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arXiv · 2605.05817

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

Callum Bell·David Sloan
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Abstract

Previous work has provided the mathematical framework within which to analyse dynamical similarities for classical theories of fields. This formalism has been extended to those theories which, in addition to scaling symmetries, also possess gauge degrees of freedom. In this article, we apply these ideas to the analysis of the first-order Palatini formulation of General Relativity. It is shown that the conformal mode of the spacetime metric may be identified as the generator of a dynamical similarity. Further, we demonstrate that the dynamical content of the Hilbert-Palatini action may be reformulated in terms of an action-dependent field theory, which makes no reference to the conformal mode. Finally, we consider the linearised limit of the equations of motion derived from the scale-reduced action. We find that, in the harmonic gauge, the first-order metric perturbations satisfy a free wave equation, as expected. However, the elimination of the conformal factor requires a qualitative reinterpretation of the physics at second order. Conventionally, one considers the second-order perturbations to be sourced by quadratic combinations of first-order terms. These are packaged into an object identified as an `effective stress-energy tensor'. This interpretation must be amended for the action-dependent theory, where the presence of terms that couple the action sector with the geometrical degrees of freedom shows that our construction is inherently non-conservative.

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Callum Bell, David Sloan. 2026-08-06. Classical General Relativity as a Non-Conservative Action-Dependent Field Theory. https://arxiv.org/abs/2605.05817

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