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Infrared Universality: The $r^{-3}$ Spectral Threshold for Coupled Gravitational and Electromagnetic Fields

Michael Wilson

TL;DR

This work identifies the $r^{-3}$ curvature-decay rate as a universal infrared boundary for Laplace-type operators in four-dimensional, asymptotically flat spacetimes, governing the transition between localized and delocalized perturbations of the coupled Einstein–Maxwell system. The authors prove essential self-adjointness of the linearized operator and show that curvature and field derivatives decaying faster than $r^{-3}$ produce relatively compact perturbations, while exact $r^{-3}$ decay yields a delocalized zero mode, placing $0$ in the essential spectrum. The analysis unifies infrared behavior across spin-1, spin-2, and mixed spin-$1\oplus2$ sectors, and connects spectral delocalization to memory phenomena, with numerical simulations reproducing the predicted quadrupolar gravitational and dipolar electromagnetic memory patterns. This spectral perspective complements asymptotic-symmetry and soft-theorem approaches, offering a geometric origin for infrared universality and guiding future studies on memory in radiative regimes and more general backgrounds.

Abstract

We identify the $r^{-3}$ curvature-decay rate as a universal geometric threshold separating compact from non-compact perturbations of Laplace-type operators on asymptotically flat manifolds. For the coupled Einstein--Maxwell system, we prove that the linearized operator $\mathcal{L}$ is essentially self-adjoint and that curvature and field strengths decaying faster than $r^{-3}$ act as relatively compact perturbations, while decay exactly at $r^{-3}$ places $0\inσ_{\mathrm{ess}}(\mathcal{L})$ through delocalized zero modes. This threshold mechanism unifies the infrared behavior of spin-1, spin-2, and mixed spin-$(1\oplus2)$ fields, linking the onset of spectral delocalization with the appearance of gravitational and electromagnetic memory. Finite-difference simulations corroborate the analytic scaling and reproduce the characteristic quadrupolar and dipolar sky maps predicted for the coupled memory fields. These results demonstrate that curvature decay at $r^{-3}$ constitutes a fundamental geometric boundary underlying infrared universality in gauge and gravitational theories, providing a spectral counterpart to the asymptotic-symmetry and soft-theorem formulations of memory.

Infrared Universality: The $r^{-3}$ Spectral Threshold for Coupled Gravitational and Electromagnetic Fields

TL;DR

This work identifies the curvature-decay rate as a universal infrared boundary for Laplace-type operators in four-dimensional, asymptotically flat spacetimes, governing the transition between localized and delocalized perturbations of the coupled Einstein–Maxwell system. The authors prove essential self-adjointness of the linearized operator and show that curvature and field derivatives decaying faster than produce relatively compact perturbations, while exact decay yields a delocalized zero mode, placing in the essential spectrum. The analysis unifies infrared behavior across spin-1, spin-2, and mixed spin- sectors, and connects spectral delocalization to memory phenomena, with numerical simulations reproducing the predicted quadrupolar gravitational and dipolar electromagnetic memory patterns. This spectral perspective complements asymptotic-symmetry and soft-theorem approaches, offering a geometric origin for infrared universality and guiding future studies on memory in radiative regimes and more general backgrounds.

Abstract

We identify the curvature-decay rate as a universal geometric threshold separating compact from non-compact perturbations of Laplace-type operators on asymptotically flat manifolds. For the coupled Einstein--Maxwell system, we prove that the linearized operator is essentially self-adjoint and that curvature and field strengths decaying faster than act as relatively compact perturbations, while decay exactly at places through delocalized zero modes. This threshold mechanism unifies the infrared behavior of spin-1, spin-2, and mixed spin- fields, linking the onset of spectral delocalization with the appearance of gravitational and electromagnetic memory. Finite-difference simulations corroborate the analytic scaling and reproduce the characteristic quadrupolar and dipolar sky maps predicted for the coupled memory fields. These results demonstrate that curvature decay at constitutes a fundamental geometric boundary underlying infrared universality in gauge and gravitational theories, providing a spectral counterpart to the asymptotic-symmetry and soft-theorem formulations of memory.

Paper Structure

This paper contains 44 sections, 6 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic structure
  4. Regularity and gauge fixing
  5. Weighted domain and Hilbert space
  6. Block operator and off–diagonal structure
  7. Explicit form of $C$ and $C^{\dagger}$.
  8. Gauge projection and compactness
  9. Matrix-valued potential and self-adjointness
  10. References.
  11. Spectral Structure and the Threshold Theorem
  12. Background-covariant decomposition
  13. Relative compactness for $p>3$
  14. Projected Weyl sequence at the critical decay
  15. Threshold theorem
  16. ...and 29 more sections

Key Result

Lemma 2.2

Let $P_{\mathrm{div}}$ be the Helmholtz projector on $(\Sigma,g^{(0)})$, a classical $\Psi$DO of order $0$ with symbol $p(x,\xi)\in S^{0}_{1,0}$ converging to the flat symbol as $r\to\infty$. Let $V(x)$ be a (matrix) multiplier with $V\in L^\infty(\Sigma)\cap C^1_{\mathrm{loc}}(\Sigma)$, $\nabla V\i and, moreover,

Figures (1)

  • Figure 1: Numerically reconstructed sky maps for the correlated Einstein–Maxwell memory of a $2.8M_\odot$ binary–neutron–star merger at $40\,\mathrm{Mpc}$. Left: Gravitational-memory strain $h_{\mathrm{mem}}(\theta,\phi)$ showing a quadrupolar $(1-\cos\theta)^2$ pattern with $h_{\max} = 3.35\times10^{-23}$. Center: Electromagnetic potential memory $A_{\mathrm{mem}}(\theta,\phi)$ in V·s·m$^{-1}$, displaying a dipolar $\cos\theta$ envelope with $A_{\max} = 7.78$. Right: Dimensionless correlation field $\mathrm{corr}(\theta,\phi) = \varepsilon_B \sin(2\theta)\cos(3\phi)$ with $|\mathrm{corr}| \le 5.8\times10^{-14}$. The mixed tensor–vector pattern confirms the perturbative, phase-locked coupling predicted at the $r^{-3}$ threshold.

Theorems & Definitions (13)

  • Remark 2.1
  • Lemma 2.2: Compactness of the gauge–projection commutator
  • proof : Proof
  • Proposition 2.3: Essential self-adjointness
  • proof : Justification
  • Proposition 3.1: Relative compactness for $p>3$
  • proof
  • Lemma 3.2: Domain-correct Weyl sequence
  • proof : Detailed proof
  • Theorem 3.3: Einstein–Maxwell spectral threshold for $p\ge3$
  • ...and 3 more