Radial canonical $\Lambda<0$ gravity

Blanca Hergueta; Nele Callebaut

arxiv: 2503.18522 · v1 · submitted 2025-03-24 · ✦ hep-th · gr-qc

Radial canonical Λ<0 gravity

Nele Callebaut , Blanca Hergueta This is my paper

Pith reviewed 2026-05-05 05:03 UTC · model claude-opus-4-7

classification ✦ hep-th gr-qc

keywords Wheeler-DeWitt equationAdS3 gravityADM deparametrizationholographyBTZ black holeTT-bar deformationYork timeconformal boundary conditions

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The pith

Radial AdS3 gravity, recast as Schrödinger evolution in slice volume, identifies CFT data as the true ADM degrees of freedom and admits a BTZ wavepacket.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the standard ADM deparametrization recipe — promote a momentum to minus a Hamiltonian, then choose a preferred time — to three-dimensional gravity with negative cosmological constant, but with the radial coordinate playing the role of time. The natural radial clock that emerges is the volume of the spatial slice; its conjugate momentum is what older literature calls York time. After the constraint is solved, the genuine "ADM" degrees of freedom that remain are a unit-determinant metric and a traceless momentum, which on the asymptotic boundary are exactly the conformal metric and the stress-tensor data of a dual CFT. The Wheeler-DeWitt equation becomes a volume-time Schrödinger equation whose near-boundary limit reproduces the holographic Weyl anomaly and whose full form reproduces the TT-bar trace-flow equation. The authors then construct the non-rotating BTZ solution by solving the radial Hamilton-Jacobi equation, build a WKB wavepacket around it, and verify that its DeWitt norm and expectation values behave as expected, recovering the classical BTZ metric in the small-width limit.

Core claim

In three-dimensional gravity with negative cosmological constant, the radial direction can be treated like a time coordinate by applying the same deparametrization procedure used in mechanics: solve the Hamiltonian constraint for a chosen momentum, then fix a "preferred time." The paper identifies the volume of the radial slices as that preferred time, recasts the Wheeler-DeWitt equation as a volume-time Schrödinger equation, and shows that its near-boundary limit is the Weyl anomaly of the dual CFT while the full equation is the trace-flow equation of a TT-bar deformed CFT. A Laplace transform in volume swaps Dirichlet for conformal boundary conditions, trading volume time for York time. Th

What carries the argument

Radial ADM deparametrization: split the spatial metric into a volume density v and a unit-determinant conformal metric, solve the Hamiltonian constraint for the volume-conjugate momentum, and read the result as a Schrödinger equation in v. This single change of variables packages the holographic Weyl anomaly, the TT-bar trace-flow equation, and the Dirichlet/conformal boundary-condition duality (via Laplace transform in v) into one canonical formalism, and provides the Hamilton-Jacobi route to BTZ and its wavepacket.

If this is right

The asymptotic radial Wheeler-DeWitt equation is literally the conformal Ward identity of the dual CFT, with the gravitational wavefunctional equal to the CFT partition function.
The full radial WdW equation away from the boundary encodes the TT-bar trace-flow equation, with TT-bar coupling λ = κl/2 fixed by the bulk Newton constant and AdS radius.
Switching from Dirichlet to conformal boundary conditions for AdS3 gravity is implemented by a Laplace transform in volume, exchanging volume time for York time (or trace of extrinsic curvature) as the radial clock.
The non-rotating BTZ black hole admits a semiclassical Gaussian wavepacket solution to the radial WdW equation with finite DeWitt norm in volume time, giving a concrete quantum state whose classical limit is BTZ.
The wavepacket's volume-time energy ⟨π_k⟩ is constant in v, so it is not the holographic TT-bar energy — the TT-bar energy must be a different combination of the ADM variables.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Treating the conformal metric and its conjugate momentum as the 'true' canonical pair makes the choice of boundary conditions look like a canonical transformation, suggesting other boundary conditions (mixed, Neumann) should appear as further integral transforms in the same family.
The constancy of ⟨π_k⟩ in volume time but not the wavepacket spread hints that finite-cutoff TT-bar energy is naturally a function of v and the conformal data together, not of π_k alone — a concrete target for matching bulk wavepackets to TT-bar spectra.
The construction is built on a mini-superspace ansatz that kills the momentum constraint by hand; lifting it to genuine field-theoretic radial slices is the obvious next test of whether 'volume time' survives as a clock.
Extending the wavepacket to rotating BTZ and to the inside-horizon region would test whether the same DeWitt norm remains positive when the radial direction becomes timelike, which is where canonical radial quantization usually breaks.

Load-bearing premise

That the slice volume can serve as a globally well-defined radial clock for the solutions of interest, including across the maximal-volume slice and into the BTZ interior where the radial direction changes character.

What would settle it

Show that the volume-time Schrödinger equation derived here fails to reproduce the TT-bar trace-flow equation at finite cutoff, or that the BTZ wavepacket's DeWitt norm and expectation values do not converge to the classical non-rotating BTZ metric (mass ε₀, with v = κlε₀ sech(k+k₀)) as the Gaussian width Δ → 0.

read the original abstract

We apply an ADM deparametrization strategy to radial canonical $\Lambda < 0$ gravity in three dimensions. It gives rise to a concise notation for previous holographic interpretations in terms of an identified radial 'volume time' and 'true' ADM degrees of freedom. We further discuss York time and conformal boundary conditions in this context, and construct a BTZ wavepacket solution to the radial WdW equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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