evident although we won t prove it that a given surface has a unique set of geodesics which are determined by its metric Now we are in the position to de ne one characteristic of a curved surface 7 it will possess geodesics which are not globally parallel unlike flat space where initially parallel geodesics straight lines remain so and never cross Finally we might ask 7 how can I tell locally from the form of the metric tensor that the surface is curved has nonparallel geodesics for example The answer is that we can compute some object called the Riemann curvature which is nonzero To compute the curvature imagine taking a small vector arrow and move it around a closed path on the surface so that it always makes a fixed angle with the tangent vector to the curve at that point this is called parallel transport Compare the final vector with the initial vector and determine the angle of rotation that has been induced by the motion Then the curvature is given by the formula R net rotation angle surface area enclosed by loop For the sphere choose the path which starts from some point on the equator moves up along a line of longitude to the North pole then returns to the equator along a similar path ending up 14 of the way round the equator Finally track back along the equator to the start point You can see that curvature 1R2 for the sphere And you can also see that curvature of flat plane is zero Final complication Since we can choose my coordinate system no need for it to be be spanned by unit vectors at 900 to each other Imagine we make a general transformation in at space from a Cartesian coordinate system x y to one x y in which the coordinate axes are not even at 900 to each other x 2 am by y 2 dy It is clear that the original distance l2 m 2 y 2 is now given by l2 a2 9932 b2 122 2ab cdxy To accommodate such coordinate systems and we must if we are to have general covariance we must think of the metric as having in general offdiagonal components 912 and 921 also Thus the metric is to be thought of as like a matrix which varies from point to point of the space Such an object on curved space is called a tensor 0 Similarly the curvature is actually a tensor also for the sphere it can be speci ed by a single number 7 special case Curved spacetime 0 Proceed by analogy with ordinary space Assume the distance in spacetime is given by an expression 2 4 4 d3 Z Z glwdxudxquot 111 121 0 Here the coordinates a have one timelike component and 3 spacelike components and gm is a metric in spacetime o Locally can always choose coordinates free fall frame in which the only nonzero elements take form 911 c 922 933 944 1 This is the metric of at Minkowski spacetime the spacetime of special relativity We see now that special relativity will apply locally provided I choose a special frame of reference 7 the approximately inertial free fall frame 0 Particles travel along geodesics in spacetime No additional force of gravity needed By analogy with the metric above we will postulate that photons massless particles will travel along null geodesics paths of zero length Remember the length is just the proper time Material particles will travel along timelikc geodesics ones with a positive squared length just like in special relativity Locally this will imply a maximum speed i 3 Of course what happens globally may depend on the nature of the curved spacetime 0 Can write down an expression for the spacetime curvature tensor analogous to the one for a curved space 0 Final ingredient need an equation which determines the metric somehow in terms of mass and energy A natural tensor arises called the stress energy tensor 0 Einstein s eld equations then take the form schematically spacetime curvature 2 Newton constantstress energy Newtonian limit 0 Why should I believe this 7 our arguments have been merely qualitative so far and while attractive could presumably lead to a variety of theories which incorporate ideas like curved space time and general covariance but differ in the details We would really like to show that Einstein s eld equations reduce to the Newtonian law of gravity in some limit where velocities are small compared to c and the deviations from the at spacetime of spe cial relativity are small ie tidal gravity is weak This is a little tricky since I haven t given you the explicit form of these equations but we will see if some kinds of arguments cannot be given which make the form of Einsteins equations more plausible To try and do this it is rst convenient to reformulate Newton s law of gravity in a more useful form by introducing the concept of gravitational potential energy This is gotten by asking how much work is done in moving a test mass m in from in nity to a point r distant from a point mass source M The answer is U GMmr as you probably know Actually it is more convenient to ask how energy per unit mass is present there which is just qt 2 U m GMr In this language the tidal gravitational effects are just given by gravity is then equivalent to the equation 1qu w where p is the density of mass at position 7 This is not quite correct but will do Newton s law of Thus we would like to see this equation fall out of the equations of GR in some limit in which all velocities are small compared to c and the tidal forces are not too strong First thing to try and gure out is what is this stress energy object that appears on the RHS of the eld equations of GR It must contain the mass of the gravitating object to reduce to Newton s laws In fact its natural to assume that it is the mass density in that limit since then it looks like p We can go from mass to energy by multiplying by c2 of course so indeed we can hope to end up with an object with dimensions of energy This tensor object must have only one large component in the nonrelativistic limit and this must be the mass density p02 What about the LHS Well this is meant to be the spacetime curvature and it should be given somehow in terms of the met ric guy Actually it is clear that for ordinary systems moving nonrelativistically and for nearly flat spacetime only the time time component 911 is signi cant since it contains a factor of 32 approximately from the Minkowski space form This must be hence identi ed with the Newtonian potential gb Thus we might guess that the curvature of spacetime in this limit reduces to two spatial derivatives of 911 Notice any time derivatives would be highly supressed because now of two factors of c in the denominator Actually you might expect that cur vature of a surface should have something to do rate of change of a tangent to the surface it must be two not derivative since I want a flat plane to yield zero curvature Thus the simplest guess would be that one has R N 2911 Thus we are led to conclude that the equations of GE indeed have the possibility of including Newton s law of gravity in the limit of small velocities and small curvature if we identify the Newtonian potential with the metric component 911 and the only nonzero component of the stress energy tensor which must also be T11 now as mass density This fixes the constant of proportional in the equations of GR also It must be up to pure numbers 0 Actually we canb a little better than this by thinking of the equation of motion of a test particle in the geometric language and comparing it with the result from Newtonian dynamics We can get the form of the equation which determines the geodesics by thinking first of the equation of motion of a free particle in flat spacetime In an inertial frame this is given simply by 12 172 Where 739 is the proper time and mi the 4vector position Now consider what this looks like from a noninertial frame of reference with coordinaites Using the chain rule we can write dxi Z 621quot 172 j By 1739 Similary the acceleration looks like 1225 Bxidgyj 622W dyj dyk 172 j By 172 Byjayk 1739 1739 This second piece is the new thing In these funny the coordi nates the simple free motion can be found by solving the equation dgyj Z dyj dyk W jkww By the principle of equivalence this equation should also hold for geodesic motion in a curved spacetime since inertial effects are equivalent to gravitational The difference between the grav itational situation and the flat space free motion is that there will not exist any global coordinate system in which all the F8 vanish 0 Now think of the limit of small velocities and an approximately inertial frame where y 2 ct Only T30 is important in this limit and the equation looks like 1122 W This will look like Newton s law for motion in a Newtonian a grav1tational eld if we take T1102 2 g2 11 3 0 But T31 should be a derivative of the metric which in this limit is just 911 Thus we can get full correspondance between Newton s laws and Einstein s theory if we take 911N1 o This allows us to x the constant in Einetein s equation to be Gc4 up to pure numbers which can be xed exactly from the full equations 0 Thus Einstein s eld equations and the geodesic rule for motion in a curved spacetime indeed contain Newton s law of gravity plus his famous second law of motion in the limit when 1 ltlt 3 and gravity is weak this was need to use the inertial frame approximation used for recovering the law of motion 0 Onward now to applications Lecture 12 Cosmology Large scale dynamics of Universe is thought to be governed only by gravitational forces In the framework of GR we thus expect the large scale features of the Universe to be described by a spacetime metric which is a solution of the field equations of GR Solutions of Einstein s equations represent all possible space times Which ones describe our Universe This is the subject of cosmology Clearly this hinges on the forms of energy momentum tensor we write for the RHS of the field equations To simplify the equations and get some handle on the possible sources of energy and momentum available consider the following experimental facts Sun is one of about 100 billion stars which are clustered together into Milky Way about 100 thousand light years across This is a typical scale for a galaxy Between galaxies there are voids in space Galaxies often group into clusters and superclusters sometimes these appear to be correlated in position large scale structure problem However it is true that if you average over large enough distances the Universe appears remarkably homoge neous and isotropic looks same at all distances and directions In searching for an understanding of the Universe s structure and dynamics these observational facts are commonly assembled into a cosmological principle which basically says that the Universe looks the same from all points within it o This assumption implies that spacetime is made up of 3d spaces each associated with a different instant of time as measured by clocks on Earth for example Furthermore this space must have constant spatial curvature since no point is distinguished There are then 3 possibilities for this space 1 Flat Euclidean space R 0 Parallel geodesics never cross ln nite in size 2 The 3d sphere with positive spatial curvature R gt O lni tially parallel geodesics converge Finite in size 3 3d hyperboloid Points in 4d at space such that t2 32 32 Z2 1 Diverging geodesics ln nite in size All these possibilities can be realized from the metric Fried mann Robertson Walker metric 1T2 1 k7quot where k O k 1 and k 1 label the three spatial topolo gies This form of the metric also incorporates the isotropy of each spatial slice at xed t 132 dt2 a2t r2 192 sin2 9M Which topology Until recently this was unclear now new observations appear to select the rst possibility a spatially at Universe on the largest scales k 0 COBEVVMAP data While we have asssumed spatial homogeneity the size of the Uni verse may depend on time This is incorporated by introducing the scale factor at which you may think of as giving a typical distance between galaxies at some instant after the Big Bang see later Einstein s equations determine this in terms of the density and pressure of matter and radiation The simplest form for these consistent with our cosmological principle is Tm p T p up p911 where p and p are the density and pressure of a perfect fluid we have set c 1 Finally Einstein s equations become d 2 as k 3 2 a 1201 2am 2 16 a2 87er A 1 87TGp A 2 The term involving A is called the cosmological constant term and was introduced by Einstein so that the equations of GR admitted static solutions However in 1929 Hubble an nounced that all the galaxies were receeding from each other with a velocity proportional to their separation just what you would have expected if at were increasing with time Thus Einstein regarded the introduction of this new term as his Greatest Mis take However recent modern observations on the apparent acceleration of the distant supernovae appear to favor such a term afterall The nature and origin of this dark energy is a source of much work right now Actually such a term is very natural when one takes into account quantum mechanics and rather ironically the real problem facing theoriest at this time is not to explain why this term is there but why it is actually so small Can rewrite the two cosmological equations as Faalt2 477G A 7 3 i a 3 5p p 3 an equation for the cosmic acceleration and dadt 2 are A 7 H 2 t 7 k 2 7 a 3 p a 3 where H is the Hubble constant governed the velocity of the scale factor Notice that a nonzero value of H indicates an expanding Universe as observed by Hubble To clarify what this means let us de ne the redshift 2 as 1t1 1t0 where t1 gt to Hence in an expanding Universe 2 gt 0 Suppose now that to and t1 are close Then we can expand 1 Mil dadtltt1 t1 to Thus 1 1t ZL6t a 12 But for light rays in a at universe 6tat 6r rays propagate radially and we nd that z H This is Hubble s law The red shift is proportional to the distance 0 It is also conventional to parametrize the socalled equation of state for the matter as p 2 mp here for radiation to 13 or extremely relativistic matter and for cold nonrelativistic Let us now restrict ourselves to the at space case A matter matter to O The cosmological constant can even be thought of as a peculiar type of matter with w 1 Notice that we can only get a accelerating expansion which we now observe if 10 lt 13 which can be done with either a pure cosmological constant or some other exotic form of matter with w lt 13 It cannot be achieved with ordinary matter Possible Universes 0 Consider a Universe made up predominately of nonrelativistic matter with p O The pressure equation can be rewritten as d 2 A 3 1tadadt ka 3a O Using the rst cosmological equation we then nd 87TG O 13 It clearly has an interpretation as the conserved amount of mass in a Universe of radius a The equation governed the Hubble parameter now looks like 1a 2 O A 2 7 7 7 k alt a T 3 a From now on we will consider the experimentally most relevant case k 0 o If we set A O as in the old style models we nd at N tg This is called the EinsteindeSitter model check this 0 Furthermore if A lt 0 then we can even nd a static Universe with da it O this was the model rst constructed by Einstein and the reason A was introduced 1 o If A lt O we see that O for a 30A Thus the expansion will halt at some point and recontraction will occur an oscillating Universe We can nd the exact solution in this case too 30 a3 1 COS3 At This solution is termed antideSitter space 0 Notice that for positive A we can nd solutions of the form at N exp the spacetime is called deS itter and describes one possible type of in ationary Universe The best t to current observation demands a small cosmological constant of this sigma o In general we might want the solution to the cosmological equa tions for matter with a general equation of state it If it is constant it is not hard to show that 2 t 31 at a0 w to where do is current scale factor and to age of Universe Big Bang Cosmology 0 Since all viable models of the Universe are dynamic and exhibit some sort of expansion it is natural to extrapolate these models back to a time when a gt 0 At this point we encounter a singularity where the curvature is in nite and spacetime ends It has been hypothesized that this event called the Big Bang corresponded to the birth of spacetime and all the matter in it 0 Given the general picture of how the scale of the Universe changed with time what can be said about the contents of that Universe as we go back in time to the Big Bang o The picture that emerges is sketched below O Big Bang spacetime comes into existence in an initial singularity place of in nite curvature and energy density It starts to expand in accord with the equations of GR t N 10 43 s Regime of quantum gravity Unknown in details but rather general arguments say that gravity becomes as strong as nuclear forces and must be combined with quan tum mechanics in some way Horizon distances distances light has travelled since t 0 are comparable to typical distances between collisions of particles Universe is Planck scale lp 10 33m Continues to expand Exotic particle regime from t 10 43 to t 10 53 Ultra relativistic matter Details depend on your favourite grand uni ed model Naively a N but see later discussion of in ation Electronpositron production regime t 10 5 to t 1 Nucleosynthesis regime t N 100 1000s Hydrogen5 helium and the lightest elements are formed Decoupling time t 100 000 years Neutral atoms form and photons propagate freely This is the surface of last scattering revealed by observations of the microwave background t gt 100 000 years Pure expansion galaxy formation Scale factor grows like t23 Timeline explanation 0 To understand these regimes we need to understand two concepts 1 Thermal equilibrium and 2 Particle Production Thresholds Thermal Equilibrium o Might think that understanding the properties of very many strongly interacting particles would be very hard such as occur during the initial expansion of the Universe after the Big Bang Fortunately it is sometimes possible to describe such systems in terms of very few parameters like the total number of particles the temperature of the system etc Such a description gives up the idea of predicting the detailed motion of all the particles but merely tries to predict average properties the mean energy the distribution of speeds etc This will be possible if the system enters a state of thermal equilibrium In this state the motions becomes randomized and the total energy is distributed in a sta tistical way over all the particles The typical thermal energy is then simply kT where k 138 x 10 23JK Providing the expansion rate is small compared to the time between collisions this will be true Furthermore the number of photons with a given energy is given by a unique function depending only on the temperature this is referred to as a black body spectrum Such a spectrum is seen in the Universe today and corresponds to the redshifted photon radiation from the decoupling time the cosmic microwave background and constitutes a big piece of the evidence for the standard big bang picture Furthermore this temperature is related to the size of the Uni verse by the simple relation T The time at which this occurs can be gotten from the relation 1 N N 3 1 pa IBMHp T m where for a flat Universe t 231w at a0 to we have assumed to is constant to get this Thus As we run time backwards the Universe will contract and heat up Certain special temperatures can be seen at which new physics happens these are the threshold temps Particle Production Thresholds As the energy of say a photon increases it becomes possible to create matter out of photon photon collisions via E mc2 The energy of such a photon must be at least this amount If we imagine a system of photons at a certain temperature this minimal energy will be reached when kT mc2 Above this temperature particles and antiparticles with mass m will be lib erated Other sorts of threshold temp occur when this thermal energy is equal to important other energy scales involved with atoms and nuclei such as the energy scales binding electrons in atoms or the binding energies in simple nuclei decoupling time and nucleosynthesis time second piece of evidence favoring the big bang primordial he lium production 25 percent of stuff in the Universe is helium left over from big bang as free neutrons are bound with free protons Problems with the standard Big Bang picture 0 Flatness problem We can rewrite the equation for the Hubble parameter in a slightly different form k 9 1 where Q and p0 So k 0 at requires 9 1 and p 2 p0 Notice though that pc is not in general a constant We can show that 19 la The problem is that if you require that Q is near 1 almost all cosmic time But this in general is hard to achieve since is not in general zero Notice though that if Q 1 e at some point in the past the derivative looks like 19 la Thus it only for 13w lt 0 will it remain so for ordinary matter it will ow away from the at point very quickly Thus for regular matter it is an amazing coincidence that Q is observed to be close to unity This is called the atness problem Notice though that cosmic acceleration would force this factor to be negative and render Q 1 natural 1 3w QQ 1 6 13 ltwgta Horizon problem Lets see how far a photon can propagate in time t from the Big Bang It is given by alt atl Using the time dependence of a given earlier this is just 1H 2 atAr at m 1H N t a 2 This is the horizon distance the maximum distance that can separate causally connected events in the Universe Notice that ordinary matter w gt 0 this always less than the size of the Universe Thus one would predict that if one looks at widely separated regions of the Universe they should show no correlation in properties if the Universe started in some general state But this is not what is observed regions that were never in causal contact appear to be almost identical ll This is the horizon problem Solution in ation We assume that early on in the Universe there was a period of very rapid accelerating expansion This requires 10 lt 13 as we saw before But in this case the solu tion 9 1 is automatically stable and the Universe is naturally fiat Also this will lead to 1H N ap where p gt 1 and the horizon is always larger than the scale factor Both problems are cured Furthermore quantum mechanical fluctuations in the early Uni verse get blown up to large sizes and offer a way of generating the density perturbations one needs to allow large scale structure formation galaxies These are robust predictions that don t depend on details of high energy physics or issues of quantum gravity PHY312 lecture 4 Simon Catterall