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arXiv · 2112.15241

General Relativistic Shock Waves that Exhibit an Accelerated Expansion

Christopher Alexander
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Abstract

This paper concerns the construction and analysis of a new family of exact general relativistic shock waves. The construction resolves the long-standing open problem of determining the expanding waves created behind a shock-wave explosion within a static isothermal sphere, akin to stellar ignition. In particular, these shock-wave spacetimes are formed through matching a family of self-similar asymptotically Friedmann spacetimes to the family of Tolman-Oppenheimer-Volkoff spacetimes. The matching is accomplished in Schwarzschild coordinates where the shock waves appear one derivative less regular than they actually are. Separately, both families contain singularities, but as matched shock-wave spacetimes, they are singularity free. It was previously unknown whether the matching of the two families could be achieved within a region where both families are nonsingular. Indeed, for a pure radiation equation of state, the matching occurs near the sonic point of the interior expanding wave and this makes the analysis quite delicate, both numerically and formally. It is for this reason the construction is accompanied by a rigorous existence proof in the pure radiation case, which forms the main result of this paper. The analysis extends to consider the matching of any self-similar expanding wave with a Tolman-Oppenheimer-Volkoff spacetime, with potentially distinct equations of state each side of the shock. The asymptotically Friedmann shock-waves exhibit an accelerated expansion, analogous to the acceleration modelled by the cosmological constant in the Standard Model of Cosmology. However, unlike in the Standard Model, these shock-wave spacetimes solve the Einstein field equations in the absence of a cosmological constant, opening up the question of whether a purely mathematical mechanism could account for the accelerated expansion observed today, rather than dark energy.

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Christopher Alexander. 2021-12-30. General Relativistic Shock Waves that Exhibit an Accelerated Expansion. https://arxiv.org/abs/2112.15241

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