[Paper Review] Status of a self-bound equations of state and analytic solutions in general relativity

P. S. Negi, M. C. Durgapal|ArXiv.org|Dec 19, 2003

Gas Dynamics and Kinetic Theory1 references6 citations

TL;DR

This paper establishes a criterion for physical consistency in general relativity: for any spherically symmetric, static configuration, the compaction parameter $ u = M/a $ must not exceed that of a homogeneous density sphere ($ u_h $) when the central pressure-to-energy-density ratio $ \sigma = P_0/E_0 $ is fixed. The key finding is that only equations of state and exact solutions with vanishing surface density (i.e., self-bound structures) satisfy this criterion, while those with finite surface density—commonly used in neutron star models—violate it and are therefore incompatible with GR's structure.

ABSTRACT

We have obtained a criterion for spherically symmetric and static structures under hydrostatic equilibrium in general relativity (GR), which states that for a given value of $σ\equiv (P_0/E_0) \equiv $ the ratio of central pressure to central energy-density], the compaction parameter $u \equiv (M/a)$, where $M$ is the total mass and $a$ is the radius of the configuration] or the surface redshift of any regular configuration cannot exceed that of the corresponding homogeneous density sphere, that is, $u \leq u_h$, where $u_h$ is the compaction parameter of the homogeneous density sphere. By examining various exact solutions and surface). On the other hand, configurations having a finite density on the surface (that is, the self-bound structures) do not fulfill this criterion. This criterion puts a severe restriction on the static structures based upon the general relativistic field equations and consequently on the upper limit of mass, surface and central redshift and other physical parameters of spherically symmetric and static configurations.

Motivation & Objective

• To identify a physical consistency criterion for spherically symmetric, static configurations in general relativity based on hydrostatic equilibrium and the Einstein field equations.
• To determine whether widely used equations of state (EOS) and exact solutions for compact objects like neutron stars are compatible with the structure of general relativity.
• To establish that configurations with finite surface density (self-bound structures) violate a fundamental GR constraint, while those with vanishing surface density do not.
• To provide a quantitative criterion for the upper limits of mass, surface redshift, and central redshift in static, spherically symmetric compact objects.

Proposed method

• Derive a criterion based on the minimum central pressure $ P_0 $ for a given compaction parameter $ u = M/a $, showing that the homogeneous density sphere minimizes $ P_0 $ for fixed $ \sigma = P_0/E_0 $.
• Use the condition $ u \leq u_h $, where $ u_h $ is the compaction parameter of the homogeneous sphere, as a necessary condition for physical consistency in GR.
• Apply this criterion to various equations of state: $ P = E - E_s $, $ P = K E^\Gamma $, $ P = K \rho^{\Gamma_1} $, and exact solutions like Durgapal-Fuloria and Tolman's type VII.
• Compute compaction parameters $ u $ for each EOS and solution at fixed $ \sigma $, and compare them to $ u_h $ to test the inequality $ u \leq u_h $.
• Analyze the continuity of metric functions and their derivatives at the surface, showing that only vanishing surface density ensures smoothness and compatibility with GR.
• Use analytical derivations and numerical comparisons (via Table 1) to demonstrate that only vanishing-surface-density models satisfy $ u \leq u_h $ for all $ \sigma $.

Experimental results

Research questions

• RQ1 Does the compaction parameter $ u = M/a $ of any regular, spherically symmetric, static configuration in GR remain bounded above by that of the homogeneous density sphere for a fixed central pressure-to-energy-density ratio $ \sigma $?
• RQ2 Are equations of state and exact solutions with finite surface density (self-bound structures) compatible with this upper bound criterion in general relativity?
• RQ3 Why do certain widely used neutron star equations of state, such as $ dP/dE = 1 $, fail to satisfy the GR consistency condition?
• RQ4 What physical conditions at the surface—specifically regarding density and pressure continuity—must be met for a solution to be consistent with the Einstein field equations?
• RQ5 Can the upper limits on mass, surface redshift, and central redshift of compact objects be redefined based on this new GR compatibility criterion?

Key findings

• For any fixed $ \sigma = P_0/E_0 $, the compaction parameter $ u $ of a regular, spherically symmetric, static configuration in GR must not exceed $ u_h $, the compaction of the homogeneous density sphere.
• Configurations with finite surface density (e.g., $ P = E - E_s $, Durgapal-Fuloria solution) violate the criterion $ u \leq u_h $, indicating incompatibility with general relativity.
• Solutions with vanishing surface density—such as polytropic EOS $ P = K E^\Gamma $, $ P = K \rho^{\Gamma_1} $, and Tolman's type VII ($ E = E_0(1 - r^2/a^2) $)—satisfy $ u \leq u_h $, confirming their GR compatibility.
• The criterion implies a severe restriction on physically viable models: only configurations with continuous density and metric derivatives at the surface (i.e., vanishing surface density) are allowed.
• The upper limit on mass, surface redshift, and central redshift of compact objects must be re-evaluated under this criterion, favoring models with zero surface density.
• The 'abnormal' behavior of the $ dP/dE = 1 $ EOS (pressure vanishing at nuclear density) is not an artifact of the EOS itself, but a consequence of violating the GR consistency criterion due to finite surface density.