Unimodular time in JT gravity: a holographic clock

A bstract 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.