Jackiw-Teitelboim gravity and near-extremal BTZ thermodynamics

We study the near horizon 2d gravity theory which captures the near extremal thermodynamics of BTZ with excess mass $\delta M$ and excess angular momentum $\delta J =\mathcal{L}\, \delta M$ over extremality. We find that the Jackiw-Teitelboim theory is able to capture such departures from extremality with an additional parameter $\mathcal{L}$ relating it to the near extremal BTZ configuration. We are able to show this by obtaining new simultaneous near horizon near extremal limits of the BTZ geometry parametrized by $\mathcal{L}$. The resulting Jackiw-Teitelboim theory captures the near extremal thermodynamics of such BTZ geometries provided we identify the temperature $T^{(2)}_H$ of the $AdS_2$ geometry in the Jackiw-Teitelboim theory to be $T^{(2)}_H=T_H/(1-\mu\,\mathcal{L})$ where $T_H$ is the BTZ temperature and $\mu$ its chemical potential with $\mu\,\mathcal{L}<1$.