The Relativity of Simultaneity — Question 7

Question 7

Light-connected events cannot be made simultaneous

A vacuum pulse travels from A to B along +x, so $\Delta x=c\Delta t$ with $\Delta t>0$. 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. Simplify the moving-frame time separation.

2. Prove its sign for every physical boost and examine the limit $v\to c$ from below.

Question 7 - Solution

Light-connected events cannot be made simultaneous

(a) Method and calculation

$$\Delta t\prime=\gamma\Delta t(1-\beta),\qquad\beta=v/c.$$$$\boxed{\Delta t\prime=\Delta t\sqrt{\frac{1-\beta}{1+\beta}}}.$$

(b) Interpretation and further calculation

$$-1<\beta<1\quad\Rightarrow\quad\boxed{\Delta t\prime>0}.$$$$\lim_{\beta\to1^-}\Delta t\prime=0,\qquad\beta=1\text{ is not an inertial frame}.$$

Physical check

A limiting value is not an attainable frame. Emission precedes absorption in every inertial frame.