Dynamics of black holes in Jackiw-Teitelboim gravity

TL;DR

This work addresses the computation of boundary operator correlators in Jackiw-Teitelboim gravity with a two-sided black hole, by leveraging a Lorentzian Hilbert space of boundary wavefunctions organized into $\tilde{SL}(2,\mathbb{R})$ irreps and the Schwarzian limit. The authors develop a representation-theoretic, diagrammatic framework in which correlators decompose into intertwiners that map boundary interactions to bulk interactions and back, with the Schwarzian density of states arising naturally as a weight in these amplitudes. They provide explicit ingredients for the amplitudes, including boundary/bulk interaction factors, gravitational scattering kernels via Wilson functions, and 6j-symbols that encode changes of basis between intertwiners, enabling a reduction of general amplitudes to products of basic components. The approach clarifies how gravity induces boundary-to-bulk transitions and connects bulk dynamics to boundary observables, offering a nonperturbative, symmetry-based tool for exploring JT gravity and its holographic (Schwarzian) limit, and suggesting paths to incorporate more general bulk topologies and interactions beyond the Schwarzian regime.

Abstract

We present a general solution for correlators of external boundary operators in black hole states of Jackiw-Teitelboim gravity. We use the Hilbert space constructed using the particle-with-spin interpretation of the Jackiw-Teitelboim action, which consists of wavefunctions defined on Lorentzian $AdS_2$. The density of states of the gravitational system appears in the amplitude for a boundary particle to emit and reabsorb matter. Up to self-interactions of matter, a general correlator can be reduced in an energy basis to a product of amplitudes for interactions and Wilson polynomials mapping between boundary and bulk interactions.

Figures (9)

• Figure 1: a) The regularization of the JT action on $\mathop{\mathrm{H}}\nolimits^2$ reveals a free particle with spin. b) We analytically continue the action to $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$, making two copies of the particle, in order to define a two-sided black hole system.
• Figure 2: a) Definition of angle $\alpha$ in \ref{['kappa']}. b) Red particles have $K=\kappa$, and blue particles $K=-\kappa$.
• Figure 3: a) The Schwarzian limit suppresses self-intersections and pushes the boundary to the asymptotic region of spacetime. b) In $\mathop{\mathrm{\widetilde{AdS}}}\nolimits_2$, a particle is effectively confined to one of asymptotic spacetimes $\mathcal{M}_{\rm L}$ or $\mathcal{M}_{\rm R}$.
• Figure 4: a,b) The time with respect to which matter fields should be quantized, determined by the direction of propagation for $\nu$- and $(-\nu)$-particles on opposite sides
• Figure 5: a) A spacetime depiction of a correlator of boundary operators. Representations in Hilbert space of b) an amplitude and c) a correlator. In general, the time-ordering of operators is different from their ordering in Hilbert space.