Gravitational Decoupling in Higher Order Theories

Joseph Sultana

doi:10.3390/sym13091598

Abstract

Gravitational decoupling via the Minimal Geometric Deformation (MGD) approach has been used extensively in General Relativity (GR), mainly as a simple method for generating exact anisotropic solutions from perfect fluid seed solutions. Recently this method has also been used to generate exact spherically symmetric solutions of the Einstein-scalar system from the Schwarzschild vacuum metric. This was then used to investigate the effect of scalar fields on the Schwarzschild black hole solution. We show that this method can be extended to higher order theories. In particular, we consider fourth order Einstein–Weyl gravity, and in this case by using the Schwarzschild metric as a seed solution to the associated vacuum field equations, we apply the MGD method to generate a solution to the Einstein–Weyl scalar theory representing a hairy black hole solution. This solution is expressed in terms of a series using the Homotopy Analysis Method (HAM).

Highlights

Deformation (MGD) technique has been proposed as a novel and simple approach to decoupling gravitational sources in General Relativity, which could lead to new static and spherically symmetric solutions having sources that are more realistic than the ideal perfect fluid
Gravitational decoupling by the minimal geometric deformation method is a simple yet powerful technique for generating anisotropic solutions from simple perfect fluid seed metrics
It can only be applied to spherically symmetric systems, and if there are multiple sources, these have to be decoupled such that no energy exchange takes place between them

Summary

Introduction

Deformation (MGD) technique has been proposed as a novel and simple approach to decoupling gravitational sources in General Relativity, which could lead to new static and spherically symmetric solutions having sources that are more realistic than the ideal perfect fluid. The main feature of this method is that one can start with a known seed solution of Einstein’s field equations having a relatively simple source Tμν , such as, for example, a perfect fluid solution, and add to this a second source θμν such that the total energy momentum tensor is Tμν = Tμν + θμν , where θμν may be a generic anisotropic source. The MGD method has been used to investigate solutions of Einstein-scalar gravity [27] In this case, the additional source θμν corresponds to the energy momentum tensor of a minimally coupled scalar field.

Minimal Geometric Deformation in General Relativity

Einstein–Weyl Gravity

Gravitational Decoupling in Einstein–Weyl Gravity

Homotopy Analysis Method

Hairy Black Hole in Einstein–Weyl Gravity

Discussion and Conclusions