It's a bit messy, and I might have made an error, even with the symbolic algebra package I'm using,but with suitable initial conditions, I get the for the solution of the geodesic equations.
Following wiki
<link>, I've set a=0 and the speed of light c=1, so the metric I'm using is:
$$-x^2 dt^2 + dx^2 + dy^2 + dz^2$$
This leads to the geodesic equations from my previous post, for completeness I'll duplicate them here:
$$\frac{d^2 t}{d\tau^2} + \frac{2}{x} \frac{dt}{d\tau}\frac{dx}{d\tau} = 0 \quad \frac{d^2x}{d\tau^2} + x \frac{dt}{d\tau}\frac{dt}{d\tau} = 0$$
Recall that t,x,y, and z are the Rindler coordinates, while ##\tau## is proper time. t,x,y,z are all functions of ##\tau##.
There are several versions of the Rindler metric. This version is one of the simplest to work with. Note that the plane given by setting the coordinate x=0 is a coordinate singularity, the metric is singular there as the coefficient of dt^2 is zero. This is often called the Rindler horizon. A particle would need infinite proper acceleration to stay on the Rindler horizon.
If one measure time t in years and distance x in light years, the plane x=1, which is one light year from the Rindler horizon, has a proper acceleration of 1 light year/year^2, which is approximately 10 m/s^2, one Earth gravity. So we can consider x=1 the location of the floor of the elevator, and with the convenient choice of units of measuring time in years and distance in light years, the proper acceleration of the floor of the elevator is 1 Earth gravity.
The solutions I'm getting are then:
$$ x(\tau) = \sqrt {{\frac {{E}^{2}}{{P}^{2}+1}}- \left( {P}^{2}+1 \right) {\tau}^{2}}$$
$$t(\tau) = -\frac{1}{2}\,\ln \left( E - \tau - \tau\,{P}^{2} \right) +\frac{1}{2}\,\ln \left( E+\tau+\tau\,{P}^{2} \right)$$
$$y(\tau) = P \, \tau$$
$$z(\tau)=0$$
P and E are constants of motion. Physically, P relates to the linear momentum in the y direction, E to the energy. I've assumed that there is no velocity or momentum in the z direction.
The maximum height (height is the x coordinate here) occurs at ##\tau=0## and is given by ##E/\sqrt{1+P^2}##. So, if one knows the maximum height, one can compute E. Seting the maximum height to occur at ##\tau=0## was a simplifying choice of the initial conditions I made.
t=0 also occurs at ##\tau=0##. ((correction from first draft)).
I'd recommend re-verify that these equations actually satisfy the geodesic equations - baring typos, though, my symbolic algebra program claims that they do.
[add]
To anticipate the next question, we can find an expression for ##\tau## as a function of t
$$\tau = \frac{E}{P^2+1} \tanh t$$
Using this, we can find x as a function of t, rather than of tau
$$x = \sqrt{\frac{E^2}{P^2+1}(1-\tanh^2 t)} = x_{max} \sqrt{1 - \tanh^2 t}$$
where ##x_{max}## is the maximum value of x, which occurs at ##\tau=0##. So we can see that as a function of coordinate time t, only the maximum height matters.
