Kruskal-Szekeres coordinates of spherically symmetric solutions in theories of gravity

A Romadani,M F Rosyid

doi:10.1088/1742-6596/1816/1/012030

Abstract

Understanding the missing matter problem in cosmological phenomena and scales of astrophysical is usually studied by modifying general relativity theory. In this article, we formulated the Kruskal-Szekeres coordinate of vacuum modified gravity model in f (R ) theory. The generalization of the field equation was obtained by generalizing Hilbert-Einstein’s action with gravitational Lagrangian in terms of f (r)function. By consider a special class of f(R) theory by taking R = R0 , we found the solution of static spherically symmetric spacetime that was known as de Sitter-Schwarzschild spacetime. The transformation rules were constructed from Kruskal-Szekeres coordinates in f (r)theory of modified general relativity to the Kruskal-Szekeres coordinate in general relativity theory. For λ ≈ 0, the Schwarzschild and Kruskal-Szekeres metric for static spherically symmetric on f (r)theory reduced to the standard Schwarzschild and Kruskal-Szekeres metric on general relativity. We also show the spacetime structure of de Sitter-Schwarzschild and Kruskal-Szekeres coordinate. This work could open a promising way to understand some features of a black hole in the f (r)theory of gravity.

Highlights

General relativity theory has been able to explain the astronomical phenomena, covering the structure of massive objects like stars, including white dwarfs, neutron stars, black hole, quasar, and as a whole of the universe [1]
Scientists have found some weakness of Einstein’s theory such as a cosmological constant that correlated with the existence of dark energy and inflation of universe that are unsolved problems recently
They began wondering whether the gravitational concepts on general relativity theory is a fundamental theory to describe the gravitational interaction [3]

Summary

Introduction

General relativity theory has been able to explain the astronomical phenomena, covering the structure of massive objects like stars, including white dwarfs, neutron stars, black hole, quasar, and as a whole of the universe [1]. Modified general relativity theories is constructed by two variational principles where it could be applied to the Einstein-Hilbert action: Palatini variation formalism and standard metric variation [9] Both of these principles are built on the field equation with the Lagrangian is linear in R [10]. General relativity contains only some of Mach’s ideas and admits solutions that are explicitly anti-Machian, such as the Godel universe and exact pp-waves [24, 8] In massive objects such as white dwarfs, neutron stars and black hole, the field equations represented strong gravitational interaction. We reviewed f R for R is a constant because the spherically symmetric solution yield generalization Schwarzschild spacetime with a cosmological constants where it was related with unsolved problems currently in general relativity theory: dark energy model and inflation of universe

Formalism Einstein-Hilbert action equation is written by

Schwarzschild Black Hole

Results and Discussions

Conclusions