Shock-Wave Solutions Of The Einstein Equations With Perfect Fluid Sources: Existence And Consistency By A Locally Inertial Glimm Scheme

Shock-Wave Solutions Of The Einstein Equations With Perfect Fluid Sources: Existence And Consistency By A Locally Inertial Glimm Scheme

Author
Jeff Groah, Blake Temple
Publisher
Amer Mathematical Society
Language
English
Year
2004
Page
84
ISBN
082183553X,9780821835531
File Type
djvu
File Size
708.8 KiB

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 $p=\\sigma^2\\rho$, $\\sigma\\equiv const$.

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