A note on the action with the Schwarzian at the stretched horizon

Abstract

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.

arXiv comment40 pages

References in corpus (18)

• Fermionic Localization of the Schwarzian Theory
• AdS${}_2$ holography and the SYK model
• The statistical mechanics of near-extremal black holes
• On the Dynamics of Near-Extremal Black Holes
• A theory of reparameterizations for AdS${}_3$ gravity
• 5D Rotating Black Holes and the nAdS${}_2$/nCFT${}_1$ Correspondence
• Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes
• Jackiw-Teitelboim Gravity and Rotating Black Holes
• Extremal and Near-Extremal Black Holes and Near-$CF{T}_1$
• Classifying boundary conditions in JT gravity: from energy-branes to $\alpha$-branes
• Geometrodynamical Formulation of Two-Dimensional Dilaton Gravity
• Near-Horizon BMS Symmetry, Dimensional Reduction, and Black Hole Entropy
• Warped Schwarzian theory
• A One-Loop Test of the near-AdS${}_2$/near-CFT${}_1$ Correspondence
• Flat JT Gravity and the BMS-Schwarzian
• Equivalence of JT Gravity and Near-extremal Black Hole Dynamics in Higher Derivative Theory
• Multiplicative anomaly and zeta factorization
• A Schwarzian on the Stretched Horizon