Spherically Symmetric C3 Matching in General Relativity

Abstract

We study the problem of matching interior and exterior solutions to Einstein’s equations along a particular hypersurface. We present the main aspects of the C3 matching approach that involve third-order derivatives of the corresponding metric tensors in contrast to the standard C2 matching procedures known in general relativity, which impose conditions on the second-order derivatives only. The C3 alternative approach does not depend on coordinates and allows us to determine the matching surface by using the invariant properties of the eigenvalues of the Riemann curvature tensor. As a particular example, we apply the C3 procedure to match the exterior Schwarzschild metric with a general spherically symmetric interior spacetime with a perfect fluid source and obtain that on the matching hypersurface, the density and pressure should vanish, which is in accordance with the intuitive physical expectation.