General relativistic shock-waves propagating at the speed of light

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 to the lightlike case. After a brief introduction to general relativity, we introduce a previously known generalization of the second fundamental form by Barrabes and Israel. Then we use this to develop an extension of a shock matching theory, which characterizes solutions of the Einstein equations when the spacetime metric is only Lipschitz continuous across a hypersurface, to include the case when the hypersurface is lightlike. The theory also demonstrates an unexpected result that the matching of the generalized second fundamental form alone is not a sufficient condition for conservation conditions to hold across the interface. Using this theory we then 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. This is done 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. Then our theory is used to show that the matched FRW, TOV metric is a solution. The pressure and density are finite on each side of the shock throughout the solution, the sound speeds, on each side of the shock, are constant and subluminous. Moreover, 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.