Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.05345 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910027552391168 |
|---|---|
| author | Wilson, Michael |
| author_facet | Wilson, Michael |
| contents | We identify a sharp geometric threshold governing the infrared spectral behavior of the spatial Lichnerowicz operator on asymptotically flat three-dimensional manifolds. Let $(M,g)$ be asymptotically flat and let $L=Δ_L$ denote the spatial Lichnerowicz operator acting on symmetric $2$-tensors. Assume \[ |{\rm Riem}(x)| \lesssim r(x)^{-p} \quad \text{as } r(x)\to\infty. \]
If $p>3$, curvature is spectrally short-range: $L$ exhibits regular low-energy scattering and zero energy is not singular. At the critical decay \[ |{\rm Riem}(x)| \sim r^{-3}, \] dispersion and curvature balance. Zero enters the essential spectrum, and the weighted resolvent develops a threshold singularity. For $s\in(1/2,1)$, \[ \|\langle r\rangle^{-s}(L-i\varepsilon)^{-1}\langle r\rangle^{-s}\| \gtrsim \varepsilon^{-(1-s)} \quad \text{as } \varepsilon \downarrow 0 . \] Thus, the limiting absorption principle fails at zero energy.
This singularity provides a spatial spectral mechanism for the infrared sector of linearized gravity. The same inverse-cube scaling governs long-range correlations, irregular low-frequency scattering, and soft gravitational modes. Numerical simulations of a radial model and the full tensor operator confirm that $p=3$ marks a sharp transition between negligible and marginal curvature.
The associated branch point at zero energy determines late-time relaxation, yielding the universal tail exponent \[ t^{-(2\ell+3)}, \] a spectral consequence of nonzero ADM mass. More generally, in $d$ spatial dimensions, the critical decay \[ |{\rm Riem}(x)| \sim r^{-d} \] forms a universal boundary for curvature-coupled Laplace-type operators, encoding the infrared structure of gravity in the spectral geometry of a Cauchy slice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Threshold Resolvent Singularities and the Infrared Structure of Linearized Gravity Wilson, Michael General Relativity and Quantum Cosmology High Energy Physics - Theory 58J50 (Primary) 35J05, 83C20, 35J25 (Secondary) We identify a sharp geometric threshold governing the infrared spectral behavior of the spatial Lichnerowicz operator on asymptotically flat three-dimensional manifolds. Let $(M,g)$ be asymptotically flat and let $L=Δ_L$ denote the spatial Lichnerowicz operator acting on symmetric $2$-tensors. Assume \[ |{\rm Riem}(x)| \lesssim r(x)^{-p} \quad \text{as } r(x)\to\infty. \] If $p>3$, curvature is spectrally short-range: $L$ exhibits regular low-energy scattering and zero energy is not singular. At the critical decay \[ |{\rm Riem}(x)| \sim r^{-3}, \] dispersion and curvature balance. Zero enters the essential spectrum, and the weighted resolvent develops a threshold singularity. For $s\in(1/2,1)$, \[ \|\langle r\rangle^{-s}(L-i\varepsilon)^{-1}\langle r\rangle^{-s}\| \gtrsim \varepsilon^{-(1-s)} \quad \text{as } \varepsilon \downarrow 0 . \] Thus, the limiting absorption principle fails at zero energy. This singularity provides a spatial spectral mechanism for the infrared sector of linearized gravity. The same inverse-cube scaling governs long-range correlations, irregular low-frequency scattering, and soft gravitational modes. Numerical simulations of a radial model and the full tensor operator confirm that $p=3$ marks a sharp transition between negligible and marginal curvature. The associated branch point at zero energy determines late-time relaxation, yielding the universal tail exponent \[ t^{-(2\ell+3)}, \] a spectral consequence of nonzero ADM mass. More generally, in $d$ spatial dimensions, the critical decay \[ |{\rm Riem}(x)| \sim r^{-d} \] forms a universal boundary for curvature-coupled Laplace-type operators, encoding the infrared structure of gravity in the spectral geometry of a Cauchy slice. |
| title | Threshold Resolvent Singularities and the Infrared Structure of Linearized Gravity |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory 58J50 (Primary) 35J05, 83C20, 35J25 (Secondary) |
| url | https://arxiv.org/abs/2511.05345 |