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This paper concerns the construction and analysis of a new family of exactgeneral relativistic shock waves. The construction resolves the open problem ofdetermining the expanding waves created behind a shock-wave explosion into astatic isothermal sphere with an inverse square density and pressure profile.The construction involves matching two self-similar families of solutions tothe perfect fluid Einstein field equations across a spherical shock surface.The matching is accomplished in Schwarzschild coordinates where the shock wavesappear one derivative less regular than they actually are. Separately, bothfamilies contain singularities, but as matched shock-wave solutions, they aresingularity free. There was no guarantee ahead of time that the matching of thetwo families could be achieved within the regions where both families arenonsingular. Indeed, for pure radiation equations of state, the matching occursnear the sonic point of the interior expanding wave and this makes the analysisquite delicate, both numerically and formally. It is for this reason theconstruction is accompanied by a rigorous existence proof in the pure radiationcase. The analysis is extended to demonstrate Lax stability in the pureradiation case and provide a criterion for Lax stability in all other cases.These shock-wave solutions represent an intriguing new mechanism in GeneralRelativity for exhibiting accelerations in asymptotically Friedmann spacetimes,analogous to the accelerations modelled by the cosmological constant in theStandard Model of Cosmology. However, unlike in the Standard Model ofCosmology, these shock-wave solutions solve the Einstein field equations in theabsence of a cosmological constant, opening up the question of whether a purelymathematical mechanism could account for the cosmic acceleration observedtoday, rather than dark energy.

General Relativistic Shock Waves that Exhibit Cosmic Acceleration