Definitions of Simultaneity Using Light Pulses

Simultaneity or lack thereof refers to the time between reception of two spatially separated events in a fixed or moving frame. In order to really define whether the two events are simultaneous we need to have an observer in the moving frame as well as a different observer in the stationary frame. We must compare the event time differences sensed by these observers. In general these time differences will not be the same. We can simplify the problem by placing the observer in the fixed frame midway between the events that occur in his frame and having the moving frame observer be coincident with fixed frame observer when light pulses are emitted from the sources in the fixed frame.

Since light from the events in any frame always travel at speed c, the fixed frame observer will conclude that the events occur at t1=t2=t=D/2c seconds after the events actually happened, where c is the speed of light and D is the spatial distance between the events (i,e, light pulse sources). The only real difference in the times the moving observer receives the light pulses is due to the relative speed between frames. This difference is the clock correction needed because of the relative speed difference. The simple Galilean transformation is

`X_0=x-vt`;
`T_0=t`

but because of the lack of simultaneity we will now have a term in T that depends on x like

`T(x,t)=b(t-ax)` (1)
`X(x,t)=b(x-vt)` (2)

where a and b are presently unknown but we know that they must involve relative speed `v` and light speed `c`.

Since the moving observer moves to the right at speed `v`, the time taken for him to receive a pulse from the right hand source is L/[2(c+v)] while the time taken for the pulse from the left hand source is L/[2(c-v)] where `L` is the distance to the source in the lab frame. The difference of these times is

`deltat=vD/(c^2-v^2)`

and this is a fair approximation of the time correction needed for the moving observer. This is the simultaneity difference and it will be used to correct both T and X. If we identify D with x', where x'=x-vt=D, in equation 1 then we would write equation 1 as

`T=b[t-a(x+vt)v/(c^2-v^2)]=`
`b[t(1+c^2/(c^2-v^2))-xv/(c^2-v^2)]`
=`b[1/(1-v^2/c^2)](t-vx/c^2)`

Since T depends on x and t so must X depend on both and with the same cofactor b multiplying them. Then we have the equations

`T(x,t)=b/(1-v^2/c^2)(t-vx/c^2)` `X(x,t)=b/(1-v^2/c^2)(x+vt)`

If we require that t(X,T) and x(X,T) be perfectly the same as T(x,t) and X(x,t) except for the sign of v, then we find that `b=sqrt(1-v^2/c^2)` and then the function multiplying all 4 equations is `gamma=1/sqrt(1-v^2/c^2)` commonly named the Greek letter gamma.

`T(x,t)=1/sqrt(1-v^2/c^2)(t-vx/c^2)`
`X(x,t)=1/sqrt(1-v^2/c^2)(x-vt)`

Similarly we can write for `t` and `x` in terms of `T` and `X`.

`t(X,T)=1/sqrt(1-v^2/c^2)(T+vX/c^2)`
`x(X,T)=1/sqrt(1-v^2/c^2)(X+vT)`

Note the change of sign of the `vX` term.

This is really a derivation of the special relativity transformations from the principles that time and space must be linked and the scale factor for that linkage just happens to be `c`, the speed of light in vacuum. The linkage scale factor did not have to be `c` but this is just as fundamental as a law of nature as is Planck's constant `ℏ` or the gravitational constant `G`.

Click Here for Discussion
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