Junction conditions in general relativity

An analytical formalism is developed to deal with the occurrence of jump discontinuities in the g_{mu nu} or their derivatives across a hypersurface Sigma . It is shown that the equations of relativity remain meaningful at Sigma , even when Sigma does not inherit a unique intrinsic geometry, so that the g_{mu nu} are discontinuous across Sigma in natural coordinates. The spherically symmetric surface layer at the Schwarzschild-Minkowski junction is used to illustrate these techniques, and to establish rigorously the existence of C⁰ solutions of the Einstein equations and the conservation equations. The possible validity of relativity at the microscopic level is examined, and it is concluded that, if relativity is valid at the microscopic level, then it is likely that the g_{mu nu} are not globally continuously differentiable.