Relativity and Cosmology Einstein and Beyond
Relativity and Cosmology Einstein and Beyond PHY 312
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Date Created: 10/21/15
Lecture 10 Recap of Schwarzschild solution and some pictures 0 Solution of GR for spherical symmetry outside an attractive mass XVI is very similar Schwarzschild 1915 AW 7quot A9 1 The function Afr is given by Afr 1 A32 62ATAt2 To write this we have necessarily adopted a speci c global co ordinate system The Tcoordinatecircumference27 is the radius from the center of the BH or star for a faraway observer Similarly the time t is a faraway time appropriate to an ob server located at spatial in nity The angle 9 is the conventional polar angle locating the position in a plane through the center we have suppressed all dependence on the third spatial coordi nate 2 say as all motions we will consider take place in a plane If we consider two simultaneous events in this spacetime which are at the same angle then we can immediately read off the expression for the distance between them from spacelike form 2 Aquot AW A3 ATshell So predicts the distance measured by an observer stationary at some r coordinate a shell observer would be greater than the difference in the r coordinate Ar 0 We may draw a picture which allows this stretching of space to be visualized I x on an instant of farway time t which reduces my expression to one involving just 7quot and 9 We are hence interested in a representation of the spatial curvature of a 2D surface This can be done by embedding the surface in a flat 3D space This extra dimension is an aid to visualizing what is going on It is not physical The resulting picture is called an embedding diagram for the Schwarzschild metric Picture The pro le of the surface is just given by the function Afr It should be clear from this picture now why the distance between two points at different r coordinate is bigger than than the mere difference in 39r The event horizon corresponds to the point where the slope of the pro le is in nite this representation does not work for points inside the event horizon The picture helps make it clear that once inside the event horizon escape is impossible Similarly if we consider a clock at that shell and imagine mea suring the time between ticks the metric timelike form gives a new shell time which is smaller than the time measured by a faraway observer At A7 Aa m 3 This is the origin of the gravitational redshift for photons escap ing from a gravitational eld Notice that if 39r TE 2rMG02 A is zero time shift factor is zer In nite red shift This is the event horizon a surface through which nothing not even light can pass out There is another way to see that the event horizon is a place of no return Consider the possible paths of a light ray They are represented by the boundaries of a light cone along which A5 O Lets just consider radial motion of light A9 0 Then we see that external to TE the outward velocity of light relative to our global coordinte system is just i CAO which indeed approaches c at spatial in nity As 7 gt TE however the velocity falls to zero and then reverses for 39r lt TE Spacetime is so curved that the ligt cones tip over and light cannot get out It must inevitably hit the singularity at 39r O at some later time Since the motion of material particles always lie within the light cone this same conclusion also applies to them Nothing with TE can get out and must proceed to 39r O in some nite proper time The event horizon divides spacetime into two regions Events in the external region cannot be affected by anything going on in the interior region They are causally disconnected Notice that other coordinate systems can also be used to view black holes and the like the shell frame the free fall frame and this global Scharwzschild rt frame GR is frame indepen dent the coordinates used to describe some motion through spacetime may yield widely different mea surements of distances times speeds and energies for bodies etc but there is only one unique physical motion of any object through the spacetime Frames of reference 0 Imagine falling into a black hole Lets describe the motion in 3 frames of reference free fall frame shell frame and Schwarzschild frame 0 FFF 0 As we fall we notice that coins keys etc remain at rest or moving at constant velocity We can use special relativity to compute distances in spacetime between 2 events This simplicity is only local we detect the ever increasing curvature of spacetime by observing tidal motions which get larger and larger eventually causing us discomfort and nally death However for sufficiently small regions of spacetime a FFF can always be de ned Shell Like the surface of the Earth Close to a black hole we may simply imagine powering a rocket such that one remains station ary at some r coordinate These frames can only exist outside the event horizon The departure from the natural motion of free float we experience as the force of gravity For ordinary motions we just add this in to compute motions For HEP eXpts it is not necessary even to do this timescales are so short we can just neglect gravity completely So shell frame looks like an inertial frame then Analyze using SR Notice that the Schwarzschild metric written in terms of Arman and Afghan looks like a at metric Since both shell and FF observer think that spacetime is locally flat they can use SR to relate their different measure ments Both shell and FF frames are physical frames that is they can be used by real observers to make real measurements Unfortunately they can only be used locally The Schwarzschild frame can be used to analyze events not nec essarily near in spacetime eg opposite sides of black hole uses the r coordinate the far away time t and an angle 9 Very important although this is global no single real observer can make the measurements In that sense its not really a reference frame but is a convenient way of plotting out trajectories made in spacetime individual increments can be measured locally by some shell observer and then converted to Schwarzschild coordi nates Suppose I am falling into a black hole I can ask now a bunch of questions like how fast am I going with respect to the Schwarzschild or other coordinates when will I hit the central singularity how much energy do I have etc etc To analyze this it is useful to think of energy In SR we have learned that energy is conserved that is the energy of an isolated system does not change with time Is there a notion of energy in GR which also satis es this principle Yes Expression 2 At E33 me E 4 In GR this is modi ed to EOE meg 1 2G1lI At 02quot AT 5 This is sometimes called the energy at in nity since it corre sponds to the energy a remote observer would ascribe to the falling body It has the property that it is conserved If goes over to SR for XVI O or 39r gt 00 It can be derived from the idea that particles follow geodesics of greatest possible proper time Aside conservation of energy from maximal ageing 0 Consider rst the case of at spacetime Imagine watching the motion of some free particle which moves from event 0 0 through an intermediate event if 3 to a nal event T X Lets think of T X a as xed and t the intermediate time as free 0 The principle of maximal ageing says that 1739 zit 0 where 739 714 T3 and A and B denote the two parts of the motion This yields the equation LL73 i612 274 dt 27 dt where in SR we have 0273 02 x2 02712 62T 02 X 2 This means that i B the trajectory with greatest proper TA TB time is the one in which the quantity the ratio of frame time to proper time is conserved Since we expect energy to be a constant of the motion for free particles we see that our principle leads to the correct de nition of relativistic energy E m dl d7 where m the mass must be an invariant This argument can be generalized to the case of radial motion in the Schwarzschild spacetime Think of events T F A7quot t r and O 39r A corresponding to the free motion of a test particle in the spacetime Again we will require that the free motion be given by a timelike geodesic on which the proper time elapsed is a maximum 1 1723 2714 E where the expression for 714 etc is given by the timelike Schwarzschild metric For this purpose let the mean position for the interval 7quot Aquot gt 39r by TA and 39r gt 39r Aquot by T3 It is straightforward to now see that the quantity 1 17393 t T t AVA4 ACTB TB Thus the expression for the conserved energy in GR is that given above Notice that we must strictly now work in the limit Aquot gt O and t gt O in order to use the metric to compute the proper time Falling into a Black Hole 0 Notice that for a particle initially at rest at in nity the energy is just the usual rest mass energy E me2 and it continues to have this energy as it falls in o For such a particle we can ask what is the velocity measured in Schwarzschild coordinates Conservation of energy leads to 2G1M 1 02391quot 2 j M m2 6 Combining with the metric formula yields 1 Aquot 2G1M 2G1M 7 1 7 At cl l l l Notice as the particle approaches the horizon its speed goes to zero It slows down as it approaches the horizon measured in Schwarzschild coordinates 02quot 02 0 You might think that a black hole as observed by a remote observer would be surrounded by images of all the junk that has every fallen into it forming a kind of halo at the event horizon it wouldn t be black Actually no the light from all this junk is in nitely red shifted as we approach the horizon and so it still looks black 0 What does locally motionless shell observer see for such a par ticle Using the expressions Arman A 12Ar and Afghan AlQAt it is easy to see that Arshell Atshell So the velocity as observed by this observer approaches the speed of light 2GM12 02quot So the closer the body falls the faster it goes for a shell observer thus its energy will increase So the shell energy will be using SR E3116 Using for v AfghanAtmng from above me 2 we d Eshell Since E me2 is the energy of the C 7 stone initially at rest as measured by the far away observer we see 9 This is a quite general expression for the energy of anything falling into the black hole as measured by a shell observer So while E is a constant Ewe is not and is the energy available to a shell observer Eshell 20M 1 7 Time to crunch Once past the horizon our fate is sealed no amount of rocket power can stop us proceeding to 39r 0 We reach it in a nite time as measured on our wristwatch We can nd this time by nding an expression for the rate of change of r coordinate with proper time g 20114 AT 10 lt gt Thus 739 This is about 10 63 for a solar mass black hole For a very massive black hole say at the center of a galaxy or a quasar this may increase to minutes As soon as we cross the horizon even if tidal forces are still quite weak it is impossible to return or send any signal to the outside world The nal conclusion is inevitable Aside energy of a stationary object at xed 7 0 Energy ascribed to an object a xed r ccordinate can also be worked out from the expression for energyat in nity E meg A39r using the relationship of xEdt 1739 dtgheu yielding 2 2G1M 1 E me 62 o This decrease over the rest mass energy can be interpreteted loosely as a result of gravitational binding negative gravitational potential energy Notice the energy approaches zero for an ob ject stationary just above the event horizon 0 Can derive the gravitational redshift this way by using the idea that the energy of a photon frequency is reduced on propagat ing out to in nity using E h f hCA Examples of radial motion 0 What if we throw the object with some initial speed into the black hole Does this change the observed speed by either global or shell observers as it crosses the event horizon 0 Conservation of energy leads to it 1 7 Y TE dT where 7 corresponds to the initial value of E meg for an object in radial motion inwards Substitute into this the expression for 1739 from the Schwarzschild metric and we nd 1 1 d7 c1 TET 1 g 1 TET We see that these limiting speeds remain the same c is indeed a local maximum speed for material objects Dropping from a nite r coordinate Use conservation of energy and the form of the metric together with your expression for the energy of a stationary object at some xed 7 to nd expressions for the Schwarzschild velocity and shell velocity for a stationary object dropped from rest at 39r To Answer 1 17 TET TETO y 17 1 TETO and a similar expression for the shell velocity Notice that the shell speed approaches the speed of light at the horizon indepen dent of the initial r coordinate To c1 39rE39r Corrections to Newtonian gravitational acceleration Consider some shell observer at r coordinate To Differentiate the above result with respect to the and reexpress the expression as a function only of shell quantities to derive an expression for the 2 locally de ned gravitational acceleration in GR shell 2 d T3116 G1M 2 dtslzell 1 2G1MTO 2 To 0 Newton approximates GR for plunging object for small when and GUM 1 dt dtmz 0 From Em 1 39rgrdT subsltute dT dim g Use the expresslon you know for T h u Use also the expresslon 8 5 1T2 dtiheu draw02 to nd idtg f Hence show that for small enough omenc and 39rgr that Gill 7 1 Em N 1 final