Weyl-transverse gravity with boundaries

Technical Summary: Weyl-transverse gravity with boundaries

Problem Statement
Weyl-transverse gravity (WTG) is a metric theory of gravity classically equivalent to General Relativity (GR) but possessing a reduced gauge symmetry: invariance under Weyl transformations and transverse diffeomorphisms, rather than general diffeomorphisms. A defining feature of WTG is the presence of a fixed, non-dynamical background volume form, $\omega$. While the bulk equations of motion in WTG reproduce the Einstein equations, the reduced symmetry and the fixed background volume form alter the variational principle and the definition of conserved quantities. Prior to this work, a covariant phase space formulation of WTG in the presence of general timelike and spacelike boundaries was lacking. This gap prevents a rigorous definition of boundary conditions, symplectic structures, and Noether charges necessary for studying thermodynamics and conserved quantities in this framework.

Methodology
The authors employ the covariant phase space formalism, adopting the framework established in [9], to analyze WTG with boundaries. The methodology proceeds through the following steps:

1. Action and Variational Principle: The theory is formulated using a dynamical metric $g_{\mu\nu}$ and an auxiliary metric $\tilde{g}_{\mu\nu}$, which is Weyl-invariant by construction. The action includes a bulk term proportional to the Ricci scalar of the auxiliary metric and a boundary term $\ell$. The variation of the action is computed to identify the bulk equations of motion and the boundary terms that must vanish for the action to be differentiable.
2. Symplectic Structure: The authors derive the symplectic potential $\Theta$ and the presymplectic current $\tilde{\omega}$. They analyze the boundary variation of the action to identify a set of boundary terms $C$ and boundary conditions that render the variational principle well-posed. This involves decomposing the boundary into timelike ($\Gamma$) and spacelike ($\Sigma_\pm$) components.
3. Boundary Conditions: The paper systematically derives admissible boundary conditions for both the auxiliary metric $\tilde{g}_{\mu\nu}$ and the dynamical metric $g_{\mu\nu}$. This includes Dirichlet and Neumann conditions for the auxiliary metric, and Dirichlet, Neumann, and York-type conditions for the dynamical metric.
4. Noether Currents and Hamiltonians: Using the derived symplectic potential, the authors construct the Noether current $J_\xi$ associated with transverse diffeomorphisms generated by a vector field $\xi$. They integrate this current to define the Hamiltonian generator $H_\xi$ and evaluate it on spacetimes containing a bifurcate Killing horizon.

Key Contributions and Results

• Symplectic Potential and Boundary Conditions:
  The authors derive the symplectic potential $\Theta_G$ for the gravitational sector. They identify a family of boundary Lagrangians $\ell_b = \frac{b}{16\pi G}\tilde{K}\omega_\Gamma$ that allows for a consistent variational principle.

  • Auxiliary Metric: They establish Dirichlet conditions (fixing the induced auxiliary metric $\tilde{q}_{\mu\nu}$) and Neumann conditions (fixing the momentum $\tilde{\Pi}_{\mu\nu}$).
  • Dynamical Metric: They show that fixing the dynamical metric requires fixing both its conformal class and the boundary volume form (Dirichlet-type) or the mean curvature (York-type).
  • York Boundary Conditions: A significant result is that York boundary conditions (fixing the conformal class of the induced metric and the trace of the extrinsic curvature) arise naturally in WTG. This is because the fundamental dynamical variable in WTG is conformally related to the metric associated with the volume form, making WTG an intrinsically adapted framework for York data.

• Noether Currents and Surface Charges:
  The Noether current $J_\xi$ is derived for transverse diffeomorphisms. Unlike in GR, the current contains an additional term proportional to the background volume form $\omega$ and the Lagrangian ambiguity parameter $\lambda$ (related to the cosmological constant). The current is found to be:
  $$J_\xi \approx dQ_{GR} - \frac{\Lambda - \lambda}{8\pi G} \xi \cdot \omega$$
  where $Q_{GR}$ is the standard GR Noether charge. The surface charge is thus modified by the integration constant $\Lambda$ arising from the traceless Einstein equations.

• First Law of Black Hole Thermodynamics:
  Evaluating the Hamiltonian identity on a spacetime with a bifurcate Killing horizon yields a first-law relation. The authors find that the variation of the cosmological constant, $\delta\Lambda$, appears explicitly in the first law:
  $$\delta M = \frac{\kappa}{8\pi G} \delta A + \Omega \delta J + \frac{V}{8\pi G} \delta \Lambda$$
  Here, $V$ is defined geometrically as the integral of the contraction of the Killing vector with the background volume form over the spatial slice ($V = \int_{\Sigma_{ext}} \xi \cdot \omega$).

Significance and Claims
The paper claims to provide the complete boundary and phase-space framework required to study conserved quantities in WTG. Its primary significance lies in:

1. Clarifying the Variational Structure: It explicitly demonstrates how the reduced gauge symmetry and fixed background volume form modify the variational principle relative to GR, particularly regarding the definition of boundary conditions.
2. Natural Implementation of York Conditions: It establishes that WTG provides a natural geometric setting for York boundary conditions, which are crucial for mathematical relativity and Hamiltonian formulations.
3. Origin of the $\delta\Lambda$ Term: The paper argues that the appearance of the $\delta\Lambda$ term in the first law of black hole thermodynamics is not an ad hoc addition (as in "black hole chemistry" where $\Lambda$ is promoted to a thermodynamic variable) but follows directly from the variational structure of WTG and the presence of the background volume form.
4. Geometric Definition of Volume: It provides an unambiguous, geometric definition of the thermodynamic volume $V$ conjugate to $\Lambda$, derived strictly from the Hamiltonian analysis, contrasting with the multiple inequivalent definitions found in standard extended phase-space thermodynamics.

The authors conclude that their construction isolates precisely which aspects of the theory depart from GR and which remain unchanged, offering a complementary viewpoint on the relationship between black hole thermodynamics and unimodular-like formulations of gravity. They suggest future work could explore null boundaries, asymptotic symmetries, and potential infrared modifications or memory effects arising from the conformal structure of WTG.