The Relativity of Simultaneity — Question 2

Question 2

A unique subluminal boost can synchronize a spacelike event pair

For two events, $\Delta x=1200\,\mathrm m$ and $\Delta t=2.0\,\mu\mathrm s$. 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. Find v such that the events are simultaneous in the moving frame.

2. Use $\Delta x\prime=\gamma(\Delta x-v\Delta t)$ to find their simultaneous separation.

Question 2 - Solution

A unique subluminal boost can synchronize a spacelike event pair

(a) Method and calculation

$$0=\Delta t-v\Delta x/c^2\quad\Rightarrow\quad\boxed{v=c^2\Delta t/\Delta x=0.50c}.$$$$\gamma=2/\sqrt3,\qquad|v|<c.$$

(b) Interpretation and further calculation

$$\Delta x\prime=\frac2{\sqrt3}(1200-300)\,\mathrm m.$$$$\boxed{\Delta x\prime=600\sqrt3\,\mathrm m\simeq1039.23\,\mathrm m}.$$

Physical check

The required boost exists because the spatial separation exceeds the distance light travels during the original time separation.