Classical Renormalization Group Equations for General Relativity

A. Codello; F. Guti\'errez; K. Falls

arxiv: 2605.22037 · v1 · pith:J6WSRUTGnew · submitted 2026-05-21 · 🌀 gr-qc · hep-th· math-ph· math.MP

Classical Renormalization Group Equations for General Relativity

F. Guti\'errez , K. Falls , A. Codello This is my paper

Pith reviewed 2026-05-22 05:56 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath-phmath.MP

keywords classical renormalization groupgeneral relativityLegendre transformPolchinski equationeffective actiontwo-body problemgravitational waves

0 comments

The pith

A Legendre transform maps the classical Polchinski equation exactly onto the classical RG equation for gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper supplies a formal derivation for a classical renormalization group flow equation previously introduced for general relativity. It demonstrates that a Legendre transform converts the classical analogue of the Polchinski equation into the authors' RG equation with no remainder. The result creates an exact duality between two equivalent flow equations for the gravitational effective action. A reader would care because the mapping turns a heuristic tool into a rigorously justified method for non-perturbative work on strongly coupled gravitational systems such as the two-body problem.

Core claim

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.

What carries the argument

Legendre transform that maps the classical Polchinski equation precisely onto the classical RG equation for the gravitational effective action

Load-bearing premise

A classical analogue of the Polchinski equation is well-defined in the gravitational setting and the Legendre transform introduces neither extra regularization nor boundary terms that would alter the resulting flow equation.

What would settle it

Deriving the classical RG equation from the Polchinski analogue via the Legendre transform in a concrete truncation such as the two-body effective action and obtaining an exact match would confirm the claim; any mismatch would falsify it.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They justify their heuristic classical RG flow for GR by Legendre-transforming a Polchinski analogue into it, but the derivation leaves open whether diffeomorphism invariance and non-compactness introduce uncancelled boundary or constraint terms.

read the letter

The main takeaway is that this paper supplies the missing formal step for the classical RG equation introduced heuristically in their companion work. They show that a Legendre transform converts the classical version of the Polchinski equation into their flow equation for the gravitational effective action, creating an exact duality between the two. That move turns the earlier heuristic into something derived rather than assumed. Combined with the companion paper's checks that the flow reproduces the Post-Minkowskian expansion and the 1PN action, the framework now rests on a clearer foundation for treating strongly coupled classical problems in GR, particularly the two-body system. The approach stays entirely classical, which sidesteps many quantum RG complications and keeps the focus on practical computations for gravitational-wave physics. The soft spot is the handling of the transform itself. Because the theory is diffeomorphism invariant and lives on non-compact spacetimes, the Legendre transform with respect to the metric source can generate surface terms at infinity or contributions tied to the Hamiltonian and momentum constraints. The abstract states the mapping is precise but gives no intermediate equations or explicit cancellation arguments. If those terms survive, the claimed exact equivalence would require additional regularization that is not mentioned. The stress-test note flags exactly this possibility, and without seeing the detailed steps it is hard to judge whether the duality holds without qualification. This work is aimed at researchers already using effective-field-theory methods or post-Minkowskian expansions for binary systems. A reader looking for a new non-perturbative classical tool would find the justification and the prior applications useful. The paper deserves peer review so that referees can examine the transform derivation and check whether boundary contributions are properly controlled.

Referee Report

1 major / 2 minor

Summary. The manuscript derives the authors' classical renormalization group (RG) flow equation for general relativity by applying a Legendre transform to the classical analogue of the Polchinski equation. This is presented as placing the heuristic RG equation from the companion paper on a rigorous footing and establishing an exact duality between two equivalent RG equations for the gravitational effective action.

Significance. If the mapping is shown to be exact, the work would strengthen the classical RG framework as a non-perturbative tool for GR, particularly for the two-body problem and gravitational-wave applications, by connecting it directly to the established Polchinski equation while building on the companion paper's reproductions of Post-Minkowskian and Post-Newtonian results.

major comments (1)

[Legendre transform derivation] The central claim of a precise, term-by-term mapping under the Legendre transform is load-bearing for the duality. In the derivation (the section performing the transform on the cutoff-dependent classical action), the manuscript must explicitly demonstrate that no residual surface terms at spatial infinity, cutoff-scale boundaries, or contributions from the Hamiltonian and momentum constraints survive in the diffeomorphism-invariant, non-compact setting. Any such terms would prevent the two flow equations from being exactly dual.

minor comments (2)

[Abstract] The abstract asserts an 'exact mapping' without indicating the key intermediate steps or the explicit form of the classical Polchinski analogue; a short parenthetical reference to the main equation would improve clarity.
[Introduction] Notation for the cutoff scale and the source term in the Legendre transform should be introduced with a brief reminder of its relation to the companion paper to aid readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a key point that requires clarification to solidify the claimed exact duality. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: The central claim of a precise, term-by-term mapping under the Legendre transform is load-bearing for the duality. In the derivation (the section performing the transform on the cutoff-dependent classical action), the manuscript must explicitly demonstrate that no residual surface terms at spatial infinity, cutoff-scale boundaries, or contributions from the Hamiltonian and momentum constraints survive in the diffeomorphism-invariant, non-compact setting. Any such terms would prevent the two flow equations from being exactly dual.

Authors: We agree that an explicit demonstration is necessary to confirm the absence of residual terms and thereby establish the exact duality. In the current derivation, the Legendre transform is applied to the cutoff-dependent classical action under the assumption of asymptotic flatness and a cutoff function with appropriate decay properties at spatial infinity. However, to directly address the referee's concern, we will revise the relevant section to include a dedicated calculation. This will show that surface terms at spatial infinity and at cutoff-scale boundaries integrate to zero by virtue of the compact support of the cutoff in momentum space combined with the fall-off conditions on the metric and its conjugate momentum. We will further demonstrate that the Hamiltonian and momentum constraints do not generate additional contributions because they are preserved by the classical flow and the transform is performed in a manner that respects the on-shell equivalence in the diffeomorphism-invariant setting. These additions will make the term-by-term mapping fully rigorous without residual terms. revision: yes

Circularity Check

0 steps flagged

Derivation via Legendre transform from classical Polchinski analogue is independent and self-contained

full rationale

The paper's central step is to show that a Legendre transform applied to the classical analogue of the Polchinski equation yields their classical RG flow equation for the gravitational effective action. This is presented as a new formal derivation that places the earlier heuristic introduction (from the companion paper) on rigorous footing. The Polchinski equation itself is an external, standard reference in the RG literature and is not defined in terms of the target equation. No quoted equations in the abstract or context reduce the claimed mapping to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation chain therefore remains independent of the authors' prior postulate and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a classical Polchinski equation analogue whose precise definition is not supplied in the abstract, plus the technical validity of the Legendre transform in the gravitational effective-action setting.

axioms (2)

domain assumption A classical analogue of the Polchinski equation exists and is well-defined for the gravitational effective action.
Invoked when the abstract refers to 'the classical analogue of the Polchinski equation' without further definition.
domain assumption The Legendre transform can be performed on the effective action without introducing extra terms that would change the resulting flow equation.
Required for the claimed precise mapping.

pith-pipeline@v0.9.0 · 5706 in / 1364 out tokens · 26012 ms · 2026-05-22T05:56:57.278515+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 14 internal anchors

[1]

˙Gab( 1 κ S b −1+S b 0+κS b 1+κ 2S b 2)+O(κ 3). This expression can be reorganised as 1 κ k∂kS−1+k∂ kS0+κ k∂ kS1+κ 2k∂kS2 =−1 2κ S a −1˙Gab S b −1−Sa 0 ˙Gab S b −1−κ 2 S a 0 ˙Gab S b 0 −κSa 1 ˙Gab S b −1−κ2 S a 0 ˙Gab S b 1−κ2 S a −1˙Gab S b 2+O(κ 3).(A1) Calculation ofS −1 Projecting theκ −1sector of (A1) gives k∂kS−1=−1 2 S a −1˙Gab S b −1.(A2) We denot...

work page
[2]

Einstein, L

A. Einstein, L. Infeld and B. Hoffmann,The Gravitational Equations and the Problem of Motion, Annals Math.39(1938), 65–100, doi:10.2307/1968714

work page doi:10.2307/1968714 1938
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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

L. Blanchet,Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel.5(2002), 3, [arXiv:gr-qc/0202016]

work page internal anchor Pith review Pith/arXiv arXiv 2002
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3rd post-Newtonian higher order Hamilton dynamics for two-body point-mass systems

P. Jaranowski and G. Sch¨ afer,Third post-Newtonian higher order ADM Hamilton dynamics for two- body point-mass systems, Phys. Rev. D57(1998), 7274–7291, [arXiv:gr-qc/9712075]

work page internal anchor Pith review Pith/arXiv arXiv 1998
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Poisson and C

E. Poisson and C. M. Will,Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge University Press, Cambridge (2014)

work page 2014
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Guti´ errez, K

F. Guti´ errez, K. Falls and A. Codello,Renormalisation Group approach to General Relativity, [arXiv:2510.27676 [gr-qc]]

work page arXiv
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Exact evolution equation for the effective potential

C. Wetterich,Exact evolution equation for the effective potential, Phys. Lett. B301(1993), 90-94 [arXiv:1710.05815 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 1993
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T. R. Morris,The Exact renormalization group and approximate solutions, Int. J. Mod. Phys. A9 (1994), 2411-2450, [arXiv:hep-ph/9308265 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 1994
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Dupuis, L

N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier and N. Wschebor,The nonperturbative functional renormalization group and its applications, Phys. Rept.910(2021), 1-114 [arXiv:2006.04853 [cond-mat.stat-mech]]

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(02)02693-x 2002
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D. F. Litim and J. M. Pawlowski,Completeness and consistency of renormalisation group flows, Phys. Rev. D66, 025030 (2002), doi:10.1103/PhysRevD.66.025030, arXiv:hep-th/0202188

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.66.025030 2002
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D. F. Litim and J. M. Pawlowski,Predictive power of renormalization group flows: A Comparison, Phys. Lett. B516, 197–207 (2001), doi:10.1016/S0370-2693(01)00922-4, arXiv:hep-th/0107020

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(01)00922-4 2001
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Polchinski,Renormalization and Effective Lagrangians, Nucl

J. Polchinski,Renormalization and Effective Lagrangians, Nucl. Phys. B231(1984), 269-295 doi:10.1016/0550-3213(84)90287-6

work page doi:10.1016/0550-3213(84)90287-6 1984
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T. R. Morris and Z. H. Slade,Solutions to the reconstruction problem in asymptotic safety, JHEP11 (2015), 094 [arXiv:1507.08657 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2015
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Guti´ errez, K

F. Guti´ errez, K. Falls and A. Codello,Post-Newtonian Derivative Expansion, In preparation

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Post-Newtonian Theory for Gravitational Waves

L. Blanchet,Post-Newtonian Theory for Gravitational Waves, Living Rev. Rel.27(2024), 4, [arXiv:1310.1528 [gr-qc]]

work page internal anchor Pith review Pith/arXiv arXiv 2024
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Effective one-body approach to general relativistic two-body dynamics

A. Buonanno and T. Damour,Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D59(1999), 084006, [arXiv:gr-qc/9811091]

work page internal anchor Pith review Pith/arXiv arXiv 1999
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Emergence of Calabi-Yau manifolds in high-precision black hole scattering

M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer and J. Usovitsch, Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature641(2025) no.8063, 603-607, [arXiv:2411.11846 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2025
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R. A. Porto,The Effective Field Theorist’s Approach to Gravitational Dynamics, Phys. Rept.633 (2016), 1–104, [arXiv:1601.04914 [gr-qc]]

work page internal anchor Pith review Pith/arXiv arXiv 2016
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Binary black holes in circular orbits. I. A global spacetime approach

E. Gourgoulhon, P. Grandcl´ ement and S. Bonazzola,Binary black holes in circular orbits. 1. A global space-time approach, Phys. Rev. D65(2002), 044020 [arXiv:gr-qc/0106015]

work page internal anchor Pith review Pith/arXiv arXiv 2002
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Binary black holes in circular orbits. II. Numerical methods and first results

P. Grandcl´ ement, E. Gourgoulhon and S. Bonazzola,Binary black holes in circular orbits. 2. Numerical methods and first results, Phys. Rev. D65(2002), 044021 [arXiv:gr-qc/0106016]

work page internal anchor Pith review Pith/arXiv arXiv 2002
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Balog, H

I. Balog, H. Chat´ e, B. Delamotte, M. Marohnic and N. Wschebor,Convergence of Nonpertur- bative Approximations to the Renormalization Group, Phys. Rev. Lett.123(2019) no.24, 240604 doi:10.1103/PhysRevLett.123.240604 [arXiv:1907.01829 [cond-mat.stat-mech]]

work page doi:10.1103/physrevlett.123.240604 2019

[1] [1]

˙Gab( 1 κ S b −1+S b 0+κS b 1+κ 2S b 2)+O(κ 3). This expression can be reorganised as 1 κ k∂kS−1+k∂ kS0+κ k∂ kS1+κ 2k∂kS2 =−1 2κ S a −1˙Gab S b −1−Sa 0 ˙Gab S b −1−κ 2 S a 0 ˙Gab S b 0 −κSa 1 ˙Gab S b −1−κ2 S a 0 ˙Gab S b 1−κ2 S a −1˙Gab S b 2+O(κ 3).(A1) Calculation ofS −1 Projecting theκ −1sector of (A1) gives k∂kS−1=−1 2 S a −1˙Gab S b −1.(A2) We denot...

work page

[2] [2]

Einstein, L

A. Einstein, L. Infeld and B. Hoffmann,The Gravitational Equations and the Problem of Motion, Annals Math.39(1938), 65–100, doi:10.2307/1968714

work page doi:10.2307/1968714 1938

[3] [4]

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

L. Blanchet,Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel.5(2002), 3, [arXiv:gr-qc/0202016]

work page internal anchor Pith review Pith/arXiv arXiv 2002

[4] [5]

3rd post-Newtonian higher order Hamilton dynamics for two-body point-mass systems

P. Jaranowski and G. Sch¨ afer,Third post-Newtonian higher order ADM Hamilton dynamics for two- body point-mass systems, Phys. Rev. D57(1998), 7274–7291, [arXiv:gr-qc/9712075]

work page internal anchor Pith review Pith/arXiv arXiv 1998

[5] [6]

Poisson and C

E. Poisson and C. M. Will,Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge University Press, Cambridge (2014)

work page 2014

[6] [7]

Guti´ errez, K

F. Guti´ errez, K. Falls and A. Codello,Renormalisation Group approach to General Relativity, [arXiv:2510.27676 [gr-qc]]

work page arXiv

[7] [8]

Exact evolution equation for the effective potential

C. Wetterich,Exact evolution equation for the effective potential, Phys. Lett. B301(1993), 90-94 [arXiv:1710.05815 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 1993

[8] [9]

T. R. Morris,The Exact renormalization group and approximate solutions, Int. J. Mod. Phys. A9 (1994), 2411-2450, [arXiv:hep-ph/9308265 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv 1994

[9] [10]

Dupuis, L

N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier and N. Wschebor,The nonperturbative functional renormalization group and its applications, Phys. Rept.910(2021), 1-114 [arXiv:2006.04853 [cond-mat.stat-mech]]

work page arXiv 2021

[10] [11]

D. F. Litim and J. M. Pawlowski,Wilsonian flows and background fields, Phys. Lett. B546, 279–286 (2002), doi:10.1016/S0370-2693(02)02693-X, arXiv:hep-th/0208216. 14

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(02)02693-x 2002

[11] [12]

D. F. Litim and J. M. Pawlowski,Completeness and consistency of renormalisation group flows, Phys. Rev. D66, 025030 (2002), doi:10.1103/PhysRevD.66.025030, arXiv:hep-th/0202188

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.66.025030 2002

[12] [13]

D. F. Litim and J. M. Pawlowski,Predictive power of renormalization group flows: A Comparison, Phys. Lett. B516, 197–207 (2001), doi:10.1016/S0370-2693(01)00922-4, arXiv:hep-th/0107020

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(01)00922-4 2001

[13] [14]

Polchinski,Renormalization and Effective Lagrangians, Nucl

J. Polchinski,Renormalization and Effective Lagrangians, Nucl. Phys. B231(1984), 269-295 doi:10.1016/0550-3213(84)90287-6

work page doi:10.1016/0550-3213(84)90287-6 1984

[14] [15]

T. R. Morris and Z. H. Slade,Solutions to the reconstruction problem in asymptotic safety, JHEP11 (2015), 094 [arXiv:1507.08657 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2015

[15] [16]

Guti´ errez, K

F. Guti´ errez, K. Falls and A. Codello,Post-Newtonian Derivative Expansion, In preparation

work page

[16] [17]

Post-Newtonian Theory for Gravitational Waves

L. Blanchet,Post-Newtonian Theory for Gravitational Waves, Living Rev. Rel.27(2024), 4, [arXiv:1310.1528 [gr-qc]]

work page internal anchor Pith review Pith/arXiv arXiv 2024

[17] [18]

Effective one-body approach to general relativistic two-body dynamics

A. Buonanno and T. Damour,Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D59(1999), 084006, [arXiv:gr-qc/9811091]

work page internal anchor Pith review Pith/arXiv arXiv 1999

[18] [19]

Emergence of Calabi-Yau manifolds in high-precision black hole scattering

M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer and J. Usovitsch, Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature641(2025) no.8063, 603-607, [arXiv:2411.11846 [hep-th]]

work page internal anchor Pith review Pith/arXiv arXiv 2025

[19] [20]

R. A. Porto,The Effective Field Theorist’s Approach to Gravitational Dynamics, Phys. Rept.633 (2016), 1–104, [arXiv:1601.04914 [gr-qc]]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[20] [21]

Binary black holes in circular orbits. I. A global spacetime approach

E. Gourgoulhon, P. Grandcl´ ement and S. Bonazzola,Binary black holes in circular orbits. 1. A global space-time approach, Phys. Rev. D65(2002), 044020 [arXiv:gr-qc/0106015]

work page internal anchor Pith review Pith/arXiv arXiv 2002

[21] [22]

Binary black holes in circular orbits. II. Numerical methods and first results

P. Grandcl´ ement, E. Gourgoulhon and S. Bonazzola,Binary black holes in circular orbits. 2. Numerical methods and first results, Phys. Rev. D65(2002), 044021 [arXiv:gr-qc/0106016]

work page internal anchor Pith review Pith/arXiv arXiv 2002

[22] [23]

Balog, H

I. Balog, H. Chat´ e, B. Delamotte, M. Marohnic and N. Wschebor,Convergence of Nonpertur- bative Approximations to the Renormalization Group, Phys. Rev. Lett.123(2019) no.24, 240604 doi:10.1103/PhysRevLett.123.240604 [arXiv:1907.01829 [cond-mat.stat-mech]]

work page doi:10.1103/physrevlett.123.240604 2019