[Paper Review] Exact Radiation Model For Perfect Fluid Under Maximum Entropy Principle

Abdul Aziz, Sourav Roy Chowdhury+4 more|arXiv (Cornell University)|Apr 12, 2015

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes an exact interior solution for a spherically symmetric radiating perfect fluid star under the maximum entropy principle (MEP) and homotopy perturbation method (HPM). By maximizing entropy to derive the mass function m(r) = ar³ − br² + cr − d, and solving the Einstein field equations, the model yields physically viable solutions for brown dwarf stars (E0 type), with finite central redshift and compactness consistent with observations, validating the radiation model for low-mass, cool stellar objects.

ABSTRACT

We find the Euler-Lagrangian equation by maximising the total entropy. Hence we obtain an expression for mass of the spherically symmetric system by solving the Euler-Lagrangian equation where the Homotopy Perturbation Method has been employed. With the help of this expression and the Einstein field equations we obtain an interior solution set. Thereafter, we explain different aspects of the solution describing the system in connection to the mass, density, pressures, energy, stability, mass-radius ratio, compactness factor and surface redshift. This analysis shows that all the physical properties, in connection to brown dwarf stars, are valid with the observed features.

Motivation & Objective

• To develop a physically valid interior solution for a spherically symmetric radiating star using the maximum entropy principle and homotopy perturbation method.
• To derive a mass function m(r) that satisfies entropy maximization and Einstein’s field equations under radiation-dominated conditions (p = 1/3 ρ).
• To test the physical viability of the solution by comparing key stellar properties—mass, density, pressure, compactness, redshift, and stability—against observed features of E0-type brown dwarfs.
• To assess the applicability of the radiation model to highly compact stars, particularly in light of negative pressure and instability in existing data sets.
• To explore the possibility of finite central density and pressure by approximating the core as a constant-density region, avoiding infinite central singularities.

Proposed method

• Apply the maximum entropy principle (MEP) to derive a Lagrangian L = (m′)^(3/4) [1 − 2m(r)/r]^(-1/2) r^(1/2), which governs the entropy of the system under spherically symmetric, radiation-dominated conditions.
• Use the homotopy perturbation method (HPM) to solve the Euler-Lagrange equation derived from the Lagrangian, yielding an approximate analytical expression for the mass function m(r) = ar³ − br² + cr − d.
• Solve the Einstein field equations using the derived m(r) to obtain the time-time component of the metric, energy density ρ, radial and tangential pressures p, and other physical quantities.
• Match the theoretical solution to observational data from Bhar et al. (2017) on compact stars to assess compatibility, particularly noting instability and negative pressure in neutron and strange stars.
• Approximate the central region as a constant-density core (ρ = 10³ g/cm³) of radius rc = 369 km to avoid infinite central density and pressure, while preserving finite central redshift.
• Validate the solution by analyzing the mass-radius ratio, compactness factor, and surface redshift, confirming physical acceptability for E0-type brown dwarfs.

Experimental results

Research questions

• RQ1 Can the maximum entropy principle and homotopy perturbation method yield a physically viable interior solution for a radiating perfect fluid star?
• RQ2 Does the derived mass function m(r) = ar³ − br² + cr − d, when used in the Einstein field equations, produce consistent and stable physical profiles for stellar structure?
• RQ3 Is the radiation model with p = 1/3 ρ compatible with E0-type brown dwarfs, given observed features such as compactness and surface redshift?
• RQ4 Can a finite core approximation resolve the issue of infinite central density and pressure while preserving physical consistency?
• RQ5 Why does the model fail to describe highly compact stars like neutron stars, and what physical limitations prevent its applicability to such systems?

Key findings

• The mass function derived via HPM is m(r) = ar³ − br² + cr − d, with coefficients a = −2.55×10⁻¹⁵ km⁻², b = 6aR³, c = 52/5 a²R⁵, d = 3aR/2 + 6aR³ + 52/5 a²R⁵, yielding a physically consistent solution.
• The central redshift is finite (1 + z ≈ 1.000007), and the surface redshift profile remains within acceptable physical bounds, as confirmed by numerical analysis and figure 8.
• The density reaches a maximum of 10³ g/cm³ at a critical radius rc = 369 km near the center, supporting a constant-density core approximation that constitutes only 0.000054311% of the star’s total volume.
• The compactness factor and mass-radius ratio are consistent with observed E0-type brown dwarfs, validating the model’s physical relevance for low-mass, cool stellar objects.
• The solution is geometrically non-singular (g_tt(0) = 0.999993 > 0, g_rr(0) = 1), but exhibits infinite central density and pressure, which is physically acceptable if central redshift is finite.
• The model is incompatible with highly compact stars such as neutron stars due to negative radial pressure and instability in existing data, suggesting radiation models are more suitable for brown dwarfs than for fuel-exhausted, dense remnants.