arXiv:2112.14089 · Physical review. D/Physical review. D.

Fast scrambling due to rotating shockwaves in BTZ

Vinay Malvimat, Rohan R. Poojary

Published Jun 21, 2022• 14 citations

Abstract

We study the perturbation due to rotating shockwaves in Banados-Teitelboim-Zanelli geometries at late times and analyze the change in mutual information between the two subsystems belonging to the dual ${\mathrm{CFT}}_{L}$ and ${\mathrm{CFT}}_{R}$. We find that the scrambling of mutual information is in general governed by the Lyapunov index ${\ensuremath{\lambda}}_{L}$ which is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}}{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}\ensuremath{\ge}\frac{2\ensuremath{\pi}}{\ensuremath{\beta}}$, where $\ensuremath{\mu}={r}_{\ensuremath{-}}/{r}_{+}$ and $\mathcal{L}$ is the angular momentum of the shockwave. For the special case of $\mathcal{L}=1$ we find the mutual information analytically and show that it is characterized by ${\ensuremath{\lambda}}_{L}=\ensuremath{\kappa}/2$ with the scrambling time for large black holes given as ${t}_{*}=\frac{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu})}{\ensuremath{\pi}}\mathrm{log}S$.

DOI: 10.1103/physrevd.105.126019

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arXiv:2112.14089 · Physical review. D/Physical review. D.

Fast scrambling due to rotating shockwaves in BTZ

Vinay Malvimat, Rohan R. Poojary

Published Jun 21, 2022• 14 citations

arXiv HTML PDF

Abstract

{\mathrm{CFT}}_{L}

and

{\mathrm{CFT}}_{R}

. We find that the scrambling of mutual information is in general governed by the Lyapunov index

{\ensuremath{\lambda}}_{L}

which is bounded by

\ensuremath{\kappa}=\frac{2\ensuremath{\pi}}{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}\ensuremath{\ge}\frac{2\ensuremath{\pi}}{\ensuremath{\beta}}

, where

\ensuremath{\mu}={r}_{\ensuremath{-}}/{r}_{+}

and

\mathcal{L}

is the angular momentum of the shockwave. For the special case of

\mathcal{L}=1

we find the mutual information analytically and show that it is characterized by

{\ensuremath{\lambda}}_{L}=\ensuremath{\kappa}/2

with the scrambling time for large black holes given as

{t}_{*}=\frac{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu})}{\ensuremath{\pi}}\mathrm{log}S

DOI: 10.1103/physrevd.105.126019

Related papers

Fast scrambling of mutual information in Kerr- AdS4 spacetime

Vinay Malvimat, Rohan R. Poojary

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-${\ensuremath{\lambda}}_{L}^{(\mathrm{min})}$ at late times and find that it is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}{T}_{H}}{(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}$, where $\ensuremath{\mu}$ 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 $\ensuremath{\kappa}$ with $\ensuremath{\kappa}{t}_{*}=\mathrm{log}\mathcal{S}$ for large black holes with a large entropy $\mathcal{S}$ . We also find a term that increases the scrambling time by $\mathrm{log}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L}{)}^{\ensuremath{-}1}$ but which does not scale with the entropy of the Kerr geometry.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-${\ensuremath{\lambda}}_{L}^{(\mathrm{min})}$ at late times and find that it is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}{T}_{H}}{(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}$, where $\ensuremath{\mu}$ 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 $\ensuremath{\kappa}$ with $\ensuremath{\kappa}{t}_{*}=\mathrm{log}\mathcal{S}$ for large black holes with a large entropy $\mathcal{S}$. We also find a term that increases the scrambling time by $\mathrm{log}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L}{)}^{\ensuremath{-}1}$ but which does not scale with the entropy of the Kerr geometry.

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Kento Watanabe

The AdS/CFT correspondence and quantum entanglement (or quantum information). This is one of topics exciting many physicists in this decade. And black holes always play a role of inspiring lab in many cases. In this thesis, we will discuss about scrambling of quantum entanglement and complexity of states in AdS/CFT. In both topics, black holes are expected to be the fastest processors of quantum information. This thesis contains brief reviews of each topics and our own works. One work is an explicit calculation of fast scrambling time in two dimensional holographic CFTs and three dimensional BTZ black holes using the two-sided mutual information. This calculation is a first CFT counterpart of Shenker-Stanford’s work in the gravity dual and an extension of their shock-wave approximation to finite time region in the gravity side. Another work is about socalled optimization procedure of Euclidean path- integrals in CFTs. The optimization procedure gives us a new way to extract dual geometries only from CFT data. And it also gives a new proposal for the definition of complexity of states in CFTs.The AdS/CFT correspondence and quantum entanglement (or quantum information). This is one of topics exciting many physicists in this decade. And black holes always play a role of inspiring lab in many cases. In this thesis, we will discuss about scrambling of quantum entanglement and complexity of states in AdS/CFT. In both topics, black holes are expected to be the fastest processors of quantum information. This thesis contains brief reviews of each topics and our own works. One work is an explicit calculation of fast scrambling time in two dimensional holographic CFTs and three dimensional BTZ black holes using the two-sided mutual information. This calculation is a first CFT counterpart of Shenker-Stanford’s work in the gravity dual and an extension of their shock-wave approximation to finite time region in the gravity side. Another work is about socalled optimization procedure of Euclidean path- integrals in CFTs. The optimization procedure gives us a new way to extract dual geometries only from CFT data. And it also gives a new proposal for the definition of complexity of states in CFTs.

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{\mathrm{CFT}}_{L}

and

{\mathrm{CFT}}_{R}

. We find that the scrambling of mutual information is in general governed by the Lyapunov index

{\ensuremath{\lambda}}_{L}

which is bounded by

\ensuremath{\kappa}=\frac{2\ensuremath{\pi}}{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}\ensuremath{\ge}\frac{2\ensuremath{\pi}}{\ensuremath{\beta}}

, where

\ensuremath{\mu}={r}_{\ensuremath{-}}/{r}_{+}

and

\mathcal{L}

is the angular momentum of the shockwave. For the special case of

\mathcal{L}=1

we find the mutual information analytically and show that it is characterized by

{\ensuremath{\lambda}}_{L}=\ensuremath{\kappa}/2

with the scrambling time for large black holes given as

{t}_{*}=\frac{\ensuremath{\beta}(1\ensuremath{-}\ensuremath{\mu})}{\ensuremath{\pi}}\mathrm{log}S

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-${\ensuremath{\lambda}}_{L}^{(\mathrm{min})}$ at late times and find that it is bounded by $\ensuremath{\kappa}=\frac{2\ensuremath{\pi}{T}_{H}}{(1\ensuremath{-}\ensuremath{\mu}\mathcal{L})}$, where $\ensuremath{\mu}$ 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 $\ensuremath{\kappa}$ with $\ensuremath{\kappa}{t}_{*}=\mathrm{log}\mathcal{S}$ for large black holes with a large entropy

\mathcal{S}

. We also find a term that increases the scrambling time by $\mathrm{log}(1\ensuremath{-}\ensuremath{\mu}\mathcal{L}{)}^{\ensuremath{-}1}$ but which does not scale with the entropy of the Kerr geometry.