Review of Special Theory of Relativity

                                                                                Akhila Raman

 

1.     Introduction

 

Special Relativity(SR) states that the physical laws of nature take the same form in all uniformly moving inertial frames and remain unchanged under a  Lorentz transformation. It also states that light speed "c" is constant in all  inertial frames and is independent of the speed of its source.

 

Let an event take place at a distance "X" and time "T" in one frame of reference F, the same event will have coordinates  X' and T' in another frame F' moving at a relative velocity "v" w.r.to F, according to the following Lorentz Transformation(LT):[1]

 

X' = ( X – v*T )/K                  --- eqn A

T' = (T – X*v/c2) /K                --- eqn B

where   K=           

 

We can see that when X=c*T, X'= c*T'.

An implication of  SR is that identical clocks in 2 inertial frames moving at a relative velocity will tick differently(time dilation). And distances will be measured differently.

 

These equations can be geometrically represented as in Fig 5. in Appendix. From eqn. B, we see that  when | X*v/c2 | > | T |, T' will have a sign opposite to that of T! This means that an event which is in the past of one frame, can appear in the future of another frame moving at  a relative velocity w.r.to the former! Say, an observer O at rest and another observer O' in motion pass each other at the origin(T=0,T'=0). An event Er which is in the past of O(T<0) will be in the future of O'(T'>0) for the region R with both T<0 and T'>0 bounded by thick lines, as depicted in Fig 5, thus allowing the possibility of time travel. However, this region R is always outside the light cone of both observers, meaning that  neither do they know of the event E, nor can they influence the event, given nothing can travel faster than light(FTL)! BUT, using concepts like wormholes, theoretically one can travel "effectively" FTL and this introduces the disturbing possibility of time travel, causality violations and unsolvable paradoxes. For example, if one could go back into the past and kill one's own grandfather when he was a child, then how does one explain one's own existence?[2]

 

2. Physical interpretation of key concepts:

 

 

2.1 Time dilation: How moving clocks tick differently

 

 

This is well explained in Fig.1 above.[1]  A photon clock apparatus is a simple setup in which a photon travels from a source to a mirror and back and everytime it completes a roundtrip, it illuminates the regular clock C1 beside the source. A similar array of apparatus is present in frame F' also along the horizontal axis, only the clock values are shown for simplicity. We wish to compare the relative clock times in the 2 frames, upon completion of a roundtrip in frame F. From the viewpoint of an observer O' in F', frame F is moving to the left with velocity "v", he sees the photon travelling a longer diagonal distance to complete a roundtrip and given the fact that photon speed is the same in all inertial frames, the roundtrip will take longer in F and by that time clock in F' will show a time greater by a factor of K(using simple pythogorean theorem).

 

The keypoint is that: any clock which is present in both events(clock C1 in this case, the events being sending and receipt of the photon) shows the minimum possible time which is termed as the "proper time". Two different clocks spaced apart measuring  spatially separated events, will show a time greater by a factor of K and is termed as "improper time".(Statement A)

 

Note that this time dilation applies not only to photon clocks, but also to any mechanical clock and our biological clocks!

 

2.2  Initial Clock Synchronisation and Relativity of Simultaneity:

 

It is impossible to synchronise  the clocks at points A and B in both the frames(totally 4 clocks for a pair of points) to zero at the same time. This is because one of the frames is moving w.r.to the other and hence light signal in the moving frame reaches one of the clocks earlier and the other clock later, while the signal reaches both clocks at the same time in the rest frame. Hence, out of the 4 clocks, while 3 read zero, the fourth will lead or lag by a value E= X*v/(c2 *K), depending on the direction of motion. This is clearly depicted in Fig 2.a through Fig 2.d in Appendix.  Hence clocks which are simultaneous in one frame, are NOT simultaneous in another frame, and this is termed as Relativity of Simultaneity, as depicted in Fig 2.e.

 

 Note that whichever method of synchronisation one chooses, whether using light signals, or by slow clock transport(synchronising them all at one location and then slowly transporting them to the desired location by say walk), this term E will be always present.[4]

 

It is interesting to note that there is no previleged frame of reference. Each frame might think it is at rest and the other frame is moving and there is no absolute validity to both claims in inertial frames. Hence when an observer O in frame F, located at point A, sees that clocks read zero at point A in both frames(T1, T1'), cannot say with absolute confidence  that the clock at point B in his frame(T2) reads zero “at the same time” as when his clock at A(T1) reads zero, simply because there is no way he can be sure whether his inertial frame is at rest or moving! However he can confidently say that when his clock at B(T2) reads zero, other frame's clock at B (T2')  leads/lags by   "E" ! And the same goes for the observer in frame F'.

 

Thus we see that, there is a synchronisation offset factor of E , between corresponding pairs of clocks in F and F'.[3]

 

 

 

Points to remember:

 

a.       Constancy of speed of light in all inertial frames is still preserved. We  have  X'=c*T' when X=c*T.

 

 

b.      In Section 2.1,  there is no reason why frame F' is previleged to be at rest. It is quite possible that F' is actually moving and F is at rest! In this case, clocks in F' will tick slower, and upon completion of roundtrip of photon in F, the clocks in F and F' will still show the same values! BUT, the actual time elapsed in F' between the clocks C1' and C2' = T1'= T/K – E1 where

 

     clock synchronisation residual offset factor between C1' and C2' =

     E1= X*v/ (c2*K ) = (vT) * v/(c2*K)= T/K* (v2/c2).

     [Note: X = vT from viewpoint of frame F in Sec 2.1]

 

      Hence T1'= (T/K) * ( 1-  (v2/c2) ) = (T/K)* K2 = T*K.

 

In either case, time dilation  is a true phenomenon, not merely an observational effect. Depending on who is moving and who is at rest, the actual time elapsed is greater or smaller by a factor K. However, the statement A given in section 2.1 is still valid, because it pertains only to times shown by clocks, not to actual time elapsed ! Infact, actual time elapsed  cannot be determined in absolute sense in inertial motion. But, we can confidently say that the actual time elapsed between 2 clocks measuring improper time differs from the proper time by factor K.We do not know if it is greater or smaller.

 

c. Twin Paradox:

 

In the case of twin paradox, the physical asymmetrical aging between twins is a physically true phenomenon. Because one of the twins who went on a spacecraft experiences acceleration(non-inertial motion) on turning back to earth and hence both know who was at rest and who was moving and there is no ambiguity about actual time elapsed. Hence both can agree that the twin on spacecraft aged slower than the twin on earth at rest.

 

 

 

References:

 

1.      James H. Smith “Introduction to Special Relativity” 1965.

2.      Jason Hinson “Relativity and FTL travel” Chapter 8:

http://www.physicsguy.com/ftl/html/FTL_part1.html#chap:srintro

 

3.      A.P.French “Special Relativity”1968  

4.      Arthur Eddington   "Mathematical Theory of relativity"

5.      Albert Einstein  “Relativity”1961