Disembodied boundary data for Einstein’s equations

A strongly well-posed initial boundary value problem based upon constraint-preserving boundary conditions of the Sommerfeld type has been established for the harmonic formulation of the vacuum Einstein’s equations. These Sommerfeld conditions have been previously presented in a four-dimensional geometric form. Here we recast the associated boundary data as three-dimensional tensor fields intrinsic to the boundary. This provides a geometric presentation of the boundary data analogous to the three-dimensional presentation of Cauchy data in terms of three-metric and extrinsic curvature. In particular, diffeomorphisms of the boundary data lead to vacuum spacetimes with isometric geometries. The proof of well-posedness is valid for the harmonic formulation and its generalizations. The Sommerfeld conditions can be directly applied to existing harmonic codes which have been used in simulating binary black holes, thus ensuring boundary stability of the underlying analytic system. The geometric form of the boundary conditions also allows them to be formally applied to any metric formulation of Einstein’s equations, although well-posedness of the boundary problem is no longer ensured. We discuss to what extent such a formal application might be implemented in a constraint-preserving manner to $3+1$ formulations, such as the Baumgarte-Shapiro-Shibata-Nakamura system which has been highly successful in binary black hole simulation.

Figure 1

Data on the three-manifolds ${\mathcal{S}}_0$ and $\mathcal{T}$, which intersect in the two-surface ${\mathcal{B}}_0$, locally determine a solution in the spacetime manifold $\mathcal{M}$