 Proper acceleration of a static Rindler observer

In $d{s}^2=(\alpha x{)}^{2}d{t}^2-d{x}^2$, a constant-$x_0>0$ observer has $t=c\tau /(\alpha x_0)$. The nonzero Christoffel symbols are ${\Gamma }^t{}_{tx}={\Gamma }^t{}_{xt}=1/x$ and ${\Gamma }^x{}_{tt}={\alpha }^{2}x$. Its covariant acceleration components are $a^t=0$, $a^x=c^2/x_0$, with invariant magnitude $c^2/x_0$. This is constant proper acceleration, not a covariantly constant four-vector. In inertial coordinates the worldline is the hyperbola $X^2-c^2{T}^2=x_0^2$.