Black Holes with Electric and Magnetic Charges in $F(R)$ Gravity

For a few decades, $F(R)$ gravity theory has attracted much attention because the theory may explain the accelerating expansion of the present universe in addition to the inflation of the early universe Capozziello:2002rd ; Capozziello:2003gx ; Carroll:2003wy ; Nojiri:2003ft ; Hu:2007nk ; Starobinsky:2007hu (for reviews, see Nojiri:2006ri ; Copeland:2006wr ; Sotiriou:2008rp ; Nojiri:2010wj ; Capozziello:2011et ; Nojiri:2017ncd ). Furthermore, the possibility that the dark matter might be explained by the $F(R)$ gravity has been investigated Nojiri:2006ri ; Copeland:2006wr ; Nojiri:2010wj ; Clifton:2011jh . The existence of higher-order curvature terms in general relativity also supplies interesting physical results, for example, it makes the condensation harder to be formed in holographic superconductivity Gregory:2009fj ; Kuang:2013oqa , it also amends the low-energy tensor perturbation spectrum in string backgrounds Gasperini:1997up , and it affects the dynamics of stellar structure Hansraj:2020xmz .

In the $F(R)$ gravity theory, the scalar curvature $R$ in the Einstein-Hilbert action is replaced by an adequate function $F(R)$ of $R$ Capozziello:2002rd ; Capozziello:2003gx ; Nojiri:2003ft ; Cognola:2007zu ; Pogosian:2007sw ; Zhang:2005vt ; Li:2007xn ; Song:2007da ; Nojiri:2007cq ; Nojiri:2007as ; Capozziello:2018ddp ; Vainio:2016qas . We may consider the higher derivative theories including the Ricci or Riemann curvatures in the actions not only in the form of the scalar curvature. Such higher-order corrections to the action of the general relativity yield a renormalizable and therefore quantizable theory of gravity Stelle:1976gc . We should note, however, that such higher-derivative theories except the $F(R)$ gravity have the Ostrogradski instability Ostrogradsky:1850fid which is problematic because there appear ghosts, which make the theory inconsistent Woodard:2006nt .

To investigate whether the $F(R)$ theory is realistic, spherically symmetric and static black hole solutions have been investigated Multamaki:2006zb ; delaCruz-Dombriz:2009pzc ; Hendi:2011hxq ; Nashed:2021sey ; Nashed:2021mpz ; Nashed:2021lzq ; Nashed:2021ffk ; Nashed:2020mnp ; Nashed:2020kdb ; Nashed:2020tbp ; Tang:2019qiy ; Nashed:2018oaf ; Nashed:2018efg ; Nashed:2018piz . In the gravitational collapse, all the matters including the charged ones are absorbed into the black hole, and therefore even in the $F(R)$ gravity, there must exist charged black hole. Thus, it is important to investigate the stationary black hole solutions in the $F(R)$ gravity coupled with the electromagnetic fields and the contribution from the electromagnetic fields to the geometry in the framework of the $F(R)$ gravity. The black hole solutions in $F(R)$ gravity in vacuum or in the case with electromagnetic fields have been studied in Multamaki:2006zb ; Nashed:2020kdb ; Sebastiani:2010kv ; Hendi:2014mba ; Multamaki:2006ym ; Nashed:2020tbp ; Nashed:2019uyi ; Nashed:2019tuk ; delaCruz-Dombriz:2009pzc ; Jaryal:2021lsu ; Eiroa:2020dip and in Tang:2020sjs ; Karakasis:2021lnq ; Karakasis:2021rpn , the scalar fields have been included as matter in three and four dimensions while the $F(R)$ gravity coupled with/without non-minimally coupled scalar fields as a matter has been studied in the frame of cosmology Pi:2017gih ; delaCruz-Dombriz:2016bjj . Capozziello et al. have used the Noether symmetry to investigate the spherically symmetric solutions Capozziello:2007wc ; Capozziello:2012iea and for the axially symmetric black hole solution Capozziello:2009jg . Dynamical spherically symmetric black hole solutions have been also presented in Elizalde:2020icc ; Nashed:2019yto ; Nashed:2019tuk for a specific form of $F(R)$. As the topics related to the strong gravitational background, not only the static spherically symmetric black holes Sultana:2018fkw ; Canate:2017bao ; Yu:2017uyd ; Canate:2015dda ; Kehagias:2015ata ; Nelson:2010ig ; delaCruz-Dombriz:2009pzc but neutron star have been investigated Feng:2017hje ; AparicioResco:2016xcm ; Capozziello:2015yza ; Staykov:2018hhc ; Doneva:2016xmf ; Yazadjiev:2016pcb ; Yazadjiev:2015zia ; Yazadjiev:2014cza ; Ganguly:2013taa ; Astashenok:2013vza ; Orellana:2013gn ; Arapoglu:2010rz ; Cooney:2009rr in the form $F(R)=R+\alpha R^{2}$. It is also well-known that the $F(R)$ gravity can be rewritten as the Brans-Dicke theories Brans:1961sx that have a scalar potential of the gravitational origin Chiba:2003ir ; OHanlon:1972xqa ; Chakraborty:2016gpg ; Chakraborty:2016ydo . In this paper, we construct a new type of spherically symmetric and static black hole with electric and magnetic charge in the framework of the $F(R)$ gravity and investigate the physical properties of such black hole solutions.

In Sec. II, we review the fundamentals of $F(R)$ gravity, and in Sec. III, we apply the field equations in the $F(R)$ gravity to a spherically symmetric space-time. There appears a system of differential equations that has three unknown functions and we derive the solutions of this system that is characterized by a convolution function. If this convolution function vanishes, we obtain the black hole solution in general relativity, that is, the Schwarzschild solution. Hence the convolution function could appear due to higher-order curvature terms that characterize the $F(R)$ gravity. By calculating the Kretschmann scalar, the Ricci tensor square, and the Ricci scalar, we show that the singularities in such invariants become weaker than those of general relativity black holes. In Sec. IV, we calculate the thermodynamical quantities in the obtained black hole solutions to compare them with those in the known solutions. In Sec. V, we use the odd-type method and study the stability of these black hole solutions. In the final section, we give our concluding remarks.

We may regard the $F(R)$ gravity as an extension of general relativity that was investigated in Buchdahl:1970ynr ; Capozziello:2011et ; Nojiri:2010wj ; Nojiri:2017ncd ; Capozziello:2003gx ; Capozziello:2002rd ; Nojiri:2003ft ; Carroll:2003wy . The action of the $F(R)$ gravity coupled with the electromagnetic fields is given by

$\displaystyle S=S_{\mathrm{g}}+S_{\mathrm{EM}}\,,$

(1)

where $S_{\mathrm{g}}$ is given by

$\displaystyle S_{\mathrm{g}}=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}F(R)\,.$

(2)

Here $\kappa$ is the gravitational constant, $R$ is the Ricci scalar, $g$ is the determinant of the metric, and $F(R)$ is an analytic function of $R$. On the other hand, $S_{\mathrm{EM}}$ is given by

$\displaystyle S_{\mathrm{EM}}=-\frac{1}{2}\int d^{4}x\sqrt{-g}F_{\mu\nu}F^{\mu\nu}\,,$

(3)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and $A_{\mu}$ is the gauge potential.

In the following, we choose the unit where $\kappa=1$. The variations of the action (1) with respect to the metric tensor $g_{\mu\nu}$ and the gauge potential $A_{\mu}$ give the field equations as in Cognola:2005de ; Koivisto:2005yc ,

	$\displaystyle 0=$	$\displaystyle\,R_{\mu\nu}F_{R}-\frac{1}{2}g_{\mu\nu}F+g_{\mu\nu}\Box F_{R}-\nabla_{\mu}\nabla_{\nu}F_{R}-8\pi T_{\mu\nu}\,,$		(4)
	$\displaystyle 0=$	$\displaystyle\,\partial_{\nu}\left(\sqrt{-g}{F}^{\mu\nu}\right)\,.$		(5)

Here $R_{\mu\nu}$ is the Ricci tensor¹¹1The Ricci tensor is defined as $$R_{\mu\nu}=R^{\rho}_{\ \mu\rho\nu}=\Gamma^{\rho}_{\mu\nu,\rho}-\Gamma^{\rho}_{\mu\rho,\nu}+\Gamma^{\rho}_{\beta\rho}\Gamma^{\beta}_{\nu\mu}-\Gamma^{\rho}_{\beta\nu}\Gamma^{\beta}_{\rho\mu}\,,$$ with e $\Gamma^{\rho}_{\mu\nu}$ being the Christoffel second kind symbols. and $\Box$ is the d’Alembertian operator which is defined as $\Box=\nabla_{\alpha}\nabla^{\alpha}$ where $\nabla_{\alpha}B^{\alpha}$ means the covariant differentiation of the vector $B^{\alpha}$ and $F_{R}=\frac{dF(R)}{dR}$. The energy-momentum tensor of the electromagnetic fields $T_{\mu\nu}$ is defined as

$\displaystyle T_{\mu\nu}:=\frac{1}{4\pi}\left(g_{\rho\sigma}{{F}_{\nu}^{\ \rho}}F_{\mu}^{\ \sigma}-\frac{1}{4}g_{\mu\nu}F^{2}\right)\,.$

(6)

The trace of the field equations (4) has the following form,

$\displaystyle 0=RF_{R}-2F+3\Box F_{R}\,.$

(7)

By rewriting Eq. (7) as $F=\frac{1}{2}\left(RF_{R}+3\Box F_{R}\right)$ and deleting $F$ in Eq. (4), we obtain

$\displaystyle 0=R_{\mu\nu}\left[1+F_{R}\right]-\frac{1}{4}g_{\mu\nu}R\left[1+F_{R}\right]+\frac{1}{4}g_{\mu\nu}\Box F_{R}-\nabla_{\mu}\nabla_{\nu}F_{R}-8\pi T_{\mu\nu}\,.$

(8)

In the next section, we apply the field equations (5) and (8) to spherically symmetric and static space-time and derive exact solutions describing the black hole with electric and/or magnetic charges.

In this section, we do not specify the function $F(R)$ as a function of $R$ but we assume the radial coordinate $r$ dependence of $F_{R}$. After solving the equations, we consider the functional form of $F(R)$. This tells that there exists a function $F(R)$ which generates the solution.

We first assume the following spherically symmetric and static space-time with two functions $h(r)$ and $h(r)$ of the radial coordinate $r$,

$\displaystyle ds^{2}=-h(r)dt^{2}+\frac{dr^{2}}{h_{1}(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\,.$

(9)

The metric (9) gives the following Ricci scalar,

$\displaystyle R(r)=\frac{r^{2}h_{1}h^{\prime 2}-r^{2}hh^{\prime}h^{\prime}_{1}-2r^{2}hh_{1}h^{\prime\prime}-4rh[h_{1}h^{\prime}-hh^{\prime}_{1}]+4h^{2}(1-h_{1})}{2r^{2}h^{2}}\,,$

(10)

where $h\equiv h(r)$, $h_{1}\equiv h_{1}(r)$, $h^{\prime}=\frac{dh}{dr}$, $h^{\prime\prime}=\frac{d^{2}h}{dr^{2}}$ and $h_{1}^{\prime}=\frac{dh}{dr}$. For the metric (9), by using Eq. (10), we find Eqs. (5) and (8) have the following forms,
The $tt$, $t\phi$, $rr$, $r\theta$, $r\phi$, $\theta\theta$, and $\phi\phi$ components of Eq. (8) are given by

	$\displaystyle 0=$	$\displaystyle\,\frac{1}{8h^{2}r^{4}\sin^{2}\theta}\left\{2h^{2}r^{4}h_{1}\sin^{2}\theta F^{\prime\prime}_{1}-2F_{1}r^{4}h_{1}h\sin^{2}\theta h^{\prime\prime}-F_{1}r^{4}h_{1}\sin^{2}\theta h^{\prime 2}+\left(3F^{\prime}_{1}h_{1}r+F_{1}\left(4h_{1}+h^{\prime}_{1}r\right)\right)hr^{3}\sin^{2}\theta h^{\prime}\right.$
		$\displaystyle\,+h\left[r^{3}h\sin^{2}\theta\left(4h_{1}+h^{\prime}_{1}r\right)F_{1}+4hF_{1}r^{3}\sin^{2}\theta h^{\prime}_{1}+8h_{1}hr^{2}n_{\phi}^{2}-16h_{1}hr^{2}n_{\phi}p^{\prime}+8h_{1}hr^{2}p^{\prime 2}+8r^{4}\sin^{2}\theta h_{1}q^{\prime 2}\right.$
		$\displaystyle\,\left.\left.+4h\left(2k_{\theta}^{2}+F_{1}r^{2}\sin^{2}\theta\left(h_{1}-1\right)\right)\right]\right\}\,,$		(11)
	$\displaystyle 0=$	$\displaystyle\,2h_{1}q^{\prime}\left(n_{\phi}-p^{\prime}\right)\,,$		(12)
	$\displaystyle 0=$	$\displaystyle\,\frac{1}{8h^{2}r^{4}\sin^{2}\theta}\left\{F_{1}r^{4}h_{1}\sin^{2}\theta h^{\prime 2}-6h^{2}r^{4}h_{1}\sin^{2}\theta F^{\prime\prime}_{1}-2F_{1}r^{4}h_{1}h\sin^{2}\theta h^{\prime\prime}+\left(F^{\prime}_{1}h_{1}r-F_{1}\left(h^{\prime}_{1}r-4h_{1}\right)\right)hr^{3}\sin^{2}\theta h^{\prime}\right.$
		$\displaystyle\,+\left[\left(4h_{1}-3h^{\prime}_{1}r\right)hr^{3}\sin^{2}\theta F^{\prime}_{1}-4hF_{1}r^{3}\sin^{2}\theta h^{\prime}_{1}-8h_{1}hr^{2}n_{\phi}^{2}+16h_{1}hr^{2}n_{\phi}p^{\prime}-8h_{1}hr^{2}p^{\prime 2}+8r^{4}h_{1}\sin^{2}\theta q^{\prime 2}\right.$
		$\displaystyle\,\left.\left.+4h\left(2k_{\theta}^{2}+F_{1}r^{2}\sin^{2}\theta\left(h_{1}-1\right)\right)\right]h\right\}\,,$		(13)
	$\displaystyle 0=$	$\displaystyle\,2k_{\theta}\left(n_{\phi}-p^{\prime}\right)\,,$		(14)
	$\displaystyle 0=$	$\displaystyle\,2h_{1}k_{\theta}\left(n_{\phi}-p^{\prime}\right)\,,$		(15)
	$\displaystyle 0=$	$\displaystyle\,\frac{1}{8h^{2}r^{4}\sin^{2}\theta}\left\{2h^{2}r^{4}h_{1}\sin^{2}\theta F^{\prime\prime}_{1}+2F_{1}r^{4}h_{1}h\sin^{2}\theta h^{\prime\prime}+\left[\left(h^{\prime}_{1}r-4h_{1}\right)h+h_{1}h^{\prime}r\right]hr^{3}\sin^{2}\theta F^{\prime}_{1}\right.$
		$\displaystyle\,-F_{1}r^{4}h_{1}\sin^{2}\theta h^{\prime 2}+F_{1}r^{4}h^{\prime}_{1}h\sin^{2}\theta h^{\prime}$
		$\displaystyle\,\left.+4\left(2h_{1}hr^{2}n_{\phi}^{2}-4h_{1}hr^{2}n_{\phi}p^{\prime}+2h_{1}hr^{2}p^{\prime 2}-2r^{4}h_{1}\sin^{2}\theta q^{\prime 2}-h\left(2k_{\theta}^{2}+F_{1}r^{2}\sin^{2}\theta\left(h_{1}-1\right)\right)\right)h\right\}\,,$		(16)
	$\displaystyle 0=$	$\displaystyle\,\frac{1}{8h^{2}r^{4}\sin^{2}\theta}\left\{2h^{2}r^{4}h_{1}\sin^{2}\theta F^{\prime\prime}_{1}+2F_{1}r^{4}h_{1}h\sin^{2}\theta h^{\prime\prime}+\left[\left(h^{\prime}_{1}r-4h_{1}\right)h+h_{1}h^{\prime}r\right]hr^{3}\sin^{2}\theta F^{\prime}_{1}\right.$
		$\displaystyle\,-F_{1}r^{4}h_{1}\sin^{2}\theta h^{\prime 2}+F_{1}r^{4}h^{\prime}_{1}h\sin^{2}\theta h^{\prime}$
		$\displaystyle\,\left.-4\left(2h_{1}hr^{2}n_{\phi}^{2}-4h_{1}hr^{2}n_{\phi}p^{\prime}+2h_{1}hr^{2}p^{\prime 2}+2r^{4}h_{1}\sin^{2}\theta q^{\prime 2}+h\left(2k_{\theta}^{2}+F_{1}r^{2}\sin^{2}\theta\left(h_{1}-1\right)\right)\right)h\right\}\,.$		(17)

The trace-component (7) has the following form,

	$\displaystyle 0=$	$\displaystyle\,\frac{1}{h^{2}r^{2}}\left\{6h^{2}r^{2}F^{\prime\prime}_{1}h_{1}-2F_{1}h_{1}h^{\prime\prime}hr^{2}+F_{1}h_{1}h^{\prime 2}r^{2}+\left[3F^{\prime}_{1}h_{1}r-F_{1}\left(4h_{1}+h^{\prime}_{1}r\right)\right]hrh^{\prime}\right.$
		$\displaystyle\,\left.+h^{2}\left[\left(3r^{2}h^{\prime}_{1}+12h_{1}r\right)F^{\prime}_{1}-4rF_{1}h^{\prime}_{1}-4\left(h_{1}-1\right)F_{1}-4Fr^{2}\right]\right\}\,.$		(18)

The non-vanishing components of the field equations (5) are $t$, $r$, and $\phi$ components which have the following forms, respectively,

	$\displaystyle 0=$	$\displaystyle\,rhh^{\prime}_{1}q^{\prime}-rh_{1}h^{\prime}q^{\prime}+2rhh_{1}q^{\prime\prime}+4hh_{1}q^{\prime}\,,$		(19)
	$\displaystyle 0=$	$\displaystyle\,n_{\phi\phi}\,,$		(20)
	$\displaystyle 0=$	$\displaystyle\,h_{1}h^{\prime}r^{2}\sin\theta n_{\phi}-h_{1}h^{\prime}r^{2}\sin\theta p^{\prime}+r^{2}h\sin\theta)h^{\prime}_{1}n_{\phi}-r^{2}A\sin\theta h^{\prime}_{1}p^{\prime}$
		$\displaystyle\,-2r^{2}h\sin\theta h_{1}p^{\prime\prime}+2hk_{\theta}\cos\theta-2hk_{\theta\theta}\sin\theta\,.$		(21)

For brevity, we put $F_{1}\equiv F_{1}(r)=\frac{dF\left(R\left(r\right)\right)}{dR(r)}$, $F^{\prime}_{1}=\frac{dF_{1}(r)}{dr}$, $F^{\prime\prime}_{1}=\frac{d^{2}F_{1}(r)}{dr^{2}}$, $n_{\phi}=\frac{dn(\phi)}{d\phi}$, $k_{\theta}=\frac{dk(\theta)}{d\theta}$, and $q^{\prime}=\frac{dq(r)}{dr}$. Here $q(r)$, $k(\theta)$, $p(r)$, and $n(\phi)$ are the components of the electric and magnetic field components defined as

$\displaystyle A_{\mu}dx^{\mu}:=q(r)dt+n(\phi)dr+\left[p(r)+k(\theta)\right]d\phi\,.$

(22)

We stress that when the magnetic field vanishes, i.e., $n=p=k=0$, the components of the field equations (8) for $\theta\theta$ and $\phi\phi$ components become identical with each other and in that case, the field equations (III), (12), (III), (14), (15), (III), (III), and (III) coincide with those derived in Nashed:2019tuk .

Now we discuss two cases that $h=h_{1}$ and that $h\neq h_{1}$²²2Because the present study deals with spherically symmetric case, we assume $F(R)$ only depends on the radial coordinate $r$, $F(R)=F(r)$.. Because we do not assume the specific form of $F(R)$ as a function of $R$, we have seven unknown functions, $h$, $h_{1}$, $q$, $n$, $k$, $p$, and $F_{1}=\frac{dF(R)}{dR}$ but we have six independent differential equations. To solve these differential equations, we assume that $F_{1}$ is given by

$\displaystyle F_{1}=1+\frac{a}{r^{3}}\,.$

(23)

Here $a$ is a constant. Eq. (23) shows that when $a=0$, we return to the case of general relativity where $F_{1}(R)=\mathrm{const}.$

In the case of $h=h_{1}$, the field equations (14), (15), (III), (III) and (III) have no solution whenever $a\neq 0$. This means that when $h=h_{1}$ and $a\neq 0$, we will not obtain any solution, and when $a=0$, we obtain the following solution,

$\displaystyle h=r^{2}c_{0}+1+\frac{c_{1}}{r}+\frac{{c_{2}}^{2}+{c_{3}}^{2}}{r^{2}}\,,\quad q(r)=\frac{c_{2}}{r}\,,\quad k(\theta)=c_{3}\cos\theta\,,\quad p(r)=c_{4}r\,,\quad n(\phi)=c_{5}\phi\,.$

(24)

Here $c_{0}$, $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ are the constants of the integration. The above discussion is consistent with the previous studies which ensure that any solution with $h=h_{1}$ in the frame of $F(R)$ will not differ from the black hole in the general relativity and $F(R)$ only play the role of a cosmological constant Multamaki:2006zb ; Nashed:2018oaf ; Nashed:2018efg ; Nashed:2018piz . The above solution (24) supports this discussion. The invariant scalars of the above solution (24) have the following form,

		$\displaystyle K=R_{abcd}R^{abcd}=48{c_{0}}^{2}+\frac{12{c_{1}}^{2}}{r^{6}}+\frac{48c_{1}({c_{2}}^{2}+{c_{3}}^{2})}{r^{7}}+\frac{56({c_{2}}^{2}+{c_{3}}^{2})^{2}}{r^{8}}\,,$
		$\displaystyle R_{ab}R^{ab}=R=36{c_{0}}^{2}+\frac{4({c_{2}}^{2}+{c_{3}}^{2})^{2}}{r^{8}}\,,\quad R=-12c_{0}\,.$		(25)

The above invariants show there are the contribution of the electric charge as well as the magnetic field. Now we are going to study the case $h\neq h_{1}$.

In this case, the system of differential equations can be solved analytically and the solution has the following form,

$\displaystyle h=256c_{6}h_{1}(r)\left(1-\frac{a}{2r^{3}}\right)^{8}\,.$

(26)

where $h_{1}$ is given by

	$\displaystyle h_{1}=$	$\displaystyle\,\frac{r^{14}\left(-972\ln\left(r\right)r^{12}+324\ln\left(a+r^{3}\right)r^{12}-260\,ar^{9}+a^{4}+66a^{2}r^{6}-12a^{3}r^{3}\right)c_{7}}{\left(a-2r^{3}\right)^{8}}+{\frac{r^{26}c_{8}}{\left(a-2r^{3}\right)^{8}}}$
		$\displaystyle\,-\frac{1390932}{455a^{\frac{4}{3}}{c_{6}}\left(a-2r^{3}\right)^{8}}\left\{\sqrt{3}\left({c_{2}}^{2}c_{6}-{\frac{507a^{\frac{2}{3}}c_{6}}{2332}}+207025{c_{3}}^{2}\right)r^{26}\arctan\left({\frac{\sqrt[3]{a}-2r}{\sqrt{3}\sqrt[3]{a}}}\right)\right.$
		$\displaystyle\,-\frac{r^{26}}{2}f_{1}\ln\left(r^{2}-r\sqrt[3]{a}+a^{\frac{2}{3}}\right)+f_{1}r^{26}\ln\left(r+\sqrt[3]{a}\right)$
		$\displaystyle\,+\frac{2035345r^{16}a^{10/3}}{122402016}\left(\frac{56207184}{10176725}{c_{2}}^{2}c_{6}+r^{2}c_{6}+\frac{35803976208}{31313}{c_{3}}^{2}\right)-\frac{1843153}{168302772}r^{13}\left({\frac{57119260}{23960989}}{c_{2}}^{2}c_{6}\right.$
		$\displaystyle\,\left.+\frac{69971093500}{141781}{c_{3}}^{2}+r^{2}c_{6}\right)a^{13/3}+{\frac{1196689r^{10}a^{16/3}}{214203528}}\left(\frac{189343}{184106}{c_{2}}^{2}c_{6}+\frac{3015287275}{14162}{c_{3}}^{2}+r^{2}c_{6}\right)$
		$\displaystyle\,-\frac{41171}{23645844}\left({\frac{407388}{782249}}{c_{2}}^{2}c_{6}+r^{2}c_{6}+{\frac{6487653900}{60173}}{c_{3}}^{2}\right)r^{7}a^{\frac{19}{3}}$
		$\displaystyle\,+\frac{20189}{61201008}\left(\frac{86504600}{1553}{c_{3}}^{2}+\frac{5432}{20189}{c_{2}}^{2}c_{6}+r^{2}c_{6}\right)r^{4}a^{\frac{22}{3}}-\frac{12805}{351905796}\left(\frac{7084}{64025}{c_{2}}^{2}c_{6}+r^{2}c_{6}\right.$
		$\displaystyle\,\left.+\frac{4512508}{197}{c_{3}}^{2}\right)ra^{\frac{25}{3}}+\frac{7416721}{30600504}\left(\frac{324676001650}{570517}{c_{3}}^{2}+r^{2}c_{6}+\frac{20387818}{7416721}{c_{2}}^{2}c_{6}\right)r^{22}a^{\frac{4}{3}}+\frac{5}{2781864}a^{\frac{28}{3}}c_{6}$
		$\displaystyle\,\left.-3r^{25}\sqrt[3]{a}\left(207025{c_{3}}^{2}+{c_{2}}^{2}c_{6}\right)-\frac{3572881}{76501260}\left(\frac{139814180}{25010167}{c_{2}}^{2}c_{6}+\frac{318077259500}{274837}{c_{3}}^{2}+r^{2}c_{6}\right)r^{19}a^{7/3}\right\}\,,$		(27)

and , $q$, $p$, $k$, and $n$ are given by

$\displaystyle q(r)=$

$\displaystyle\,\frac{c_{2}}{r}\,,\quad p(r)=c_{4}r\,,\quad k(\theta)=c_{3}\cos\theta\,,\quad n(\phi)=c_{5}\phi\,.$

(28)

Here $c_{2}$, $c_{3}$, $c_{4}$, $c_{5}$, $c_{6}$, and $c_{7}$ are the constants of the integration, again, and $f_{1}$ is a constant that is defined as $f_{1}={c_{2}}^{2}c_{6}+\frac{507a^{\frac{2}{3}}c_{6}}{2332}+207025{c_{3}}^{2}$. In spite that Eq. (26) seems to tell that we could obtain $h\propto h_{1}$ when $a=0$, this is not true because the third term of $h_{1}$ in (III) includes the inverse power of the dimensional parameter $a$ and therefore $a$ cannot vanish.

By substituting Eqs. (26), (III), and (28) into the trace equation (III), we obtain a very lengthy expression $f(r)$ whose asymptotic form up to $\mathcal{O}\left(\frac{1}{r^{6}}\right)$ is given by

	$\displaystyle F(r)=$	$\displaystyle\,\frac{1043199\pi\sqrt{3}}{14560a^{1/3}r^{3}}\left({c_{2}}^{2}+52998400{c_{3}}^{2}-\frac{507}{2332}a^{\frac{2}{3}}\right)\pi$
		$\displaystyle\,-\frac{1}{5r^{5}}\left\{\frac{9388791\sqrt{3}}{14560a^{1/3}L}\left({c_{2}}^{2}+52998400{c_{3}}^{2}-\frac{507}{2332}a^{\frac{2}{3}}\right)\pi\right.$
		$\displaystyle\,\left.+{\frac{-15300252\pi\sqrt{3}\left({c_{2}}^{2}+52998400{c_{3}}^{2}\right)a^{\frac{2}{3}}-2038400a^{2}L+3326427a^{\frac{4}{3}}\sqrt{3}\pi}{29120aL}}\right\}+c_{9}\,,$		(29)

where $L$ is a constant defined as $L=\frac{10010c_{8}a^{\frac{4}{3}}-810888875596800\sqrt{3}\pi{c_{3}}^{2}-15300252\sqrt{3}\pi{c_{2}}^{2}+3326427\sqrt{3}\pi a^{\frac{2}{3}}}{2562560a^{\frac{4}{3}}}$. Using Eqs. (26), (III), and (28), we obtain a lengthy form of the Ricci scalar in (10) whose asymptotic form up to $\mathcal{O}\left(\frac{1}{r^{6}}\right)$ is given by

	$\displaystyle R\approx$	$\displaystyle\,-{\frac{1}{640640L\sqrt[3]{a}r^{5}}}\left\{7687680\,L^{2}\sqrt[3]{a}r^{5}-2432666626790400\,\sqrt{3}\pi\,Lr^{2}{c_{3}}^{2}-45900756\sqrt{3}\pi\,Lr^{2}{c_{2}}^{2}\right.$
		$\displaystyle\,\left.+9979281\sqrt{3}\pi Lr^{2}a^{\frac{2}{3}}-8968960a^{\frac{4}{3}}C+810888875596800\sqrt{3}\pi{c_{3}}^{2}+15300252\sqrt{3}\pi{c_{2}}^{2}-3326427\sqrt{3}\pi a^{\frac{2}{3}}\right\}\,.$		(30)

From Eq. (III), we obtain $r(R)$ as

$\displaystyle r\approx\frac{27\sqrt[3]{33647995180800}}{40040R+480480L}\sqrt[3]{\sqrt{3}\left(\frac{1}{12}R+L\right)^{2}\pi\left({c_{2}}^{2}+52998400{c_{3}}^{2}-\frac{507}{2332}a^{\frac{2}{3}}\right)\frac{1}{\sqrt[3]{a}}}\,.$

(31)

Using Eq. (31) in Eq. (III), we obtain the following expression of $F(R)$

$\displaystyle F(R)=C_{0}+C_{1}R+C_{2}R^{2}+C_{3}R^{3}\cdots\,,$

(32)

where $C_{i}$, $i=0,1,\cdots$ are constants defined as

	$\displaystyle C_{0}=$	$\displaystyle\,{\frac{{2332}^{\frac{2}{3}}\sqrt[3]{33647995180800}}{2803999598400}}$
		$\displaystyle\,\times\left\{{14428814400}^{\frac{2}{3}}\left({\frac{\pi\sqrt{3}\left(507a^{\frac{2}{3}}-2332{c_{2}}^{2}-123592268800{c_{3}}^{2}\right)L^{2}}{\sqrt[3]{a}}}\right)^{\frac{5}{3}}\left(\frac{c_{9}}{12}+L\right)\sqrt[3]{a}\right.$
		$\displaystyle\,\left.-{\frac{4335057126400}{81}}L^{4}a^{\frac{2}{3}}\sqrt{3}\pi+{\frac{230064816357376000}{1594323}}L^{4}\left({\frac{347733}{203840}}\pi\sqrt{3}\left({c_{2}}^{2}+52998400{c_{3}}^{2}\right)+La^{\frac{4}{3}}\right)\right\}$
		$\displaystyle\,\times\left(2332\sqrt{3}\left({\frac{507}{2332}}a^{\frac{2}{3}}-{c_{2}}^{2}-52998400{c_{3}}^{2}\right){\frac{L^{2}\pi}{\sqrt[3]{a}}}\right)^{-\frac{5}{3}}{\frac{1}{\sqrt[3]{a}}}\,,$
	$\displaystyle C_{1}=$	$\displaystyle\,-\frac{16307200}{2281476213}{2332}^{\frac{2}{3}}\sqrt[3]{33647995180800}\left\{\frac{255879}{689920}L^{3}a^{\frac{2}{3}}\sqrt{3}\pi-\frac{4782969}{1150324081786880000}\sqrt[3]{(14428814400)^{2}}\right.$
		$\displaystyle\,\times\left(\frac{\sqrt{3}\pi\left(-123592268800{c_{3}}^{2}-2332{c_{2}}^{2}+507a^{\frac{2}{3}}\right)L^{2}}{\sqrt[3]{a}}\right)^{\frac{5}{3}}\sqrt[3]{a}$
		$\displaystyle\,\left.+\left(-\frac{347733}{203840}\pi\left(52998400{c_{3}}^{2}+{c_{2}}^{2}\right)\sqrt{3}+La^{\frac{4}{3}}\right)L^{3}\right\}$
		$\displaystyle\times\left(-2332\sqrt{3}\left(52998400{c_{3}}^{2}+{c_{2}}^{2}-\frac{507}{2332}a^{\frac{2}{3}}\right){\frac{\pi L^{2}}{\sqrt[3]{a}}}\right)^{-\frac{5}{3}}{\frac{1}{\sqrt[3]{a}}}\,,$
	$\displaystyle C_{2}=$	$\displaystyle\,-{\frac{4076800}{20533285917}}\left({\frac{255879}{689920}}\,a^{\frac{2}{3}}\sqrt{3}\pi-{\frac{347733}{203840}}\pi\left(52998400{c_{3}}^{2}+{c_{2}}^{2}\right)\sqrt{3}+La^{\frac{4}{3}}\right){2332}^{\frac{2}{3}}\sqrt[3]{33647995180800}L^{2}$
		$\displaystyle\,\times\left(-2332\sqrt{3}\left(52998400{c_{3}}^{2}+{c_{2}}^{2}-{\frac{507}{2332}}a^{\frac{2}{3}}\right){\frac{\pi C^{2}}{\sqrt[3]{a}}}\right)^{-\frac{5}{3}}{\frac{1}{\sqrt[3]{a}}}\,,$
	$\displaystyle C_{3}=$	$\displaystyle\,{\frac{1019200}{554398719759}}\left({\frac{255879}{689920}}a^{\frac{2}{3}}\sqrt{3}\pi-{\frac{347733}{203840}}\pi\left(52998400{c_{3}}^{2}+{c_{2}}^{2}\right)\sqrt{3}+La^{\frac{4}{3}}\right){2332}^{\frac{2}{3}}\sqrt[3]{33647995180800}L$
		$\displaystyle\,\times\left(-2332\sqrt{3}\left(52998400{c_{3}}^{2}+{c_{2}}^{2}-{\frac{507}{2332}}c^{\frac{2}{3}}\right){\frac{\pi C^{2}}{\sqrt[3]{a}}}\right)^{-\frac{5}{3}}{\frac{1}{\sqrt[3]{a}}}\,.$		(33)

Eqs. (26), (III), (28), (III), and (III) show that the dimensional parameter $a$ cannot vanish. Thus we have a new charged black hole solution that does not coincide with any charged black hole solution of general relativity. When the constants $c_{2}$ and $c_{3}$ vanish, however, we can put the dimensional parameter $a$ to vanish, and in this case, we obtain the Schwarzschild black hole of general relativity, i.e.,

$\displaystyle h(r)=h_{1}(r)=r^{2}c_{0}+1+\frac{c_{1}}{r}\quad\mbox{and}\quad F(r)=1\,.$

(34)

We should note that if the form of $F(R)$ is given as in (32) with (III), it is difficult to find the general solution. It is clear, however, that there should really exist a model realizing the black hole in (26), (III), and (28).

In the next section, we investigate the physical properties of the black hole given by (26), (III), and (28).