Abstract
The results on the initial boundary value problem for Einstein’s vacuum field equation obtained in Friedrich and Nagy Commun. Math. Phys. 201 619–655 rely on an unusual gauge. One of the defining gauge source functions represents the mean extrinsic curvature of the time-like leaves of a foliation that includes the boundary and covers a neighbourhood of it. The others steer the development of a frame field and coordinates on the leaves. In general their combined action is needed to control in the context of the reduced field equations the evolution of the leaves. In this article are derived the hyperbolic equations implicit in that gauge. It is shown that the latter are independent of the Einstein equations and well defined on arbitrary space-times. The analysis simplifies if boundary conditions with constant mean extrinsic curvature are stipulated. It simplifies further if the boundary is required to be totally geodesic.
Highlights
In this article we consider a question that arises in the context of the initial boundary value problem for Einstein’s vacuum field equation formulated in [7]
The two functions F A, A = 1, 2, correspond to connection coefficients that control the evolution of a time-like vector field tangential to the hypersurfaces Tc, that serves to define coordinates on these hypersurfaces. With these particular fields singled out as the gauge source functions there can be extracted from the Einstein equations, in the representation used in [7], a hyperbolic system of reduced equations that allows one to formulate with suitably prescribed data a well-posed initial boundary value problem
Its final justification follows from the existence of a hyperbolic subsidiary system which implies that the reduced system preserves the constraints and gauge conditions and that the latter do what they have been chosen for. This leads to a well-posedness result, local in time, for the initial boundary value problem for Einstein’s field equations and one could leave it at that
Summary
In this article we consider a question that arises in the context of the initial boundary value problem for Einstein’s vacuum field equation formulated in [7]. The two functions F A, A = 1, 2, correspond to connection coefficients that control the evolution of a time-like vector field tangential to the hypersurfaces Tc, that serves to define coordinates on these hypersurfaces With these particular fields singled out as the gauge source functions there can be extracted from the Einstein equations, in the representation used in [7], a hyperbolic system of reduced equations that allows one to formulate with suitably prescribed data a well-posed initial boundary value problem. Its final justification follows from the existence of a hyperbolic subsidiary system which implies that the reduced system preserves the constraints and gauge conditions and that the latter do what they have been chosen for This leads to a well-posedness result, local in time, for the initial boundary value problem for Einstein’s field equations and one could leave it at that.
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