Weyl-Transverse Gravity (WTG)

• Weyl-Transverse Gravity is a generally covariant metric theory defined by local Weyl invariance and transverse diffeomorphisms that preserve a fixed volume form.
• Its variational principle leads to traceless field equations where the cosmological constant emerges as an integration constant, differing from its treatment in General Relativity.
• The canonical structure and boundary formulation of WTG incorporate unique gauge symmetries and ADM splitting, ensuring two propagating graviton modes in four dimensions.

Weyl-Transverse Gravity (WTG) is a generally covariant metric theory of gravity characterized by invariance under transverse diffeomorphisms (TDiff) and local Weyl transformations. In contrast to standard General Relativity (GR), whose local gauge group is the full diffeomorphism group Diff($M$), WTG restricts diffeomorphism invariance to the subgroup that preserves a fixed background volume form and supplements it with pointwise conformal invariance. The theory is classically equivalent to GR but exhibits distinct gauge structure that deeply modifies the variational setup, the role of the cosmological constant, canonical formalism, and the formulation with boundaries.

1. Dynamical Fields, Auxiliary Metric, and Action Principles

WTG is formulated on a $D$-dimensional Lorentzian manifold $M$ with metric $g_{\mu\nu}$ (determinant $g = \det(g_{\mu\nu})$) and a fixed, non-dynamical $D$-form volume density $\bm\omega = \omega(x) d^Dx$ (Odak et al., 22 Jan 2026). The local symmetry group is Weyl × TDiff:

• Weyl transformations: $g_{\mu\nu} \mapsto e^{2\sigma(x)} g_{\mu\nu}$, $\bm\omega$ fixed.
• Transverse diffeomorphisms: Generated by $\xi^\mu(x)$ with $\Lie_\xi \bm\omega = 0$, i.e., $\tilde\nabla_\mu \xi^\mu = 0$ for the Levi-Civita connection $\tilde\nabla$ of an auxiliary, Weyl-invariant metric.

The auxiliary metric is defined by

$$\tilde{g}_{\mu\nu} = \left( \frac{\sqrt{-g}}{\omega} \right)^{-2/D} g_{\mu\nu}$$

ensuring $\sqrt{-\tilde{g}} = \omega$ and full invariance under the above gauge symmetries.

The bulk-plus-boundary action is: $$S[g; \bm\omega] = \frac{1}{16\pi G} \int_M \bm\omega\, \tilde{R} + \int_{\partial M} \ell_G - \frac{\lambda}{8\pi G} \int_M \bm\omega$$ where $\tilde{R}$ is the Ricci scalar of $\tilde{g}_{\mu\nu}$ and $\lambda$ is a Lagrangian ambiguity (cosmological constant term not present in the field equations but contributing to Noether charges) (Odak et al., 22 Jan 2026, Álvarez et al., 2010).

2. Gauge Symmetries: Weyl and Transverse Diffeomorphisms

Weyl-transverse gauge transformations form a nontrivial algebra:

• Weyl: For scalar $\sigma(x)$, $\delta_\sigma g_{\mu\nu} = 2\sigma g_{\mu\nu}$, $\delta_\sigma \tilde{g}_{\mu\nu}=0$.
• TDiff: For $\xi^\mu$ with $\tilde{\nabla}_\mu \xi^\mu = 0$, $\delta_\xi g_{\mu\nu}=2\nabla_{(\mu} \xi_{\nu)}$, and $\omega$ is fixed.

Gauge redundancy is such that both TDiff and Weyl each remove $D$ parameters from the $D(D+1)/2$ components of $g_{\mu\nu}$, yielding two propagating graviton degrees of freedom in $D=4$ (Alonso-Serrano et al., 6 Feb 2025). The algebra closes under Lie bracket, and local physical observables are invariant under the combined symmetry.

3. Variational Principle, Field Equations, and Conserved Quantities

The variation of the action leads to traceless field equations: $$\tilde{R}_{\mu\nu} - \frac{1}{D} \tilde{R}\, \tilde{g}_{\mu\nu} = 8\pi G\, \left( \tilde{T}_{\mu\nu} - \frac{1}{D}\tilde{T}\,\tilde{g}_{\mu\nu} \right)$$ where the energy-momentum tensor is defined with respect to $\tilde{g}_{\mu\nu}$ (Alonso-Serrano et al., 6 Feb 2025, Alonso-Serrano et al., 2022). Contracted Bianchi identities ensure that $\tilde{R}$ must be a constant, allowing the equations to be recast in standard Einstein form with a cosmological constant that emerges as an integration constant: $$\tilde{R}_{\mu\nu} - \tfrac{1}{2} \tilde{R}\, \tilde{g}_{\mu\nu} + \Lambda \tilde{g}_{\mu\nu} = 8\pi G\, \tilde{T}'_{\mu\nu}$$ The absence of a bare cosmological constant reflects the rigid Weyl invariance and renders vacuum energy contributions radiatively irrelevant (Álvarez et al., 2010). Noether currents for both gravity and matter fields can be constructed via the Iyer-Wald formalism, with explicit formulas for charge densities and surface charges (Odak et al., 22 Jan 2026, Alonso-Serrano et al., 2022).

4. Boundary Formulation and Classification of Boundary Conditions

The formulation of WTG with boundaries requires careful treatment of the variational principle and associated surface terms. Boundary Lagrangians and conditions are classified as follows (Odak et al., 22 Jan 2026):

• Auxiliary Dirichlet: Fix $\tilde{q}_{\mu\nu}$ on boundary, $\delta \tilde{q}_{\mu\nu}|_\Gamma = 0$.
• Auxiliary Neumann: Fix conjugate momentum $\delta \tilde{\Pi}^{\mu\nu}|_\Gamma = 0$.
• Conformal Dirichlet (dynamical metric): Fix conformal class and volume for $q_{\mu\nu}$.
• York boundary conditions: Fix conformal class and mean curvature $\tilde{K}$.

Each choice corresponds to a specific form of counterterm and uniquely guarantees a differentiable action under variation. York conditions are particularly natural in WTG, enabling a transparent geometric formulation.

5. Canonical Structure and Hamiltonian Analysis

WTG admits a well-defined canonical structure, distinctly different from GR owing to the gauge principle (Kluson, 2023). In the ADM splitting, the primary constraint associated with Weyl invariance leads to a traceless momentum tensor, and the Hamiltonian constraints are preserved up to a global mode which becomes the dynamical cosmological constant. The constraint algebra (spatial TDiff, Weyl, local Hamiltonian minus global zero mode) closes up to itself, with the cosmological constant emerging as an undetermined global Hamiltonian mode. Gauge fixing Weyl invariance reduces WTG to unimodular gravity, preserving equivalence with GR at the classical level.

6. Black Hole Mechanics, Thermodynamics, and First Laws

The phase space and Noether-charge formalism allows for comprehensive treatment of black hole mechanics in WTG (Odak et al., 22 Jan 2026, Alonso-Serrano et al., 2022, Alonso-Serrano et al., 2024). The Wald entropy is well-defined and the extended first law incorporates variation of the cosmological constant: $$\delta E = \frac{\kappa}{8\pi G}\, \delta A + \Omega\, \delta J + \frac{V}{8\pi G}\, \delta \Lambda$$ with $V$ the thermodynamic volume determined by background $\bm\omega$. In vacuum, horizon entropy is proportional to area as in GR, but the $\delta \Lambda$ pressure term appears whenever the phase space includes cosmological constant variations. This reflects the fact that $\bm\omega$ is non-dynamical and the cosmological constant enters only through global integration. The first law for causal diamonds and black holes in AdS/dS is unified through this approach, and the Smarr relation naturally follows (Alonso-Serrano et al., 2022, Alonso-Serrano et al., 2024).

7. Physical Consequences, Equivalence Principle, and Observational Status

WTG predicts universal free-fall and retains local Lorentz invariance for both test particles and self-gravitating bodies, satisfying all standard forms of the Equivalence Principle (weak, Einstein, strong, gravitational EP) (Alonso-Serrano et al., 6 Feb 2025). Matter must couple minimally to $\tilde{g}_{\mu\nu}$; non-minimal couplings may break energy–momentum conservation but do not violate the fundamental Equivalence Principle. The theory is classically equivalent to GR (solutions, degrees of freedom, black holes, cosmology) but can differ quantum mechanically due to the vanishing of the Weyl Noether current (the symmetry is "fake" at the quantum level (Oda, 2016)). Cosmological models require spatially flat slices with lapse fixed by unimodular constraint, reproducing Friedmann dynamics with undetermined initial $\Lambda$ (Oda, 2016). Physical predictions in realistic contexts may require explicit breaking of Weyl symmetry or inclusion of anomalous terms.

Table: Comparison of Principal Features in WTG and GR

Feature	WTG	GR
Gauge group	Weyl × TDiff (volume-preserving & Weyl)	Full Diff
Cosmological constant	Integration constant	Lagrangian parameter
Physical DOFs ($D=4$)	2 gravitational polarizations	2 gravitational polarizations
Black hole entropy	Wald formula (area law, $\tilde{g}$)	Bekenstein-Hawking
Boundary conditions	Dirichlet, Neumann, York for $\tilde{g}$ or $g$	Gibbons-Hawking-York
Classical equivalence	Yes (with appropriate gauge fixing)	Yes

The essential distinguishing feature of Weyl-Transverse Gravity is its gauge symmetry principle, which modifies the variational setup, eliminates a dynamical cosmological constant at the level of the action, and imposes distinctive structure on the canonical and boundary formalisms, while ensuring classical phenomenological equivalence with General Relativity (Odak et al., 22 Jan 2026, Alonso-Serrano et al., 2022, Alonso-Serrano et al., 6 Feb 2025, Kluson, 2023).