Generalized Lemaitre-Tolman-Bondi Solutions with Pressure

TL;DR

This paper derives a unified spherically symmetric solution describing a perfect fluid with pressure, generalizing LTB models to include pressure and connecting interior and exterior spacetimes in a single coordinate system.

Contribution

It introduces a generalized LTB solution with pressure using ADM equations, unifying interior perfect fluid and exterior Schwarzschild spacetimes in a single coordinate framework.

Findings

Derived a metric describing both interior and exterior regions in one coordinate patch.

Reduced to TOV equations in the static limit.

Provided coordinate transformations to comoving coordinates for dynamic cases.

Abstract

Utilizing the ADM equations, we derive a metric and reduced field equations describing a general, spherically symmetric perfect fluid. The metric describes both the interior perfect fluid region and exterior vacuum Schwarzschild spacetime in a single coordinate patch. The exterior spacetime is in generalized Painleve-Gullstrand coordinates which is an infinite class of coordinate systems. In the static limit the system reduces to a Tolman-Oppenheimer-Volkoff equation on the interior with the exterior in Schwarzschild coordinates. We show the coordinate transformation for the non-static cases to comoving coordinates, where the metric is seen to be a direct generalization of the Lemaitre-Tolman-Bondi spacetime to include pressures.