= Solution
For coordinates $(t,x)$ the inverse metric is $g^{tt}=1/(\alpha^2x^2)$, $g^{xx}=-1$. Only $\partial_xg_{tt}=2\alpha^2x$ is nonzero, giving the <Christoffel symbols>
$$
\boxed{\Gamma^t{}_{tx}=\Gamma^t{}_{xt}=\frac1x,\qquad
\Gamma^x{}_{tt}=\alpha^2x,}
$$
with 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:
$$
\boxed{a^t=0,\qquad a^x=\Gamma^x{}_{tt}(u^t)^2=\frac{c^2}{x_0}.}
$$
The <proper acceleration of a static Rindler observer> is
$$
\boxed{\sqrt{-g_{\mu\nu}a^\mu a^\nu}=\frac{c^2}{x_0}=\text{constant}.}
$$
The 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.
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