Light Clock and Length Contraction in General Relativity

General relativity doesn't have a fundamental notion of distance, but the same is true of special relativity. The big problem being that, unlike inertial observers in special relativity, there's no "canonical" foliation of the spacetime into spacelike hypersurfaces.

In other words, for some inertial observer $gamma$ in special relativity, there is a canonical foliation of Minkowski space $M$ into the spacelike hypersurfaces $Sigma_t$, with a time function $t$, which is defined by the inertial frame in which $gamma$ is at rest.

This doesn't entirely fix the foliation, but if we add the Einstein synchronization condition, which is that for a light clock emitting at point $p_1$, reflecting at $q$, and received at $p_2$, then we have a time function such that

$$t(q) = frac{1}{2}(t(p_1) + t(p_2))$$

which leads to the Synge formula for distance, where the distance (at time $t(q)$) of the observer to the point $q$ is

$$d(q) = c sqrt{(t - t_1)(t_2 - t)}$$

There is no such canonical foliation in general relativity. If our spacetime is nice enough, we can define a time function which assigns a time value to every point, and then compute the distance by simply considering the usual distance function on a spacelike hypersurface, with the usual formula for distances in Riemannian manifolds :

$$d(p,q) = min_{gamma}(int_{p}^{q} sqrt{g(dot{gamma}(lambda), dot{gamma}(lambda))}dlambda)$$

ie the smallest proper length to that point. But of course, that distance is somewhat arbitrary, as we can choose another time function to give us a different distance. Note also that this distance is dynamic : even if we consider two objects at rest, they may still drift apart, because the metric tensor may change with time (this happens for the expansion of the universe, for instance).

Light clocks can be used to define such canonical coordinates, called radar coordinates. After all, these correspond to physical measurements, and these are coordinate invariants. But beware that you can't actually make an entire map of a spacetime with radar coordinates in general, as they may not work out for very large distances. On the other hand, there is always some neighbourhood where those coordinates exist, and on a more practical level, they tend to be valid at pretty large distances if the gravity is low enough.

You can then define a synchronization, a time function, and a distance as in the case of special relativity, with which you can check for length contraction. As general relativity has to reduce to special relativity locally, length contraction does indeed happen, and you could if you wish check the length contraction occuring between two "inertial" (unaccelerated) observers. The distance will indeed change depending on the speed of the observer, but also depend on the curvature of the spacetime.

Locally, the geometry of times and distances of general relativity is almost entirely identical to special relativity. If we have three points $p, q, r$ connected by three geodesics, their distances are related by

begin{equation} d^2(p,q) = d^2(r,p) + d^2(r,q) - 2 d(r,p) d(r,q) cos(theta) + phi end{equation}

where the distance $d$ can be positive, negative or null, depending on how those points are organized, and $theta$ is the angle between $rp$ and $rq$. This is exactly the same formula as the elementary trigonometry formula for Minkowski space, except for the last term $phi$, which depends on the spacetime curvature. It's a rather complicated term but it can be roughly considered as the integral of a series of products of the Riemann tensor.

So, for $phi approx 0$, you will get your usual length contraction, implying a very small distortion due to the curvature. For a larger $phi$, the difference of length between two points will not be solely due to length contraction as it is understood in special relativity : the change of the metric tensor will also play a role, making that distance shorter or longer (usually even shorter).

If you'd like more derivations regarding distances in general relativity, there is quite a lot in Synge's book "Relativity, the general theory".