Hypersurface Data: General Properties and Birkhoff Theorem in Spherical Symmetry
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Hypersurface Data: General Properties and Birkhoff Theorem in Spherical Symmetry Marc Mars Abstract. The notions of (metric) hypersurface data were introduced in Mars (Gen Relativ Gravit 45:2175–2221, 2013) as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudoRiemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived. Mathematics Subject Classification. 53B05, 53B25, 53C50, 83C05. Keywords. Abstract hypersurface, metric data, hypersurface data, Gauge group, spherical symmetry, Birkhoff theorem.
1. Introduction The geometric character of the gravitational field requires that the initial value problem for the gravitational field is quite different than for other evolutionary systems. The initial data do not consist simply in providing the initial values of the fields (and as many initial time derivatives as needed) at a given initial initial hypersurface. Instead, one provides an (n − 1)-manifold and geometric data on that manifold (usually satisfying suitable constraint equations, depending on the specific gravity theory at hand) and asks for the existence of a spacetime where these data can be embedded so that the geometry of the embedded hypersurface suitably agrees with the prescribed geometric data. Sometimes this essentially geometric nature of the problem may be somewhat obscured by the way how the evolution problem is set up.
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For instance, in the case of the standard Cauchy problem for the Einstein (say vacuum) field equations, one knows by fundamental existence theorems, that there exists a globally hyperbolic Ricci flat spacetime where the abstract initial manifold (and data) can be embedded as a spacelike hypersurface. One can then adapt coordinates, say {t, xi } so that {t = 0} corresponds to this embedded hypersurface. By this method, the problem has been converted into a more standard initial value problem (although the diffeomorphism freedom remains, and needs to be addressed, e.g. by imposing coordinate conditions such as, for instance, harmonic coordinates). In this case, however, it is clear and well understood that the initial data are given in a a fully abstract manifold, completely detached from the spacetime one intends to construct. Another insta
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