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 δJ−LδM=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta J-\mathcal{L}\delta M=0 $$\end{document} while for Myers-Perry(MP) type Kerr black hole in AdS5 we hold δJφ1,2−Lφ1,2δM=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta {J}_{\varphi 1,2}-{\mathcal{L}}_{\varphi 1,2}\delta M=0 $$\end{document}. 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 TH. We show this by generalizing the near horizon limits found in literature by parameters L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{L} $$\end{document} and Lφ1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{L}}_{\varphi 1,2} $$\end{document} for the two geometries. The resulting JT theory captures the near extremal thermodynamics of such geometries provided we identify the temperature TH2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_H^{(2)} $$\end{document} of the near horizon AdS2 geometry to be TH2=TH/1−μL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_H^{(2)}={T}_H/\left(1-\mu \mathcal{L}\right) $$\end{document} for 4d Kerr and Th2=TH/−μLφ1+Lφ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {T}_h^2={T}_H/\left(-\mu \left({\mathcal{L}}_{\varphi 1}+{\mathcal{L}}_{\varphi 2}\right)\right) $$\end{document} for 5d Kerr μ is their chemical potential, with μL<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu \mathcal{L}<1 $$\end{document} and μLφ1+Lφ2<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu \left({\mathcal{L}}_{\varphi 1}+{\mathcal{L}}_{\varphi 2}\right)<1 $$\end{document} respectively. We also argue that such a theory embeds itself non-trivially in the higher dimensional theory