Proving that test particles in GR, follow spacetime geodesics

Question

My question is pretty much in the title. According to this paper, this is not exactly proven rigorously yet. What I dont understand is what exactly is not proven. If I'm not too wrong, a test particle is just a particle, which does not affect the ambient gravitational field.

According to this Paper, it can be easily shown that the eqn. Of motion for a point particle in a curved space (having metric $g_{\mu \nu}$), can be found by considering the action to be just, $$S = -mc \int ds \, ,$$ where $$ds^2 = -g_{\mu \nu}dx^{\mu}dx^{\nu}$$ This would give the geodesic eq. As the eq. Of motion.

Now, if we assume that the particle is somewhat massive, couldn't we just modify the metric linearly, by superposing a mass coupled metric as, $$g'_{\mu \nu} = g_{\mu \nu} + m h_{\mu \nu}$$

Considering the same form of the action above, we should get the same geodesic eq. Along with a subsidiary eq.(which has a mass coupling). In the limit $m \rightarrow 0$, the subsidiary term should vanish, leaving the original geodesic eq.

So what I want to know, is what exactly is the unsolved part of this problem ? I failed to understand it from the text itself. It would be great if someone can point that out.

Answer 1

Proving that test particles in GR follow geodesics is far from trivial. A core aspect of the problem is that point particles do not make sense in general relativity (or similarly non-linear theories). Hence argument that starts by assuming a point particle (such as given by the OP) cannot be fully trusted.

That being said I know of (at least) two methods the rigorously prove that massive objects follow geodesics in the limit that their mass goes to zero.

The first starts by assuming an object that is sufficiently compact compared to the curvature length scale(s) of the background spacetime in which it moves (in particular if the background spacetime is a black hole, you assume that the object's mass is much smaller than that of the background black hole and that the size of the small object is of a similar order of magnitude as its Schwarzschild radius. I.e. its size scales linearly with its mass). You can than use Einstein's equation and matched asymptotic expansions to find a systematic expansion of the equations of motion for the object's worldline as an expansion in the ratio of these length scales (i.e. the ratio of the masses in case of a black hole background). The zeroth order term in this expansion is simply the geodesic equation in the background spacetime. This general framework for solving the equations of motion of a 2-body system in GR is known as the "gravitational self-force" approach. A good recent review (that also deals with the problem of point particles in GR) was published last year by Barack and Pound (arXiv:1805.10385, See section 3 and specifically 3.5).

The second approach (developed by Weatherall and Geroch), starts from energy-momentum distributions that are a) conserved b) satisfy the dominant. One can then show that sufficiently localized concentrations of energy "track" geodesics. (See arXiv:1707.04222, Theorem 3 gives a precise statement of their result).