Is a beam of light stationary in Rindler coordinates?

Question

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1. What the question is asking (in plain language)

A uniformly‑accelerated observer (constant proper acceleration (a_{0})) can be described with Rindler coordinates ((\eta ,\xi )).
The observer’s own world‑line is (\xi =0); his proper time equals the Rindler time (\eta).

The claim we have to test is:

“A beam of light that is emitted from the origin (the point where the observer’s world‑line meets the Minkowski‑space origin) appears to be *still in the accelerated observer’s frame; i.e. in Rindler coordinates the light’s spatial coordinate does not change while the time coordinate runs.”*

Our job is to write the light‑ray world‑line in inertial ((t,x)) coordinates, transform it to ((\eta ,\xi )), and see what the (\xi)‑coordinate does as (\eta) increases.

2. Step‑by‑step derivation

2.1 Minkowski description of the uniformly accelerated observer

In an inertial frame ((ct,x)) the world‑line of an observer with constant proper acceleration (a_{0}>0) is

[ \boxed{ \begin{aligned} ct(\tau) &= \frac{c^{2}}{a_{0}}\;\sinh!\left(\frac{a_{0}\tau}{c}\right),\[2mm] x(\tau) &= \frac{c^{2}}{a_{0}}\;\cosh!\left(\frac{a_{0}\tau}{c}\right), \end{aligned}} \tag{1} ]

where (\tau) is the observer’s proper time.
These equations describe the right‑hand Rindler wedge ((x>|ct|)).

2.2 Definition of Rindler coordinates

Introduce coordinates ((\eta ,\xi )) that cover the same wedge:

[ \boxed{ \begin{aligned} ct &= \frac{c^{2}}{a_{0}}\;e^{a_{0}\xi/c^{2}}\;\sinh!\left(\frac{a_{0}\eta}{c}\right),\[2mm] x &= \frac{c^{2}}{a_{0}}\;e^{a_{0}\xi/c^{2}}\;\cosh!\left(\frac{a_{0}\eta}{c}\right). \end{aligned}} \tag{2} ]

The Rindler time (\eta) measures the proper time of the observer sitting at (\xi =0) (compare (1) with (2) and set (\xi=0) → (\eta=\tau)).
The Rindler space coordinate (\xi) is a logarithmic measure of the distance from the observer’s world‑line; surfaces of constant (\xi) are hyperbolae of constant proper acceleration.

The metric obtained from (2) is

[ ds^{2}= -\left( a_{0}\, \xi\right)^{2} d\eta^{2}+ d\xi^{2}, \qquad (\xi>0). \tag{3} ]

2.3 The light ray in the inertial frame

A light signal that is emitted from the Minkowski origin ((t=0,x=0)) and travels in the (+x) direction satisfies

[ x = c t . \tag{4} ]

(It is a null line, (ds^{2}=0).)

2.4 Transform the light ray to Rindler coordinates

Insert the expressions (2) for (ct) and (x) into (4):

[ \frac{c^{2}}{a_{0}}\;e^{a_{0}\xi/c^{2}}\;\cosh!\left(\frac{a_{0}\eta}{c}\right) = \frac{c^{2}}{a_{0}}\;e^{a_{0}\xi/c^{2}}\;\sinh!\left(\frac{a_{0}\eta}{c}\right). ]

The overall factor (\dfrac{c^{2}}{a_{0}}e^{a_{0}\xi/c^{2}}) is non‑zero

Original question: Is a beam of light stationary in Rindler coordinates? on Physics Stack Exchange, licensed CC BY-SA.

1. What the question is asking (in plain language)

2. Step‑by‑step derivation