Renormalisation Group approach to General Relativity

Abstract

The detection of gravitational waves has intensified the need for efficient, high-precision modeling of the two-body problem in General Relativity. Current analytical methods, primarily the Post-Minkowskian and Post-Newtonian expansions, are inherently perturbative, while numerical relativity remains computationally expensive. In this Letter, we introduce a middle path: an exact renormalization group (RG) equation for classical gravitational systems. We demonstrate that our equation correctly reproduces the first three orders of the Post-Minkowskian expansion. Crucially, it provides a framework for non-perturbative approximations. As a first application, we show that our method efficiently recovers the 1PN two-body action, bypassing the need for complex three-graviton vertex calculations by leveraging the intrinsic nonlinearity of the RG flow. This establishes the exact RG as a powerful new tool for tackling strong-field dynamics in gravity.

Figures (2)

• Figure 1: Diagrammatic representation of the functional RG equation for GR. Each big dot over the double lines represent the first functional derivative of the running effective action $S_k$ for the two-particle system. The thin line corresponds to the regularized propagator $G_k$, and the dot above indicates the derivative $\partial_k$ acting on it.
• Figure 2: Diagrammatic conventions used in the PM expansion. Thick double lines denote the worldlines of the sources and the single dashed lines denote graviton propagators