[Paper Review] An Investigation of AdS$_2$ Backreaction and Holography

Julius Engelsöy, Thomas G. Mertens+1 more|arXiv (Cornell University)|Jun 10, 2016

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates the Almheiri-Polchinski dilaton gravity model in AdS₂, showing that its boundary dynamics reduces to a 1D effective theory governed by a Schwarzian derivative action. The model exhibits maximal chaos via exponentially growing commutators, matches shockwave-induced time shifts, and describes black hole evaporation through a non-linear effective action, linking AdS₂ holography to conformal mechanics and the SYK model.

ABSTRACT

We investigate a dilaton gravity model in AdS$_2$ proposed by Almheiri and Polchinski and develop a 1d effective description in terms of a dynamical boundary time with a Schwarzian derivative action. We show that the effective model is equivalent to a 1d version of Liouville theory, and investigate its dynamics and symmetries via a standard canonical framework. We include the coupling to arbitrary conformal matter and analyze the effective action in the presence of possible sources. We compute commutators of local operators at large time separation, and match the result with the time shift due to a gravitational shockwave interaction. We study a black hole evaporation process and comment on the role of entropy in this model.

Motivation & Objective

• To develop a 1D effective description of AdS₂ boundary dynamics in the Almheiri-Polchinski dilaton gravity model.
• To establish a connection between the boundary time reparametrization and the Schwarzian action, analogous to the SYK model.
• To analyze the dynamics and symmetries of the system using a canonical Hamiltonian framework.
• To compute commutators of local operators and match them to gravitational shockwave effects, probing quantum chaos.
• To model black hole evaporation in the effective theory and examine entropy and energy loss mechanisms.

Proposed method

• Derive the boundary equations of motion from the bulk action, identifying the time coordinate as a dynamical variable governed by a reparametrization constraint.
• Construct a 1D effective action based on the Schwarzian derivative, showing equivalence to a 1D Liouville theory.
• Implement a canonical Hamiltonian formulation to analyze the system's symmetries, including SL(2,R) and Virasoro algebra structures.
• Couple the boundary theory to conformal matter fields and compute the holographic stress tensor expectation value.
• Use the effective action to compute time-ordered commutators of local operators, revealing maximal chaos via exponential growth.
• Derive a non-linear effective action for quantum processes and apply it to model black hole evaporation with energy loss via Hawking emission.

Experimental results

Research questions

• RQ1 How does the boundary time coordinate become dynamical in the Almheiri-Polchinski AdS₂ model, and what effective action describes its dynamics?
• RQ2 To what extent does the 1D boundary theory exhibit maximal quantum chaos, as measured by commutator growth?
• RQ3 How do gravitational shockwave interactions in the bulk manifest as time shifts in the boundary commutators?
• RQ4 What is the role of the Schwarzian action in connecting AdS₂ holography to the Sachdev-Ye-Kitaev model?
• RQ5 How can the effective action describe black hole evaporation, including energy and entropy evolution?

Key findings

• The boundary dynamics of the AdS₂ dilaton gravity model is fully described by a 1D effective action with a Schwarzian derivative, equivalent to a 1D Liouville theory.
• The commutator $[ au(t_1), au(t_2)]$ grows exponentially at late times, matching the time shift induced by a gravitational shockwave, confirming maximal chaos.
• The system exhibits $SL(2,bR)$ symmetry, and the boundary stress tensor expectation value is given by $\langle \hat{T}_{tt} \rangle = -\frac{1}{2\kappa} \{\tau, t\}$, where $\{\cdot, \cdot\}$ is the Schwarzian derivative.
• For a single pulse with $\kappa\omega > 1$, the solution exhibits a black hole formation via $\tan y = \sqrt{1/(\kappa\omega - 1)} \tanh(\sqrt{\kappa\omega - 1}\, t)$, confirming the black hole mass as $E = \kappa\omega - 1$.
• Multi-pulse dynamics are described by an $SL(2,\bbR)$ transfer matrix, showing that all such frames are related by Möbius transformations and unitarily equivalent.
• The non-linear effective action successfully models black hole evaporation, with energy decreasing due to Hawking emission, and provides a framework for studying entropy dynamics in the model.