The Relativity of Simultaneity — Question 4

Question 4

A moving frame takes a tilted slice through laboratory clocks

A set of events is simultaneous at $t\prime=0$ and includes the common origin event. Let $v=0.40c$. 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 the laboratory equation of this simultaneity line.

2. Find the laboratory times sampled at $x=\pm750\,\mathrm m$.

Question 4 - Solution

A moving frame takes a tilted slice through laboratory clocks

(a) Method and calculation

$$t\prime=\gamma(t-vx/c^2)=0\quad\Rightarrow\quad\boxed{ct=(v/c)x}.$$$$\frac{d(ct)}{dx}=\boxed{0.40}.$$

(b) Interpretation and further calculation

$$t(+750)=0.40(750/c)=\boxed{+1.00\,\mu\mathrm s}.$$$$t(-750)=\boxed{-1.00\,\mu\mathrm s}.$$

Physical check

A simultaneous snapshot in the moving frame uses different laboratory times at different laboratory positions.