Unimodular time in JT gravity: a holographic clock

TL;DR

This paper explores how a bulk physical clock is holographically encoded in JT gravity by promoting the vacuum energy to a dynamical variable, establishing a connection between boundary dynamics and bulk time.

Contribution

It introduces a novel formulation of bulk time in JT gravity through a top form conjugate to spacetime volume, linking boundary and bulk clocks explicitly.

Findings

Boundary dynamics match the Schwarzian mode coupled to a $U(1)$ particle.

JT gravity with Maxwell theory is equivalent to 2D HT gravity.

The boundary clock is identified with the $U(1)$ phase mode.

Abstract

How is a ''bulk clock'' encoded holographically? We address this in Jackiw-Teitelboim (JT) gravity, where a natural physical clock emerges by promoting the vacuum energy to a dynamical variable: the vacuum cosmological constant becomes a top form degree of freedom conjugate to spacetime volume, thereby defining a notion of bulk physical time. This construction is naturally formulated in the Henneaux-Teitelboim (HT) framework. We show that the boundary dynamics is the Schwarzian mode coupled to a free particle on $U(1)$, matching the universal low-energy effective action of the complex SYK model. By further clarifying the role of the vacuum cosmological constant as a top form, we establish the equivalence between JT gravity coupled to two-dimensional Maxwell theory and 2d HT gravity via an explicit field redefinition. The initial question is addressed: we show that the resulting boundary theory can itself be rewritten as an observer action, equivalently a $(0+1)$-dimensional HT theory. This yields a direct identification of the boundary clock with the $U(1)$ phase mode, and makes its relation to the bulk clock explicit.