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Register a new userThermodynamics of black hole in $D$-dimensional $f(R)$ theory
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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.
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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.
We investigate whether the new horizon first law still holds in $f(R,R^{mu u}R_{mu u})$ theory. For this complicated theory, we first determine the entropy of black hole via Wald method, then we derive the energy by using the new horizon first law, the degenerate Legendre transformation, and the gravitational field equations. For application, we consider the quadratic-curvature gravity and firstly calculate the entropy and the energy for a static spherically symmetric black hole, which reduces to the results obtained in literatures for a Schwarzschild-(A)dS black hole.
We investigate whether the new horizon first law proposed recently still work in $f(R)$ theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in $f(R)$ theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in $f(R)$ theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.
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.
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.
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