The Relativity of Simultaneity — Question 8

Question 8

Simultaneity in two distinct collinear frames forces coincidence in x

Two events are simultaneous in S and also in a frame $S\prime$ moving at nonzero v along x. They may differ in y and z. Use $\Delta t\prime=\gamma(\Delta t-v\Delta x/c^2)$, $\gamma=(1-v^2/c^2)^{-1/2}$; $S\prime$ moves at +v along x. Take $c=3.00\times10^8\,\mathrm{m/s}$.

1. Determine what this implies about $\Delta x$.

2. Decide whether the events must be the same spacetime point, giving an explicit counterexample if not.

Question 8 - Solution

Simultaneity in two distinct collinear frames forces coincidence in x

(a) Method and calculation

$$\Delta t=0,\quad\Delta t\prime=0\quad\Rightarrow\quad0=-\gamma v\Delta x/c^2.$$$$v\ne0\quad\Rightarrow\quad\boxed{\Delta x=0}.$$

(b) Interpretation and further calculation

$$A:(t,x,y,z)=(0,0,0,0),\quad B:(0,0,d,0),\quad d\ne0.$$$$\boxed{\Delta t\prime=0,\quad\Delta y\prime=d\ne0}.$$

Physical check

A boost along x mixes only x and time. Agreement on simultaneity does not erase transverse separation.