Integration of the Lane-Emden equation for relativistic anisotropic polytropes through Gravitational Decoupling: a novel approach

TL;DR

This paper introduces a new method using Gravitational Decoupling to efficiently solve the Lane-Emden equations for relativistic anisotropic polytropes, simplifying calculations and extending known solutions.

Contribution

The novel approach leverages Gravitational Decoupling to reduce degrees of freedom and extend solutions for relativistic anisotropic polytropes using the Lane-Emden equations.

Findings

Successfully applied to Tolman IV, Durgapal IV, and Wymann IIa solutions

Simplifies computation of the Tolman mass with minimal geometric deformation

Extends isotropic and anisotropic solutions efficiently

Abstract

In this work we propose a novel approach to integrate the Lane-Emden equations for relativistic anisotropic polytropes. We take advantage of the fact that Gravitational Decoupling allows to decrease the number of degrees of freedom once a well known solution of the Einstein field equations is provided as a seed so after demanding the polytropic equation for the radial pressure the system is automatically closed. The approach not only allows to extend both isotropic or anisotropic known solutions but simplifies the computation of the Tolman mass whenever the Minimal Geometric Deformation is considered given that the $g_{tt}$ component of the metric remains unchanged. We illustrate how the the method works by analyzing the solutions obtained from Tolman IV, Durgapal IV and Wymann IIa isotropic systems as a seed for the integration.