Great Ideas in Science
Great Ideas in Science PHYS 2018
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Introduction to Special Relativity Dr Bob Gardner Great Ideas in Science PHYS 2018 Notes based on Dijj e rential Geometry and Relativity Theory An Introduction by Richard Faber DIFFERENTIAL GEOMETRY AND RELATIVITY THEORY An Introduction Richard L Faber Chapter 2 Special Relativity The Geometry of Flat Spacetime Note Classically ie in Newtonian mechanics space is thought of as 1 unbounded and in nite 2 3 dimensional and explained by Euclidean geometry and 3 always similar and immovable Newton Principia Mathematical 1687 This would imply that one could set up a system of spatial coordinates xyz and describe any dynamical event in terms of these spatial coordinates and time t Note Newton s Three Laws of Motion 1 The Law of Inertia A body at rest remains at rest and a body in motion remains in motion with a constant speed and in a straight line unless acted upon by an outside force 2 The acceleration of an object is proportional to the force acting upon a it and is directed in the direction of the force That is F m5 3 To every action there is an equal and opposite reaction Note Newton also stated his Law of Universal Gravitation in Prin ctpta Every particle in the universe attracts every other particle in such a way that the force between the two is directed along the line between them and has a magnitude propor tional to the product of their masses and inversely propor tional to the square of the distance between them where F is the magnitude of the force 7 the Syrnbolically F Gjlgm distance between the two bodies M and m are the masses of the bodies involved and G is the gravitational constant 667 X 10 8 cm g sec2 Assurning only Newton s Law of Universal Gravitation and Newton s Second Law of Motion one can derive Kepler s Laws of Planetary Mo tion 21 Inertial Frames of Reference De nition A frame of reference is a system of spatial coordinates and possibly a temporal coordinate A frame of reference in which the Law of Inertia holds is an inertial frame or inertial system An observer at rest ie with zero velocity in such a system is an inertial observer Note The main idea of an inertial observer in an inertial frame is that the observer experiences no acceleration and therefore no net force If S is an inertial frame and S is a frame ie coordinate system moving uniformly relative to S then S is itself an inertial frame Frames S and S are equivalent in the sense that there is no mechanical experiment that can be conducted to determine whether either frame is at rest or in uniform motion that is there is no preferred frame This is called the Galilean or classical Principle of Relativity Note Special relativity deals with the observations of phenomena by inertial observers and with the comparison of observations of inertial observers in equivalent frames ie NO ACCELERATION General relativity takes into consideration the effects of acceleration and there fore gravitation on observations 22 The MichelsonsMorley Experiment Note Sound waves need a medium though which to travel In 1864 James Clerk Maxwell showed that light is an electromagnetic wave Therefore it was assumed that there is an ether which propagates light waves This ether was assumed to be everywhere and unaffected by matter This ether could be used to determine an absolute reference frame with the help of observing how light propagates through the ether Note The MichelsoneMorley experiment circa 1885 was performed to detect the Earth s motion through the ether as follows Mirror Ether Wind 4 Arm 2 I Arm 1 nght 7 Source Beam Splitter MHTOT The Viewer will see the two beams of light which have traveled along different arms display some interference pattern If the system is rotated then the in uence of the ether wind should change the time the beams of light take to travel along the arms and therefore should change the interference pattern The experiment was performed at different times of the day and of the year NO CHANGE IN THE INTERFERENCE PATTERN WAS OBSERVED Note In 1892 Fitzgerald proposed that an object moving through the ether wind with velocity v experiences a contraction in the direc tion of the ether wind of W That is in the diagram above L1 is contracted to inm and then we get t1 t2 when L1 L2 potentially explaining the results of the Michelson Morley experiment This is called the Lorentz Fitzgerald contraction Even under this as sumption it turns out that the Michelson Morley apparatus with unequal arms will exhibit a pattern shift over a 6 month period as the Earth changes direction in its orbit around the Sun In 1932 Kennedy and Thorndike performed such an experiment and detected no such shift Conclusion The speed of light is constant and the same in all direc tions and in all inertial frames 23 The Postulates of Relativity Note Albert Einstein published Zur Elektrodynamz39k bewegter Korper On the Electrodynamics of Moving Bodies in Annalen der Physik An nals of Physics 17 1905 In this paper he established the SPECIAL THEORY OF RELATIVITY I quote from The Principles of Relativity by H A Lorenz A Einstein H Minkowski and H Weyl published by Dover Publications the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of me chanics hold good We raise this conjecture the purport of which will hereafter be called the Principle of Relativity to the status of a postulate and also introduce another postulate which is only apparently irreconcilable with the former namely that light is always propagated in empty space with a de nite velocity 0 which is independent of the state of motion of the emitting body In short P1 All physical laws valid in one frame of reference are equally valid in any other frame moving uniformly relative to the rst P2 The speed of light in a vacuum is the same in all inertial frames of reference regardless of the motion of the light source From these two simple and empirically vari ed assumptions arises the beginning of the revolution that marks our transition from classical to modern physics 24 Relativity of Simultaneity Note Suppose two trains T and T pass each other traveling in op posite directions this is equivalent to two inertial frames moving uni formly relative to one another Also suppose there is a ash of lighten ing an emission of light at a certain point Mark the points on trains T and T where this ash occurs at A and A respectively Next suppose there is another ash of lightning and mark the points B and B Suppose point 0 on train T is midway between points A and B7 AND that point 0 on train T is midway between points A and B An outsider might see Suppose an observer at point 0 sees the ashes at points A and B occur at the same time From the point of view of O the sequence of events is 1 Both ashes occur7 A O B i T A7 07 B opposite A7O7 BC resp x T39 gt 2 v 3 v 4 Wavefront from BB meets O T Both wavefronts meet 0 T Wavefront from AA meets O From the point of vieW of an observer at 0 7 the following sequence of events are observed 1 Flash occurs at BB 4 T A 0 B39 2 Flash occurs A O B at AA T A O B 3 Wavefront from A O B BB meets O T 4 Wavefronts from AA and BB meet 0 A O B 5 Wavefront from AA meets O A O B Notice that the speed of light is the same in both frames of reference However7 the observer on train T sees the ashes occur simultaneously7 Whereas the observer on train Tl sees the ash at BB occur before the ash at AA Therefore7 events that appear to be simultaneous in one frame of reference7 may not appear to be simultaneous in another This is the relativity 0f simultaneity Note The relativity of simultaneity has implications for the mea surements of lengths In order to measure the length of an object we must measure the position of both ends of the object simultaneously Therefore if the object is moving relative to us there is a problem In the above example observer 0 sees distances AB and AB equal but observer 0 sees AB shorter than AB Therefore we see that measurements of lengths are relative 25 Coordinates De nition In 3 dimensional geometry positions are represented by points x y In physics we are interested in events which have both time and position txyz The collection of all possible events is spacettme De nition With an event txy z in spacetime we associate the units of cm with coordinates x y z In addition we express t time in terms of cm by multiplying it by c In fact many texts use coordinates ctxyz for events These common units cm for us are called geometric um39ts Note We express velocities in dimensionless units by dividing them by 0 So for velocity v in cm sec say we associate the dimensionless velocity vc Notice that under this convention the speed of light is 1 Note In an inertial frame S we can imagine a grid laid out with a clock at each point of the grid The clocks can by synchronized see page 118 for details When we mention that an object is observed in frame S we mean that all of its parts are measured simultaneously using the synchronized clocks This can be quite different from what an observer at a point actually sees Note From now on when we consider two inertial frames S and S moving uniformly relative to each other we adopt the conventions 1 The are and ziaxes and their positive directions coincide 2 Relative to S S is moving in the positive z direction with velocity 3 3 The y7 and yiaxes are always parallel 4 The 27 and 2Laxes are always parallel We call S the laboratory frame and S the rocket frame Assumptions We assume space is homogeneous and isotropic that is space appears the same at all points on a suf ciently large scale and appears the same in all directions Note In the next section we ll see that things are much different in the direction of motion 26 Invariance of the Interval Note In this section we de ne a quantity called the interval be tween two events which is invariant under a change of spacetime coor dinates from one inertial frame to another analogous to distance in geometry We will also derive equations for time and length dilationi Note Consider the experiment described in Figure II S Mirror 9 M b l B B 839 Point of View 3 Point of View In inertial frame 5quot a beam of light is emitted from the origin travels a distance L hits a mirror and returns to the origini If At39 is the amount of time it takes the light to return to the origin then L A1572 recall that t is multiplied by c in order to put it in geometric units An observer in frame 5 sees the light follow the path of Figure II Sb in time At Notice that the situation here is not symmetric since the laboratory observer requires two clocks at two positions to determine At whereas the rocket observer only needs one clock so the Principle 15 of Relativity does not apply In geometric units we have At22 AtlZ2 Age22 or Atl2 At2 A062 with the velocity of 3 relative to S we have AgeAt and so A06 At and At2 At2 At2 or At xl mm 78 Therefore we see that under the hypotheses of relativity time is not absolute and the time between events depends on an observer s motion relative to the events Note You might be more familiar with equation 78 in the form At Atl W where At is an interval of time in the rocket frame and At is how the laboratory frame measures this time interval Notice At 2 At so that time is dilated lengthened Note Since vc for 1 ltlt 0 m 0 and At m At De nition Suppose events A and B occur in inertial frame S at t1x1y1z1 and t2x2y222 respectively where yl yg and 21 22 Then de ne the interval or proper time between A and B as A7 At2 A062 where At t2 t1 and A06 062 061 Note As shown above in the 3 frame At2 Ax2 A152 A062 recall Ax 0 So AT is the same in 3 That is the interval is invariant from S to 3 As the text says The interval is to spacetirne geometry what the distance is to Euclidean geometry Note We could extend the de nition of interval to motion more com plicated than motion along the x axis as follows AT KN A062 A112 MVP2 01quot interval2 tirne separation2 space separation Note Let s explore this time dilation in more detail In our example we have events A and B occuring in the S frame at the same position At 0 but at different times Suppose for example that events A and B are separated by one time unit in the S frame At 1 We could then represent the ticking of a second hand on a watch which is stationary in the S frame by these two events An observer in the S frame then measures this At l as At At W That is an observer in the S frame sees the one time unit stretched dilated to a length of 1 2 1 time unit So the factor W shows how much slower a moving clock ticks in comparison to a sta tionary clock The Principle of Relativity implies that on observer in frame S will see a clock stationary in the S frame tick slowly as well However the Principle of Relativity does not apply in our example above see p 123 and both an observer in S and an observer in S agree that At and At are related by At At M So both agree that At 2 At m this case This seems strange ini tially but will make more sense when we explore the interval below Remember A06 y 0 De nition An interval in which time separation dominates and AT2 gt 0 is ttmeltke An interval in which space separation dominates and AT2 lt 0 is spaceltke An interval for which AT 0 is lightltke Note If it is possible for a material particle to be present at two events then the events are separated by a timelike interval No material object can be present at two events which are separated by a spacelike interval the particle would have to go faster than light If a ray of light can travel between two events then the events are separated by an interval which is lightlike We see this in more detail when we look at spacetime diagrams Section 28 Note If an observer in frame 3 passes a platform all the train talk is due to Einstein s original work of length L in frame S at a speed of then a laboratory observer on the platform sees the rocket observer pass the platform in a time At L As argued above the rocket observer measures this time period as At AIRWE Therefore the rocket observer sees the platform go by in time A17 and so measures the length of the platform as L At At 1 2 L 1 2 Therefore we see that the time dilation also implies a length contraction L L 1 g 83 Note Equation 83 implies that lengths are contracted when an ob ject is moving fast relative to the observer Notice that with m 0 L m L Example Exercise 262 Pions are subatomic particles which de cay radioactively At rest they have a half life of 18 X 10 8 sec A pion beam is accelerated to 099 According to classical physics this beam should drop to one half its original intensity after traveling for 0993 gtlt 10818 X 10 s 53m However it is found that it drops to about one half intensity after traveling 38m Explain using either time dilation or length contraction Solution Time is not absolute and a given amount of time A17 in one inertial frame the pion s frame say is observed to be dilated in another inertial frame the particle accelerator s to At A17 m So with A17 18 X 10 8sec and 099 At ns gtlt 108sec 128 x 10 7sec Now with 99 the speed of the pion is 993 gtlt 108msec 297 X 108msec and in the inertial frame of the accelerator the pion travels 297 x 108msec128 gtlt 107sec 38m In terms of length contraction the accelerator s length of 38m is con tracted to a length of L L 1 52 38m1 992 53m in the pion s frame With 1 990 the pion travels this distance in 53m 993 gtlt 108msec This is the half life and therefore the pion drops to 12 its intensity 18 x 10 8sec after traveling 38m in the accelerator s frame 20 27 The Lorentz Transformation Note We seek to nd the transformation of the coordinates x y 21 in an inertial frame S to the coordinates 06 7 2 15 in inertial frame 3 Throughout this section we assume the x and 06 axes coincide 3 moves with velocity in the direction of the positive x axis and the origins of the systems coincide at t t 0 See Figure 11 9 page 128 Note Classically we have the relations xx t 212 22 2527 De nition The assumption of homogeneity says that there is no pre ferred location in space that is space looks the same at all points on a suf ciently large scale The assumption of isotropy says that there is no preferred direction in space that is space looks the same in every direction Note Under the assumptions of homogeneity and isotropy the rela tions between xyzt and x yzt must be linear throughout everything is done in geometric unitsl 06 11106 1121 1132 11417 y 12106 1221 1232 12417 Z 13106 1321 1332 13417 t 141 my 1432 14417 If not say y 1062 then a rod lying along the x axis of length 06 and would get longer as we moved it out the x axis contradicting homo geneity Similarly relationships involving time must be linear since the length of a time interval should not depend on time itself nor should the length of a spatial interval Note We saw in Section 25 that lengths perpendicular to the direction of motion are invariant Therefore Note 06 does not depend on y and 2 Therefore the coef cients on and 113 are 0 Similarly isotropy implies 142 143 0 We have reduced the system of equations to x a11xla14tl 85 t a41xla44tl 86 De nition The transformation relating coordinates x y z t in S to coordinates x y 2 15 in 3 given by i xl tl Vl g y 2 z 2 ti an17 W is called the Lorentz Transformation Note With ltlt 1 and g m 0 we have x 58 t t 15 in geometric units 06 and 06 are small compared to t and t and xl is negligible compared to 15 but t is NOT negligible compared to 06 Note By the Principle of Relativity we can invert the Lorentz Trans formation simply by interchanging x and t with 06 and 15 respectively and replacing with l