 Solution

For coordinates $(t,x)$ the inverse metric is $g^{tt}=1/({\alpha }^2{x}^2)$, $g^{xx}=-1$. Only ${\partial }_x{g}_{tt}=2{\alpha }^{2}x$ is nonzero, giving the Christoffel symbolswith all remaining components zero.

On the specified constant-$x_0$ worldline, assume $x_0>0$. The four-velocity is $u^t=c/(\alpha x_0)$, $u^x=0$, and its norm is $c^2$, so $\tau$ is indeed proper time. The coordinate components are constant, but the covariant acceleration is nonzero:The proper acceleration of a static Rindler observer isThe acceleration has constant components in this accelerated chart and constant invariant magnitude. It is not a covariantly constant vector: its inertial-frame components change along the hyperbolic trajectory. For $x_0<0$, the magnitude is $c^2/|x_0|$, with the appropriate choice of future time orientation.