Gravitational multipole moments from Noether charges

TL;DR

<3-5 sentence high-level summary describing the problem, approach, contributions, and impact>: We address intrinsic definitions of gravitational mass and current multipole moments in arbitrary diffeomorphism-invariant theories by formulating them as canonical Noether charges associated with multipole symmetries in canonical harmonic gauge. The method reproduces Thorne's $I^{lm}$ and $S^{lm}$ in Einstein gravity and unifies near-zone source information with surface charges at spatial infinity and a regularization at null infinity, yielding a conservation law that relates source multipole variations to radiative flux $F^{+-}$ through null infinity. This Noether-charge framework generalizes multipole structure to broader gravity theories and provides a concrete, gauge-structured route to connect near-zone sources with far-zone radiation, with potential observational implications for gravitational-wave sources and no-hair tests.

Abstract

We define the mass and current multipole moments for an arbitrary theory of gravity in terms of canonical Noether charges associated with specific residual transformations in canonical harmonic gauge, which we call multipole symmetries. We show that our definition exactly matches Thorne's mass and current multipole moments in Einstein gravity, which are defined in terms of metric components. For radiative configurations, the total multipole charges -- including the contributions from the source and the radiation -- are given by surface charges at spatial infinity, while the source multipole moments are naturally identified by surface integrals in the near-zone or, alternatively, from a regularization of the Noether charges at null infinity. The conservation of total multipole charges is used to derive the variation of source multipole moments in the near-zone in terms of the flux of multipole charges at null infinity.

Figures (1)

• Figure 1: (Left) Two constant time slices $\Sigma_{\pm}$ both asymptote to the sphere $S^\infty$ at spatial infinity. The coloured region $r \leq a$, where $a$ is the size of the source, is not described by the linear solution and contains the source. The inner boundaries are denoted as $S^0_\pm$. (Right) One can smoothly deform $\Sigma_{\pm}$ such that the difference of $Q^{\text{rad}}_{\mathcal{I}^{+-}}$ between the two advanced times $u_+$ and $u_-$ represents the multipole moment flux through null infinity.