SHOCK-WAVE SOLUTIONS IN CLOSED-FORM AND THE OPPENHEIMER-SNYDER LIMIT IN GENERAL REALITY

In earlier work the authors derived a set of ODEs that describe a clas s of spherically symmetric, fluid dynamical shock-wave solutions of th e Einstein equations. These solutions model explosions in a general re lativistic setting. The theory is based on matching Friedmann-Robertso n-Walker (FRW) metrics (models for the expanding universe) to Oppenhei mer-Tolman (OT) metrics, (models for the interior of a star) Lipschitz -continuously across a surface that represents a time-irreversible, ou tgoing shock-wave. In the limit when the outer OT solution reduces to the empty space Schwarzschild metric and the inner FRW metric is restr icted to the case of bounded expansion (k > 0), our equations reproduc e the well-known solution of Oppenheimer and Snyder in which the press ure p = 0. In this article we derive closed form expressions for solut ions 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 s olution when the inner FRW metric is restricted to the case of critica l expansion (k = 0). This produces a large class of shock-wave solutio ns given by explicit formulas. Among other things, these formulas can be useful in testing numerical shock-wave codes in general relativity.