The Lemaitre-Schwarzschild Problem Revisited

TL;DR

This paper revisits the classical Lemaitre and Schwarzschild solutions for relativistic spheres, linking them via the Weyl invariant and analyzing how the critical collapse radius varies with internal pressure anisotropy.

Contribution

It establishes a continuous connection between the Lemaitre and Schwarzschild solutions using the Weyl invariant and explores the effects of pressure anisotropy on gravitational collapse.

Findings

Critical radius varies from 9/8 to 1 times the Schwarzschild radius.

Internal pressures become anisotropic during collapse.

The solutions are linked through the Weyl quadratic invariant.

Abstract

The Lemaitre and Schwarzschild analytical solutions for a relativistic spherical body of constant density are linked together through the use of the Weyl quadratic invariant. The critical radius for gravitational collapse of an incompressible fluid is shown to vary continuously from 9/8 of the Schwarzschild radius to the Schwarzschild radius itself while the internal pressures become locally anisotropic.