JT gravity and near-extremal thermodynamics for Kerr black holes in AdS4,5 for rotating perturbations

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 δJ and excess mass δM 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 AdS₄ we hold $$ \delta J-\mathcal{L}\delta M=0 $$δJ − L δM = 0 while for Myers-Perry(MP) type Kerr black hole in AdS₅ we hold $$ \delta {J}_{\varphi 1,2}-{\mathcal{L}}_{\varphi 1,2}\delta M=0 $$δ J φ 1 , 2 − L φ 1 , 2 δM = 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 $$ \mathcal{L} $$L and $$ {\mathcal{L}}_{\varphi 1,2} $$L φ 1 , 2 for the two geometries. The resulting JT theory captures the near extremal thermodynamics of such geometries provided we identify the temperature $$ {T}_H^{(2)} $$T H 2 of the near horizon AdS₂ geometry to be $$ {T}_H^{(2)}={T}_H/\left(1-\mu \mathcal{L}\right) $$T H 2 = T H / 1 − μ L for 4d Kerr and $$ {T}_h^2={T}_H/\left(-\mu \left({\mathcal{L}}_{\varphi 1}+{\mathcal{L}}_{\varphi 2}\right)\right) $$T h 2 = T H / − μ L φ 1 + L φ 2 for 5d Kerr μ is their chemical potential, with $$ \mu \mathcal{L}<1 $$μ L < 1 and $$ \mu \left({\mathcal{L}}_{\varphi 1}+{\mathcal{L}}_{\varphi 2}\right)<1 $$μ L φ 1 + L φ 2 < 1 respectively. We also argue that such a theory embeds itself non-trivially in the higher dimensional theorydual to the Kerr geometries.