A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity

Abstract

To cite this version:Herbert Pfister. A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity. Classical and Quantum Gravity, IOP Publishing, 2011, 28 (7) AbstractIn comparison to previous existence proofs for static and spherically symmetric perfect fluid stars in general relativity the new proof applies to a more general class of equations of state: In the star's interior we allow for piecewise Lipschitz continuous functions, in this way including the physically important case of phase transitions. Near the star's surface we allow for even more general functions, thereby including a large class of polytropic equations of state. Furthermore, the proof technique proceeds along standard techniques of functional analysis (Banach's fixed point theorem), and therefore applies in a similar manner to static stars in Newtonian gravity, and perhaps to rotating Newtonian and Einsteinian stars. In detail, the Einstein field equations for static perfect fluid stars are transformed to a system of coupled nonlinear integral equations being valid equally in the matter region and in the vacuum exterior. These integral equations are interpreted as a mapping in a Banach space. With the standard iteration technique, beginning with appropriate start functions, it is proven that the mapping has a unique fixed point, and that the solutions have appropriate regularity properties determined by the properties of the equation of state. The Introduction gives an overview of earlier work on such systems, on the question of sphericity of static fluid stars, and on possible extensions of the above methods to rotating Newtonian

Are there any models with homogeneous energy density?

“…First, we have shown that if the regularity condition at the center of the distribution and some other physical reasonable boundary condition at the surface of the distribution are to be satisfied, then the only static solution for a spherically symmetric matter distribution with homogeneous energy density is the Schwarzschild isotropic solution. This rules out any anisotropic generalization for ρ = const found in the literature [8,11] and complements the proof for the classic problem that a static perfect fluid star should be spherically symmetric for physically reasonable isotropic equation of state [52][53][54][55]. More over, we have shown that even for the static homogeneous Schwarzschild solution the center of the matter distribution has to be excluded because it does not satisfy the Euclidean condition.…”

Spherical inhomogeneous solutions of Einstein and scalar-tensor gravity: a map of the land

“…Stars are configurations of equilibrium and the search for static perfect fluid spheres amounts to solving the Einstein equations for hydrostatic and thermodynamic equilibrium. Aside from rotation, spherical symmetry is not just a convenient approximation because static perfect fluid configurations on stellar scales are expected to smooth out "mountains" and other deviations from sphericity (see (Masood-ul Alam, 2007;Pfister, 2011) for a partial proof in GR).…”

“…There are a few spherical and time-dependent solutions of the Einstein equations that describe central objects (which could possibly be black holes, as defined by the notion of apparent horizon instead of event horizon (Booth, 2005;Faraoni, 2015Faraoni, , 2018Nielsen, 2009)) embedded in FLRW universes. The most well-known is without doubt the 1933 McVittie solution, which has been the subject of renewed attention during the past decade and is also a solution of alternative theories of gravity (Abdalla et al, 2014;Afshordi, 2009;Afshordi et al, 2007Afshordi et al, , 2014Aghili et al, 2017;Arakida, 2011;Faraoni and Lapierre-Léonard, 2017;Faraoni and Moreno, 2013;Faraoni et al, 2012aFaraoni et al, , 2014Gibbons et al, 2008;Guariento et al, 2012;Kaloper et al, 2010;Lake and Abdelqader, 2011;Le Delliou et al, 2011, 2013Maciel et al, 2015a,b;Mello et al, 2017;Mimoso et al, 2010Mimoso et al, , 2013Nandra et al, 2012a,b;Nolan, 1999;Piattella, 2016;da Silva et al, 2013). The charged version of the McVit-tie solution is known (Mashhoon and Partovi, 1979;Shah and Vaidya, 1968), as well as generalized McVittie solutions (Castelo Ferreira, 2009, 2013Faraoni and Jacques, 2007;Gao et al, 2008;Li and Wang, 2007).…”

OV or TOV?

“…For Newtonian gravity, the proof was given in 1919 [1,2]; for General Relativity (GR), a complete proof still does not exist. The conjecture was first explicitly stated in 1955 [3], proven for a certain class of equations-of-state (EoS -pressure-density relationships) in 2007 [4], and for a wider class in 2011 [5].…”