Semi-classical dilaton gravity and the very blunt defect expansion

TL;DR

The paper investigates semi-classical dilaton gravity with generalized dilaton potentials by contrasting an exact defect-gas description with canonical geodesic-gauge quantization and a dense gas of very blunt defects. It uncovers non-perturbative, horizon-related saddles that reshape the density of states and ground-state energy, and reveals a breakdown of geodesic gauge at disk level due to pacifier spacetimes. A defector-density (very blunt) expansion reproduces smooth classical geometries and yields a bulk interpretation of new saddles, connecting to matrix-model descriptions and Weil-Peterson volumes in a controlled limit. The study suggests profound interior geometry implications, including potential firewall-like behavior and a richer non-perturbative structure that bridges JT gravity, minimal strings, and dilaton gravity with exponential deformations.

Abstract

We explore dilaton gravity with general dilaton potentials in the semi-classical limit viewed both as a gas of blunt defects and also as a semi-classical theory in its own right. We compare the exact defect gas picture with that obtained by naively canonically quantizing the theory in geodesic gauge. We find a subtlety in the canonical approach due to a non-perturbative ambiguity in geodesic gauge. Unlike in JT gravity, this ambiguity arises already at the disk level. This leads to a distinct mechanism from that in JT gravity by which the semi-classical approximation breaks down at low temperatures. Along the way, we propose that new, previously un-studied saddles contribute to the density of states of dilaton gravity. This in particular leads to a re-interpretation of the disk-level density of states in JT gravity in terms of two saddles with fixed energy boundary conditions: the disk, which caps off on the outer horizon, and another, sub-leading complex saddle which caps off on the inner horizon. When the theory is studied using a defect expansion, we show how the smooth classical geometries of dilaton gravity arise from a dense gas of very blunt defects in the $G_N \to 0$ limit. The classical saddle points arise from a balance between the attractive force on the defects toward negative dilaton and a statistical pressure from the entropy of the configuration. We end with speculations on the nature of the space-like singularity present inside black holes described by certain dilaton potentials.

Figures (16)

• Figure 1: We illustrate what the other saddle point is that contributes to the density of states in JT gravity. This amounts to a complex geometry that is the AdS$_2$ disk. As we will describe in Sec. \ref{['sec:exactdos']}, the radial coordinate, which can be taken to be the dilaton, follows a complex contour that avoids the pole in the metric at the outer horizon and ends on the negative root of the equation $\Phi^2 = E$. The metric in the range $-\sqrt{E}< \Phi< \sqrt{E}$ looks like that of an anti-sphere, whose metric is Euclidean up to an overall minus sign. This solution has constant curvature $R = -2$.
• Figure 2: We give an example of what the pacifier spacetimes look like qualitatively, illustrating that there can be multiple geodesics connecting the same pair of boundary points. There exists a closed geodesic of length $b$ and a horizon at $\Phi = \Phi_h$.
• Figure 3: This figure illustrates how a gas of very blunt defects can approximate a smooth spacetime. On the left, we start with a few sharp defects, with the arrows indictating that in the classical limit the defect gas becomes simultaneously denser and blunter.
• Figure 4: This figure illustrates the pre-potential $W_{\lambda}(\Phi)$ for $\alpha = 0$ and $\lambda = .001$. We label the energy of the zero temperature classical solution, which is set by the value of $W_{\lambda}(\Phi)$ at its outermost minimum, $E_0$. We also label the critical energy $E_c$. Solutions with $E>E_c$ have one horizon while solutions with $E_c>E>E_0$ have three.
• Figure 5: This figure illustrates the Penrose diagram for solutions with prepotential $W_{\lambda}(\Phi)$ with $\lambda_c>\lambda >0$. The diagram on the left shows the solution at low energies $E$ where $W_{\lambda}(\Phi) = E$ has three real roots. The diagram at the right where there is just one real real root. The jagged lines denote regions where both the curvature blows up and $\Phi \to - \infty$. The solid lines denote asymptotically AdS boundaries and the dashes lines are horizons. Note that we have drawn the singularities as horizontal but they will have an energy dependent shape.