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The $\boldsymbol{r^{-3}}$ Curvature Decay and the Infrared Structure of Linearized Gravity

Michael Wilson

TL;DR

The paper identifies a sharp spectral threshold for linearized gravity on asymptotically flat spaces, showing that curvature decay faster than $r^{-3}$ leaves the Lichnerowicz operator $L$ spectrally radiative with $\sigma_{\mathrm{ess}}(L)=[0,\infty)$. At the critical inverse-cube rate $|{\rm Riem}|\sim r^{-3}$, zero energy enters $\sigma_{\mathrm{ess}}(L)$, yielding marginally bound, finite-energy tensor modes that are spatially extended and serve as precursors to gravitational memory and soft gravitons. A complementary radial-model analysis $L_p=-\frac{d^2}{dr^2}+\frac{\ell(\ell+1)}{r^2}+\frac{C}{r^p}$ confirms the transition at $p=3$, with a continuous approach to the flat-space limit for $p>3$ and stronger infrared effects for $p<3$. The results establish a universal $r^{-3}$ scaling as the infrared threshold, paralleling similar thresholds in non-Abelian gauge theory and linking the geometric decay of curvature to the infrared structure of both gauge and gravitational fields.

Abstract

We identify curvature decay $|\mathrm{Riem}| \sim r^{-3}$ as a sharp spectral threshold in linearized gravity on asymptotically flat manifolds. For faster decay, the spatial Lichnerowicz operator possesses a purely continuous spectrum $σ_{\mathrm{ess}}(L) = [0,\infty)$, corresponding to freely radiating tensor modes. At the inverse-cube rate, compactness fails and zero energy enters $σ_{\mathrm{ess}}(L)$, yielding marginally bound, finite-energy configurations that remain spatially extended. These static modes constitute the linear precursors of gravitational memory and soft-graviton phenomena, delineating the geometric boundary between dispersive and infrared behavior. A complementary numerical study of the radial model \[ L_p = -\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{r^2} + \frac{C}{r^p} \] confirms the analytic scaling law, locating the same transition at $p = 3$. The eigenvalue trends approach the flat-space limit continuously for $p > 3$ and strengthen progressively for $p < 3$, demonstrating a smooth yet sharp spectral transition rather than a discrete confinement regime. The result parallels the critical threshold of the non-Abelian covariant Laplacian~[18], indicating a common $r^{-3}$ scaling that governs the infrared structure of gauge and gravitational fields.

The $\boldsymbol{r^{-3}}$ Curvature Decay and the Infrared Structure of Linearized Gravity

TL;DR

The paper identifies a sharp spectral threshold for linearized gravity on asymptotically flat spaces, showing that curvature decay faster than leaves the Lichnerowicz operator spectrally radiative with . At the critical inverse-cube rate , zero energy enters , yielding marginally bound, finite-energy tensor modes that are spatially extended and serve as precursors to gravitational memory and soft gravitons. A complementary radial-model analysis confirms the transition at , with a continuous approach to the flat-space limit for and stronger infrared effects for . The results establish a universal scaling as the infrared threshold, paralleling similar thresholds in non-Abelian gauge theory and linking the geometric decay of curvature to the infrared structure of both gauge and gravitational fields.

Abstract

We identify curvature decay as a sharp spectral threshold in linearized gravity on asymptotically flat manifolds. For faster decay, the spatial Lichnerowicz operator possesses a purely continuous spectrum , corresponding to freely radiating tensor modes. At the inverse-cube rate, compactness fails and zero energy enters , yielding marginally bound, finite-energy configurations that remain spatially extended. These static modes constitute the linear precursors of gravitational memory and soft-graviton phenomena, delineating the geometric boundary between dispersive and infrared behavior. A complementary numerical study of the radial model confirms the analytic scaling law, locating the same transition at . The eigenvalue trends approach the flat-space limit continuously for and strengthen progressively for , demonstrating a smooth yet sharp spectral transition rather than a discrete confinement regime. The result parallels the critical threshold of the non-Abelian covariant Laplacian~[18], indicating a common scaling that governs the infrared structure of gauge and gravitational fields.

Paper Structure

This paper contains 41 sections, 15 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral Scaling and Structural Parallels Across Spin-1 and Spin-2 Fields
  3. Geometric setup and harmonic gauge
  4. Shared scaling structure
  5. Weighted Sobolev spaces and integration identity
  6. Self-adjoint realization and spectral framework
  7. Critical scaling and spectral threshold
  8. Infrared Spectrum and Marginal Modes
  9. Fredholm and gauge framework
  10. Weyl sequence at the critical decay
  11. Physical interpretation
  12. Dimensional Scaling and Spectral Phase Structure
  13. Dimensional analysis of curvature potentials
  14. Interpretation and structural implications
  15. Numerical Verification of the Spectral Threshold
  16. ...and 26 more sections

Key Result

Lemma 1

For $-1<\delta<0$, the operator is Fredholm with bounded inverse. Hence, for each $f\in L^2_{\delta-2}$ there exists a unique $X\in H^2_\delta$ satisfying $\Delta_V X=f$ and $\|X\|_{H^2_\delta}\le C\|f\|_{L^2_{\delta-2}}$.

Figures (2)

  • Figure 1: Log–log scaling of $|\Delta E(R,p)|$ for representative $p$. Measured slopes $\alpha(p)\!\approx\!-(p-2)$ agree with the analytic prediction.
  • Figure 2: Convergence of $\lambda_1$ with domain size $R_{\max}$ for several decay exponents $p$. The flattening of $\lambda_1(R_{\max})$ for $p\!\ge\!3$ shows that curvature becomes spectrally negligible beyond the inverse–cube rate, while slower decay $(p<3)$ yields progressively deeper infrared shifts.

Theorems & Definitions (40)

  • Lemma 1: Fredholm property
  • proof
  • Lemma 2: Approximate zero modes
  • proof
  • Theorem 1: Onset of the infrared continuum
  • Proposition 1: Dimensional criterion for the critical decay rate
  • proof
  • Remark 1: On sharpness
  • Definition 1: Numerical operators
  • Remark 2: Radial correspondence
  • ...and 30 more