Specific neutral and charged black holes in $f(R)$ gravitational theory. (arXiv:2103.02382v3 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Nashed_G/0/1/0/all/0/1">G.G.L. Nashed</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Nojiri_S/0/1/0/all/0/1">Shin'ichi Nojiri</a>
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
$ds^2=-U(r)dt^2+frac{dr^2}{V(r)}+r^2 left( dtheta^2+sin^2theta dphi^2
right)$ 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 $crightarrow 0$ which
means that the only case that can generate GR BHs is $n=2$. In this paper, we
assume another form, i.e., $F_1(r)= 1-frac{F_0-left(n-3right)}{r^n}$ 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 $F_0to
n-3$. We show that the form of the electric charge depends on $n$ and that
$nneq 2$. 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.
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
$ds^2=-U(r)dt^2+frac{dr^2}{V(r)}+r^2 left( dtheta^2+sin^2theta dphi^2
right)$ 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 $crightarrow 0$ which
means that the only case that can generate GR BHs is $n=2$. In this paper, we
assume another form, i.e., $F_1(r)= 1-frac{F_0-left(n-3right)}{r^n}$ 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 $F_0to
n-3$. We show that the form of the electric charge depends on $n$ and that
$nneq 2$. 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.
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