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These are quite orthogonal conditions. To start, one is a global condition, while the other is a local one. Every point has a small enough neighborhood that is strongly causal (even globally hyperbol …

The dependence on AC through the use of Zorn's lemma in the proof of the Choquet-Bruhat–Geroch theorem on the existence of a maximal globally hyperbolic development for solutions of the Einstein equat …

Have a look at Ringström's The Cauchy Problem in General Relativity (EMS 2009). He spends several chapters building up the analytical material of the kind that you are asking about.

I presume the formula you are asking about is the long one highlighed by $*$ $*$ in your question, while the standard "contracted Bianchi identity" $\delta \operatorname{Ein}\left({ }^{(4)} g\right)=0 …

Trying to recover as much of the topology/geometry from the causal order as possible has been studied quit a bit since the early paper of Hawking et al that you cite. A quick summary of my understandi …

Although the focus of the original question was on conformal compactification, a necessary step along the way is an introduction of double-null coordinates that are regular on the horizons and bifurca …

For a finite dimensional inner product space $(V,\eta)$, $\bigwedge^2 V \cong_\eta \mathfrak{so}(\eta) \subset \operatorname{End}(V) \cong V\otimes V^* \cong_\eta V\otimes V$. The antisymmetry conditi …

The answer is Yes, at least under the reasonable conditions that (i) the number of conformal Killing vectors locally admitted by $(M,g)$ is constant and that (ii) the de Rham cohomology $H^{n-1}(M)=0$ …

There is a quite detailed pedagogical presentation of both the Einstein-Hilbert and the Palatini variational principles for the Einstein equations in §III.3 Lagrangians for General Relativity of Baez …

I will add a pessimistic answer. You are right that Choque-Bruhat's (and any related local-in-time) existence result only guarantees that the solution metric exists and is sufficiently regular (includ …

If I understood your question correctly, the answer indeed is due to Lovelock. I think it's important to state all the hypotheses clearly, because they are not always reported accurately. Theorem. (Lo …

In Riemannian geometry, the largest such $r$ is the injectivity radius. And there are curvature based bounds for it. To make sense of such a radius in Lorentzian geometry, you need also some reference …

An infinitesimal diffeomorphism, generated by $\xi^a$, acts on the metric as $g_{ab} \mapsto g_{ab} + 2 \nabla_{(a} \xi_{b)}$. The last term is zero precisely when $\nabla_{(a} \xi_{b)} = 0$, that is, …

I will add here some more details to expand my comment. Any two functions $Y_1(x^3,x^4)$ and $Y_2(x^3,x^4)$ give local coordinates on any open domain of the $(x^3,x^4)$-plane where their Jacobian dete …

A first remark is that in many spacetimes of interest, it is possible to choose a global tetrad (or frame field). So the need to lift everything from the spacetime to the frame bundle to have globally …