Classical Renormalization Group Equations for General Relativity

TL;DR

This paper establishes a rigorous theoretical foundation for a classical renormalization group approach to general relativity, enhancing the analysis of the two-body problem and gravitational waves.

Contribution

It rigorously derives the classical RG flow equation for gravity from a Legendre transform, confirming its validity and duality with the Polchinski equation.

Findings

Demonstrates the Legendre transform maps the Polchinski equation to the classical RG equation.

Establishes a duality between the classical RG flow and the effective gravitational action.

Solidifies the classical RG framework as a rigorous tool for gravitational physics.

Abstract

In a companion paper arXiv:2510.27676, we introduced a non-perturbative classical renormalisation group (RG) flow equation as a novel method for treating strongly interacting problems in general relativity, with a prominent application to the two-body problem. While we demonstrated that it reproduces perturbation theory, via the Post-Minkowskian (PM) expansion, and its computational efficiency in reproducing the 1PN Post-Newtonian action, its derivation was heuristic. In this work, we place this flow equation on a firm formal foundation. In particular, we demonstrate that a Legendre transform maps the classical analogue of the Polchinski equation precisely to our classical RG equation. This establishes a duality between equivalent, exact RG equations for the gravitational effective action. The result, combined with the successful applications in arXiv:2510.27676, solidifies the classical RG framework as a powerful and rigorous new approach to the general relativistic two-body problem and gravitational wave physics.