Asymptotically flat vacuum solution in modified theory of Einstein\u2019s gravity

Surajit Kalita,Banibrata Mukhopadhyay

doi:10.1140/epjc/s10052-019-7396-x

Surajit Kalita, Banibrata Mukhopadhyay

Open Access

https://doi.org/10.1140/epjc/s10052-019-7396-x

Copy DOI

Abstract

A number of recent observations have suggested that the Einstein’s theory of general relativity may not be the ultimate theory of gravity. The f(R) gravity model with R being the scalar curvature turns out to be one of the best bet to surpass the general relativity which explains a number of phenomena where Einstein’s theory of gravity fails. In the f(R) gravity, behaviour of the spacetime is modified as compared to that of given by the Einstein’s theory of general relativity. This theory has already been explored for understanding various compact objects such as neutron stars, white dwarfs etc. and also describing evolution of the universe. Although researchers have already found the vacuum spacetime solutions for the f(R) gravity, yet there is a caveat that the metric does have some diverging terms and hence these solutions are not asymptotically flat. We show that it is possible to have asymptotically flat spherically symmetric vacuum solution for the f(R) gravity, which is different from the Schwarzschild solution. We use this solution for explaining various bound orbits around the black hole and eventually, as an immediate application, in the spherical accretion flow around it.

Highlights

general relativity (GR) is one of the most efficient and powerful theories, a number of recent observations have suggested that it may fall short in very high density regions [3,4,5]
We show that the solution for f (R) gravity in vacuum, and for black holes, can be obtained which behaves as the Schwarzschild/ Minkowski metric at asymptotic limit and this solution can be used in accretion physics effectively
A better exploration in a realistic model containing angular momentum profile, e.g. accretion discs, will be carried out in future. It has already been discussed about the behaviour of vacuum spacetime as well as various marginal orbits in the context of f (R) gravity

Summary

Introduction

GR is one of the most efficient and powerful theories, a number of recent observations have suggested that it may fall short in very high density regions [3,4,5]. The vacuum solution of f (R) gravity is an interesting problem and the solutions for a static, spherically symmetric spacetime in f (R) gravity were first obtained by Multamäki and Vilja [27] They showed that for a large class models, Schwarzschild-de Sitter metric is an exact solution of the field equations. Where c is the speed of light, G the Newton’s gravitational constant, LM the Lagrangian of the matter field and g = det(gμν) is the determinant of the metric gμν Varying this action with respect to gμν and equating to zero with appropriate boundary conditions, we obtain the Einstein’s field equation for general relativity, which is given by.

Solution for vacuum spacetime

FX and

Temporal and spatial components of the metric

Conclusion