Static geodesics in GR Can we find static geodesics of the type
$$x^{\nu}=x_0^{\nu}+\delta_0^{\nu}\tau$$
in some space-time other than Minkowski's?
 A: I think Jerry's point is that if you have a metric that looks like:
$$ ds^2 = a(dx^0)^2 + f(dx^1, dx^2, dx^3) $$
where $a$ is a constant and $f$ is any function, then an observer moving in the $x^0$ direction follows a geodesic of the type you describe. An obvious example is the FLRW metric where a static observer follows the geodesic $t = \tau$, $x = y = z = 0$.
Although I can't think of an example, a metric of the form:
$$ ds^2 = a(dx^0)^2 + b(dx^1)^2 + f(dx^2, dx^3) $$
would have such geodesics in the $x^0, x^1$ plane, and so on.
A: Consider any given spacetime $M$ and a given timelike geodesic $\gamma$ therein, exiting from an event $p\in M$, and parametrized by means of its proper time $\tau$ with origin fixed at $p$ itself.
Next consider the coordinate system constructed as follows (it is possible to prove that it is well defined, see e.g., O'Neill's textbook).
Fix a pseudo orthonormal basis of $T_pM$ at $p$, $e_0,e_1,e_2,e_3$ such that $e_0$ is tangent to $\gamma$ at $p$. Parallely transport this basis along $\gamma$ determining a pseudo orthonormal basis $\{e_a(\tau)\}_{a=0,1,2,3}$ at each $\gamma(t)$ (thus, $e_a(\tau)$ is spacelike for $a=1,2,3$ and is timelike for $a=0$). Finally consider the map, $$(x^0,x^1,x^3,x^3)\mapsto \exp_{\gamma(\tau= x^0)}\left(\sum_{a=1}^3 x^a e_a(\tau)\right) \in M$$
It is possible to prove that this map defines a diffeomorphism from a neighborhhod of a segment of the line $x^1=x^1=x^3=0$ in $\mathbb R^4$ including the origin and a corresponding neighborhood of $\gamma$. In other words we have constructed a coordinate system around $\gamma$ such that $\gamma$ is described by $$x^a = x_0^a + \delta^a_0 \tau$$
with $x^a=0$ for $a=0,1,2,3$. However,  an obvious translation of these coordinates fixes $x^a_0$ as you prefer, preserving the equation of the geodesic. Finally notice that $x^0$, exactly on $\gamma$, is timelike. However it remains timelike in a neighborhood of $\gamma$ by continuity. An analogous result is true for the spacelike coordinates $x^1,x^2,x^3$. So, for every spacetime, for every timelike geodesic it is always possible arrange a local coordinate system (three spatial coordinates $x^1,x^2,x^3$ and $x^0$ temporal) such that the geodesic, described in that coordinate system, takes just the form you stated in your question.
I forgot to say that, exactly on $\gamma$, referring to the constructed coordinates $g_{ab}=\eta_{ab}$ and $\partial_cg_{ab}=0$
