General Relativistic Shock-Waves Propagating at the Speed of Light

Abstract

We investigate shock-wave solutions of the Einstein equations in the case when the speed of propagation is equal to the speed of light. The work extends the shock matching theory of Smoller and Temple, which characterizes solutions of the Einstein equations when the spacetime metric is only Lipschitz continuous across a hypersurface, to the lightlike case. After a brief introduction to general relativity, we develop the extension of a shock matching theory after introducing a previously known generalization of the second fundamental form by Barrabes and Israel. In the development of the theory we demonstrate that the matching of the generalized second fundamental form alone is not a sufficient condition for conservation conditions to hold across the interface. We then use this theory to construct a new exact solution of the Einstein equations that can be interpreted as an outgoing spherical shock wave that propagates at the speed of light. The solution is constructed by matching a Friedman Robertson Walker (FRW) metric, which is a geometric model for the universe, to a Tolman Oppenheimer Volkoff (TOV) metric, which models a static isothermal spacetime. The sound speeds, on each side of the shock, are constant and sub-luminous. Furthermore, the pressure and density are smaller at the leading edge of the shock, which is consistent with the Lax entropy condition in classical gas dynamics. However, the shock speed is greater than all the characteristic speeds. The solution also yields a surprising result in that the solution is not equal to the limit of previously known subluminous solutions as they tend to the speed of light.