JT gravity and near-extremal thermodynamics for Kerr black holes in $Ad{S}_{4,5}$ for rotating perturbations

Abstract

We study the near horizon 2d gravity theory which captures the near extremal thermodynamics of Kerr black holes where a linear combination of excess angular momentum $\delta J$ and excess mass $\delta M$ is held fixed. These correspond to processes where both the mass and the angular momenta of extremal Kerr black holes are perturbed leaving them near extremal. For the Kerr $Ad{S}_4$ we hold $\delta J-\mathcal{L}\delta M=0$ while for Myers-Perry(MP) type Kerr black hole in $Ad{S}_5$ we hold $\delta J_{{\phi }_{1,2}}-{\mathcal{L}}_{{\phi }_{1,2}}\delta M=0$. We show that in near horizon, the 2d Jackiw-Teitelboim theory is able to capture the thermodynamics of the higher dimensional black holes at small near extremal temperatures $T_H$. We show this by generalizing the near horizon limits found in literature by parameters $\mathcal{L}$ and ${\mathcal{L}}_{{\phi }_{1,2}}$ for the two geometries. The resulting JT theory captures the near extremal thermodynamics of such geometries provided we identify the temperature $T_H^{(2)}$ of the near horizon $Ad{S}_2$ geometry to be $T_H^{(2)}=T_H/(1-\mu \mathcal{L})$ for 4d Kerr and $T_H^{(2)}=T_H/(1-\mu ({\mathcal{L}}_{{\phi }_1}+{\mathcal{L}}_{{\phi }_2}))$ for 5d Kerr where $\mu$ is their chemical potential, with $\mu \mathcal{L}<1$ and $\mu ({\mathcal{L}}_{{\phi }_1}+{\mathcal{L}}_{{\phi }_2})<1$ respectively. We also argue that such a theory embeds itself non-trivially in the higher dimensional theory dual to the Kerr geometries.

arXiv comment27-pages, Appendix-A added

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