ar X iv : g r - qc / 0 10 90 97 v 1 2 8 Se p 20 01 The Lemâıtre - Schwarzschild Problem Revisited

We present here an analysis based on the Weyl tensor of the Lemaˆıtre-Schwarzschild problem of finding the equilibrium conditions of an anisotrop-ically sustained spherical body (with thus different radial and tangential pressure) under its relativistic gravitational field. An overview of the problem is presented as well as a justification of the unbounded surface redshift when the radial pressure vanishes (i.e., in the Lemaˆıtre case). Then, we show how the criterion of the Weyl quadratic invariant brings more physical insight in the problem, as an indicator of stability regions. Finally, we apply it to a quasi-static gravitational collapse and prove that the boundary of the sphere can pass from 9/8 of the Schwarzschild radius to the strict value of the Schwarzschild radius itself without breaking the equilibrium conditions by infinite tidal forces.