A hierarchical, boundary-regular affine-null form of the conformal Einstein-scalar equations is derived, shown equivalent to compactified physical-space equations, and used to extract the news function at null infinity.
Characteristic initial value problems for the Einstein-Maxwell-scalar field equations in spherical symmetry
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abstract
The characteristic initial boundary problem is discussed in spherical symmetry for the Einstein-Maxwell-scalar field equations. It is formulated for an affine-null metric and the resulting field equations are cast into a hierarchical system of partial differential equations. The initial boundary value problem for a family of null hypersurfaces is specified for a timelike-null foliation at the central geodesic of spherical symmetry as well as for a double-null foliation where the corresponding boundary is a null hypersurface. For the latter, two distinct boundary value formulations arise -- one where the null boundary has zero Misner-Sharp mass and another one where the corresponding Misner-Sharp mass is nonzero. As an application, the nonextremal and the extremal Reissner-Nordstr\"om solution in null coordinates for a charged black hole and the Fisher-Janis-Newman-Winicour solution are derived.
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Conformal compactification and affine-null metric formulation of the Einstein equations
A hierarchical, boundary-regular affine-null form of the conformal Einstein-scalar equations is derived, shown equivalent to compactified physical-space equations, and used to extract the news function at null infinity.