Kerr-NUT black hole thermodynamics in f(T) gravity theories

Abstract

An axially symmetric tetrad field is applied to the field equation of f(T) gravity theory, where T is the scalar torsion tensor. A solution, with three constants of integration, for the linear case, $f(T)=T$, is derived. Then, we substitute this solution in the tetrad field and change the angle $\phi$ to $\Phi$, such that $\Phi$ is a function of r , $\theta$ and $\phi$, i.e. $\Phi=N(r,\theta,\phi)$. We calculate the scalar torsion and its derivatives, then, a constraint on the scalar torsion is assumed to be constant. From this assumption the value of $N(r,\theta,\phi)$ is derived. We show that the value of $N(r,\theta,\phi)$ is a solution to the f(T) gravitational theories provided some restrictions on the form of f(T) and its first derivatives. The associated metric of the derived solution is shown to represent the Kerr-NUT spacetime. The energy of the derived solution is calculated using the Euclidean continuation method and we show that the NUT parameter may be related to the electromagnetic field which is a consistent value with what was obtained before from other definitions. Finally, we calculate the thermodynamical quantities of the Kerr-NUT space times and investigate the first law of thermodynamics and quantum statistical relation.