A shock-wave formulation of the Einstein equations

Jeffrey M. Groah, Blake Temple
2000 Methods and Applications of Analysis  
We derive a weak (shock wave) formulation of the Einstein equations for a perfect fluid assuming spherical symmetry. Our purpose is to provide a framework for the weak equations that makes them amenable to mathematical methods developed for the study of shock waves in nonlinear conservation laws. Introduction. We derive a weak formulation of the Einstein equations for a perfect fluid under the assumption of spherical symmetry. The weak formulation is required in order to allow for the presence
more » ... f shock wave discontinuities in the fluid variables. We assume the standard gauge, and we show that in the resulting coordinates, the weak formulation of the initial value problem for the Einstein equations is valid for metrics that are only Lipschitz continuous: that is, valid for metrics that lie in the space C 0,1 , the space of continuous functions with bounded difference quotients, (and hence Holder exponent one, [13] ). Moreover, we show that in the standard gauge, the metric can in general be no smoother than Lipschitz continuous when shock waves are present. To clarify this, note that if T is discontinuous across a smooth 3-dimensional surface S, then the Einstein equations G = KT imply that the curvature tensor G will also have discontinuities across the surface. Since G involves second derivatives of the metric tensor #, one expects that g should be continuously differentiable at shock waves, with bounded second derivatives on either side, (that is, g G C 1,1 ), in order that the equation G = KT will hold in the classical, pointwise a.e. sense at the shocks. However, it is known that shock-wave solutions of the Einstein equations make sense under the assumption that the metrics match only Lipschitz continuously at a shock surface, that is, g E C 0,1 . But in this case, the Lipschitz continuous matching of the metric alone is not enough to guarantee conservation at a shock, and an additional condition must be imposed to rule out the possibility that there are delta function sources in T on the shock surface, [8, 15] . Our analysis shows that for spherically symmetric solutions of G = ACT, it is in general not possible to have metrics smoother than Lipschitz continuous, (that is, smoother than C 0,1 at shocks), when the metric is written in the standard gauge. In this paper, we show that the weak formulation is nonetheless consistent for metrics in the lower smoothness class C 0,1 . This helps explain why the Oppenheimer-Snyder solution, [12] , and its shock wave generalizations, [15, 16] , involve metrics that are matched only Lipschitz continuously at an interface. Thus our results imply that when shock-waves are present, we cannot expect metrics to be smoother than these examples in the standard gauge. The aim of this paper is to formulate the Einstein equations as a system of conservation laws with source terms. For this reason, the discussion is written to be accessible to experts in the mathematical theory of shock waves and conservation laws, [9, 5, 13] . The work here is preparatory for a subsequent paper in which the authors will give a rigorous local existence theory for shock wave solutions of these * Received
doi:10.4310/maa.2000.v7.n4.a10 fatcat:d5ikpmme55dlhdfisjs52cn7ti