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Balance flux laws beyond general relativity

David Maibach, Jann Zosso

TL;DR

This work develops a non-perturbative framework to derive balance flux laws at null infinity for beyond-GR theories, focusing on luminal Horndeski gravity. Using the covariant phase space formalism and Wald–Zoupas, the authors derive a general flux expression that cleanly splits into a GR-like metric part and a Horndeski scalar contribution, and validate it by Brans–Dicke gravity as a concrete instance. They show that the cross terms vanish in the asymptotic limit, pull back to $ scri^+$, and connect the flux to gravitational memory, including a scalar-driven component, with explicit formulas for memory in terms of the Bondi news and shear. The results provide a practical, implementable guide for computing balance laws in generic diffeomorphism-invariant theories and establish a bridge between non-perturbative covariant phase space methods and the Isaacson separation-of-scales approach, enabling more robust beyond-GR waveform analyses and memory studies.

Abstract

Balance flux laws of asymptotic symmetries in general relativity provide fully non-perturbative constraint equations on gravitational strain. They have proven useful for constructing numerical gravitational waveforms and for characterizing gravitational memory. As the precision of current and future detectors continues to improve, such constraints become increasingly important for high-precision tests of gravity, including searches for deviations from general relativity. This motivates a systematic understanding of analogous balance laws in theories beyond general relativity. In this work, we investigate the existence and structure of flux laws at null infinity in diffeomorphism-invariant extensions of general relativity. Our analysis is based on the covariant phase space formalism and the definition of conserved quantities, as presented by Wald and Zoupas. For a particularly relevant class of Horndeski theories, we derive a general expression for the flux and formulate the corresponding balance equation via the associated non-conserved charges. We cross-check our general results by comparing them with previous studies of Brans-Dicke gravity. Furthermore, we demonstrate that the employed methods extend straightforwardly to a broader class of diffeomorphism-invariant theories. The null part of the resulting flux laws associated with null memory is compared with and validated against the alternative derivation based on the Isaacson approach to gravitational radiation. Beyond the specific results obtained, this work is intended to serve as a practical guide for computing balance laws in generic diffeomorphism-invariant theories of gravity and paves the way for an in-depth comparison between the Isaacson approach and the covariant phase space formalism.

Balance flux laws beyond general relativity

TL;DR

This work develops a non-perturbative framework to derive balance flux laws at null infinity for beyond-GR theories, focusing on luminal Horndeski gravity. Using the covariant phase space formalism and Wald–Zoupas, the authors derive a general flux expression that cleanly splits into a GR-like metric part and a Horndeski scalar contribution, and validate it by Brans–Dicke gravity as a concrete instance. They show that the cross terms vanish in the asymptotic limit, pull back to , and connect the flux to gravitational memory, including a scalar-driven component, with explicit formulas for memory in terms of the Bondi news and shear. The results provide a practical, implementable guide for computing balance laws in generic diffeomorphism-invariant theories and establish a bridge between non-perturbative covariant phase space methods and the Isaacson separation-of-scales approach, enabling more robust beyond-GR waveform analyses and memory studies.

Abstract

Balance flux laws of asymptotic symmetries in general relativity provide fully non-perturbative constraint equations on gravitational strain. They have proven useful for constructing numerical gravitational waveforms and for characterizing gravitational memory. As the precision of current and future detectors continues to improve, such constraints become increasingly important for high-precision tests of gravity, including searches for deviations from general relativity. This motivates a systematic understanding of analogous balance laws in theories beyond general relativity. In this work, we investigate the existence and structure of flux laws at null infinity in diffeomorphism-invariant extensions of general relativity. Our analysis is based on the covariant phase space formalism and the definition of conserved quantities, as presented by Wald and Zoupas. For a particularly relevant class of Horndeski theories, we derive a general expression for the flux and formulate the corresponding balance equation via the associated non-conserved charges. We cross-check our general results by comparing them with previous studies of Brans-Dicke gravity. Furthermore, we demonstrate that the employed methods extend straightforwardly to a broader class of diffeomorphism-invariant theories. The null part of the resulting flux laws associated with null memory is compared with and validated against the alternative derivation based on the Isaacson approach to gravitational radiation. Beyond the specific results obtained, this work is intended to serve as a practical guide for computing balance laws in generic diffeomorphism-invariant theories of gravity and paves the way for an in-depth comparison between the Isaacson approach and the covariant phase space formalism.
Paper Structure (34 sections, 124 equations, 1 figure)

This paper contains 34 sections, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Intuition and Notation
  3. Symplectic structure in classical mechanics
  4. Covariant phase space formalism
  5. From mechanical phase space to covariant phase space
  6. Symplectic form
  7. Relation to Noether current
  8. Radiative boundaries and Wald--Zoupas formalism
  9. Balance laws of luminal Horndeski gravity
  10. Luminal Horndeski gravity
  11. Symplectic potential and current
  12. The Wald--Zoupas flux in the limit to $\mathscr{I}^{+}$
  13. Asymptotics in luminal Horndeski theory
  14. Pulling back to $\mathscr{I}^{+}$
  15. Conformal completion and field expansions.
  16. ...and 19 more sections

Figures (1)

  • Figure 1: Sketch of $\mathscr{I}^{+}$ as the future boundary of the Penrose diagram of Minkowski spacetimes Maibach:2025iku. In red, we denote cross sections of constant $u$. The spiral in the center of the diagram represents a event emitting gravitational waves in null direction toward $\mathscr{I}^{+}$.