What Is the Relativity of Simultaneity?

The relativity of simultaneity means that two spatially separated events judged to occur at the same time in one inertial reference frame may occur at different times in another frame moving relative to the first. Simultaneity is therefore not absolute.

The Reasoning

This is an Additional Higher Level (AHL) concept in A.5 Galilean and special relativity. It follows from the two postulates of special relativity, especially that the speed of light in a vacuum has the same value, $c$, for every inertial observer.

Observers in each frame synchronize clocks at different positions using light signals and the assumption that light travels at $c$ in both directions. Because moving frames disagree about which distant clocks are synchronized, they also disagree about whether separated events share the same time coordinate.

Suppose two flashes occur simultaneously at different positions in frame $S$. Their separations satisfy $\Delta t=0$ and $\Delta x\neq 0$. For a frame (S') moving at velocity $v$ relative to $S$, the Lorentz transformation gives

$$\Delta t'=\gamma\left(\Delta t-\frac{v\Delta x}{c^2}\right), \qquad \gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$$

Because $\Delta t=0$,

$$\Delta t'=-\gamma\frac{v\Delta x}{c^2}.$$

Unless $v=0$ or $\Delta x=0$, the second observer measures $\Delta t'\neq 0$. For example, if and , then , giving

$$\Delta t'=-1.5\times10^{-6}\ \text{s}.$$

Thus, events simultaneous in $S$ are separated by $1.5\ \mu\text{s}$ in (S'). The negative sign indicates which event occurs first according to the chosen coordinate direction.

Such events are spacelike separated, so no signal travelling at or below $c$ can connect them. Their order may therefore reverse between inertial frames without violating causality.

A common misconception is that differing observations result from light taking time to reach the observers. That is not the relativity of simultaneity: even after correcting for light-travel time, the frames assign different times to the events.

Exam Technique

For an IB Explain question, state that simultaneity depends on the inertial reference frame, link this to the invariance of $c$, and use the Lorentz time transformation. Mention that event order can differ only for events that cannot be causally connected.