The Lemaître-Schwarzschild Problem Revisited

Abstract

The Lemaître 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. *

Analytical study of anisotropic compact star models

“…They can serve as an approximation to some features of the stars. The same may be said for models with p r = 0 and any ρ [3,57,58,64]. Because of C4 we should have p t0 = 0, and because of C5 p t ≤ 0.…”

“…It has vanishing p r and constant ρ and was found independently by Florides [57]. It was studied further in [58]. A very important equation of hydrostatic equilibrium exists, involving only ρ, p r and p t in canonical coordinates.…”

“…Conditions C7-C9 become singular and make no sense. The first interior solution was such a model [2] and others followed [3,5,57,58,[61][62][63]76]. They can serve as an approximation to some features of the stars.…”

Maximum bounds on the surface redshift of anisotropic stars

“…This model was generalized for variable density by Florides [4] and was recently revisited in Ref. [5]. The application of anisotropic fluid models to neutron stars began with the pioneering work of Bowers and Liang [6] and was done both analytically and numerically [7][8][9][10][11][12][13][14][15].…”

“…In many papers it is stressed that arbitrarily big redshifts are obtained when t grows to infinity [5][6][7]10,12,16]. However, in realistic models t should be finite, positive, and should satisfy the dominant energy condition (DEC) t ≤ ρ or even the strong energy condition (SEC) 2t + p ≤ ρ.…”