Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim Gravity

TL;DR

The paper develops a covariant Peierls bracket framework to compute the algebra of a broad set of diffeomorphism-invariant observables in Jackiw-Teitelboim gravity with matter. It shows that traversable wormholes, $SL(2, ext{R})$-type algebras acting on matter, and scrambling time follow directly from this algebra, without introducing unphysical degrees of freedom. The authors analyze energy changes when creating bulk excitations, revealing scenarios where excitations behind horizons carry negative energy, with implications for firewall arguments; they also demonstrate robustness when generalizing to higher dimensions. The work provides a comprehensive, gravity-based, gauge-invariant approach to black hole interior dynamics, connecting interior observables to boundary data and offering explicit calculations in JT gravity and beyond.

Abstract

In this paper we use the covariant Peierls bracket to compute the algebra of a sizable number of diffeomorphism-invariant observables in classical Jackiw-Teitelboim gravity coupled to fairly arbitrary matter. We then show that many recent results, including the construction of traversable wormholes, the existence of a family of $SL(2,\mathbb{R})$ algebras acting on the matter fields, and the calculation of the scrambling time, can be recast as simple consequences of this algebra. We also use it to clarify the question of when the creation of an excitation deep in the bulk increases or decreases the boundary energy, which is of crucial importance for the "typical state" versions of the firewall paradox. Unlike the "Schwarzian" or "boundary particle" formalism, our techniques involve no unphysical degrees of freedom and naturally generalize to higher dimensions. We do a few higher-dimensional calculations to illustrate this, which indicate that the results we obtain in JT gravity are fairly robust.

Figures (15)

• Figure 1: Different kinds of bulk time evolution in the AdS-Schwarzschild geometry (figure adapted from Harlow:2018tqv). The ADM Hamiltonians $H_{\pm}$ move the boundary edges of a bulk Cauchy slice up and down, as in the left two diagrams, but the evolution of the interior of the slice, as in going from the red slice to the green slice in the right diagram, is generated by the Hamiltonian constraint and thus is trivial on gauge-invariant states.
• Figure 2: The linearized solution $\delta_g \phi=\delta\phi_R-\delta\phi_A$ as a function of space and time. The dotted lines indicate the locations of the shockwaves which restore the boundary conditions.
• Figure 3: The geometry of the two-sided geodesic $y^\mu_{t_-t_+}$, shown in blue, at finite $\epsilon$. The length $L(t_-,t_+)$ of this geodesic and the relative boosts $\eta_{\pm}(t_-,t_+)$ between its rest frame and those of the left and right boundaries $\Gamma_\pm$ define three diffeomorphism-invariant observables.
• Figure 4: Construction of a diffeomorphism-invariant "one-sided" dressed matter observable $\psi_{s\eta}(t_-)$: we fire a geodesic $y^\mu_{t_-\eta}(s)$, shown in blue, from the left boundary at time $t_-$ with relative boost $\eta$, and after a proper distance $s$ we evaluate the scalar matter field $\psi$.
• Figure 5: Some examples of the one-sided geodesic $y^\mu_{t_-\widetilde{\eta}}(\widetilde{s})$ fired from the left boundary at $t_-=0$ in the pure JT solution \ref{['nomattsol']}. Geodesics for other starting times are obtained by acting with the boost isometry of this solution. The dashed lines are horizons, and the spacetime region whose properties are determined by the initial data on any Cauchy slice connecting the two boundaries and the boundary conditions \ref{['BC']} is shaded green.