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# General Relativity and Quantum Cosmology

# Title: Specific neutral and charged black holes in $f(R)$ gravitational theory

(Submitted on 3 Mar 2021 (v1), last revised 17 Aug 2021 (this version, v5))

Abstract: With the successes of $f(R)$ theory as a neutral modification of Einstein's general relativity (GR), we continue our study in this field and attempt to find general natural and charged black hole (BH) solutions. In the previous papers \cite{Nashed:2020mnp,Nashed:2020tbp}, we applied the field equation of the $f(R)$ gravity to a spherically symmetric space-time $ds^2=-U(r)dt^2+\frac{dr^2}{V(r)}+r^2 \left( d\theta^2+\sin^2\theta d\phi^2 \right)$ with unequal metric potentials $U(r)$ and $V(r)$ and with/without electric charge. Then we have obtained equations which include all the possible static solutions with spherical symmetry. To ensure the closed form of system of the resulting differential equations in order to obtain specific solutions, we assumed the derivative of the $f(R)$ with respect to the scalar curvature $R$ to have a form $F_1(r)=\frac{df(R(r))}{dR(r)} \propto \frac{c}{r^n}$ but in case $n>2$, the resulting black hole solutions with/without charge do not generate asymptotically GR BH solutions in the limit $c\rightarrow 0$ which means that the only case that can generate GR BHs is $n=2$. In this paper, we assume another form, i.e., $F_1(r)= 1-\frac{F_0-\left(n-3\right)}{r^n}$ with a constant $F_0$ and show that we can generate asymptotically GR BH solutions for $n>2$ but we show that the $n=2$ case is not allowed. This form of $F_1(r)$ could be the most acceptable physical form that we can generate from it physical metric potentials that can have a well-known asymptotic form and we obtain the metric of the Einstein general relativity in the limit of $F_0\to n-3$. We show that the form of the electric charge depends on $n$ and that $n\neq 2$. Our study shows that the power $n$ is sensitive and why we should exclude the case $n=2$ for the choice of $F_1(r)$ presented in this study.

## Submission history

From: Shin'ichi Nojiri [view email]**[v1]**Wed, 3 Mar 2021 13:12:22 GMT (343kb)

**[v2]**Fri, 30 Apr 2021 11:55:56 GMT (344kb)

**[v3]**Tue, 25 May 2021 14:32:55 GMT (444kb)

**[v4]**Tue, 29 Jun 2021 12:07:25 GMT (444kb)

**[v5]**Tue, 17 Aug 2021 08:57:41 GMT (444kb)

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