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The Initial Boundary Value Problem for the Einstein Equations with Totally Geodesic Timelike Boundary

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Abstract

We prove the well-posedness of the initial boundary value problem for the Einstein equations with sole boundary condition the requirement that the timelike boundary is totally geodesic. This provides the first well-posedness result for this specific geometric boundary condition and the first setting for which geometric uniqueness in the original sense of Friedrich holds for the initial boundary value problem. Our proof relies on the ADM system for the Einstein vacuum equations, formulated with respect to a parallelly propagated orthonormal frame along timelike geodesics. As an independent result, we first establish the well-posedness in this gauge of the Cauchy problem for the Einstein equations, including the propagation of constraints. More precisely, we show that by appropriately modifying the evolution equations, using the constraint equations, we can derive a first order symmetric hyperbolic system for the connection coefficients of the orthonormal frame. The propagation of the constraints then relies on the derivation of a hyperbolic system involving the connection, suitably modified Riemann and Ricci curvature tensors and the torsion of the connection. In particular, the connection is shown to agree with the Levi-Civita connection at the same time as the validity of the constraints. In the case of the initial boundary value problem with totally geodesic boundary, we then verify that the vanishing of the second fundamental form of the boundary leads to homogeneous boundary conditions for our modified ADM system, as well as for the hyperbolic system used in the propagation of the constraints. An additional analytical difficulty arises from a loss of control on the normal derivatives to the boundary of the solution. To resolve this issue, we work with an anisotropic scale of Sobolev spaces and exploit the specific structure of the equations.

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Notes

  1. See also [4, 7] for extensions and other proofs of well-posedness in the Anti-de-Sitter case.

  2. To be more precise, the boundary data in [1] relies on an auxiliary wave map equation akin to generalized wave coordinates. This introduces a geometric framework to address the IBVP, albeit for the Einstein equations coupled to the auxiliary wave map equation.

  3. We would like to thank M.T. Anderson and E. Witten for this suggestion.

  4. In fact, the system (2.21) corresponds exactly to (2.19)–(2.20) up to zeroth order terms.

  5. We are grateful to an anonymous referee for this observation.

  6. Since the domain of dependence depends on the spacetime metric, in a Picard iteration the actual region of spacetime is not known until after a solution has been found, but one can enlarge slightly the domain to guarantee that in the end the resulting region includes the true domain of dependence of \(U_0\).

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Acknowledgements

G.F. would like to thank Jonathan Luk for useful discussions. We would also like to thank M.T. Anderson for several interesting comments on our work. Both authors are supported by the ERC grant 714408 GEOWAKI, under the European Union’s Horizon 2020 research and innovation program.

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  1. CNRS, Sorbonne Université, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), 75005, Paris, France

    Grigorios Fournodavlos & Jacques Smulevici

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  1. Grigorios Fournodavlos
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  2. Jacques Smulevici
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Correspondence to Jacques Smulevici.

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Communicated by P. Chrusciel

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Fournodavlos, G., Smulevici, J. The Initial Boundary Value Problem for the Einstein Equations with Totally Geodesic Timelike Boundary. Commun. Math. Phys. 385, 1615–1653 (2021). https://doi.org/10.1007/s00220-021-04141-8

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  • DOI: https://doi.org/10.1007/s00220-021-04141-8

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