Problem — Simultaneity (true or false)

Problem

True or false? “Two events that are simultaneous in one inertial reference frame are simultaneous in any other inertial frame.” Justify briefly.

Solution

Answer: False.

Simultaneity is not absolute: it depends on the reference frame. This is the relativity of simultaneity.

Justification. In the Lorentz transformations, the time of an event in a frame $S'$ moving at velocity $v$ relative to $S$ is

$t' = \gamma\left(t - \frac{v\,x}{c^2}\right)$

Consider two events simultaneous in $S$ (same $t$) but at different positions $x_1 \neq x_2$. In $S'$:

$\Delta t' = t'_2 - t'_1 = -\gamma\,\frac{v\,(x_2 - x_1)}{c^2} \neq 0$

Since $x_2 \neq x_1$ and $v \neq 0$, we have $\Delta t' \neq 0$: in frame $S'$ the two events are not simultaneous.

$\ev{\text{False: two events simultaneous in }S\text{ are in general not simultaneous in }S'}$

The only exception is when the events occur at the same point ($x_1 = x_2$): in that case $\Delta t' = 0$ in every frame.

Links

Topics: Special relativity Concepts: Relativity of simultaneity Skills: Changing reference frame