Kerr-dS/AdS null tetrads in modified teleparallel gravity theories and thermodynamics

Abstract

We apply a diagonal axially symmetric tetrad to the field equations of the lowest order of $f(T)$, “where $T$ is the teleparallel torsion scalar”, and get an exact vacuum solution which describes Kerr-dS/AdS spacetime. We rewrite the tetrad field of the derived solution in a null tetrad form and multiply it by a boost local Lorentz transformation, which contains unknown function of generalized coordinates, and get a new tetrad field. An exact vacuum solutions, which represents Kerr-dS/AdS spacetime, are derived under the following constraints: (i) The scalar torsion, $T ={T^{\alpha}}_{\mu \nu}{S_{\alpha}}^{\mu \nu}=T_{0}$, where $T_{0}$ is constant. (ii) The total derivative term is constant. (iii) The combination of the scalar torsion and the total derivative term is constant. We show that all the derived solutions coincide with general relativity solution when the constants of the torsion scalar, the total derivative term and the combination of the torsion scalar and total derivative term are vanishing. When the rotation parameter vanishing, the solution in the lower order of $f(T)$ approaches asymptotically de Sitter spacetime, and show the existence of the Nariai spacetime as a background solution. The scalar torsion of Nariai spacetime is vanishing naturally without any constraint. Considering the perturbation around the Nariai spacetime up to the first order, we show the behavior of black hole horizon. We investigate that the anti-evaporation does not exist on the classical level. Finally, we used the total derivative term and the Euclidean continuation method to calculate the energy of Kerr-dS/AdS, entropy and temperature. The mass of a Kerr-dS/AdS black hole is interpreted, in terms of thermodynamic potentials, as being the enthalpy, with the pressure given by the cosmological constant.

Appendix

Using Eq. (3), we get the non-vanishing components of the torsion ${T^{\mu}}_{\nu \rho}$ in the form

$$\begin{aligned} & {T^{0}}_{01}=\frac{2S[r^{2}+a^{2}]\xi_{t}+S_{5}S_{r}-S_{5r}[r^{2}+a^{2}]}{2SS_{5}}, \\ & {T^{0}}_{02}=\frac{SS_{6}S_{\theta}+a^{2}SS_{6}\sin2\theta+a^{2}SS_{6\theta}\sin^{2}\theta}{2S^{2}S_{6}}, \\ & {T^{0}}_{21}=\frac{(r^{2}+a^{2})S \xi_{\theta}}{SS_{5}}, \\ & {T^{0}}_{32}=\frac{\varOmega a (r^{2}+a^{2})\sin\theta[2S_{6}\cos\theta-\sin\theta S_{6_{\theta}}]}{2S S_{6}}, \\ & {T^{0}}_{13}=\frac{\varOmega a\sin^{2}\theta [4rS_{5}S_{6}+(r^{2}+a^{2})(S_{5}S_{6r}-S_{6}S_{5r})]}{2SS_{5}S_{6}}, \\ & {T^{1}}_{21}=\frac{S_{\theta}}{2S},\qquad {T^{1}}_{31}=\frac{\varOmega a\sin^{2}\theta S_{5} \xi_{r} }{S}, \\ & {T^{1}}_{10}=\frac{S_{5} \xi_{r}}{S},\qquad {T^{1}}_{32}=\frac{\varOmega a\sin^{2}\theta S_{5} \xi_{\theta}}{S}, \\ & {T^{1}}_{20}=\frac{S_{5} \xi_{\theta}}{S},\qquad {T^{1}}_{30}=\frac{\varOmega a\sin^{2}\theta S_{5} \xi_{t} }{S}, \\ & {T^{2}}_{12}=\frac{S_{r}}{2S }, \qquad {T^{3}}_{21}=\frac{a S \xi_{\theta}}{\varOmega S S_{5}}, \\ & {T^{3}}_{13}=\frac{4rS_{5}-a^{2}\sin^{2}\theta S_{5r}-S_{5}S_{r}}{2S S_{5}}, \\ & {T^{3}}_{01}=\frac{a[2S\xi_{t}-S_{5r}]}{2\varOmega SS_{5}}, \\ &{T^{3}}_{02}= \frac{a[2S_{6}\cos\theta+\sin\theta S_{6\theta}]}{2\varOmega SS_{6} \sin\theta}, \\ & {T^{3}}_{23}=\bigl\{ 2a^{2}SS_{6}\cos\theta \cos2\theta -\sin\theta SS_{6}S_{\theta}+2r^{2}SS_{6}\cos\theta \\ &\phantom{{T^{3}}_{23}=\,}+\sin\theta\bigl(r^{2}+a^{2}\bigr)SS_{6\theta}\bigr\} \bigl\{ 2S^{2} S_{6}\sin\theta\bigr\} ^{-1}, \end{aligned}$$

(67)

and the non-vanishing components of the superpotential tensor ${S_{\mu}}^{\nu \rho}$

$$\begin{aligned} & {S_{0}}^{01}=\bigl\{ a^{2}\bigl(S S_{5r}+2r\bigl[r^{2}+a^{2}\bigr]S_{6}\bigr)\sin^{2}\theta \\ &\phantom{{S_{0}}^{01}=\,}-2S_{5}\bigl[2r^{2}+a^{2}\bigl(1+\cos^{2}\theta\bigr)\bigr]\bigr\} \bigl\{ 4S^{3}\bigr\} ^{-1}, \\ & {S_{0}}^{02}=-\bigl\{ 2\cos\theta\bigl\{ S_{6}\bigl(2a^{4}\cos^{4}\theta+2a^{2}\cos^{2}\theta\bigl[ r^{2}-a^{2}\bigr] \\ &\phantom{{S_{0}}^{02}=\,}+r^{4}+a^{4}\bigr) -a^{2}S_{5}\sin^{2}\theta\bigr\} +S\bigl(r^{2}+a^{2}\bigr)S_{6\theta}\sin\theta \bigr\} \\ &\phantom{{S_{0}}^{02}=\,}\times\bigl\{ 4S^{3}\sin\theta\bigr\} ^{-1}, \\ & {S_{0}}^{03}=-\frac{a \xi_{r}}{2\varOmega S}, \\ & {S_{0}}^{13}=\frac{aS_{6}(2S^{2}\xi_{t}-[SS_{5r}+2r\{a^{2}S_{6}\sin^{2}\theta-S_{5}\}])}{4S^{3} S \varOmega}, \\ & {S_{0}}^{23}=\frac{ a [S S_{6\theta}\sin\theta +2\cos\theta(S_{6}(r^{2}+a^{2})-S_{5})]}{4S^{3}\varOmega \sin\theta}, \\ & {S_{1}}^{30}=\frac{a r S}{2\varOmega S^{2} S_{5}},\qquad {S_{2}}^{03}= \frac{a \cos\theta}{2\varOmega S S_{6} \sin\theta}, \\ & {S_{1}}^{21}=\bigl\{ S^{2} S_{6\theta}\sin\theta+2S_{6}\bigl[S\bigl(r^{2}+a^{2}\bigl\{ 3\cos^{2}\theta-2\bigr\} \bigr) \\ &\phantom{{S_{1}}^{21}=\,} -S S_{\theta}\sin\theta\bigr]\bigr\} \bigl\{ 4S^{3}\sin\theta\bigr\} ^{-1}, \\ & {S_{2}}^{13}={S_{2}}^{32}=\frac{a \xi_{t}}{2\varOmega S },\\ & {S_{2}}^{21}=\frac{2S^{2}S_{6}(r^{2}+a^{2})\xi_{t}-SS_{6}[ SS_{5r}+4rS_{5}-2S_{5}S_{r} ]}{4S^{3}S_{6}}, \\ & {S_{2}}^{10}={S_{2}}^{02}={S_{3}}^{03}=\frac{(r^{2}+a^{2}) \xi_{t}}{2 S }, \\ & {S_{3}}^{31}=\bigl\{ 2S S_{6}\xi_{t}\bigl(r^{2}+a^{2}\bigr)S-S_{6}\bigl\{ S\bigl(r^{2}+a^{2}\bigr)S_{5r} \\ &\phantom{{S_{3}}^{31}=\,} +2r a^{2}\sin^{2}\theta\bigl[S_{6}\bigl(r^{2}+a^{2}\bigr)-S_{8}\bigr]\bigr\} \bigr\} \bigl\{ 4S^{3}S_{6}\bigr\} ^{-1}, \\ & {S_{3}}^{10}=\bigl\{ a\varOmega \sin^{2}\theta S_{6}\bigl\{ SS_{5r}\bigl(r^{2}+a^{2}\bigr)+2rS_{6}\bigl(r^{2}+a^{2}\bigr)^{2} \\ &\phantom{{S_{3}}^{10}=\,} -2rS_{5}\bigl(2r^{2}+a^{2}\bigl[1+\cos^{2}\theta\bigr]\bigr)\bigr\} \bigr\} \bigl\{ 4S^{3}S_{6}\bigr\} ^{-1}, \\ & {S_{3}}^{32}=\bigl\{ a^{2}\bigl\{ SS_{6\theta}\sin^{2}\theta+\sin2\theta\bigl[\bigl(2r^{2}+a^{2}\bigl[1+\cos^{2}\theta\bigr]\bigr) S_{6} \\ &\phantom{{S_{3}}^{32}=\,} -S_{5}\bigr]\bigr\} \bigr\} \bigl\{ 4S^{3}\bigr\} ^{-1}, \\ & {S_{3}}^{02}=\bigl\{ a \varOmega \sin^{2}\theta \bigl\{ SS_{6\theta}\bigl(r^{2}+a^{2}\bigr)+a^{2}\sin2\theta \bigl[\bigl(r^{2}+a^{2}\bigr)S_{6} \\ &\phantom{{S_{3}}^{02}=\,}-S_{5}\bigr]\bigr\} \bigr\} \bigl\{ 4S^{3}\bigr\} ^{-1}. \end{aligned}$$

(68)