Specific neutral and charged black holes in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>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 neutral and charged black hole (BH) solutions. In the previous papers [Phys. Rev. D 102, 124022 (2020); Phys. Lett. B 820, 136475 (2021)], we applied the field equation of the $f(R)$ gravity to a spherically symmetric space-time $d{s}^{2}=\ensuremath{-}U(r)d{t}^{2}+\frac{d{r}^{2}}{V(r)}+{r}^{2}(d{\ensuremath{\theta}}^{2}+{\mathrm{sin}}^{2}\ensuremath{\theta}d{\ensuremath{\phi}}^{2})$ with unequal metric potentials $U(r)$ and $V(r)$ and with and 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 ${F}_{1}(r)=\frac{df(R(r))}{dR(r)}=1\ensuremath{-}\frac{{F}_{0}\ensuremath{-}(n\ensuremath{-}3)}{{r}^{n}}$ with a constant ${F}_{0}$ and show that we can generate asymptotically GR BH solutions for $n&gt;2$ 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 ${F}_{0}\ensuremath{\rightarrow}n\ensuremath{-}3$. We show that the form of the electric charge depends on $n$ and that $n\ensuremath{\ne}2$. 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. We also study the physics of these black hole solutions by calculating their thermodynamical quantities, like entropy, the Hawking temperature and Gibb's free energy, and derive the stability conditions by using geodesic deviations. In the standard Reissner-Nordstr\"om space-time which is the charged black hole solution in GR, there appear two black hole horizons, that is, inner horizon and outer horizon. When the radii of the two horizons coincide with each other, which is called the extremal limit, the absolute value of the charge equals to the mass and the Hawking temperature vanishes. In our model, however, the absolute value of the charge is not equal to the mass in the limit although the Hawking temperature vanishes.

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