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Register a new userNon-trivial black hole solutions in $mathit{f(R)}$ gravitational theory
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Recent observation shows that general relativity (GR) is not valid in the strong regime. $mathit{f(R)}$ gravity where $mathit{R}$ is the Ricci scalar, is regarded to be one of good candidates able to cure the anomalies appeared in the conventional general relativity. In this realm, we apply the equation of motions of $mathit{f(R)}$ gravity to a spherically symmetric spacetime with two unknown functions and derive original black hole (BH) solutions without any constrains on the Ricci scalar as well as on the form of $mathit{f(R)}$ gravity. Those solutions depend on a convolution function and are deviating from the Schwarzschild solution of the Einstein GR. These solutions are characterized by the gravitational mass of the system and the convolution function that in the asymptotic form gives extra terms that are responsible to make such BHs different from GR. Also, we show that these extra terms make the singularities of the invariants much weaker than those of the GR BH. We analyze such BHs using the trend of thermodynamics and show their consistency with the well known quantities in thermodynamics like the Hawking radiation, entropy and quasi-local energy. We also show that our BH solutions satisfy the first law of thermodynamics. Moreover, we study the stability analysis using the odd-type mode and shows that all the derived BHs are stable and have radial speed equal to one. Finally, using the geodesic deviations we derive the stability conditions of these BHs.
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We show, in detail, that the only non-trivial black hole (BH) solutions for a neutral as well as a charged spherically symmetric space-times, using the class ${textit F(R)}={textit R}pm{textit F_1 (R)} $, must-have metric potentials in the form $h(r)=frac{1}{2}-frac{2M}{r}$ and $h(r)=frac{1}{2}-frac{2M}{r}+frac{q^2}{r^2}$. These BHs have a non-trivial form of Ricci scalar, i.e., $R=frac{1}{r^2}$ and the form of ${textit F_1 (R)}=mpfrac{sqrt{textit R}} {3M} $. We repeat the same procedure for (Anti-)de Sitter, (A)dS, space-time and got the metric potentials of neutral as well as charged in the form $h(r)=frac{1}{2}-frac{2M}{r}-frac{2Lambda r^2} {3} $ and $h(r)=frac{1}{2}-frac{2M}{r}+frac{q^2}{r^2}-frac{2Lambda r^2} {3} $, respectively. The Ricci scalar of the (A)dS space-times has the form ${textit R}=frac{1+8r^2Lambda}{r^2}$ and the form of ${textit F_1(R)}=mpfrac{textit 2sqrt{R-8Lambda}}{3M}$. We calculate the thermodynamical quantities, Hawking temperature, entropy, quasi-local energy, and Gibbs-free energy for all the derived BHs, that behaves asymptotically as flat and (A)dS, and show that they give acceptable physical thermodynamical quantities consistent with the literature. Finally, we prove the validity of the first law of thermodynamics for those BHs.
With the successes of $f(R)$ theory as a neutral modification of Einsteins 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 $n eq 2$. 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 consider whether the new horizon-first law works in higher-dimensional $f(R)$ theory. We firstly obtain the general formulas to calculate the entropy and the energy of a general spherically-symmetric black hole in $D$-dimensional $f(R)$ theory. For applications, we compute the entropies and the energies of some black hokes in some interesting higher-dimensional $f(R)$ theories.
We consider the new horizon first law in $f(R)$ theory with general spherically symmetric black hole. We derive the general formulas to computed the entropy and energy of the black hole. For applications, some nontrivial black hole solutions in some popular $f(R)$ theories are investigated, the entropies and the energies of black holes in these models are first calculated.
The detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory opens a new era to use gravitational waves to test alternative theories of gravity. We investigate the polarizations of gravitational waves in $f(R)$ gravity and Horndeski theory, both containing scalar modes. These theories predict that in addition to the familiar $+$ and $times$ polarizations, there are transverse breathing and longitudinal polarizations excited by the massive scalar mode and the new polarization is a single mixed state. It would be very difficult to detect the longitudinal polarization by interferometers, while pulsar timing array may be the better tool to detect the longitudinal polarization.
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