Chevalley–Warning theorem

Summary

This is an illustration of a three-dimensional hyperslice of a simple example of static spherically symmetric perfect fluid solution, with one dimension suppressed, embedded in a flat euclidean spacetime for ease of visualizing the geometry.

The figure shows a perfect fluid interior region (gold) matched (with continuous metric and extrinsic curvature tensor) across a spherical surface (represented here as a circle) to a vacuum exterior region (khaki), which is a portion of the Schwarzschild vacuum solution.

In this example, the interior region is Schwarzschild's fluid, in which the hyperslices have the geometry of a three-spherical cap. In this example, the embedding is

$z(r)=10-{\sqrt {125-r^{2}}},\;0<r<5$

$z(r)=-4+2{\sqrt {r-1}},\;5<r<\infty$

All static spherically symmetric stellar models resemble the Schwarzschild solution near the center, and are matched to a Schwarzschild vacuum exterior, but other choices of fluid solution for the interior region may differ from the geometry shown here in ways difficult to depict without restoring the suppressed dimension.

Licensing

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