Sometime ago, I posted here that General Relativity is a Machian theory, in the sense that matter causes spacetime to curve, and the spacetime causes matter to move, and at the time there were some objections to this, mostly stating that I don't know some concepts of differential geometry (disclaimer: I really don't), and a more interesting part relating to the fact that the existence of nontrivial vacuum solutions to the field equations (Schwarzschild, Kerr, gravitational waves, etc.) rules out this idea, because different matter configurations are generating different spacetimes.
I would like to share a
paper which confirms what I'm saying in a completely rigorous way. This paper is a mathematical proof (notice that it was published in a math journal, not a physics one) that General Relativity makes complete sense as a constrained initial value problem, in the sense that field configurations at a certain time completely determine fields in the future.
I'll summarize the results the best I can. One good context to start is the
ADM formulation of general relativity. Essentially, the equations are decomposed in a time part and a space part. The time part says how the fields should evolve, and the space part constrains how the fields should behave. For this to make sense, the time evolution should be consistent with the constraint (you cannot impose a constraint on the fields at some point in time, and then later they evolve and no longer respect it), in GR this is guaranteed by the Bianchi identities. Anyway, in the ADM formulation, to simulate the system you need to provide four arbitrary functions, the so-called lapse and shift vectors. These functions will tell spacetime how to deform. Different choices will give different results and it's not clear what GR is really predicting.
Einstein was well aware of this, and his thought about this issue goes by the name of the
hole argument. In any case, Einstein's solution is that the only thing meaningful about the coordinate is if the points are different or not. I think this is a good solution, but at our current knowledge of physics, completely impractical. In order to make any physical prediction, all of physics should be cast in a general covariant form, and we are still far from this. Einstein tried a similar idea, with the so-called
unified field theory, but didn't have much success.
Given this, how does the author of the 1952 paper get around this? Simple, the idea is to not allow the lapse and drift vectors of the ADM formalism to have a general form. The group of coordinate transformation has a well understood structure as a Lie group, and so has a dimension. That means, that if we can constrain the lapse and shift vectors with a given number of equations as to cancel the dimension of this Lie group, and prove that the initial value problem is well-posed by any means, we are done. We necessarily have a theory with meaningful physical predictions. Now, there are some technical complications when you do this, the route followed in the paper is to impose the conditions we know as
harmonic gauge. What this does is: it breaks the general covariance of the Einstein equations, and they are now only Lorentz covariant (they are invariant with respect to boosts and rotations, but not general coordinate transformations).
This introduces some technical difficulties, because now it's inconsistent with the ADM. However, they can be overcome, you can still prove that it's a constrained initial value problem, where the constraint is consistent with the time evolution, and you can prove that a solution exists using standard PDE techniques. The Lorentz covariance of the resulting equations is actually a good thing because the rest of physics is cast in a Lorentz (not general) covariant way.
This result is important because, historically, relativists had a hard time settling the issue of whether gravitational waves were real. Although you can find wave solutions in linearized GR, it's not clear why they could not be simply a changing coordinate chart as time passes. The following theorem proves that General Relativity in the harmonic gauge is a well defined initial value problem, even when all nonlinearities are taken into account, so that the wave solutions describe real physical waves. By the way, none of this uses any differential geometry. All of this can be derived using knowledge of partial differential equations and Lie groups, and everything I said can be done with some modification for Maxwell's equations, Yang-Mills equations, and whatever. In this aspect, General Relativity is really not different from these other theories. Besides that, numerical GR calculations use these formalism and correctly predicted the form of gravitational waves coming from a black hole merger. I still haven't heard any reasonable explanation from physicists who claim GR is not Machian, about how this prediction makes sense for them.
As to the other objection, that there are several vacuum solutions to the Einstein equations, it's not relevant for the following reason. An initial value problem is given by the differential equations plus the boundary conditions. It's perfectly fine to study solutions of a differential equation without worrying about boundary conditions, but often what happens is that you end up with topological invariants. What the enormous amount of vacuum solutions mean is that there are several nontrivial topologies for the spacetime manifold. That does not mean GR should not be well posed as an initial value problem, all that this means is that if we take two topologically distinct solutions, it's impossible that the time evolution of one of them can take us to the other. That also happens with lots of partial differential equations, and is not exclusive to general relativity.
