  We employ an ADM deparametrization strategy to discuss the radial canonical
formalism of asymptotically AdS$_3$ 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 $T\bar T$ theory as a Schr\"odinger
wavefunctional satisfying a Schr\"odinger evolution equation in volume time,
and the $T\bar T$ 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 $T\bar T$ interpretation, as well as semi-classical
Wheeler-DeWitt states for that solution.
