AcademyRelativity
Academy
Simultaneity
Level 1 - Physics topic page in Relativity.
Principle
Events simultaneous in one inertial frame are not necessarily simultaneous in another.
Notation
\(\Delta t\)
time separation in frame \(S\)
\(\mathrm{s}\)
\(\Delta x\)
position separation in frame \(S\)
\(\mathrm{m}\)
\(\Delta t'\)
time separation in frame \(S'\)
\(\mathrm{s}\)
\(v\)
speed of \(S'\) relative to \(S\)
\(\mathrm{m\,s^{-1}}\)
\(\gamma\)
Lorentz factor
Method
Derivation 1: Compare event times
The Lorentz time transformation gives the time coordinate of each event. Subtract the two event equations.
Time transform
\[t'=\gamma\left(t-\frac{vx}{c^2}\right)\]
Time separation
\[\Delta t'=\gamma\left(\Delta t-\frac{v\Delta x}{c^2}\right)\]
Derivation 2: Test simultaneity
If two events are simultaneous in \(S\), then \(\\Delta t=0\). For separated events, the moving frame usually assigns different times.
Simultaneous in S
\[\Delta t=0\]
Moving-frame time gap
\[\Delta t'=-\gamma\frac{v\Delta x}{c^2}\]
Derivation 3: Separate cause from ordering
Only spacelike-separated events can have their time order reversed by changing frame. Timelike cause-and-effect order is preserved.
Rules
Time difference
\[\Delta t'=\gamma\left(\Delta t-\frac{v\Delta x}{c^2}\right)\]
Simultaneous in S
\[\Delta t'= -\gamma\frac{v\Delta x}{c^2}\]
Same place test
\[\Delta x=0\Rightarrow \Delta t'=\gamma\Delta t\]
Examples
Question
Two flashes are simultaneous in \(S\), separated by
\[900\,\mathrm m\]
Frame \[S'\]
moves at \[0.600c\]
Find \[\Delta t'\]
Answer
\[\gamma=1.25\]
\[\Delta t'=-\gamma\frac{v\Delta x}{c^2}=-1.25\frac{(0.600c)(900)}{c^2}=-2.25\,\mu\mathrm s\]
Checks
- Always state which frame says the events are simultaneous.
- Use signed \(\Delta x\); reversing event order changes the sign.
- Relativity of simultaneity is not signal delay.
- Causal order cannot be reversed for events linked by a material object or light signal.