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ID: past-exam-of-the-relativity-course-of-the-university-of-cambridge/2021/section-b/6/a/solution

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

Codex  0 2026-10-18

Take

α > 0

and work in a connected patch with

x \neq = 0

. Setting the line element to zero gives

\frac{d x}{d t} = \pm αx, x (t) = C e^{\pm α t}, C \neq = 0.

These are the two families of Rindler null geodesics. They remain on the same side of

x = 0

and approach it only at infinite coordinate time.

x = 0

itself is not an additional geodesic within this chart, since the metric is degenerate there.

The paths can also be written

t = \pm α^{- 1} lo g ∣ x / C ∣

. In two dimensions a null direction has only one independent orthogonal null direction, namely itself, so each smooth null curve is a geodesic after reparametrization. More explicitly, the transformation in part (d) makes these paths straight null lines. The coordinate

t

is generally not an affine parameter.

Figure 1.
Rindler light rays and the reception horizon of a uniformly accelerated observer
. Original diagrams. The left panel shows the two exponential null families on both sides of the excluded $x = 0$ chart boundary. The right panel uses inertial coordinates: a hyperbolic observer receives a signal emitted within its horizon, but cannot receive the horizon ray or a ray emitted farther behind. Length and time are normalized by $R = c^{2} / a$ . Details.

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