The Lorentz Transformation — Question 7

Question 7

The frame in which two timelike events share a position

A pair of events has $c\Delta t=1000\,\mathrm m$, $\Delta x=600\,\mathrm m$, and no transverse displacement. Use $ct\prime=\gamma(ct-\beta x)$, $x\prime=\gamma(x-\beta ct)$, $y\prime=y$, $z\prime=z$, where $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$.

1. Find the boost that makes $\Delta x\prime=0$.

2. Find the transformed time and identify its invariant meaning.

Question 7 - Solution

The frame in which two timelike events share a position

(a) Method and calculation

$$0=\gamma(\Delta x-\beta c\Delta t)\quad\Rightarrow\quad\boxed{\beta=0.600}.$$$$\gamma=1.250.$$

(b) Interpretation and further calculation

$$c\Delta t\prime=1.25(1000-0.60\cdot600)=\boxed{800\,\mathrm m}.$$$$\boxed{c\Delta\tau=\sqrt{(c\Delta t)^2-\Delta x^2}=800\,\mathrm m}.$$

Physical check

This boost follows the inertial worldline through both events. Their proper time is the interval measured by that single clock.