Rohan R. Poojary

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 $\mu\,\mathcal{L}<1$.

We study the near horizon 2D gravity theory which captures the near extremal thermodynamics of Banados-Teitelboim-Zanelli (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 $T_H^{(2)}$ of the ${\mathit{AdS}}_2$ geometry in the Jackiw-Teitelboim theory to be $T_H^{(2)}=T_H/(1-\mu \mathcal{L})$, where $T_H$ is the BTZ temperature and $\mu$ its chemical potential with $\mu \mathcal{L}<1$.

We compute the disruption of mutual information in a TFD state dual to a Kerr black hole with equal angular momenta in $AdS_5$ due to an equatorial shockwave. The shockwave respects the axi-symmetry of the Kerr geometry with specific angular momenta $\mathcal{L}_{\phi_1}$ & $\mathcal{L}_{\phi_2}$. The sub-systems considered are hemispheres in the $left$ and the $right$ dual CFTs with the equator of the $S^3$ as their boundary. We compute the change in the mutual information by determining the growth of the HRT surface at late times. We find that at late times leading upto the scrambling time the minimum value of the instantaneous Lyapunov index $\lambda_L^{(min)}$ is bounded by $\kappa=\frac{2\pi T_H}{(1-\mu \,\mathcal{L}_+)}$ and is found to be greater than $2\pi T_H$ in certain regimes with $T_H$ and $\mu$ denoting the black hole's temperature and the horizon angular velocity respectively while $\mathcal{L}_+=\mathcal{L}_{\phi_1}+\mathcal{L}_{\phi_2}$. We also find that for non-extremal geometries the null perturbation obeys $\mathcal{L}_+<\mu^{-1}$ for it to reach the outer horizon from the $AdS$ boundary. The scrambling time at very late times is given by $\kappa\tau_*\approx\log \mathcal{S}$ where $\mathcal{S}$ is the Kerr entropy. We also find that the onset of scrambling is delayed due to a term proportional to $\log(1-\mu\,\mathcal{L}_+)^{-1}$ which is not extensive and does not scale with the entropy of Kerr black hole.

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 $AdS_4$ we hold $\delta J-\mathcal{L}\,\delta M=0 $ while for Myers-Perry(MP) type Kerr black hole in $AdS_5$ we hold $\delta J_{\varphi_{1,2}}\hspace{-0.2cm}-\mathcal{L}_{\varphi_{1,2}}\,\delta M=0$. 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 $\mathcal{L}_{\varphi_{1,2}}$ for the two geometries. The resulting JT theory captures the near extremal thermodynamics of such geometries provided we identify the temperature $T^{(2)}_H$ of the near horizon $AdS_2$ geometry to be $T^{(2)}_H=T_H/(1-\mu\,\mathcal{L})$ for 4d Kerr and $T^{(2)}_H=T_H/(1-\mu\,(\mathcal{L}_{\varphi_1}+\mathcal{L}_{\varphi_2}))$ for 5d Kerr where $\mu$ is their chemical potential, with $\mu\,\mathcal{L}<1$ and $\mu\,(\mathcal{L}_{\varphi_1}+\mathcal{L}_{\varphi_2})<1$ respectively. We also argue that such a theory embeds itself non-trivially in the higher dimensional theory dual to the Kerr geometries.

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.

We compute the disruption of mutual information between the hemispherical subsystems on the left and right conformal field theories of a thermofield double state described by a Kerr geometry in ${\mathrm{AdS}}_4$ due to shock waves along the equatorial plane. The shock waves and the subsystems considered respect the axisymmetry of the geometry. At late times, the disruption of the mutual information is given by the lengthening of the Hubeny-Rangamani-Takayanagi surface connecting the two subsystems; we compute the minimum value of the Lyapunov index-${\lambda }_L^{(\min )}$ at late times and find that it is bounded by $\kappa =\frac{2\pi T_H}{(1-\mu \mathcal{L})}$, where $\mu$ is the horizon velocity, and $\mathcal{L}$ is the angular momentum per unit energy of the shock wave. At very late times, we find the scrambling time for such a system is governed by $\kappa$ with $\kappa t_{*}=\log \mathcal{S}$ for large black holes with a large entropy $\mathcal{S}$. We also find a term that increases the scrambling time by $\log (1-\mu \mathcal{L}{)}^{-1}$ but which does not scale with the entropy of the Kerr geometry.

Recently introduced reparametrization mode operators in CFTs have been shown to govern stress tensor interactions $via$ the shadow operator formalism and seem to govern the effective dynamics of chaotic systems. We initiate a study of Ward identities of reparametrization mode operators $i.e.$ how two dimensional CFT Ward identities govern the behaviour of insertions of reparametrization modes $\epsilon$ in correlation functions: $\langle\epsilon\epsilon\phi\phi\rangle$. We find that in the semi-classical limit of large $c$ they dictate the leading $\mathcal{O}(c^{-1})$ behaviour. While for the $4$pt function this reproduces the same computation as done by Heahl, Reeves \& Rozali in \cite{Haehl:2019eae}, in the case of 6pt function of pair-wise equal operators this provides an alternative way of computing the Virasoro block in stress-tensor comb channel. We compute a maximally out of time ordered correlation function in a thermal background and find the expected behaviour of an exponential growth governed by Lyapunov index $\lambda_L=2\pi/\beta$ lasting for twice the scrambling time of the system $t^*=\frac{\beta}{2\pi}\log\,c$ for the maximally braided type of $out$-$of$-$time$-$ordering$. However when only the internal operators of the comb channel are \emph{out-of-time-ordered}, the correlator sees no exponential behaviour despite the inclusion of the Virasoro contribution. From a bulk perspective for the \emph{out-of-time-ordered} $4$pt function we find that the Casimir equation for the stress tensor block reproduces the linearised back reaction in the bulk.

We compute the disruption of mutual information between the hemispherical subsystems on the left and right CFT$s$ of a Thermofield Double state described by a Kerr geometry in $AdS_4$ due to shockwaves along the equatorial plane. The shockwaves and the subsystems considered respect the axi-symmetry of the geometry. At late times the disruption of the mutual information is given by the lengthening of the HRT surface connecting the two subsystems, we compute the minimum value of the Lyapunov index-$\lambda_L^{(min)}$ at late times and find that it is bounded by $\kappa=\frac{2\pi T_H}{(1-\mu\, \mathcal{L})}$ where $\mu$ is the horizon velocity and $\mathcal{L}$ is the angular momentum per unit energy of the shockwave. At very late times we find the the scrambling time for such a system is governed by $\kappa$ with $\kappa t_*=\log \mathcal{S}$ for large black holes with large entropy $\mathcal{S}$. We also find a term that increases the scrambling time by $\log(1-\mu\,\mathcal{L})^{-1}$ but which does not scale with the entropy of the Kerr geometry.