Radial canonical AdS₃ gravity and TT̄

We employ an ADM deparametrization strategy to discuss the radial canonical formalism of asymptotically AdS₃ gravity. It leads to the identification of a radial 'time' before quantization, which is the volume time, canonically conjugate to York time. Holographically, this allows to interpret the semi-classical path integral of T T theory as a Schr\"odinger wavefunctional satisfying a Schr\"odinger evolution equation in volume time, and the T T operator expectation value in terms of the Hamiltonian that generates volume time translations -- both consistent with cut-off holography. We make use of the canonical perspective to construct the rotating BTZ solution from the Hamilton-Jacobi equation, with a finite cut-off energy spectrum that has a known holographic T T interpretation, as well as semi-classical Wheeler-DeWitt states for that solution.