⚛️ general relativity

Internal translation gauging and local gravitational energy in Weyl gauge theory

This paper investigates local gravitational energy and charges in Weyl gauge theory by treating translations as an internal Stückelberg redundancy, deriving gauge-invariant excitation equations and superpotentials that recover known results like the Schwarzschild ADM energy in the teleparallel limit while providing specific charge predictions for New General Relativity.

Original authors: Michael Hobson, Will Barker, Anthony Lasenby

Published 2026-10-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Hobson, Will Barker, Anthony Lasenby

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the force that holds our feet to the ground and keeps the planets in their orbits, yet it remains the most elusive of nature's fundamental interactions. Unlike electricity or magnetism, which can be measured with simple instruments and described by clear, local rules, gravity resists being pinned down to a specific point in space. Physicists have long struggled to define exactly how much energy a gravitational field contains at any single location. In the standard view of general relativity, gravity is not a force acting within space, but rather the shape of space itself. This makes the idea of a local "gravitational energy" difficult to grasp, because you cannot easily separate the shape of the stage from the actors performing on it. To solve this, some researchers have tried to treat gravity like other forces, imagining it as a field that exists on a fixed, flat background, much like how light waves travel through empty space. This approach allows them to ask a specific question: if we treat the geometry of space as a field with its own internal rules, can we identify a precise source of energy that behaves consistently, even when we change our perspective?

A team of researchers has taken a fresh look at this problem by introducing a new way of thinking about the internal symmetries of gravity. They worked within a framework known as Weyl gauge theory, which extends the standard models of gravity by including a specific type of symmetry related to stretching and shrinking. In their new approach, they treated the translations of space—moving from one point to another—not as movements of the coordinate grid itself, but as an internal redundancy, similar to how a clock can be reset without changing the time it actually tells. By introducing a compensating field to handle this redundancy, they were able to rewrite the equations of gravity in a way that separates the gravitational field from the matter it acts upon. This separation allowed them to identify a specific source term for the gravitational potential, a mathematical object that acts as the "fuel" for the gravitational field, without inventing any new physical laws or dynamics.

The researchers found that this internal source is not a new discovery but is actually the same mathematical object that has been used for decades to describe how matter influences geometry, just viewed through a different lens. They derived a balance law that describes how this gravitational source interacts with matter and with the field itself. Crucially, they showed that while this source is invariant under the internal rules they introduced, it does not automatically behave like a conserved quantity in the traditional sense. In other words, the total amount of this gravitational energy does not necessarily stay constant in the way that electric charge does, unless specific conditions are met. This distinction is vital: the team demonstrated that having a well-defined local source does not guarantee that it represents a universal, observer-independent energy density.

To test whether their new definition of gravitational energy made physical sense, the team applied it to several well-known scenarios. They first looked at the Schwarzschild solution, which describes the space around a non-rotating star or black hole. When they calculated the total energy of this system using their new method, they found that it matched the famous ADM mass, a standard value used by physicists to describe the total energy of a system in general relativity. This was a significant confirmation, showing that their abstract construction could reproduce the correct, known answer for a real astrophysical object. They also checked their method against a flat, expanding universe and against gravitational waves, finding that the energy density they calculated behaved consistently with established theories in those contexts as well.

However, the study also revealed limits to how far this definition can be stretched. When the researchers applied their method to a different, more complex theory of gravity that deviates from the standard model, they found that the calculated energy of a star-like object differed from its expected mass. The difference depended on how much the theory's parameters deviated from the standard model, specifically related to how space curves around matter. This result suggests that while the method works perfectly for the standard theory of gravity, it does not automatically provide a single, universal answer for every possible theory of gravity. The energy assigned to a system depends on the specific rules of the theory being used.

The paper concludes by clarifying the relationship between this new internal view and the traditional external view of gravity. The researchers showed that their approach does not contradict the standard understanding of general relativity but rather offers a different way to access the same underlying physics. They proved that their definition of gravitational energy is stable under certain mathematical changes, meaning that the total energy value remains consistent even if the description of the field is slightly altered. Yet, they also emphasized that this local energy density is not an absolute, observer-independent quantity. It remains tied to the specific way the gravitational field is split from the matter and the specific reference frame chosen. The work provides a rigorous, mathematically sound way to talk about local gravitational energy, but it also reinforces the idea that in the complex world of gravity, energy is a relational concept, defined by the interaction between the field, the matter, and the observer's perspective, rather than a fixed property that exists in isolation.

Technical Summary: Internal Translation Gauging and Local Gravitational Energy in Weyl Gauge Theory

Problem and Motivation
The localization of gravitational energy remains a subtle issue in gauge theories of gravity, particularly regarding the existence of strictly invariant local energy-momentum tensors. Previous work by the authors established that for linearized metric theories (Fierz-Pauli) and Weyl-squared gravity, no preferred strictly invariant local representative of the translation class exists under standard assumptions of locality and polynomiality. In those contexts, translations are realized externally as general coordinate transformations, with the tetrad acting as a soldering map rather than an internal gauge connection.

This paper investigates whether introducing translations as an internal Stueckelberg redundancy within Weyl Gauge Theory (WGT) alters this conclusion. Specifically, the authors ask if treating the translational potential as an internal gauge field on a fixed Minkowski background allows for the definition of a local gravitational source and charge that is strictly invariant under the internal gauge group, without introducing new dynamics.

Methodology
The authors work within the framework of Weyl Gauge Theory (WGT), which includes Lorentz, dilation, and translation sectors. They introduce an internal $T(4)$ symmetry acting on the tangent-space (fibre) coordinates $\xi^a$ via a compensator (Stueckelberg) field. The dressed coframe is defined as $b^a_\mu = D^*_\mu \xi^a + C^a_\mu$ , where $C^a_\mu$ is the translational potential and $D^*_\mu$ is the homogeneous covariant derivative.

Key methodological steps include:

Internal Gauging: Treating $C^a_\mu$ as an internal gauge potential. The compensator $\xi^a$ ensures that the dressed coframe $b^a_\mu$ and its field strengths (torsion $T^{*a}_{\mu\nu}$ , Lorentz curvature $R^{ab}_{\mu\nu}$ , and dilation strength $H_{\mu\nu}$ ) are strictly invariant under internal translations.
Variational Analysis: Deriving the field equations by varying the action with respect to the translational potential $C^a_\mu$ . Since $C^a_\mu$ enters the action only through the coframe, the source conjugate to $C^a_\mu$ is identified with the existing coframe variational derivative, $\delta L / \delta b^a_\mu$ .
Constitutive Families: Explicitly analyzing the parity-even quadratic torsion family, defined by the Lagrangian $L_g \propto \alpha A_1 + \beta A_2 + \gamma A_3$ , which includes the Teleparallel Equivalent of General Relativity (TEGR) as a specific point.
Charge Constructions: Developing two distinct sufficient conditions for defining scalar charges:
- Contraction of the source with a covariantly constant internal direction $n^a$ .
- Generalized reducibility of the full diffeomorphism-Lorentz-dilation configuration, yielding a three-sector Noether superpotential.

Key Results

Excitation/Source Equation: The authors derive a Yang-Mills-type source equation for the coframe: $(D^*_\nu + T^*_\nu) H^a_{\nu\mu} = J^a_\mu$ , where $H^a_{\nu\mu}$ is the translational excitation and $J^a_\mu$ is the total source (gravitational self-current plus matter source). In the translations-only sector, the total source is ordinarily conserved ( $\partial_\mu J^a_\mu = 0$ ). In the full theory with non-vanishing homogeneous curvatures, the source satisfies a balance law involving exchange terms with the Lorentz and dilation sectors.
Gravitational Self-Current: A local gravitational self-current $(j_g)^a_\mu$ is defined relative to a specified gravitational/matter split. Contracting this with the coframe yields a mixed coordinate density $b^a_\nu (j_g)^a_\mu$ that is strictly invariant under internal $T(4)$ transformations. However, the authors emphasize that this invariance does not imply ordinary conservation of the density, nor does it identify the density with the external translation Noether current.
Constitutive Dependence: The algebraic form of the self-current is common to the entire parity-even quadratic torsion family, though its magnitude depends on the constitutive coefficients $(\alpha, \beta, \gamma)$ .
Calibration at TEGR: At the TEGR point, the excitation reduces to the standard teleparallel superpotential. Evaluating the charge for the Schwarzschild solution using a covariant coframe-inertial connection pair yields the ADM mass $M$ without reference subtraction. The construction also reproduces the Friedmann balance for spatially flat cosmology and the Isaacson energy density for transverse-traceless waves.
Beyond TEGR:
- An axial deformation of the constitutive law (changing the torsion trace) leaves the Schwarzschild solution and its charge unchanged, as the solution possesses no totally antisymmetric torsion.
- For an exact solution in New General Relativity (NGR), the charge $Q$ differs from the Kepler mass $M$ . The fractional difference is found to be $Q/M - 1 = (\gamma_{PPN} - 1)/2$ , where $\gamma_{PPN}$ is the post-Newtonian spatial-curvature parameter. This links the charge assignment directly to the deviation of the constitutive parameters from the TEGR point.
Hamiltonian Stability: The authors prove that the integrable Hamiltonian charge is invariant under the addition of covariant Lagrangian boundary terms, provided the presymplectic representative and reference normalisation are fixed. This establishes "improvement stability" as a condition for physical admissibility.

Significance and Claims
The paper claims to establish a precise sense in which a local gravitational density is available in WGT when translations are treated as an internal redundancy. The construction:

Identifies the source of the translational potential with the existing coframe variational derivative, introducing no new dynamics.
Provides a strictly internal-gauge-invariant excitation/source system.
Reproduces known results (ADM energy, cosmological balance, wave density) at the TEGR point, validating the construction against established physics.
Clarifies the distinction between internal gauge invariance and external diffeomorphism invariance. The authors explicitly state that their construction does not identify the internal $T(4)$ with spacetime diffeomorphisms, nor does it resolve the metric no-go theorems regarding external translation classes.

The authors remain modest regarding the physical interpretation of the soldered density. They assert that while the density is gauge-invariant, it is not an observer-independent physical energy density, nor is its ordinary conservation guaranteed. The work defines a class of charges and sources but does not claim to exhaust all possible charges or to provide a unique, preferred energy observable independent of the specific constitutive choice and boundary representatives. The extension to extended Weyl Gauge Theory (eWGT) is noted as requiring separate treatment due to modifications in the translational sector.

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