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Shock-wave solutions in closed form and the Oppenheimer-Snyder limit in general relativity

By Joel Smoller and Blake Temple


Abstract. In earlier work the authors derived a set of ODEs that describe a class of spherically symmetric, fluid dynamical shock-wave solutions of the Einstein equations. These solutions model explosions in a general relativistic setting. The theory is based on matching Friedmann–Robertson– Walker (FRW) metrics (models for the expanding universe) to Oppenheimer–Tolman (OT) metrics, (models for the interior of a star) Lipschitz-continuously across a surface that represents a timeirreversible, outgoing shock-wave. In the limit when the outer OT solution reduces to the empty space Schwarzschild metric and the inner FRW metric is restricted to the case of bounded expansion (k>0), our equations reproduce the well-known solution of Oppenheimer and Snyder in which the pressure p ≡ 0. In this article we derive closed form expressions for solutions of our ODEs in all cases (k>0,k < 0,k = 0) when the outer OT solution is Schwarzschild, as well as in the case of an arbitrary OT solution when the inner FRW metric is restricted to the case of critical expansion (k = 0). This produces a large class of shock-wave solutions given by explicit formulas. Among other things, these formulas can be useful in testing numerical shock-wave codes in general relativity

Topics: Key words. general relativity, differential geometry, hyperbolic conservation laws/shock-waves
Year: 1998
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