Past exam of the relativity course of the University of Cambridge / 2021 / Section B / 6 / d / Solution

Differentiating the supplied coordinates givesSubtract the squares. The cross terms cancel and ${\cosh }^2-{\sinh }^2=1$, yieldingThe spacetime is flat Minkowski spacetime in Rindler coordinates. $x>0$ covers $X>|cT|$, the right Rindler wedge, while $x<0$ covers the left wedge $X<-|cT|$; neither is the whole spacetime.

The null combinations areFor $x=C{e}^{-\alpha t}$ the first is constant; for $x=C{e}^{\alpha t}$ the second is constant. Thus part (a)'s exponentials are simply the inertial light rays $X\pm cT=\text{constant}$. Their coordinate-time asymptotes reflect the chart's Rindler horizon, not a failure of light to move at $c$ in an inertial frame.