Einstein's equations as a Dirichlet boundary problem

Question

Can Einstein's equations in vacuum $R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab}= 0$ be treated as a Dirichlet problem?

I am thinking of something along those lines: Consider a compact manifold $M$ with boundary $\partial M$. For a given metric $g_{ab}$ on $\partial M$, does a unique (up to diffeomorphisms) solution to Einstein's equations on $M$ exist? Are there any constraints on the metric on the boundary?

Einstein's equation can be recovered by looking for critical points in the Einstein-Hilbert action, but I don't know enough about variational problems to deduce anything about existence or uniqueness. The fact that general relativity is generally covariant also seems to complicate things, since this leads to a huge coordinate freedom.

Does this depend on the dimension of the manifold or can a general statement be made? I am mostly interested in the three and four dimensional cases.

Note that I am not interested in the initial value formulation usually considered.

Answer 1

In the Lorentzian case: I am not aware of anyone studying it, and don't know explicit counterexamples off-hand. But I have doubts on the uniqueness. With the Lorentzian case, the nature candidate to draw comparisons with is the wave equation. And we see that on something as simple as the unit square $[0,1]\times [0,1]$, the wave equation with vanishing Dirichlet boundary condition admits solutions $$ (\partial_t^2 - \partial_x^2)u_k(t,x) = 0 \qquad u_k(t,x) = \sin (2\pi k x)\sin(2\pi k t) $$ Similar constructions can be made in arbitrary dimensions for the wave equation. In general for hyperbolic PDEs, the Dirichlet boundary value problem tend not to be well-posed.

In the Riemannian case: there has been a lot of interest in the mathematical community about this. To the best of my knowledge the Dirichlet problem for Einstein manifolds are still poorly understood. A starting place to do literature search would be Michael Anderson's paper on the subject.