Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries

Juan A. Valiente Kroon

Juan A. Valiente Kroon works on various aspects of mathematical Relativity and, in particular, on applications of conformal methods to analyse the global properties of spacetimes.

The construction of physically realistic data for the Einstein field equations is one of the great challenges of the Cauchy problem in General Relativity. In this paper C. Meier and M. Holst show how to construct solutions to the constraint equations of General Relativity representing data which will evolve, assuming that a certain form of weak cosmic censorship holds, into a spacetime containing one or more black holes.

The most studied procedure for solving the constraint equations is the so-called conformal method. This approach can be traced back to the pioneering work of Licnerowicz, Choquet-Bruhat and York among others. It allows an identification of the freely specifiable data in the equations and leads to a reformulation of the constraint equations as a system of coupled non-linear partial differential elliptic equations. The traditional approach to the solution of these equations assumes that the initial data has a constant mean curvature (CMC). This is a technical assumption which allows to decouple the equations but, otherwise, has a limited physical motivation. Under suitable boundary conditions, it can be shown that the resulting equations have a unique solution. Several authors have considered the issue of the type of conditions required on an inner boundary so that it describes a “marginally trapped surface” —this type of surface is the appropriate description of a black hole in the context of the constraint equations of General Relativity. In this article, the authors revisit this question and crucially remove the technical assumption of the constancy of the mean curvature. In doing so, they pave the road for the construction of more physically realistic initial data. At the same time they expand our knowledge of the space of initial data for the Einstein equations.

Read the full article in Classical and Quantum Gravity:
Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries
Michael Holst and Caleb Meier
2015 Class. Quantum Grav. 32 025006

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