Eur. Phys. J. C (2024) 84: 5
https://doi.org/10.1140/epjc/s10052-023-12349-2

Regular Article - Theoretical Physics

Extension of the Schwarzschild black hole solution in f(R) gravitational theory and its physical properties

Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43, 11837, El Sherouk City, Cairo, Egypt

^a nashed@bue.edu.eg

Received: 16 August 2023
Accepted: 8 December 2023
Published online: 5 January 2024

Abstract

The successes of f(R) gravitational theory as a logical extension of Einstein’s theory of general relativity (GR) encourage us to delve deep into this theory and continue our study to attempt to derive an extension of the Schwarzschild black hole (BH) solution. In this study, in order to solve the output nonlinear differential equation, we closed the form of the system by assuming the derivative of f(R) with respect to the scalar curvature R to have the form $F(r)=\frac{\text{d}f(R(r))}{\text{d}R(r)}=1-\frac{\alpha }{r^4}$, where $\alpha$ is a dimensional constant. Our study shows that when $\alpha \to 0$, we obtain the Schwarzschild BH solution of GR assuming some constraints on the constant of integration, and if these constraints are bounded, we obtain the anti-de Sitter (AdS)/de Sitter (dS) spacetime. For the general case, i.e., when $\alpha \ne 0$, we obtain a BH solution that tends asymptotically to AdS/dS spacetime. Moreover, we derive the timelike and null particle geodesics of the BH solution studied in this article. The equation of motion and effective potential of test particles are calculated to study the stability of radial orbits (trajectories). The energy and angular momentum are calculated to study the circular motion and stability of circular orbits. We also derive the stability condition using the geodesic deviation. Moreover, we discuss the physics of the output BH solutions through calculation of the thermodynamic quantities including entropy, the Hawking temperature, and Gibbs free energy. Finally, we check the validity of the first law of thermodynamics applied to the BH of this study. Although we can derive a Schwarzschild black hole solution in the lower order of f(R), specifically when $f(R)=R$, where the gravitational mass is generated from the source of gravity, we demonstrate that in the higher orders of f(R), when $f(R)\ne R$, the source of gravity is attributed primarily to higher-order corrections, and the source of gravity that was originally derived from the Schwarzschild black hole has ceased to be dominant.