Time Correction Using Simultaneity Requirement

 

Introduction

            There is something very confusing about relativity.  We have several possibilities for interpreting experiments.  First we need to ask about results in the proper frames of both fixed and moving observers.  Then we need to ask about how a fixed observer views the results of an experiment done in a moving frame.  In a perfectly asymmetric manner we need to ask about how the moving observer views results of an experiment in the fixed frame.  The latter question is what I will consider here.

Obtaining the time difference for a simple experiment MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOaa@35EA@       

            We will use the convention that lower case (x,t) are the parameters used by the observer in the fixed system and upper case (X,T) are the parameters used by the moving system.  

            First, let's assume the simplest case scenario where both observers are coincident at times T=0 and t=0 when both events occur (in the stationary frame) and that the events are separated in the stationary observer's proper frame by distance D. Also assume that one of the events in the stationary system is at x=-D/2 and the other event is at x=+D/2.

For the events imagine simultaneous (in the fixed frame) lightning strikes at -D/2 and D/2.  These will lead to light rays as well as thunder sound which arrives much later but, if in still air, will result in the same equations as light.  The symbol c will stand for the speed of sound or speed of light.

Obviously, the stationary observer hears or sees the events  at the same time

t= D 2c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabg2da9maalaaabaGaamiraaqaaiaaikdacaWGJbaaaaaa @3A87@  

(1.1)

so for him ΔT=0.  The moving observer hears or sees the +D/2  event sooner at

T Front = D 2(c+v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaa Wcbaaeaaaaaaaaa8qacaWGgbGaamOCaiaad+gacaWGUbGaamiDaaWd aeqaaOWdbiabg2da9maalaaabaGaamiraaqaaiaaikdacaGGOaGaam 4yaiabgUcaRiaadAhacaGGPaaaaaaa@4294@  

(1.2)

and the -D/2 event later at

T Back = D 2(cv) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaa WcbaGaamOqaiaadggacaWGJbGaam4AaaqabaGcqaaaaaaaaaWdbiab g2da9maalaaabaGaamiraaqaaiaaikdacaGGOaGaam4yaiabgkHiTi aadAhacaGGPaaaaaaa@4163@  

(1.3)

The difference between these two receptions is

ΔT= Dv c 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaamivaiabg2da9iabgkHiTmaalaaabaGaamiraiaadAha aeaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamODamaaCa aaleqabaGaaGOmaaaaaaaaaa@40BD@  

(1.4)

We can now generalize the expression for time transformation taking this difference into account

T=tx'v/( c 2 v 2 ) X=xvt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivai abg2da9iaadshacqGHsislcaWG4bGaai4jaiaadAhacaGGVaGaaiik aiaadogadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG2bWaaWbaaS qabeaacaaIYaaaaOGaaiykaaqaaiaadIfacqGH9aqpcaWG4bGaeyOe I0IaamODaiaadshaaaaa@48DD@  

(1.5)

Up to now the equations are okay but we have to take into account that x' in the first of equations (1.5) is really x from a different reference frame and must be transformed according to (see Appendix)

x'=xvt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGNa Gaeyypa0JaamiEaiabgkHiTiaadAhacaWG0baaaa@3C77@  

(1.6)

Collecting terms in t we get:

T=t(xvt)xv/( c 2 v 2 )=[t(1+ v 2 c 2 v 2 )xv 1 c 2 v 2 ] = c 2 c 2 v 2 ( txv/ c 2 )= γ 2 (txv/ c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivai abg2da9iaadshacqGHsislcaGGOaGaaiiEaiabgkHiTiaadAhacaWG 0bGaaiykaiaadIhacaWG2bGaai4laiaacIcacaWGJbWaaWbaaSqabe aacaaIYaaaaOGaeyOeI0IaamODamaaCaaaleqabaGaaGOmaaaakiaa cMcacqGH9aqpcaGGBbGaamiDaiaacIcacaaIXaGaey4kaSYaaSaaae aacaWG2bWaaWbaaSqabeaacaaIYaaaaaGcbaGaam4yamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaadAhadaahaaWcbeqaaiaaikdaaaaaaO GaaiykaiabgkHiTiaadIhacaWG2bWaaSaaaeaacaaIXaaabaGaam4y amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadAhadaahaaWcbeqaai aaikdaaaaaaOGaaiyxaaqaaiabg2da9maalaaabaGaam4yamaaCaaa leqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaGccq GHsislcaWG2bWaaWbaaSqabeaacaaIYaaaaaaakmaabmaabaGaamiD aiabgkHiTiaadIhacaWG2bGaai4laiaadogadaahaaWcbeqaaiaaik daaaaakiaawIcacaGLPaaacqGH9aqpcqaHZoWzdaahaaWcbeqaaiaa ikdaaaGccaGGOaGaaiiDaiabgkHiTiaacIhacaGG2bGaai4laiaaco gadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaaa@7873@  

(1.7)

where

γ= 1 1 v 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNjabg2 da9maalaaabaGaaGymaaqaamaakaaabaGaaGymaiabgkHiTmaalaaa baGaamODamaaCaaaleqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbe qaaiaaikdaaaaaaaqabaaaaaaa@3EEB@  .

Since the expression of X also involves time the same factor γ2 must precede it as well.

T(x,t)= γ 2 ( t vx c 2 ) X(x,t)= γ 2 (xvt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivai aacIcacaWG4bGaaiilaiaadshacaGGPaGaeyypa0Jaeq4SdC2aaWba aSqabeaacaaIYaaaaOWaaeWaaeaacaWG0bGaeyOeI0YaaSaaaeaaca WG2bGaamiEaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaaGccaGL OaGaayzkaaaabaGaamiwaiaacIcacaWG4bGaaiilaiaadshacaGGPa Gaeyypa0Jaeq4SdC2aaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIha cqGHsislcaWG2bGaamiDaiaacMcaaaaa@536F@  

(1.8)

When we solve for x and t in terms of X and T we need to obtain expressions that look the same as equations (1.8) except for the sign of v.  If the expressions looked different then one of the two frames would be "special" and that has been proven not to be valid.  We can write the following matrix expression for X and T

γ 2 ( 1 v/ c 2 v 1 )( t x )=( T X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNnaaCa aaleqabaGaaGOmaaaakmaabmaabaqbaeqabiGaaaqaaiaaigdaaeaa cqGHsislcaWG2bGaai4laiaadogadaahaaWcbeqaaiaaikdaaaaake aacqGHsislcaWG2baabaGaaGymaaaaaiaawIcacaGLPaaadaqadaqa auaabeqaceaaaeaacaWG0baabaGaamiEaaaaaiaawIcacaGLPaaacq GH9aqpdaqadaqaauaabeqaceaaaeaacaWGubaabaGaamiwaaaaaiaa wIcacaGLPaaaaaa@49D0@  

(1.9)

To solve for t and x we multiply both sides of (1.9) by the inverse of the matrix on the left and obtain

 

γ 2 ( 1 v/ c 2 v 1 ) 1 v 2 c 2 ( 1 v/ c 2 v 1 )( t x )=( 1 0 0 1 )( t x )=( 1 v/ c 2 v 1 )( T X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNnaaCa aaleqabaGaaGOmaaaakmaalaaabaWaaeWaaeaafaqabeGacaaabaGa aGymaaqaaiaadAhacaGGVaGaam4yamaaCaaaleqabaGaaGOmaaaaaO qaaiaadAhaaeaacaaIXaaaaaGaayjkaiaawMcaaaqaaiaaigdacqGH sisldaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGJb WaaWbaaSqabeaacaaIYaaaaaaaaaGcdaqadaqaauaabeqaciaaaeaa caaIXaaabaGaeyOeI0IaamODaiaac+cacaWGJbWaaWbaaSqabeaaca aIYaaaaaGcbaGaeyOeI0IaamODaaqaaiaaigdaaaaacaGLOaGaayzk aaWaaeWaaeaafaqabeGabaaabaGaamiDaaqaaiaadIhaaaaacaGLOa GaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaaGymaaqaaiaa icdaaeaacaaIWaaabaGaaGymaaaaaiaawIcacaGLPaaadaqadaqaau aabeqaceaaaeaacaWG0baabaGaamiEaaaaaiaawIcacaGLPaaacqGH 9aqpdaqadaqaauaabeqaciaaaeaacaaIXaaabaGaamODaiaac+caca GGJbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamODaaqaaiaaigdaaaaa caGLOaGaayzkaaWaaeWaaeaafaqabeGabaaabaGaamivaaqaaiaadI faaaaacaGLOaGaayzkaaaaaa@679B@  

(1.10)

Which results in the expressions:

t=[T+vX/ c 2 ] x=(X+vT) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiDai abg2da9iaacUfacaWGubGaey4kaSIaamODaiaadIfacaGGVaGaai4y amaaCaaaleqabaGaaGOmaaaakiaac2faaeaacaWG4bGaeyypa0Jaai ikaiaadIfacaWGRaGaamODaiaadsfacaGGPaaaaaa@468E@  

(1.11)

Obviously the factor γ 2 = 1 1 v 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNnaaCa aaleqabaGaaGOmaaaakiabg2da9maalaaabaGaaGymaaqaaiaaigda cqGHsisldaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaaca WGJbWaaWbaaSqabeaacaaIYaaaaaaaaaaaaa@3FCE@   is missing in front of these expressions and they do not look like equations (1.8).  For small v/c this doesn't seem important but it means that either equations (1.11) or equations (1.8) are wrong. We can correct both equation pairs by multiplying them by the factor γ= 1 1 v 2 /( c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo7aNjabg2 da9maalaaabaGaaGymaaqaamaakaaabaGaaGymaiabgkHiTiaadAha daahaaWcbeqaaiaaikdaaaGccaGGVaGaaiikaiaadogadaahaaWcbe qaaiaaikdaaaGccaGGPaaaleqaaaaaaaa@40FC@  at the outset.

Then equations (1.8) and equations (1.11) become:

 

T=γ[txv/ c 2 ] X=γ(xvt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivai abg2da9iabeo7aNjaacUfacaWG0bGaeyOeI0IaamiEaiaadAhacaGG VaGaam4yamaaCaaaleqabaGaaGOmaaaakiaac2faaeaacaWGybGaey ypa0Jaeq4SdCMaaiikaiaadIhacqGHsislcaWG2bGaamiDaiaacMca aaaa@4A65@  

(1.12)

and

t=γ[T+vX/ c 2 ] x=γ(X+vT) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiDai abg2da9iabeo7aNjaacUfacaWGubGaey4kaSIaamODaiaadIfacaGG VaGaai4yamaaCaaaleqabaGaaGOmaaaakiaac2faaeaacaWG4bGaey ypa0Jaeq4SdCMaaiikaiaadIfacaWGRaGaamODaiaadsfacaGGPaaa aaa@49DC@  

(1.13)

We might think of the factor γ' as an expansion factor since it is always greater than 1.  It is important to understand that γ' arose only because simultaneity required that we add a spatial dependence to time measurements.       

When applied to the time equation, γ' is called the time dilation factor.

Seat of the Pants Length Conversion

 

            There is something tricky about how to apply these Lorentz equations to the measurements of the lengths of moving objects.  Since distance is linked with time, we must be sure that we measure the two ends of the object simultaneously.  Think of the two measurements as events.  We can send light pulses from the center toward the ends and observe the elapsed time required to arrive. If the length is L, then in a frame where the object is stationary, the times will both be t=L/2c so that the apparent length is ct=L. In the frame viewing the moving object, the times will be from equations (1.12)

                                                                       

T Front =γ[ L 2c Lv 2 c 2 ]= L(cv) 2 c 2 T Back =γ[ L 2c + Lv 2 c 2 ]= L(c+v) 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivam aaBaaaleaacaWGgbGaamOCaiaad+gacaWGUbGaamiDaaqabaGccqGH 9aqpcqaHZoWzdaWadaqaamaalaaabaGaamitaaqaaiaaikdacaWGJb aaaiabgkHiTmaalaaabaGaamitaiaadAhaaeaacaaIYaGaam4yamaa CaaaleqabaGaaGOmaaaaaaaakiaawUfacaGLDbaacqGH9aqpdaWcaa qaaiaadYeacaGGOaGaam4yaiabgkHiTiaadAhacaGGPaaabaGaaGOm aiaadogadaahaaWcbeqaaiaaikdaaaaaaaGcbaGaamivamaaBaaale aacaWGcbGaamyyaiaadogacaWGRbaabeaakiabg2da9iabeo7aNnaa dmaabaWaaSaaaeaacaWGmbaabaGaaGOmaiaadogaaaGaey4kaSYaaS aaaeaacaWGmbGaamODaaqaaiaaikdacaWGJbWaaWbaaSqabeaacaaI YaaaaaaaaOGaay5waiaaw2faaiabg2da9maalaaabaGaamitaiaacI cacaWGJbGaey4kaSIaamODaiaacMcaaeaacaaIYaGaam4yamaaCaaa leqabaGaaGOmaaaaaaaaaaa@6A6D@  

(1.14)

These times are not the same so we must correct the apparent distance measurement for this difference which is presently:

l=c( T Back T Front )= γLv c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYgacqGH9a qpcaWGJbGaaiikaiaadsfadaWgaaWcbaGaamOqaiaadggacaWGJbGa am4AaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaadAeacaWGYbGaam 4Baiaad6gacaWG0baabeaakiaacMcacqGH9aqpdaWcaaqaaiabeo7a NjaadYeacaWG2baabaGaam4yaaaaaaa@4AC6@  

(1.15)

To make the correction, multiply the first equation in(1.14) by (1+v/c) and the second equation by (1-v/c):

T ' Front =γ L 2 c 2 (cv)(1+v/c)=γL/c(1 v 2 / c 2 )=L/(2cγ) T ' Back =γ L(c+v)(1v/c) 2 c 2 =γL/c(1 v 2 / c 2 )=L/(2cγ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamivai aacEcadaWgaaWcbaGaamOraiaadkhacaWGVbGaamOBaiaadshaaeqa aOGaeyypa0Jaeq4SdC2aaSaaaeaacaWGmbaabaGaaGOmaiaadogada ahaaWcbeqaaiaaikdaaaaaaOGaaiikaiaadogacqGHsislcaWG2bGa aiykaiaacIcacaaIXaGaey4kaSIaamODaiaac+cacaWGJbGaaiykai abg2da9iabeo7aNjaadYeacaGGVaGaam4yaiaacIcacaaIXaGaeyOe I0IaamODamaaCaaaleqabaGaaGOmaaaakiaac+cacaGGJbWaaWbaaS qabeaacaaIYaaaaOGaaiykaiabg2da9iaadYeacaGGVaGaaiikaiaa ikdacaWGJbGaeq4SdCMaaiykaaqaaiaadsfacaGGNaWaaSbaaSqaai aadkeacaWGHbGaam4yaiaadUgaaeqaaOGaeyypa0Jaeq4SdC2aaSaa aeaacaWGmbGaaiikaiaadogacqGHRaWkcaWG2bGaaiykaiaacIcaca aIXaGaeyOeI0IaaiODaiaac+cacaGGJbGaaiykaaqaaiaaikdacaWG JbWaaWbaaSqabeaacaaIYaaaaaaakiabg2da9iabeo7aNjaadYeaca GGVaGaam4yaiaacIcacaaIXaGaeyOeI0IaamODamaaCaaaleqabaGa aGOmaaaakiaac+cacaGGJbWaaWbaaSqabeaacaaIYaaaaOGaaiykai abg2da9iaadYeacaGGVaGaaiikaiaaikdacaWGJbGaeq4SdCMaaiyk aaaaaa@88C5@  

(1.16)

The length L'=c(T ' Front +T ' Back )/γ=L/γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaGGNa Gaeyypa0Jaam4yaiaacIcacaWGubGaai4jamaaBaaaleaacaWGgbGa amOCaiaad+gacaWGUbGaamiDaaqabaGccqGHRaWkcaWGubGaai4jam aaBaaaleaacaWGcbGaamyyaiaadogacaWGRbaabeaakiaacMcacaGG VaGaeq4SdCMaeyypa0Jaamitaiaac+cacqaHZoWzaaa@4DB7@   in this frame is the length in the stationary frame divided by γ and is therefore longer since γ is always greater than 1 when v is not zero.

Appendix What does the transformation x'=x-vt mean?

This transformation is used when converting the  lack of simultaneity of light pulse receptions for an event separation D into a x-dependent clock correction.  Essentially in our expression for time difference D becomes x'=x-vt which results in the relativistic expression for time:

T(x,t)=α γ 2 (t-xv/ c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfacaGGOa GaamiEaiaacYcacaWG0bGaaiykaiabg2da9iabeg7aHjabeo7aNnaa CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaamylaiaadIhacaWG2b Gaam4laiaadogadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@478D@  

(1.17)

where α is to be determined by requiring that t(X,T) look the same as T(x,t) except for the sign of the term xv/c2.

The first comment we can make is that it postulates that x' is zero when x and t are zero. It also stipulates, assuming v>0,  that x' becomes more negative when t becomes more positive while x remains constant.  For an observer starting at x'=0 to move at time t=0 at positive velocity v in the x direction looking at a particular position x in a stationary frame, x'=x-vt  would be the x' coordinate that he would observe.  Note that this value is fixed in the moving observer's frame of reference and therefore requires no relativistic correction. 

 

I feel more comfortable computing the observations of a fixed observer who sees a moving frame moving to the right at speed v.  It turns out that the x dependence of time is almost the same if we use x'=x+vt except for the sign of the vx/c2 term .  

T'(x,t)=α γ 2 (t+xv/ c 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfacaGGNa GaaiikaiaadIhacaGGSaGaamiDaiaacMcacqGH9aqpcqaHXoqycqaH ZoWzdaahaaWcbeqaaiaaikdaaaGccaGGOaGaamiDaiabgUcaRiaadI hacaWG2bGaam4laiaadogadaahaaWcbeqaaiaaikdaaaGccaGGPaaa aa@4868@  

(1.18)