York time in JT gravity

TL;DR

This paper investigates the role of York time as an internal clock in JT gravity, explicitly calculating the Hartle-Hawking wavefunction's dependence on York time and deriving a Hermitian York Hamiltonian.

Contribution

It provides a detailed analysis of York time in JT gravity, deriving a Schrödinger equation and Hamiltonian that are free of operator ordering ambiguities.

Findings

Wavefunction satisfies Schrödinger equation in York time

York Hamiltonian is Hermitian

Wavefunction dependence arises from unitary transformation, not physical evolution

Abstract

The notion of time in general relativity must arise from an internal clock, i.e., a degree of freedom in the gravitational theory internal to the system that can serve the role of a physical clock. One such internal notion of time is the York time, corresponding to constant extrinsic curvature slicing of spacetime. We study the Hartle-Hawking wavefunction of asymptotically $AdS_2$ JT gravity as a function of York time. Using both canonical quantization and the JT gravity path integral, we explicitly calculate this wavefunction and show that it satisfies a Schrodinger equation with respect to York time. We find the corresponding York Hamiltonian, which turns out to be manifestly Hermitian. Our analysis cleanly avoids operator ordering ambiguities. The dependence of the wavefunction on York time should be thought of as emerging from a unitary transformation of the gravitational length basis states, and not from a physical time evolution of the state in the dual boundary theory.