Narrow Results

We construct a static black hole solution of Gauss–Bonnet-massive gravity coupled to nonlinear Maxwell and Yang–Mills fields in higher dimensions. Then we calculate related thermodynamic quantities, check the validity of the first law of black hole thermodynamics and analyze the phase transition behaviors of the black hole in extended phase space.

We find an exact black hole solution with a minimally coupled scalar field. The corresponding spacetime has two horizons and one of them is the black hole event horizon and the other is the cosmic horizon. In this sense, the solution is analogous to the Schwarzschild-de Sitter (or anti-de Sitter) spacetime. We investigate the thermodynamics and construct the first law of thermodynamics. At the same time, we make a study on the shadow and quasinormal modes of this black hole solution.

In this paper, we find an exact black hole solution for the Einstein gravity in the presence of Ayón–Beato–García nonlinear electrodynamics and a cloud of strings. The resulting black hole solution is singular, and the solution becomes nonsingular when gravity is coupled with Ayón–Beato–García nonlinear electrodynamics only. This solution interpolates between Ayón–Beato–García black hole, Letelier black hole and Schwarzschild black hole in the absence of cloud of strings parameter, magnetic monopole charge and both of them, respectively. We also discuss the thermal properties of this black hole and find that the solution follows the modified first law of black hole thermodynamics. Furthermore, we estimate the solution’s black hole shadow and quasinormal modes.

We study the modified Reissner–Nordstrom (RN) metric in the unimodular gravity. So far the spherical symmetric Einstein field equation in unimodular gravity has been studied in the absence of any source. We consider static electric and magnetic charge as source. We solve for Maxwell equations in unimodular gravitational background. We show that in unimodular gravity, the electromagnetic field strength tensor is modified. We also show that the solution in unimodular gravity differs from the usual RN metric in Einstein gravity with some corrections. We further study the thermodynamical properties of the RN black hole solution in this theory.

In this paper, we present a detailed study of both nonrotating and rotating black hole solutions in $f(R)=R-2\alpha \sqrt{R}$ model. The complicated expression of Ricci scalar, which appears in the vacuum field equations, brings the main difficulty of searching analytic solutions. In the case of searching nonrotating black hole solution, we use the 00 component minus 11 component of field equations to constrain the Ricci scalar. In order to obtain analytic solutions of rotating black hole, two strategies are used. One is the simplification of the line element using Carter’s consideration that the Klein–Gordon equation is separable. The other is that we give the general solutions of the simultaneous equations of 00 minus 11 component equation and 01 component of the field equations. These strategies greatly simplify the process and difficulty of solving the axisymmetric solutions. The calculation process is more friendly to generalize to other modified gravities.

This work provides the exact solution of the Bardeen black hole in association with $4D$ Gauss–Bonnet massive gravity in Anti-de-Sitter $(AdS)$ space–time. It is a modification of the Gauss–Bonnet when gravity couples with nonlinear matter fields which is the function of the electromagnetic field. The obtained solution gives rise to $4D$ EGB Bardeen black holes when the massive gravity parameter is set to zero and it yields a $4D$ Gauss–Bonnet black hole in the absence of magnetic monopole charge. Further, we analyze and adopt the thermodynamic quantities like mass ($M_{+}$), temperature $(T_{+})$ and heat capacity $(C_{+})$ in the presence of massive gravity and nonlinear electrodynamics. In addition, we extend our results by considering the cosmological constant $(\mathrm{\Lambda })$ as a thermodynamical variable $(P=-\mathrm{\Lambda }/8\pi )$ and obtain the critical values of pressure, temperature, horizon radius and analyze the behavior of the global parameter $P_c{v}_c/T_c$. The effect of a massive parameter ($m$) of the critical exponent is opposite to the magnetic monopole charge ($e$) and Gauss–Bonnet parameter ($\alpha$). According to our analysis the phase transition between a small and large black hole and van der Waals phase transition are analogous to each other.