Classical General Relativity as a Non-Conservative Action-Dependent Field Theory

Abstract: Scaling symmetries have previously been examined for classical field theories described by singular Lagrangians; in this article, we apply these results to the first-order formulation of General Relativity. It is shown that the dynamical content of the Hilbert action may be formulated in terms of the conformal spacetime geometry, together with a dissipative sector, which is required in order to compensate the elimination of the notion of scale encoded by the conformal factor. Further, we consider the linearisation of the equations of motion of the scale-invariant action, demonstrating that the first-order metric perturbations satisfy a free wave equation, as expected. The second-order dynamics, describing gravitational backreaction, are found to be sourced by quadratic combinations of the first-order perturbations. However, these dynamics are non-conservative, as is made manifest by the presence of terms which couple the action sector with the geometrical degrees of freedom.

• The paper demonstrates that reformulating GR by eliminating the conformal mode yields a non-conservative, action-dependent theory dynamically equivalent to standard GR.
• It employs multisymplectic and contact geometric frameworks with Herglotz-type Lagrangians to systematically reduce gauge redundancy and modify conservation laws.
• The weak-field analysis confirms the preservation of two graviton polarizations while introducing dissipative corrections and non-conservative dynamics in gravitational wave propagation.

Reformulating General Relativity via Non-Conservative Action-Dependent Field Theory

Introduction and Motivation

The paper "Classical General Relativity as a Non-Conservative Action-Dependent Field Theory" (2605.05817) develops a reformulation of classical General Relativity (GR) in which the conformal degree of freedom associated with the spacetime metric is systematically eliminated as physically redundant. This is achieved through the framework of scaling symmetry and non-conservative variational principles, notably those inspired by Herglotz's generalization of the classical action principle. The resulting theory, although dynamically equivalent to standard GR, is recast as a fundamentally action-dependent, non-conservative field theory with reduced gauge symmetry.

The motivation for this reformulation is rooted in a combination of mathematical and philosophical considerations. From a physical perspective, the conformal factor—encoding local spacetime scale—is empirically inaccessible and introduces a gauge redundancy. Philosophically, the approach aligns with Leibniz's Principle of the Identity of Indiscernibles, advocating the elimination of unobservable degrees of freedom. This leads to a representation of GR that only depends on conformal geometry and is supplemented by a "dissipative" action-density sector necessitated by the symmetry reduction.

Geometrical and Variational Formalism

The authors employ the multisymplectic and multicontact geometric frameworks for field theory, with a careful treatment of the jet bundles and Lagrangian densities. The core procedure leverages dynamical scaling symmetries: when the Lagrangian admits such a symmetry, the canonical variables can be reduced by quotienting out the scale direction—a process termed contact reduction. Mathematically, this leads from a multisymplectic formulation to a contact geometry, in which the configuration space is extended to include components of the action density as new dynamical variables. This transition is formalized using Herglotz-type Lagrangians, yielding intrinsically non-conservative dynamics.

In the context of GR, the first-order Palatini formalism is chosen for technical reasons: the reduction is naturally formulated on the first jet bundle, and the frame (vielbein) + spin connection variables are more amenable to explicit decomposition and symmetry analysis than the metric alone. The tetrad fields are decomposed into a conformal factor and a unimodular part. The scaling symmetry associated with rescalings of the conformal factor is then used to effect the reduction, resulting in a theory with variables $\tilde{e}_\mu^I$ (the unimodular tetrads) and a vector-valued action density $s^\mu$.

Construction of the Action-Dependent Theory

Applying contact reduction to the Hilbert-Palatini action, the conformal mode $\phi$ is eliminated, with its derivatives replaced by the components of $s^\mu$. The resulting multicontact Herglotz Lagrangian is structurally more elaborate, but its first term is a manifestly scale-invariant adaptation of the Palatini action, while additional terms encode quadratic and higher-order couplings in $s^\mu$, typically of dissipative character.

Notably, the dynamical content of standard GR is exactly retained: the new degrees of freedom (the action density) are not independent propagating fields but serve to compensate for the removal of the conformal mode, enforcing a non-conservative structure in the evolution equations. The formalism is shown to reproduce the original Einstein field equations modulo the elimination of the volume/scale.

A passage to the metric (second-order) formalism gives a particularly transparent Lagrangian:

$$L^H = \frac{1}{2\kappa}\tilde{R}(G) - \frac{\kappa}{3}G_{\mu\nu}s^\mu s^\nu$$

where $G_{\mu\nu}$ has fixed determinant and $\tilde{R}(G)$ is its Ricci scalar. The Herglotz constraint enforces that $\partial_\mu s^\mu = L^H$.

Gauge Symmetry and Field Equations

A key consequence of the reduction is that the gauge symmetry of the resulting field theory is diminished: diffeomorphism invariance is reduced to unimodular diffeomorphisms (volume-preserving). The loss of full Diff$(M)$ symmetry is compensated by the explicit presence of the action-density variables, which absorb the otherwise gauge-trivial dynamics associated with scale.

The frictional (action-dependent) field equations derived from the Herglotz Lagrangian closely track those of GR, but with non-conservative modifications. Specifically, at the level of field equations, conservation laws (such as energy-momentum conservation in the gravitational sector) are replaced by evolution equations expressing the interaction between geometric and action-density degrees of freedom. This is a sharp deviation from the standard conserved current structure of GR; however, since the non-conservativity arises from the reorganization and redistribution of constraints and gauge, no new physics is introduced.

Weak-Field Analysis and Propagation

The paper provides a detailed analysis of the linearized (weak-field) limit around Minkowski space. In this regime:

• The first-order metric perturbations in the scale-invariant theory obey a free wave equation as in standard GR.
• The gauge symmetry is now restricted to unimodular diffeomorphisms, constraining the possible gauge choices (e.g., fully fixing the harmonic gauge as in ordinary GR is not possible, but an adapted gauge fixing exists).
• The counting of physical degrees of freedom is unchanged: two propagating graviton polarizations remain, with residual gauge and dynamic constraints enforcing this.
• At second order, describing gravitational wave backreaction, the theory exhibits genuinely non-conservative dynamics: the quadratic contributions to the field equations involve detailed couplings between the metric perturbations and action-density sector. The absence of an effective conserved energy-momentum tensor for gravitational waves necessitates a new interpretation of energy exchange in the system.

Theoretical and Practical Implications

The presented formalism provides a mathematically robust and physically well-motivated avenue for re-expressing GR as a non-conservative, scale-invariant field theory. This has several implications:

• Redundant degrees of freedom are formally excised in alignment with foundational philosophical principles, potentially clarifying the ontology of gravity.
• The action-dependent (dissipative) structure offers a potentially advantageous framework for handling cosmological and black hole singularities. Prior work indicates that, in certain symmetry-reduced spacetimes, the non-conservative theory avoids breakdowns at singularities, permitting predictive evolution.
• The approach draws a sharp contrast with Unimodular Gravity: while both pass to a fixed-determinant metric, only the action-dependent reduction maintains dynamical equivalence with GR, as the compensation via $s^\mu$0 restores the full set of equations.
• The formalism is compatible with standard matter couplings and extends naturally to minimally coupled scalar fields, with possible applications in cosmology and early universe dynamics.

Potential future directions include:

• Systematic study of initial singularities and their resolution in the action-dependent theory.
• canonical analysis and $s^\mu$1 decomposition of the non-conservative theory, possibly simplifying constraint analysis.
• Extension to quantum gravity proposals where redundant degrees of freedom complicate canonical quantization.

Conclusion

This work establishes a coherent and technically rigorous reformulation of classical General Relativity as a non-conservative, action-dependent field theory via contact reduction and the Herglotz variational approach (2605.05817). The framework achieves the elimination of empirically intractable scale degrees of freedom, maintaining the core dynamical content of GR through intrinsic non-conservativity. The analysis clarifies the interplay between gauge symmetry, energy conservation, and the physical degrees of freedom of the gravitational field. These results invite further mathematical and physical exploration, particularly in the contexts of cosmological singularities, gravitational wave non-linearity, and the ongoing development of generally covariant action-dependent theories.