A.5.12 (HL)—Relativity of simultaneity

State the idea

Two events that are simultaneous in one inertial reference frame need not be simultaneous in another frame moving relative to it. Simultaneity is therefore not an absolute property of separated events.

Use the transformed time

For two events, $\Delta t'=\gamma(\Delta t-v\Delta x/c^2)$. If $\Delta t=0$ in S but $\Delta x\neq0$, then $\Delta t'\neq0$ in a relatively moving frame.

Worked example from local Question Bank row 30453

Two lamps are simultaneous in S and separated by $9.00\times10^3\,\mathrm{m}$. For $v=0.80c$ and $\gamma=5/3$,

$$\Delta t'=\frac53\left(0-\frac{(0.80c)(9.00\times10^3)}{c^2}\right)=-4.0\times10^{-5}\,\mathrm{s}$$

The negative sign fixes the event order in S′; it is not an error.

Keep the order test local

To decide which event occurs first in a frame, calculate or read the sign of $\Delta t′$ using the same pair of events. A negative time difference means the event assigned as the second reference event occurs earlier in that frame.

Common trap

The relativity of simultaneity concerns spatially separated events. Events at the same place cannot be simultaneous in one frame and ordered differently in another inertial frame.