special relativity - Is there an equivalent of Rindler coordinates for an object in centripetal motion?

Rindler coordinates are a parametrization of (a subset of) Minkowski space that are "natural" for an object experiencing constant acceleration - more specifically, an object experiencing constant acceleration parallel to its velocity. Centripetal motion is of course the related case where the object's acceleration is perpendicular to its velocity. Is there an equivalent natural set of coordinates for this case? If there is, is it trivial (i.e. no different than polar coordinates for Minkowski space)? I'm also wondering if there is an equivalent of the Unruh effect.

A starting point: I went through some of the initial derivation, and (unless I'm mistaken) here are some results. Assuming an ansatz circular path for the object in Minkowski space of radius $R$ and tangential velocity $v$, it works out that the object's acceleration in it's local frame is (in units where c = 1) $$ a = - \hat r \cdot \frac{d^2 \vec x}{dt'^2} = \frac{1}{1-v^2} \frac{v^2}{R} = \gamma^2 \frac{v^2}{R} $$ The proper time of the object in motion is, of course, $\tau = t/\gamma$. That this is independent of $x$, unlike the case in Rindler coordinates where $\tau = a^{-1} \arctan(t/x)$, makes me think that the end result might be trivial, but I'm not sure.