= Proper acceleration of a static Rindler observer
{title2=$a=c^2/|x_0|$}
In $ds^2=(\alpha x)^2dt^2-dx^2$, 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 $\Gamma^x{}_{tt}=\alpha^2x$. 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 $X^2-c^2T^2=x_0^2$.
Back to article page
