# Shock-Wave Solutions of the Einstein Equations with Perfect Fluid Sources: Existence and Consistency by a Locally Inertial Glimm Scheme

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*Jeff Groah; John Blake Temple*

We demonstrate the consistency of the Einstein equations at the
level of shock-waves by proving the existence of shock wave solutions of the
spherically symmetric Einstein equations for a perfect fluid, starting from
initial density and velocity profiles that are only locally of bounded total
variation. For these solutions, the components of the gravitational metric
tensor are only Lipschitz continuous at shock waves, and so it follows that
these solutions satisfy the Einstein equations, as well as the relativistic
compressible Euler equations, only in the weak sense of the theory of
distributions. The analysis introduces a locally inertial Glimm scheme
that exploits the locally flat character of spacetime, and relies on special
properties of the relativistic compressible Euler equations when
\(p=\sigma^2\rho\), \(\sigma\equiv const\).

We demonstrate the consistency of the Einstein equations at the
level of shock-waves by proving the existence of shock wave solutions of the
spherically symmetric Einstein equations for a perfect fluid, starting from
initial density and velocity profiles that are only locally of bounded total
variation. For these solutions, the components of the gravitational metric
tensor are only Lipschitz continuous at shock waves, and so it follows that
these solutions satisfy the Einstein equations, as well as the relativistic
compressible Euler equations, only in the weak sense of the theory of
distributions. The analysis introduces a locally inertial Glimm scheme
that exploits the locally flat character of spacetime, and relies on special
properties of the relativistic compressible Euler equations when
\(p=\sigma^2\rho\), \(\sigma\equiv const\).