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

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 { neutral} 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 $d{s}^2=-U(r)d{t}^2+\frac{d{r}^2}{V(r)}+r^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right)$ 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 %$F_1(r)=\frac{df(R(r))}{dR(r)}\propto$ but in case $n>2$, the resulting black hole solutions with/without charge do not %generate asymptotically GR BH solutions in the limit $c\to 0$ which means that the only case that can generate GR BHs is $n=2$. %In this paper, we assume another form, i.e., $F_1(r)=\frac{df(R(r))}{dR(r)}=1-\frac{F_0-\left(n-3\right)}{r^n}$ 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 $n\ne 2$.}

LaTeX 18 pages, 16 figures, several revisions

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