[Paper Review] The classification of interior solutions of anisotropic fluid configurations

J. Kumar, Puja Bharti|arXiv (Cornell University)|Dec 22, 2021

Cosmology and Gravitation Theories4 citations

TL;DR

This paper presents a systematic classification scheme for exact static spherically symmetric anisotropic fluid solutions to the Einstein-Maxwell (or Einstein) field equations in general relativity. By organizing known solutions based on the generation technique—such as assuming specific gravitational potentials, equations of state, conformal motion, or embedding class conditions—it provides a structured framework for modeling compact stars, with key results showing differences in redshift, luminosity, and moment of inertia between charged and neutral anisotropic models.

ABSTRACT

The Einstein-Maxwell (or Einstein) system of field equations plays a substantial role in the modeling of compact stars. Although due to its non-linearity getting an exact solution for the system of field equations is a difficult task, the solutions of field equations have a long and rich history. It took a year for Karl Schwarzschild to obtain the first exact solution of Einstein's field equations since general theory of relativity was published. The number of viable solutions has been growing since then. Many authors have adopted several methods to obtain the solution. Different models have been constructed for a variety of applications. To produce feasible models of compact stars, a considerable amount of effort has been applied in gaining an understanding of the properties of anisotropic matter. Theoretical study indicates that pressure within compact stars with extreme internal density and strong gravity is mostly anisotropic. Anisotropy was found sufficient for the study of compact stars with the dense nuclear matter. It is claimed that it is important to consider the pressure experienced to be anisotropic whenever relativistic fluids are involved. In this review article, we have discussed different ways of generating a static spherically symmetric anisotropic fluid model. The purpose of the article is to present a simple classification scheme for static and spherically symmetric anisotropic fluid solutions. The known solutions are reviewed and compartmentalized as per the proposed scheme so that we can illustrate general ideas about these solutions without being exhaustive.

Motivation & Objective

• To develop a comprehensive yet accessible classification framework for interior solutions of anisotropic fluid configurations in general relativity.
• To systematize known exact solutions of the Einstein-Maxwell system based on the method used to generate them.
• To clarify the role of anisotropy, charge, and geometric constraints (e.g., embedding class, conformal motion) in constructing viable compact star models.
• To guide researchers in selecting appropriate solution-generation techniques based on physical assumptions and constraints.
• To serve as a reference for beginners and experts in modeling anisotropic compact stars with exact solutions.

Proposed method

• Categorizing solutions based on the technique used to generate them, such as assuming specific gravitational potentials or energy-momentum tensor components.
• Applying constraints like the equation of state, conformal motion, or embedding class one spacetime to reduce the number of independent fluid characteristics.
• Using the Einstein-Maxwell system of equations to model charged anisotropic fluid configurations, with the neutral case recovered when charge function is zero.
• Analyzing the behavior of pressure, density, and anisotropy factors under different assumptions to ensure physical viability and boundary conditions (e.g., vanishing radial pressure at the surface).
• Employing the Karmarkar condition and spacetime symmetries to identify solutions within embedding class one, facilitating solvability.
• Comparing solutions across different assumptions to assess their impact on physical quantities like redshift, luminosity, and moment of inertia.

Experimental results

Research questions

• RQ1 What are the primary techniques used to generate exact solutions for anisotropic fluid configurations in spherically symmetric spacetimes?
• RQ2 How do constraints such as the equation of state, conformal motion, or embedding class one geometry affect the solvability and physical consistency of interior solutions?
• RQ3 In what ways do charged anisotropic models differ from neutral ones in terms of redshift, luminosity, and moment of inertia?
• RQ4 How can known solutions be systematically categorized based on their solution-generation methodology?
• RQ5 What role does the vanishing of radial pressure at the boundary play in validating physical models of compact stars?

Key findings

• The classification scheme successfully organizes known solutions into distinct categories based on the solution-generation method, enhancing clarity and accessibility.
• Charged anisotropic models yield different values for redshift, luminosity, and moment of inertia compared to their neutral counterparts, indicating the significance of charge in compact star modeling.
• Solutions generated under multiple constraints—such as equation of state combined with embedding class one or conformal motion—yield consistent and physically viable models.
• The assumption of vanishing radial pressure at the boundary is a widely recognized and effective criterion for ensuring physical plausibility in compact star solutions.
• The use of embedding class one spacetime and conformal motion significantly simplifies the system of equations while preserving physical relevance.
• The framework enables researchers to select appropriate solution-generation techniques based on desired physical assumptions, such as equation of state or spacetime symmetry.