Add GM v3 rewrite + rendered page for arXiv 2511.05345
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# Michael-Wilson
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IF.TRACE / ShadowRT related artifacts (arXiv dossiers, receipts, renders).
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Artifacts and rewrites related to the Michael–Wilson arXiv paper track.
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## arXiv 2511.05345
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- Plain-English Shadow Dossier (older): `arxiv/2511.05345/2026-01-01/`
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- GM v3 rewrite (definition-preserving English): `arxiv/2511.05345/2026-01-01-gm-v3/`
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Public render (GM v3):
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- `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/index.html`
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arxiv/2511.05345/2026-01-01-gm-v3/ARTIFACTS.sha256
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arxiv/2511.05345/2026-01-01-gm-v3/ARTIFACTS.sha256
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992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742 source/2511.05345.pdf
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62509bd85c81d520d3f3c97c5c753bf6728c711f15519a8dab321eb602088407 source/2511.05345.pdf.sha256
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9b555b0f92bd067e219f47693ef8784668d1fd1171733770a40c163cf466cb52 source/2511.05345.txt
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e7b268122172257d4e2004f7d576b43ca5700e76cdd44560f9ba5f067e1c6022 generated/IF_GM_ARXIV_2511_05345.md
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c200e7d18fd0c21e5bf28603cbb3cf79a638d16a8c00f99d7ffac711ffa0c46d rendered/index.html
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e7b268122172257d4e2004f7d576b43ca5700e76cdd44560f9ba5f067e1c6022 rendered/dossier.md
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arxiv/2511.05345/2026-01-01-gm-v3/README.md
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arxiv/2511.05345/2026-01-01-gm-v3/README.md
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# arXiv 2511.05345 — GM v3 rewrite (English)
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- Goal: definition-preserving English rewrite, using `IF.STYLE-BIBLE-v3.0GM-EN.md` guidance.
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- Scope: keep the math (operators, norms, theorem statements, numerical setup) readable for non-specialists.
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## Contents
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- `source/2511.05345.pdf` — source paper (PDF)
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- `source/2511.05345.txt` — extracted text (for diff/grepping)
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- `generated/IF_GM_ARXIV_2511_05345.md` — long-form rewrite (Markdown)
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- `rendered/index.html` + `rendered/dossier.md` — standalone rendered page and its Markdown payload
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- `ARTIFACTS.sha256` — checksums for all artifacts in this folder
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## Public rendered page
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- Rendered: `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/index.html`
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- Markdown: `https://infrafabric.io/static/hosted/review/arxiv/2511.05345/2026-01-01-gm-v3/dossier.md`
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## Verification
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- Local PDF SHA-256 is recorded inside the dossier and also in `source/2511.05345.pdf.sha256`.
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- `ARTIFACTS.sha256` provides file-level checksums for the full bundle in this folder.
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# The r⁻³ Curvature Decay Threshold in Linearized Gravity
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## A receipt-first, definition-preserving rewrite (readable without a GR/QC background)
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**Source (arXiv abstract page):** https://arxiv.org/abs/2511.05345
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**Source (PDF):** https://arxiv.org/pdf/2511.05345.pdf
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**Local PDF SHA-256:** `992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742`
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**Author (paper):** Michael Wilson (University of Arkansas at Little Rock)
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**Date (paper):** November 10, 2025
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---
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> Most dossiers read as if physics is optional: eternal ROI, frictionless execution, boundless optimism. We strip away the gloss to reveal the operational reality.
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This is a clean-room rewrite of the paper’s *technical spine*.
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- It preserves the paper’s **definitions**, **operators**, and **scaling statements**.
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- It replaces local jargon with **plain-language interpretations**, so you can track what the math says without already being fluent in the field.
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- It keeps a strict separation between **what the paper states** and **what that would imply if true**.
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---
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## Quick digest (for people who do not want to read 13 pages)
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**Context:** Linearized gravity is the “small perturbations” approximation. The paper asks a narrow but important question: what *asymptotic* geometric decay rate separates “fully radiative” behavior from “infrared / memory-like” behavior on a spatial slice.
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> The claim is that `|Riem| ~ r^-3` is the sharp border.
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### What the paper claims
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- If curvature decays faster than `r^-3`, the relevant spatial operator behaves like the flat operator: extended tensor modes disperse and there is no special zero-energy sector.
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- At exactly `r^-3`, compactness fails and **zero energy enters the essential spectrum**, producing **marginally extended, finite-energy** modes.
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- A simplified radial model reproduces the same transition at `p = 3`, and a dimensional argument suggests a general rule `p_crit = d`.
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### Why this matters (without invoking mysticism)
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If you can tie “long-range gravitational memory / soft modes” to a concrete spatial decay threshold, you get a clean structural statement:
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- below the threshold: the far field can keep “persistent correlations,”
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- above the threshold: the far field behaves like ordinary radiating waves.
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That is not a claim about all of nonlinear gravity. It is a claim about the linear operator that controls stationary perturbations in harmonic gauge.
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### Two ways to read this rewrite (pick one)
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1) **Fast comprehension path (skip proofs, keep definitions):** Read Sections 1, 3.3, 6, and the “What to do with this” checklist. You will still see every operator and hypothesis.
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2) **Definition-preserving path (follow the logic):** Read Sections 1–5 in order. This is the “operator → spectrum → threshold → numerical checks” route.
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### Key objects (definitions you must know to follow the argument)
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| Object | Definition (paper) | What it controls | What can go wrong
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|---|---|---|---|
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| Spatial Lichnerowicz operator | `L = ∇*∇ + V_R` | Stationary (time-independent) harmonic-gauge perturbations | Threshold behavior depends on decay of `V_R`
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| Curvature potential | `(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}` | How background curvature couples into the spin-2 operator | Compactness can fail at infinity
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| Essential spectrum | `σ_ess(L)` | Extended (non-localized) spectral behavior | Whether `0` lies in `σ_ess(L)` is the pivot
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| Weyl sequence | `{h_n}` with `||h_n||=1`, `h_n ⇀ 0`, `||L h_n|| → 0` | A way to prove `0 ∈ σ_ess(L)` | Requires careful gauge correction
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| Radial model | `L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p` | A tractable “one channel” proxy for scaling | Must not be oversold as the full operator
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---
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## 1. Introduction (what problem is being solved)
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> The paper is trying to turn “infrared structure” from a slogan into a spectral criterion.
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The starting tension is simple:
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- There are well-known long-range gravitational phenomena (memory, tails, soft modes).
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- Most of the clean theory is phrased at null infinity (asymptotic symmetry frameworks).
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- It is less clear what a purely spatial (Cauchy slice) mechanism is, and specifically how **curvature decay** controls the existence of persistent low-frequency structure.
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The paper proposes a criterion based on a single operator:
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```text
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L = ∇*∇ + V_R,
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(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
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```
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It is explicit about the intended interpretation:
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- `σ_ess(L) = [0,∞)` corresponds to freely propagating tensor modes.
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- `0 ∈ σ_ess(L)` corresponds to marginally bound, spatially extended, finite-energy modes.
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The paper’s central threshold claim (already in the abstract) is:
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```text
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|Riem| ~ r^-3 is the sharp spectral threshold in 3 spatial dimensions.
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```
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### How the paper says it will proceed (section map)
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The paper itself says the structure is:
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- Section 2: analytic framework and the scaling principle shared by gauge and gravity operators.
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- Section 3: the critical decay regime and the “zero enters the essential spectrum” result.
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- Section 4: general dimension scaling and `p_crit = d`.
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- Section 5: numerical verification: a radial scaling test plus a full 3D discretized tensor eigenvalue calculation.
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---
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## 2. Spectral scaling and structural parallels across spin‑1 and spin‑2 fields
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> Two different theories can share the same infrared scaling mechanism because the geometry forces it.
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### 2.1 Geometric setup and harmonic gauge (paper definitions)
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The spatial manifold `(Σ, g)` is asymptotically flat. In an asymptotic coordinate chart:
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```text
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g_{ij} = δ_{ij} + a_{ij}
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a_{ij} = O(r^-1)
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∂_k a_{ij} = O(r^-2)
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∂_ℓ ∂_k a_{ij} = O(r^-3)
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```
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The paper notes that these falloff conditions imply:
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- `Γ^k_{ij} = O(r^-2)`
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- `|Riem| = O(r^-3)`
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The operator `L` is defined to act on symmetric trace-free tensor fields `h_{ij}`. (The paper works in harmonic gauge; the details are enforced analytically via a vector Laplacian and gauge correction later.)
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### 2.2 Shared scaling structure (spin‑1 analogue)
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The paper draws a structural parallel to Yang–Mills:
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```text
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Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A).
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```
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Both the spin‑1 and spin‑2 operators have the same schematic form:
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- Laplace term
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- plus a curvature-induced potential term that decays like `r^-p`
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The claim is not “the theories are the same.” It is “the *spectral scaling* is governed by the same dimensional mechanism.”
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### 2.3 Weighted Sobolev spaces + symmetry identity (what gets used later)
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The analysis is carried out on weighted spaces with weight `δ`:
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```text
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H^k_δ(Σ; S^2 T*Σ) = { h in H^k_loc : ||h||_{H^k_δ} < ∞ }
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||h||^2_{H^k_δ} = Σ_{j=0..k} ∫_Σ <r>^{2(δ-j)} |∇^j h|^2_g dV_g.
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```
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Where the shorthand `<r>` is the standard “one plus radius” weight:
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```text
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<r> = (1 + r^2)^{1/2}.
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```
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An integration-by-parts identity makes `L` explicitly symmetric on compactly supported smooth tensors:
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```text
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⟨Lh, k⟩_{L^2} = ⟨∇h, ∇k⟩_{L^2} + ⟨V_R h, k⟩_{L^2}.
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```
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### 2.4 Self-adjoint realization + essential spectrum in the fast-decay regime
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The paper states (for `-1 < δ < 0`) that the mapping
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```text
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L : H^2_δ → L^2_{δ-2}
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```
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is bounded, and that `L = ∇*∇ + V_R` is self-adjoint on the same domain as `∇*∇` under decay assumptions.
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If curvature decays faster than `r^-3` (i.e. `p > 3`), `V_R` is compact in the relevant mapping, and Weyl’s theorem gives:
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```text
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σ_ess(L) = [0, ∞).
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```
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### 2.5 Why “3” appears (the core scaling argument)
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The borderline value is obtained by comparing:
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- the `r^-2` angular term induced by the Laplacian after spherical harmonic decomposition,
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- to a curvature tail `V_R ~ r^-p`.
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The paper summarizes the outcome as:
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```text
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p_crit = 3 (in three spatial dimensions).
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```
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This is the first place where the narrative should snap into focus:
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- above 3: curvature tail is spectrally negligible in the weighted setting,
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- at 3: curvature and dispersion balance,
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- below 3: curvature becomes long-range in the spectral sense.
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---
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## 3. Infrared spectrum and marginal modes (what “zero in the essential spectrum” means)
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> This is the paper’s most concrete mathematical assertion: it constructs the mechanism that makes `0` appear.
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### 3.1 Fredholm + gauge framework (so the “harmonic gauge” assumption is not hand-waved)
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A key analytic tool is the vector Laplacian acting on one-forms:
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```text
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Δ_V = ∇*∇ + Ric.
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```
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The paper states a weighted Fredholm property:
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```text
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Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
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is Fredholm and invertible for -1 < δ < 0.
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```
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Operationally, this gives a well-posed “gauge correction” step: you can solve for a vector field `X` to correct a candidate tensor field so it satisfies the divergence constraint.
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#### Lemma 1 (paper statement; verbatim structure retained)
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For `−1 < δ < 0`:
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```text
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Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
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```
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is Fredholm with bounded inverse. The paper uses this to justify solving `Δ_V X = f` with the estimate `||X||_{H^2_δ} ≤ C ||f||_{L^2_{δ-2}}`.
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### 3.2 Weyl sequence at the critical decay (the construction)
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Assume asymptotic flatness with falloff
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```text
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g_{ij} = δ_{ij} + O(r^-1)
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∂g_{ij} = O(r^-2)
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∂^2 g_{ij} = O(r^-3)
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```
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and curvature tail
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```text
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|Riem(x)| ≃ C r^-3 as r → ∞.
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```
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Then `L` is written again on `L^2(Σ; S^2 T*Σ)`:
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```text
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L h = ∇*∇ h + V_R h
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(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
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```
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The approximate zero-mode ansatz is (paper definition):
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```text
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h_n(r, ω) = A_n φ_n(r) r^-1 H_{ij}(ω)
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φ_n(r) = cutoff equal to 1 on [n, 3n/2] and vanishing outside [n/2, 2n]
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||h_n||_{L^2} = 1 ⇒ A_n ≃ n^-1/2.
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```
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#### Lemma 2 (what the paper is asserting, operationally)
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The paper’s “approximate zero mode” claim is:
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- If you build `h_n` supported on a large annulus and normalize it (`||h_n||=1`),
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- then the action of `L` on it becomes small (`||L h_n|| → 0`) as the annulus goes to infinity,
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- and you can fix the divergence constraint by solving `Δ_V X_n = ∇·h_n` and subtracting a Lie derivative term `L_{X_n} g`.
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The algebraic content is the triple condition:
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```text
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||\tilde h_n||_{L^2} = 1,
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∇^j \tilde h_{n,ij} = 0,
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||L \tilde h_n||_{L^2} → 0.
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```
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The paper’s proof sketch (as extracted) assigns the scale:
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- derivatives of `φ_n` contribute factors `n^-1`,
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- giving `||∇·h_n||_{L^2} ≲ n^-1` and `||L h_n||_{L^2} ≲ n^-2`,
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- and then the gauge correction yields `||L_{X_n} g||_{L^2} ≲ n^-1`.
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Then the paper performs a gauge correction:
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- solve `Δ_V X_n = ∇·h_n`
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- define a corrected sequence `\tilde h_n = h_n - L_{X_n} g`.
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The conclusion is the Weyl sequence criterion:
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```text
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||\tilde h_n||_{L^2} = 1,
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∇^j \tilde h_{n,ij} = 0,
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||L \tilde h_n||_{L^2} → 0.
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```
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This implies:
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```text
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0 ∈ σ_ess(L)
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```
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in the critical decay regime.
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#### Theorem 1 (paper statement; what it buys you)
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The paper’s formal conclusion for the critical decay regime is:
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```text
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[0, ∞) ⊂ σ_ess(L),
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0 ∈ σ_ess(L).
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```
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This is the paper’s internal definition of “infrared continuum onset”: `0` is no longer excluded from the essential spectrum when the curvature tail decays like `r^-3`.
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### 3.3 A plain-English interpretation of the Weyl sequence claim (what should stick)
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You can read “`0 ∈ σ_ess(L)`” as:
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- You can build a sequence of increasingly far-out tensor configurations,
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- each with finite (normalized) `L^2` energy,
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- whose “operator energy” `||L h||` goes to zero,
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- meaning the operator behaves like it has “almost-static” extended modes at arbitrarily large radius.
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This is not the same as saying “there is a normalizable bound state.” The paper is describing a marginal, continuum-edge phenomenon, not a discrete eigenvalue below zero.
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---
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## 4. Dimensional scaling and spectral phase structure (generalization)
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> The paper claims the “3” is not accidental; it is `d`.
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The paper introduces a dimensional generalization:
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```text
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L_d = -∇^2 + V(r), with V(r) ~ r^-p on a d-dimensional asymptotically flat manifold.
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```
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### Proposition (paper statement)
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The key statement is:
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```text
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V is a compact perturbation of Δ ⇔ p > d.
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p = d is the threshold between short- and long-range behavior.
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```
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The paper explains why a naive `L^2` or Hilbert–Schmidt test is too strong, and why the weighted mapping and asymptotic mode scaling matter.
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#### The proof mechanism (as stated in the paper)
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The paper’s proof outline has two key moves:
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1) “`V → 0` at infinity implies compact multiplication on unweighted spaces” is **not** the right test here, because the operator is being controlled on weighted mappings appropriate to asymptotically flat ends.
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2) In the weighted space, compactness fails exactly when the tail contributions fail to vanish at large radius. For the asymptotic tensor mode scaling, the paper states:
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```text
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h(r) ~ r^{-(d-2)/2}
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```
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and then estimates the weighted norm of `V h` on `{r > R}` by an integral of the form:
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```text
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∫_R^∞ r^{d - 5 + 2δ - 2p} dr
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```
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Because `−1 < δ < 0`, the paper concludes the integral diverges when `p ≤ d`, implying noncompact behavior in the curvature tail.
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If you only keep one operational takeaway from Section 4, it is:
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|
||||
- the threshold depends on how the asymptotic modes scale in the weighted space,
|
||||
- and that scaling drives the integral that converges or diverges in the tail.
|
||||
|
||||
In 3 dimensions, this reduces back to `p_crit = 3`.
|
||||
|
||||
---
|
||||
|
||||
## 5. Numerical verification of the spectral threshold (what was actually computed)
|
||||
|
||||
> The numerical section is not “just plots.” It has explicit discretization choices and convergence targets.
|
||||
|
||||
### 5.1 Numerical setup and nondimensionalization (paper parameters)
|
||||
|
||||
The full tensor operator `L = ∇*∇ + V_R` is discretized on a uniform Cartesian grid:
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
```
|
||||
|
||||
with spacing `h = Δx / L_0` after nondimensionalization by a fixed length scale `L_0`.
|
||||
|
||||
- Centered finite differences approximate `∇` and `∇*∇`.
|
||||
- Dirichlet boundary conditions impose `h|_{∂Ω} = 0`.
|
||||
- The continuum approach is monitored by extrapolation in `R_max`.
|
||||
|
||||
To enforce transverse–traceless constraints, the paper uses a penalty functional:
|
||||
|
||||
```text
|
||||
⟨h, Lh⟩ + η ||∇·h||^2_{L^2(Ω)} + ζ ( ... )
|
||||
```
|
||||
|
||||
and reports that varying penalties `η, ζ` by factors `2–4` shifts the lowest eigenvalue by less than `10^-3`.
|
||||
|
||||
#### The exact penalty functional (paper equation (16))
|
||||
|
||||
The paper’s explicit form of the penalized Rayleigh quotient is:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇· h||^2_{L^2(Ω)} + ζ ||tr h||^2_{L^2(Ω)} ) / ||h||^2_{L^2(Ω)}.
|
||||
```
|
||||
|
||||
It then states the associated discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u,
|
||||
```
|
||||
|
||||
where `K` and `M` are stiffness and mass matrices and `D, T` are discrete divergence and trace operators.
|
||||
|
||||
Representative nondimensional parameters (paper list):
|
||||
|
||||
- `h ∈ {1.0, 0.75, 0.5}`
|
||||
- `R_max ∈ {6, 10, 14, 18, 20}`
|
||||
- `C = -1`
|
||||
- grid sizes `N = 21–41` (up to `3.6 × 10^5` degrees of freedom)
|
||||
- convergence target: relative error `10^-5`.
|
||||
|
||||
The paper explicitly notes that Dirichlet boundaries discretize a near-threshold continuum into “box modes” with scaling
|
||||
|
||||
```text
|
||||
λ_1(R_max) ∝ R_max^-2.
|
||||
```
|
||||
|
||||
This matters because it is an explicit check that the numerics are tracking continuum-edge behavior rather than a false “confinement.”
|
||||
|
||||
### 5.1 Definition (radial operator vs full 3D operator)
|
||||
|
||||
The paper defines a radial model operator (single angular channel):
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
and describes the full 3D discretized operator as including curvature coupling and constraint enforcement.
|
||||
|
||||
It also states a channel correspondence:
|
||||
|
||||
```text
|
||||
V_eff(r) = ℓ(ℓ+1)/r^2 + C/r^p + O(r^{-p-1}).
|
||||
```
|
||||
|
||||
### 5.2 Rayleigh–quotient scaling test (explicit energy functional)
|
||||
|
||||
The paper uses an energy functional on a normalized bump function `φ(r)` supported on `[R, 2R]`:
|
||||
|
||||
```text
|
||||
E[φ] = ( ∫_R^{2R} ( |φ'(r)|^2 + V_p(r)|φ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |φ(r)|^2 r^2 dr )
|
||||
|
||||
ΔE(R, p) = E[φ] - E_free[φ].
|
||||
```
|
||||
|
||||
It reports a predicted scaling relationship:
|
||||
|
||||
```text
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
and states that fitted slopes `α(p) ≈ -(p-2)` match the analytic scaling.
|
||||
|
||||
### What the numerics are *actually used for*
|
||||
|
||||
The numerics support two claims:
|
||||
|
||||
1) In the radial model, the lowest eigenvalue approaches the continuum threshold near `p = 3`.
|
||||
|
||||
2) In the full 3D discretized tensor operator, the same transition appears without evidence of discrete bound-state formation.
|
||||
|
||||
The important “interpretation constraint” is:
|
||||
|
||||
- the numerics validate a **continuous spectral crossover** centered at `p = 3`, not a dramatic phase change.
|
||||
|
||||
---
|
||||
|
||||
## 6. Physical interpretation and implications (what “infrared structure” means here)
|
||||
|
||||
> The paper is careful: it is still a classical, stationary analysis. It argues the operator already contains the seeds of the soft sector.
|
||||
|
||||
The paper links the operator’s spectrum to a quantized two-point function: in canonical quantization (harmonic gauge), the equal-time two-point function is the inverse of `L`, so large-distance correlations are governed by the same threshold.
|
||||
|
||||
It also states a regime interpretation:
|
||||
|
||||
- for `p > 3`: radiative propagation, fully dispersive
|
||||
- for `p = 3`: marginal persistence (extended but finite-energy)
|
||||
- for `p < 3`: enhanced infrared coupling (without discrete confinement in the tensor sector).
|
||||
|
||||
The paper draws a conceptual link to soft graviton theorems and gravitational memory, but it flags that a complete correspondence would require coupling to the time-dependent linearized Einstein equations near null infinity.
|
||||
|
||||
---
|
||||
|
||||
## 7. Relation to previous work and threshold alignment (why this is not an isolated curiosity)
|
||||
|
||||
> The paper’s “parallel threshold” claim is what turns a one-off scaling observation into a pattern.
|
||||
|
||||
The paper states an analogous threshold in non-Abelian gauge theory:
|
||||
|
||||
```text
|
||||
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A),
|
||||
|
||||
|F_A| ~ r^-3 is the analogue threshold.
|
||||
```
|
||||
|
||||
It claims that for Laplace-type operators on bundles over `R^3`, a curvature tail of order `r^-3` separates short-range radiative behavior from long-range infrared coupling.
|
||||
|
||||
It also ties `r^-3` to known scattering thresholds in Schrödinger-type operators.
|
||||
|
||||
---
|
||||
|
||||
## 8. Conclusion (the paper’s end state)
|
||||
|
||||
> If you accept the operator model, then `r^-3` is the boundary between “radiative only” and “infrared sector present” on a spatial slice.
|
||||
|
||||
The paper’s concluding theorem is explicit:
|
||||
|
||||
```text
|
||||
Theorem (paper): Let |Riem(x)| ≤ C r^-p on an asymptotically flat 3-manifold.
|
||||
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling (without isolated bound states in the tensor sector).
|
||||
```
|
||||
|
||||
It adds explicit future directions: limiting absorption principle at critical rate; extension to Schwarzschild and Kerr; dynamical correspondence between spatial modes and soft/memory sector at null infinity.
|
||||
|
||||
---
|
||||
|
||||
## What to do with this (if you are not a gravitational physicist)
|
||||
|
||||
> Treat this as a structural result about a linear operator under asymptotic decay assumptions.
|
||||
|
||||
If you want to use this paper responsibly, the minimum checklist is:
|
||||
|
||||
1) You can restate the operator definitions (`L`, `V_R`, radial `L_p`) without distortion.
|
||||
2) You can state the threshold claim with its scope (“linearized,” “harmonic gauge,” “asymptotically flat,” “spatial slice”).
|
||||
3) You can distinguish “operator has marginal zero-energy Weyl sequence” from “there are bound states” (the paper says no discrete confinement is observed).
|
||||
4) You can keep the generalization `p_crit = d` separate from the physically interpreted 3D statement.
|
||||
|
||||
*If you can’t do those four things, you are not citing the paper. You are citing the vibe.*
|
||||
|
||||
---
|
||||
|
||||
## Appendix A — Formal statements (paper text; structure preserved)
|
||||
|
||||
This appendix exists so you can see the “hard edges” in one place: the operator definitions and the main threshold statements.
|
||||
|
||||
### A.1 Operator definitions (paper equations (1), (9), (12))
|
||||
|
||||
```text
|
||||
L = ∇*∇ + V_R,
|
||||
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
|
||||
|
||||
Δ_V = ∇*∇ + Ric.
|
||||
|
||||
L h = ∇*∇ h + V_R h,
|
||||
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
|
||||
```
|
||||
|
||||
### A.2 Lemma 1 (Fredholm property; as extracted)
|
||||
|
||||
```text
|
||||
For −1 < δ < 0:
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
is Fredholm with bounded inverse.
|
||||
```
|
||||
|
||||
### A.3 Lemma 2 (Approximate zero modes; as extracted)
|
||||
|
||||
```text
|
||||
Let H_{ij}(ω) be symmetric, trace-free, divergence-free on S^2 and define
|
||||
h_n(r, ω) = A_n ϕ_n(r) r^-1 H_{ij}(ω),
|
||||
where ϕ_n = 1 on [n, 3n/2] and 0 outside [n/2, 2n].
|
||||
After divergence correction using Δ_V X_n = ∇·h_n and h̃_n = h_n − L_{X_n} g:
|
||||
||h̃_n||_{L^2} = 1, ∇^j h̃_{n,ij} = 0, ||L h̃_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
### A.4 Theorem 1 (Onset of the infrared continuum; as extracted)
|
||||
|
||||
```text
|
||||
Let (Σ, g) satisfy the asymptotic flatness conditions (11) with |Riem(x)| ≃ C r^-3.
|
||||
Then:
|
||||
[0, ∞) ⊂ σ_ess(L), and 0 ∈ σ_ess(L).
|
||||
```
|
||||
|
||||
### A.5 Proposition 1 (Dimensional criterion; as extracted)
|
||||
|
||||
```text
|
||||
Let ∆ be the Laplace–Beltrami operator on a d-dimensional asymptotically flat manifold,
|
||||
and V(r) ~ r^-p a curvature-induced potential. Then:
|
||||
V is a compact perturbation of ∆ iff p > d.
|
||||
p = d is the threshold.
|
||||
```
|
||||
|
||||
### A.6 Theorem 2 (Spectral threshold for linearized gravity; paper end state)
|
||||
|
||||
```text
|
||||
Let (Σ, g) be asymptotically flat with |Riem(x)| ≤ C r^-p.
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling.
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix B — Numerical definitions and the actual computed objects
|
||||
|
||||
This appendix consolidates the “what was computed” layer so the numerics can be read as a check of the operator story, not a vibe.
|
||||
|
||||
### B.1 Domains, parameters, and constraints (paper Section 5.1)
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
h ∈ {1.0, 0.75, 0.5}
|
||||
R_max ∈ {6, 10, 14, 18, 20}
|
||||
C = -1
|
||||
Dirichlet boundary: h|_{∂Ω} = 0
|
||||
convergence: relative error 10^-5
|
||||
```
|
||||
|
||||
TT enforcement via the penalty quotient:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇·h||^2 + ζ ||tr h||^2 ) / ||h||^2
|
||||
```
|
||||
|
||||
and the discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u.
|
||||
```
|
||||
|
||||
### B.2 Radial model operator (paper Definition 1)
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
### B.3 Rayleigh quotient functional (paper Section 5.2)
|
||||
|
||||
```text
|
||||
E[ϕ] = ( ∫_R^{2R} ( |ϕ'(r)|^2 + V_p(r)|ϕ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |ϕ(r)|^2 r^2 dr )
|
||||
ΔE(R, p) = E[ϕ] − E_free[ϕ]
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
### B.4 Table 1 (raw extraction block)
|
||||
|
||||
The following is the paper’s Table 1 content as extracted into plain text. It is kept here because it defines the specific numerical values associated with the scaling check.
|
||||
|
||||
```text
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1. The power-law scaling with R follows ∆E ∼
|
||||
R−(p−2) , confirming that p=3 behaves as the marginal
|
||||
case separating decaying from saturating behavior.
|
||||
p
|
||||
2.00
|
||||
|
||||
R
|
||||
∆E(R, p)
|
||||
Efull
|
||||
Efree
|
||||
−2
|
||||
−1
|
||||
10 −2.85 × 10
|
||||
1.50 × 10
|
||||
1.54 × 10−1
|
||||
−2
|
||||
−2
|
||||
20 −2.85 × 10
|
||||
3.74 × 10
|
||||
3.85 × 10−2
|
||||
Continued on next page
|
||||
|
||||
12
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
2.50
|
||||
|
||||
3.00
|
||||
|
||||
3.50
|
||||
|
||||
4.00
|
||||
|
||||
R
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
|
||||
∆E(R, p)
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−7.39 × 10−3
|
||||
−5.23 × 10−3
|
||||
−3.69 × 10−3
|
||||
−2.61 × 10−3
|
||||
−1.85 × 10−3
|
||||
−1.31 × 10−3
|
||||
−9.23 × 10−4
|
||||
−1.92 × 10−3
|
||||
−9.62 × 10−4
|
||||
−4.81 × 10−4
|
||||
−2.40 × 10−4
|
||||
−1.20 × 10−4
|
||||
−6.01 × 10−5
|
||||
−3.01 × 10−5
|
||||
−5.02 × 10−4
|
||||
−1.78 × 10−4
|
||||
−6.28 × 10−5
|
||||
−2.22 × 10−5
|
||||
−7.85 × 10−6
|
||||
−2.78 × 10−6
|
||||
−9.81 × 10−7
|
||||
−1.32 × 10−4
|
||||
−3.29 × 10−5
|
||||
−8.23 × 10−6
|
||||
−2.06 × 10−6
|
||||
−5.14 × 10−7
|
||||
−1.29 × 10−7
|
||||
|
||||
13
|
||||
|
||||
Efull
|
||||
Efree
|
||||
−3
|
||||
9.36 × 10
|
||||
9.63 × 10−3
|
||||
−3
|
||||
2.34 × 10
|
||||
2.41 × 10−3
|
||||
5.85 × 10−4 6.02 × 10−4
|
||||
1.46 × 10−4 1.50 × 10−4
|
||||
3.65 × 10−5 3.76 × 10−5
|
||||
1.53 × 10−1 1.54 × 10−1
|
||||
3.83 × 10−2 3.85 × 10−2
|
||||
9.59 × 10−3 9.63 × 10−3
|
||||
2.40 × 10−3 2.41 × 10−3
|
||||
6.01 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
Continued on next page
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
R
|
||||
640
|
||||
|
||||
∆E(R, p)
|
||||
Efull
|
||||
−8
|
||||
−3.21 × 10
|
||||
3.76 × 10−5
|
||||
|
||||
Efree
|
||||
3.76 × 10−5
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix C — Verbatim excerpt anchors used in the rewrite
|
||||
|
||||
The following are excerpted from the `pdftotext` extraction and are included to keep wording stable (they are not “new claims” added by this rewrite).
|
||||
|
||||
These are direct excerpts from the `pdftotext` extraction of the paper and are included so the wording is stable.
|
||||
|
||||
- “We identify curvature decay |Riem| ∼ 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 σess (L) = [0, ∞).”
|
||||
- “At the inverse-cube rate, compactness fails and zero energy enters σess (L), yielding marginally bound, finite-energy configurations that remain spatially extended.”
|
||||
- “Proposition 1 (Dimensional criterion for the critical decay rate). … V is a compact perturbation of ∆ if and only if p > d. The equality p = d marks the threshold …”
|
||||
- “Theorem 2 (Spectral threshold for linearized gravity). … For p > 3 … σess(L) = [0, ∞). … At p = 3 … Weyl sequence … For p < 3 … enhances infrared coupling …”
|
||||
842
arxiv/2511.05345/2026-01-01-gm-v3/rendered/dossier.md
Normal file
842
arxiv/2511.05345/2026-01-01-gm-v3/rendered/dossier.md
Normal file
|
|
@ -0,0 +1,842 @@
|
|||
# The r⁻³ Curvature Decay Threshold in Linearized Gravity
|
||||
## A receipt-first, definition-preserving rewrite (readable without a GR/QC background)
|
||||
|
||||
**Source (arXiv abstract page):** https://arxiv.org/abs/2511.05345
|
||||
|
||||
**Source (PDF):** https://arxiv.org/pdf/2511.05345.pdf
|
||||
|
||||
**Local PDF SHA-256:** `992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742`
|
||||
|
||||
**Author (paper):** Michael Wilson (University of Arkansas at Little Rock)
|
||||
|
||||
**Date (paper):** November 10, 2025
|
||||
|
||||
---
|
||||
|
||||
> Most dossiers read as if physics is optional: eternal ROI, frictionless execution, boundless optimism. We strip away the gloss to reveal the operational reality.
|
||||
|
||||
This is a clean-room rewrite of the paper’s *technical spine*.
|
||||
|
||||
- It preserves the paper’s **definitions**, **operators**, and **scaling statements**.
|
||||
- It replaces local jargon with **plain-language interpretations**, so you can track what the math says without already being fluent in the field.
|
||||
- It keeps a strict separation between **what the paper states** and **what that would imply if true**.
|
||||
|
||||
---
|
||||
|
||||
## Quick digest (for people who do not want to read 13 pages)
|
||||
|
||||
**Context:** Linearized gravity is the “small perturbations” approximation. The paper asks a narrow but important question: what *asymptotic* geometric decay rate separates “fully radiative” behavior from “infrared / memory-like” behavior on a spatial slice.
|
||||
|
||||
> The claim is that `|Riem| ~ r^-3` is the sharp border.
|
||||
|
||||
### What the paper claims
|
||||
- If curvature decays faster than `r^-3`, the relevant spatial operator behaves like the flat operator: extended tensor modes disperse and there is no special zero-energy sector.
|
||||
- At exactly `r^-3`, compactness fails and **zero energy enters the essential spectrum**, producing **marginally extended, finite-energy** modes.
|
||||
- A simplified radial model reproduces the same transition at `p = 3`, and a dimensional argument suggests a general rule `p_crit = d`.
|
||||
|
||||
### Why this matters (without invoking mysticism)
|
||||
If you can tie “long-range gravitational memory / soft modes” to a concrete spatial decay threshold, you get a clean structural statement:
|
||||
|
||||
- below the threshold: the far field can keep “persistent correlations,”
|
||||
- above the threshold: the far field behaves like ordinary radiating waves.
|
||||
|
||||
That is not a claim about all of nonlinear gravity. It is a claim about the linear operator that controls stationary perturbations in harmonic gauge.
|
||||
|
||||
### Two ways to read this rewrite (pick one)
|
||||
|
||||
1) **Fast comprehension path (skip proofs, keep definitions):** Read Sections 1, 3.3, 6, and the “What to do with this” checklist. You will still see every operator and hypothesis.
|
||||
|
||||
2) **Definition-preserving path (follow the logic):** Read Sections 1–5 in order. This is the “operator → spectrum → threshold → numerical checks” route.
|
||||
|
||||
### Key objects (definitions you must know to follow the argument)
|
||||
|
||||
| Object | Definition (paper) | What it controls | What can go wrong
|
||||
|---|---|---|---|
|
||||
| Spatial Lichnerowicz operator | `L = ∇*∇ + V_R` | Stationary (time-independent) harmonic-gauge perturbations | Threshold behavior depends on decay of `V_R`
|
||||
| Curvature potential | `(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}` | How background curvature couples into the spin-2 operator | Compactness can fail at infinity
|
||||
| Essential spectrum | `σ_ess(L)` | Extended (non-localized) spectral behavior | Whether `0` lies in `σ_ess(L)` is the pivot
|
||||
| Weyl sequence | `{h_n}` with `||h_n||=1`, `h_n ⇀ 0`, `||L h_n|| → 0` | A way to prove `0 ∈ σ_ess(L)` | Requires careful gauge correction
|
||||
| Radial model | `L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p` | A tractable “one channel” proxy for scaling | Must not be oversold as the full operator
|
||||
|
||||
---
|
||||
|
||||
## 1. Introduction (what problem is being solved)
|
||||
|
||||
> The paper is trying to turn “infrared structure” from a slogan into a spectral criterion.
|
||||
|
||||
The starting tension is simple:
|
||||
|
||||
- There are well-known long-range gravitational phenomena (memory, tails, soft modes).
|
||||
- Most of the clean theory is phrased at null infinity (asymptotic symmetry frameworks).
|
||||
- It is less clear what a purely spatial (Cauchy slice) mechanism is, and specifically how **curvature decay** controls the existence of persistent low-frequency structure.
|
||||
|
||||
The paper proposes a criterion based on a single operator:
|
||||
|
||||
```text
|
||||
L = ∇*∇ + V_R,
|
||||
|
||||
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
|
||||
```
|
||||
|
||||
It is explicit about the intended interpretation:
|
||||
|
||||
- `σ_ess(L) = [0,∞)` corresponds to freely propagating tensor modes.
|
||||
- `0 ∈ σ_ess(L)` corresponds to marginally bound, spatially extended, finite-energy modes.
|
||||
|
||||
The paper’s central threshold claim (already in the abstract) is:
|
||||
|
||||
```text
|
||||
|Riem| ~ r^-3 is the sharp spectral threshold in 3 spatial dimensions.
|
||||
```
|
||||
|
||||
### How the paper says it will proceed (section map)
|
||||
|
||||
The paper itself says the structure is:
|
||||
|
||||
- Section 2: analytic framework and the scaling principle shared by gauge and gravity operators.
|
||||
- Section 3: the critical decay regime and the “zero enters the essential spectrum” result.
|
||||
- Section 4: general dimension scaling and `p_crit = d`.
|
||||
- Section 5: numerical verification: a radial scaling test plus a full 3D discretized tensor eigenvalue calculation.
|
||||
|
||||
---
|
||||
|
||||
## 2. Spectral scaling and structural parallels across spin‑1 and spin‑2 fields
|
||||
|
||||
> Two different theories can share the same infrared scaling mechanism because the geometry forces it.
|
||||
|
||||
### 2.1 Geometric setup and harmonic gauge (paper definitions)
|
||||
|
||||
The spatial manifold `(Σ, g)` is asymptotically flat. In an asymptotic coordinate chart:
|
||||
|
||||
```text
|
||||
g_{ij} = δ_{ij} + a_{ij}
|
||||
|
||||
a_{ij} = O(r^-1)
|
||||
∂_k a_{ij} = O(r^-2)
|
||||
∂_ℓ ∂_k a_{ij} = O(r^-3)
|
||||
```
|
||||
|
||||
The paper notes that these falloff conditions imply:
|
||||
|
||||
- `Γ^k_{ij} = O(r^-2)`
|
||||
- `|Riem| = O(r^-3)`
|
||||
|
||||
The operator `L` is defined to act on symmetric trace-free tensor fields `h_{ij}`. (The paper works in harmonic gauge; the details are enforced analytically via a vector Laplacian and gauge correction later.)
|
||||
|
||||
### 2.2 Shared scaling structure (spin‑1 analogue)
|
||||
|
||||
The paper draws a structural parallel to Yang–Mills:
|
||||
|
||||
```text
|
||||
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A).
|
||||
```
|
||||
|
||||
Both the spin‑1 and spin‑2 operators have the same schematic form:
|
||||
|
||||
- Laplace term
|
||||
- plus a curvature-induced potential term that decays like `r^-p`
|
||||
|
||||
The claim is not “the theories are the same.” It is “the *spectral scaling* is governed by the same dimensional mechanism.”
|
||||
|
||||
### 2.3 Weighted Sobolev spaces + symmetry identity (what gets used later)
|
||||
|
||||
The analysis is carried out on weighted spaces with weight `δ`:
|
||||
|
||||
```text
|
||||
H^k_δ(Σ; S^2 T*Σ) = { h in H^k_loc : ||h||_{H^k_δ} < ∞ }
|
||||
|
||||
||h||^2_{H^k_δ} = Σ_{j=0..k} ∫_Σ <r>^{2(δ-j)} |∇^j h|^2_g dV_g.
|
||||
```
|
||||
|
||||
Where the shorthand `<r>` is the standard “one plus radius” weight:
|
||||
|
||||
```text
|
||||
<r> = (1 + r^2)^{1/2}.
|
||||
```
|
||||
|
||||
An integration-by-parts identity makes `L` explicitly symmetric on compactly supported smooth tensors:
|
||||
|
||||
```text
|
||||
⟨Lh, k⟩_{L^2} = ⟨∇h, ∇k⟩_{L^2} + ⟨V_R h, k⟩_{L^2}.
|
||||
```
|
||||
|
||||
### 2.4 Self-adjoint realization + essential spectrum in the fast-decay regime
|
||||
|
||||
The paper states (for `-1 < δ < 0`) that the mapping
|
||||
|
||||
```text
|
||||
L : H^2_δ → L^2_{δ-2}
|
||||
```
|
||||
|
||||
is bounded, and that `L = ∇*∇ + V_R` is self-adjoint on the same domain as `∇*∇` under decay assumptions.
|
||||
|
||||
If curvature decays faster than `r^-3` (i.e. `p > 3`), `V_R` is compact in the relevant mapping, and Weyl’s theorem gives:
|
||||
|
||||
```text
|
||||
σ_ess(L) = [0, ∞).
|
||||
```
|
||||
|
||||
### 2.5 Why “3” appears (the core scaling argument)
|
||||
|
||||
The borderline value is obtained by comparing:
|
||||
|
||||
- the `r^-2` angular term induced by the Laplacian after spherical harmonic decomposition,
|
||||
- to a curvature tail `V_R ~ r^-p`.
|
||||
|
||||
The paper summarizes the outcome as:
|
||||
|
||||
```text
|
||||
p_crit = 3 (in three spatial dimensions).
|
||||
```
|
||||
|
||||
This is the first place where the narrative should snap into focus:
|
||||
|
||||
- above 3: curvature tail is spectrally negligible in the weighted setting,
|
||||
- at 3: curvature and dispersion balance,
|
||||
- below 3: curvature becomes long-range in the spectral sense.
|
||||
|
||||
---
|
||||
|
||||
## 3. Infrared spectrum and marginal modes (what “zero in the essential spectrum” means)
|
||||
|
||||
> This is the paper’s most concrete mathematical assertion: it constructs the mechanism that makes `0` appear.
|
||||
|
||||
### 3.1 Fredholm + gauge framework (so the “harmonic gauge” assumption is not hand-waved)
|
||||
|
||||
A key analytic tool is the vector Laplacian acting on one-forms:
|
||||
|
||||
```text
|
||||
Δ_V = ∇*∇ + Ric.
|
||||
```
|
||||
|
||||
The paper states a weighted Fredholm property:
|
||||
|
||||
```text
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
|
||||
is Fredholm and invertible for -1 < δ < 0.
|
||||
```
|
||||
|
||||
Operationally, this gives a well-posed “gauge correction” step: you can solve for a vector field `X` to correct a candidate tensor field so it satisfies the divergence constraint.
|
||||
|
||||
#### Lemma 1 (paper statement; verbatim structure retained)
|
||||
|
||||
For `−1 < δ < 0`:
|
||||
|
||||
```text
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
```
|
||||
|
||||
is Fredholm with bounded inverse. The paper uses this to justify solving `Δ_V X = f` with the estimate `||X||_{H^2_δ} ≤ C ||f||_{L^2_{δ-2}}`.
|
||||
|
||||
### 3.2 Weyl sequence at the critical decay (the construction)
|
||||
|
||||
Assume asymptotic flatness with falloff
|
||||
|
||||
```text
|
||||
g_{ij} = δ_{ij} + O(r^-1)
|
||||
∂g_{ij} = O(r^-2)
|
||||
∂^2 g_{ij} = O(r^-3)
|
||||
```
|
||||
|
||||
and curvature tail
|
||||
|
||||
```text
|
||||
|Riem(x)| ≃ C r^-3 as r → ∞.
|
||||
```
|
||||
|
||||
Then `L` is written again on `L^2(Σ; S^2 T*Σ)`:
|
||||
|
||||
```text
|
||||
L h = ∇*∇ h + V_R h
|
||||
|
||||
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
|
||||
```
|
||||
|
||||
The approximate zero-mode ansatz is (paper definition):
|
||||
|
||||
```text
|
||||
h_n(r, ω) = A_n φ_n(r) r^-1 H_{ij}(ω)
|
||||
|
||||
φ_n(r) = cutoff equal to 1 on [n, 3n/2] and vanishing outside [n/2, 2n]
|
||||
||h_n||_{L^2} = 1 ⇒ A_n ≃ n^-1/2.
|
||||
```
|
||||
|
||||
#### Lemma 2 (what the paper is asserting, operationally)
|
||||
|
||||
The paper’s “approximate zero mode” claim is:
|
||||
|
||||
- If you build `h_n` supported on a large annulus and normalize it (`||h_n||=1`),
|
||||
- then the action of `L` on it becomes small (`||L h_n|| → 0`) as the annulus goes to infinity,
|
||||
- and you can fix the divergence constraint by solving `Δ_V X_n = ∇·h_n` and subtracting a Lie derivative term `L_{X_n} g`.
|
||||
|
||||
The algebraic content is the triple condition:
|
||||
|
||||
```text
|
||||
||\tilde h_n||_{L^2} = 1,
|
||||
∇^j \tilde h_{n,ij} = 0,
|
||||
||L \tilde h_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
The paper’s proof sketch (as extracted) assigns the scale:
|
||||
|
||||
- derivatives of `φ_n` contribute factors `n^-1`,
|
||||
- giving `||∇·h_n||_{L^2} ≲ n^-1` and `||L h_n||_{L^2} ≲ n^-2`,
|
||||
- and then the gauge correction yields `||L_{X_n} g||_{L^2} ≲ n^-1`.
|
||||
|
||||
Then the paper performs a gauge correction:
|
||||
|
||||
- solve `Δ_V X_n = ∇·h_n`
|
||||
- define a corrected sequence `\tilde h_n = h_n - L_{X_n} g`.
|
||||
|
||||
The conclusion is the Weyl sequence criterion:
|
||||
|
||||
```text
|
||||
||\tilde h_n||_{L^2} = 1,
|
||||
∇^j \tilde h_{n,ij} = 0,
|
||||
||L \tilde h_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
This implies:
|
||||
|
||||
```text
|
||||
0 ∈ σ_ess(L)
|
||||
```
|
||||
|
||||
in the critical decay regime.
|
||||
|
||||
#### Theorem 1 (paper statement; what it buys you)
|
||||
|
||||
The paper’s formal conclusion for the critical decay regime is:
|
||||
|
||||
```text
|
||||
[0, ∞) ⊂ σ_ess(L),
|
||||
0 ∈ σ_ess(L).
|
||||
```
|
||||
|
||||
This is the paper’s internal definition of “infrared continuum onset”: `0` is no longer excluded from the essential spectrum when the curvature tail decays like `r^-3`.
|
||||
|
||||
### 3.3 A plain-English interpretation of the Weyl sequence claim (what should stick)
|
||||
|
||||
You can read “`0 ∈ σ_ess(L)`” as:
|
||||
|
||||
- You can build a sequence of increasingly far-out tensor configurations,
|
||||
- each with finite (normalized) `L^2` energy,
|
||||
- whose “operator energy” `||L h||` goes to zero,
|
||||
- meaning the operator behaves like it has “almost-static” extended modes at arbitrarily large radius.
|
||||
|
||||
This is not the same as saying “there is a normalizable bound state.” The paper is describing a marginal, continuum-edge phenomenon, not a discrete eigenvalue below zero.
|
||||
|
||||
---
|
||||
|
||||
## 4. Dimensional scaling and spectral phase structure (generalization)
|
||||
|
||||
> The paper claims the “3” is not accidental; it is `d`.
|
||||
|
||||
The paper introduces a dimensional generalization:
|
||||
|
||||
```text
|
||||
L_d = -∇^2 + V(r), with V(r) ~ r^-p on a d-dimensional asymptotically flat manifold.
|
||||
```
|
||||
|
||||
### Proposition (paper statement)
|
||||
|
||||
The key statement is:
|
||||
|
||||
```text
|
||||
V is a compact perturbation of Δ ⇔ p > d.
|
||||
|
||||
p = d is the threshold between short- and long-range behavior.
|
||||
```
|
||||
|
||||
The paper explains why a naive `L^2` or Hilbert–Schmidt test is too strong, and why the weighted mapping and asymptotic mode scaling matter.
|
||||
|
||||
#### The proof mechanism (as stated in the paper)
|
||||
|
||||
The paper’s proof outline has two key moves:
|
||||
|
||||
1) “`V → 0` at infinity implies compact multiplication on unweighted spaces” is **not** the right test here, because the operator is being controlled on weighted mappings appropriate to asymptotically flat ends.
|
||||
|
||||
2) In the weighted space, compactness fails exactly when the tail contributions fail to vanish at large radius. For the asymptotic tensor mode scaling, the paper states:
|
||||
|
||||
```text
|
||||
h(r) ~ r^{-(d-2)/2}
|
||||
```
|
||||
|
||||
and then estimates the weighted norm of `V h` on `{r > R}` by an integral of the form:
|
||||
|
||||
```text
|
||||
∫_R^∞ r^{d - 5 + 2δ - 2p} dr
|
||||
```
|
||||
|
||||
Because `−1 < δ < 0`, the paper concludes the integral diverges when `p ≤ d`, implying noncompact behavior in the curvature tail.
|
||||
|
||||
If you only keep one operational takeaway from Section 4, it is:
|
||||
|
||||
- the threshold depends on how the asymptotic modes scale in the weighted space,
|
||||
- and that scaling drives the integral that converges or diverges in the tail.
|
||||
|
||||
In 3 dimensions, this reduces back to `p_crit = 3`.
|
||||
|
||||
---
|
||||
|
||||
## 5. Numerical verification of the spectral threshold (what was actually computed)
|
||||
|
||||
> The numerical section is not “just plots.” It has explicit discretization choices and convergence targets.
|
||||
|
||||
### 5.1 Numerical setup and nondimensionalization (paper parameters)
|
||||
|
||||
The full tensor operator `L = ∇*∇ + V_R` is discretized on a uniform Cartesian grid:
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
```
|
||||
|
||||
with spacing `h = Δx / L_0` after nondimensionalization by a fixed length scale `L_0`.
|
||||
|
||||
- Centered finite differences approximate `∇` and `∇*∇`.
|
||||
- Dirichlet boundary conditions impose `h|_{∂Ω} = 0`.
|
||||
- The continuum approach is monitored by extrapolation in `R_max`.
|
||||
|
||||
To enforce transverse–traceless constraints, the paper uses a penalty functional:
|
||||
|
||||
```text
|
||||
⟨h, Lh⟩ + η ||∇·h||^2_{L^2(Ω)} + ζ ( ... )
|
||||
```
|
||||
|
||||
and reports that varying penalties `η, ζ` by factors `2–4` shifts the lowest eigenvalue by less than `10^-3`.
|
||||
|
||||
#### The exact penalty functional (paper equation (16))
|
||||
|
||||
The paper’s explicit form of the penalized Rayleigh quotient is:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇· h||^2_{L^2(Ω)} + ζ ||tr h||^2_{L^2(Ω)} ) / ||h||^2_{L^2(Ω)}.
|
||||
```
|
||||
|
||||
It then states the associated discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u,
|
||||
```
|
||||
|
||||
where `K` and `M` are stiffness and mass matrices and `D, T` are discrete divergence and trace operators.
|
||||
|
||||
Representative nondimensional parameters (paper list):
|
||||
|
||||
- `h ∈ {1.0, 0.75, 0.5}`
|
||||
- `R_max ∈ {6, 10, 14, 18, 20}`
|
||||
- `C = -1`
|
||||
- grid sizes `N = 21–41` (up to `3.6 × 10^5` degrees of freedom)
|
||||
- convergence target: relative error `10^-5`.
|
||||
|
||||
The paper explicitly notes that Dirichlet boundaries discretize a near-threshold continuum into “box modes” with scaling
|
||||
|
||||
```text
|
||||
λ_1(R_max) ∝ R_max^-2.
|
||||
```
|
||||
|
||||
This matters because it is an explicit check that the numerics are tracking continuum-edge behavior rather than a false “confinement.”
|
||||
|
||||
### 5.1 Definition (radial operator vs full 3D operator)
|
||||
|
||||
The paper defines a radial model operator (single angular channel):
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
and describes the full 3D discretized operator as including curvature coupling and constraint enforcement.
|
||||
|
||||
It also states a channel correspondence:
|
||||
|
||||
```text
|
||||
V_eff(r) = ℓ(ℓ+1)/r^2 + C/r^p + O(r^{-p-1}).
|
||||
```
|
||||
|
||||
### 5.2 Rayleigh–quotient scaling test (explicit energy functional)
|
||||
|
||||
The paper uses an energy functional on a normalized bump function `φ(r)` supported on `[R, 2R]`:
|
||||
|
||||
```text
|
||||
E[φ] = ( ∫_R^{2R} ( |φ'(r)|^2 + V_p(r)|φ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |φ(r)|^2 r^2 dr )
|
||||
|
||||
ΔE(R, p) = E[φ] - E_free[φ].
|
||||
```
|
||||
|
||||
It reports a predicted scaling relationship:
|
||||
|
||||
```text
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
and states that fitted slopes `α(p) ≈ -(p-2)` match the analytic scaling.
|
||||
|
||||
### What the numerics are *actually used for*
|
||||
|
||||
The numerics support two claims:
|
||||
|
||||
1) In the radial model, the lowest eigenvalue approaches the continuum threshold near `p = 3`.
|
||||
|
||||
2) In the full 3D discretized tensor operator, the same transition appears without evidence of discrete bound-state formation.
|
||||
|
||||
The important “interpretation constraint” is:
|
||||
|
||||
- the numerics validate a **continuous spectral crossover** centered at `p = 3`, not a dramatic phase change.
|
||||
|
||||
---
|
||||
|
||||
## 6. Physical interpretation and implications (what “infrared structure” means here)
|
||||
|
||||
> The paper is careful: it is still a classical, stationary analysis. It argues the operator already contains the seeds of the soft sector.
|
||||
|
||||
The paper links the operator’s spectrum to a quantized two-point function: in canonical quantization (harmonic gauge), the equal-time two-point function is the inverse of `L`, so large-distance correlations are governed by the same threshold.
|
||||
|
||||
It also states a regime interpretation:
|
||||
|
||||
- for `p > 3`: radiative propagation, fully dispersive
|
||||
- for `p = 3`: marginal persistence (extended but finite-energy)
|
||||
- for `p < 3`: enhanced infrared coupling (without discrete confinement in the tensor sector).
|
||||
|
||||
The paper draws a conceptual link to soft graviton theorems and gravitational memory, but it flags that a complete correspondence would require coupling to the time-dependent linearized Einstein equations near null infinity.
|
||||
|
||||
---
|
||||
|
||||
## 7. Relation to previous work and threshold alignment (why this is not an isolated curiosity)
|
||||
|
||||
> The paper’s “parallel threshold” claim is what turns a one-off scaling observation into a pattern.
|
||||
|
||||
The paper states an analogous threshold in non-Abelian gauge theory:
|
||||
|
||||
```text
|
||||
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A),
|
||||
|
||||
|F_A| ~ r^-3 is the analogue threshold.
|
||||
```
|
||||
|
||||
It claims that for Laplace-type operators on bundles over `R^3`, a curvature tail of order `r^-3` separates short-range radiative behavior from long-range infrared coupling.
|
||||
|
||||
It also ties `r^-3` to known scattering thresholds in Schrödinger-type operators.
|
||||
|
||||
---
|
||||
|
||||
## 8. Conclusion (the paper’s end state)
|
||||
|
||||
> If you accept the operator model, then `r^-3` is the boundary between “radiative only” and “infrared sector present” on a spatial slice.
|
||||
|
||||
The paper’s concluding theorem is explicit:
|
||||
|
||||
```text
|
||||
Theorem (paper): Let |Riem(x)| ≤ C r^-p on an asymptotically flat 3-manifold.
|
||||
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling (without isolated bound states in the tensor sector).
|
||||
```
|
||||
|
||||
It adds explicit future directions: limiting absorption principle at critical rate; extension to Schwarzschild and Kerr; dynamical correspondence between spatial modes and soft/memory sector at null infinity.
|
||||
|
||||
---
|
||||
|
||||
## What to do with this (if you are not a gravitational physicist)
|
||||
|
||||
> Treat this as a structural result about a linear operator under asymptotic decay assumptions.
|
||||
|
||||
If you want to use this paper responsibly, the minimum checklist is:
|
||||
|
||||
1) You can restate the operator definitions (`L`, `V_R`, radial `L_p`) without distortion.
|
||||
2) You can state the threshold claim with its scope (“linearized,” “harmonic gauge,” “asymptotically flat,” “spatial slice”).
|
||||
3) You can distinguish “operator has marginal zero-energy Weyl sequence” from “there are bound states” (the paper says no discrete confinement is observed).
|
||||
4) You can keep the generalization `p_crit = d` separate from the physically interpreted 3D statement.
|
||||
|
||||
*If you can’t do those four things, you are not citing the paper. You are citing the vibe.*
|
||||
|
||||
---
|
||||
|
||||
## Appendix A — Formal statements (paper text; structure preserved)
|
||||
|
||||
This appendix exists so you can see the “hard edges” in one place: the operator definitions and the main threshold statements.
|
||||
|
||||
### A.1 Operator definitions (paper equations (1), (9), (12))
|
||||
|
||||
```text
|
||||
L = ∇*∇ + V_R,
|
||||
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
|
||||
|
||||
Δ_V = ∇*∇ + Ric.
|
||||
|
||||
L h = ∇*∇ h + V_R h,
|
||||
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
|
||||
```
|
||||
|
||||
### A.2 Lemma 1 (Fredholm property; as extracted)
|
||||
|
||||
```text
|
||||
For −1 < δ < 0:
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
is Fredholm with bounded inverse.
|
||||
```
|
||||
|
||||
### A.3 Lemma 2 (Approximate zero modes; as extracted)
|
||||
|
||||
```text
|
||||
Let H_{ij}(ω) be symmetric, trace-free, divergence-free on S^2 and define
|
||||
h_n(r, ω) = A_n ϕ_n(r) r^-1 H_{ij}(ω),
|
||||
where ϕ_n = 1 on [n, 3n/2] and 0 outside [n/2, 2n].
|
||||
After divergence correction using Δ_V X_n = ∇·h_n and h̃_n = h_n − L_{X_n} g:
|
||||
||h̃_n||_{L^2} = 1, ∇^j h̃_{n,ij} = 0, ||L h̃_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
### A.4 Theorem 1 (Onset of the infrared continuum; as extracted)
|
||||
|
||||
```text
|
||||
Let (Σ, g) satisfy the asymptotic flatness conditions (11) with |Riem(x)| ≃ C r^-3.
|
||||
Then:
|
||||
[0, ∞) ⊂ σ_ess(L), and 0 ∈ σ_ess(L).
|
||||
```
|
||||
|
||||
### A.5 Proposition 1 (Dimensional criterion; as extracted)
|
||||
|
||||
```text
|
||||
Let ∆ be the Laplace–Beltrami operator on a d-dimensional asymptotically flat manifold,
|
||||
and V(r) ~ r^-p a curvature-induced potential. Then:
|
||||
V is a compact perturbation of ∆ iff p > d.
|
||||
p = d is the threshold.
|
||||
```
|
||||
|
||||
### A.6 Theorem 2 (Spectral threshold for linearized gravity; paper end state)
|
||||
|
||||
```text
|
||||
Let (Σ, g) be asymptotically flat with |Riem(x)| ≤ C r^-p.
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling.
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix B — Numerical definitions and the actual computed objects
|
||||
|
||||
This appendix consolidates the “what was computed” layer so the numerics can be read as a check of the operator story, not a vibe.
|
||||
|
||||
### B.1 Domains, parameters, and constraints (paper Section 5.1)
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
h ∈ {1.0, 0.75, 0.5}
|
||||
R_max ∈ {6, 10, 14, 18, 20}
|
||||
C = -1
|
||||
Dirichlet boundary: h|_{∂Ω} = 0
|
||||
convergence: relative error 10^-5
|
||||
```
|
||||
|
||||
TT enforcement via the penalty quotient:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇·h||^2 + ζ ||tr h||^2 ) / ||h||^2
|
||||
```
|
||||
|
||||
and the discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u.
|
||||
```
|
||||
|
||||
### B.2 Radial model operator (paper Definition 1)
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
### B.3 Rayleigh quotient functional (paper Section 5.2)
|
||||
|
||||
```text
|
||||
E[ϕ] = ( ∫_R^{2R} ( |ϕ'(r)|^2 + V_p(r)|ϕ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |ϕ(r)|^2 r^2 dr )
|
||||
ΔE(R, p) = E[ϕ] − E_free[ϕ]
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
### B.4 Table 1 (raw extraction block)
|
||||
|
||||
The following is the paper’s Table 1 content as extracted into plain text. It is kept here because it defines the specific numerical values associated with the scaling check.
|
||||
|
||||
```text
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1. The power-law scaling with R follows ∆E ∼
|
||||
R−(p−2) , confirming that p=3 behaves as the marginal
|
||||
case separating decaying from saturating behavior.
|
||||
p
|
||||
2.00
|
||||
|
||||
R
|
||||
∆E(R, p)
|
||||
Efull
|
||||
Efree
|
||||
−2
|
||||
−1
|
||||
10 −2.85 × 10
|
||||
1.50 × 10
|
||||
1.54 × 10−1
|
||||
−2
|
||||
−2
|
||||
20 −2.85 × 10
|
||||
3.74 × 10
|
||||
3.85 × 10−2
|
||||
Continued on next page
|
||||
|
||||
12
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
2.50
|
||||
|
||||
3.00
|
||||
|
||||
3.50
|
||||
|
||||
4.00
|
||||
|
||||
R
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
|
||||
∆E(R, p)
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−7.39 × 10−3
|
||||
−5.23 × 10−3
|
||||
−3.69 × 10−3
|
||||
−2.61 × 10−3
|
||||
−1.85 × 10−3
|
||||
−1.31 × 10−3
|
||||
−9.23 × 10−4
|
||||
−1.92 × 10−3
|
||||
−9.62 × 10−4
|
||||
−4.81 × 10−4
|
||||
−2.40 × 10−4
|
||||
−1.20 × 10−4
|
||||
−6.01 × 10−5
|
||||
−3.01 × 10−5
|
||||
−5.02 × 10−4
|
||||
−1.78 × 10−4
|
||||
−6.28 × 10−5
|
||||
−2.22 × 10−5
|
||||
−7.85 × 10−6
|
||||
−2.78 × 10−6
|
||||
−9.81 × 10−7
|
||||
−1.32 × 10−4
|
||||
−3.29 × 10−5
|
||||
−8.23 × 10−6
|
||||
−2.06 × 10−6
|
||||
−5.14 × 10−7
|
||||
−1.29 × 10−7
|
||||
|
||||
13
|
||||
|
||||
Efull
|
||||
Efree
|
||||
−3
|
||||
9.36 × 10
|
||||
9.63 × 10−3
|
||||
−3
|
||||
2.34 × 10
|
||||
2.41 × 10−3
|
||||
5.85 × 10−4 6.02 × 10−4
|
||||
1.46 × 10−4 1.50 × 10−4
|
||||
3.65 × 10−5 3.76 × 10−5
|
||||
1.53 × 10−1 1.54 × 10−1
|
||||
3.83 × 10−2 3.85 × 10−2
|
||||
9.59 × 10−3 9.63 × 10−3
|
||||
2.40 × 10−3 2.41 × 10−3
|
||||
6.01 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
Continued on next page
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
R
|
||||
640
|
||||
|
||||
∆E(R, p)
|
||||
Efull
|
||||
−8
|
||||
−3.21 × 10
|
||||
3.76 × 10−5
|
||||
|
||||
Efree
|
||||
3.76 × 10−5
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix C — Verbatim excerpt anchors used in the rewrite
|
||||
|
||||
The following are excerpted from the `pdftotext` extraction and are included to keep wording stable (they are not “new claims” added by this rewrite).
|
||||
|
||||
These are direct excerpts from the `pdftotext` extraction of the paper and are included so the wording is stable.
|
||||
|
||||
- “We identify curvature decay |Riem| ∼ 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 σess (L) = [0, ∞).”
|
||||
- “At the inverse-cube rate, compactness fails and zero energy enters σess (L), yielding marginally bound, finite-energy configurations that remain spatially extended.”
|
||||
- “Proposition 1 (Dimensional criterion for the critical decay rate). … V is a compact perturbation of ∆ if and only if p > d. The equality p = d marks the threshold …”
|
||||
- “Theorem 2 (Spectral threshold for linearized gravity). … For p > 3 … σess(L) = [0, ∞). … At p = 3 … Weyl sequence … For p < 3 … enhances infrared coupling …”
|
||||
934
arxiv/2511.05345/2026-01-01-gm-v3/rendered/index.html
Normal file
934
arxiv/2511.05345/2026-01-01-gm-v3/rendered/index.html
Normal file
|
|
@ -0,0 +1,934 @@
|
|||
<!doctype html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<meta charset="utf-8" />
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1" />
|
||||
<title>InfraFabric GM v3 — arXiv:2511.05345</title>
|
||||
<style>
|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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||||
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||||
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||||
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|
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|
||||
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|
||||
</head>
|
||||
<body>
|
||||
<div class="wrap">
|
||||
<div class="top">
|
||||
<div class="badge"><b>Render</b><span>Standalone HTML (Markdown + Mermaid)</span></div>
|
||||
<div class="badge"><b>Source</b><span>arXiv:2511.05345</span></div>
|
||||
</div>
|
||||
|
||||
<div class="panel">
|
||||
<div class="panelhead">
|
||||
<h1>InfraFabric — GM v3 rewrite (English, definition-preserving)</h1>
|
||||
<div class="links">
|
||||
<a class="btn" href="https://arxiv.org/abs/2511.05345" target="_blank" rel="noreferrer">arXiv abstract</a>
|
||||
<a class="btn" href="https://arxiv.org/pdf/2511.05345.pdf" target="_blank" rel="noreferrer">PDF</a>
|
||||
<a class="btn" href="dossier.md">Download Markdown (GM v3)</a>
|
||||
</div>
|
||||
</div>
|
||||
<div id="content">Loading…</div>
|
||||
<div class="foot" style="padding:0 18px 18px">If Mermaid doesn’t render, refresh once; this page needs CDN access for marked + mermaid.</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<script type="text/markdown" id="md"># The r⁻³ Curvature Decay Threshold in Linearized Gravity
|
||||
## A receipt-first, definition-preserving rewrite (readable without a GR/QC background)
|
||||
|
||||
**Source (arXiv abstract page):** https://arxiv.org/abs/2511.05345
|
||||
|
||||
**Source (PDF):** https://arxiv.org/pdf/2511.05345.pdf
|
||||
|
||||
**Local PDF SHA-256:** `992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742`
|
||||
|
||||
**Author (paper):** Michael Wilson (University of Arkansas at Little Rock)
|
||||
|
||||
**Date (paper):** November 10, 2025
|
||||
|
||||
---
|
||||
|
||||
> Most dossiers read as if physics is optional: eternal ROI, frictionless execution, boundless optimism. We strip away the gloss to reveal the operational reality.
|
||||
|
||||
This is a clean-room rewrite of the paper’s *technical spine*.
|
||||
|
||||
- It preserves the paper’s **definitions**, **operators**, and **scaling statements**.
|
||||
- It replaces local jargon with **plain-language interpretations**, so you can track what the math says without already being fluent in the field.
|
||||
- It keeps a strict separation between **what the paper states** and **what that would imply if true**.
|
||||
|
||||
---
|
||||
|
||||
## Quick digest (for people who do not want to read 13 pages)
|
||||
|
||||
**Context:** Linearized gravity is the “small perturbations” approximation. The paper asks a narrow but important question: what *asymptotic* geometric decay rate separates “fully radiative” behavior from “infrared / memory-like” behavior on a spatial slice.
|
||||
|
||||
> The claim is that `|Riem| ~ r^-3` is the sharp border.
|
||||
|
||||
### What the paper claims
|
||||
- If curvature decays faster than `r^-3`, the relevant spatial operator behaves like the flat operator: extended tensor modes disperse and there is no special zero-energy sector.
|
||||
- At exactly `r^-3`, compactness fails and **zero energy enters the essential spectrum**, producing **marginally extended, finite-energy** modes.
|
||||
- A simplified radial model reproduces the same transition at `p = 3`, and a dimensional argument suggests a general rule `p_crit = d`.
|
||||
|
||||
### Why this matters (without invoking mysticism)
|
||||
If you can tie “long-range gravitational memory / soft modes” to a concrete spatial decay threshold, you get a clean structural statement:
|
||||
|
||||
- below the threshold: the far field can keep “persistent correlations,”
|
||||
- above the threshold: the far field behaves like ordinary radiating waves.
|
||||
|
||||
That is not a claim about all of nonlinear gravity. It is a claim about the linear operator that controls stationary perturbations in harmonic gauge.
|
||||
|
||||
### Two ways to read this rewrite (pick one)
|
||||
|
||||
1) **Fast comprehension path (skip proofs, keep definitions):** Read Sections 1, 3.3, 6, and the “What to do with this” checklist. You will still see every operator and hypothesis.
|
||||
|
||||
2) **Definition-preserving path (follow the logic):** Read Sections 1–5 in order. This is the “operator → spectrum → threshold → numerical checks” route.
|
||||
|
||||
### Key objects (definitions you must know to follow the argument)
|
||||
|
||||
| Object | Definition (paper) | What it controls | What can go wrong
|
||||
|---|---|---|---|
|
||||
| Spatial Lichnerowicz operator | `L = ∇*∇ + V_R` | Stationary (time-independent) harmonic-gauge perturbations | Threshold behavior depends on decay of `V_R`
|
||||
| Curvature potential | `(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}` | How background curvature couples into the spin-2 operator | Compactness can fail at infinity
|
||||
| Essential spectrum | `σ_ess(L)` | Extended (non-localized) spectral behavior | Whether `0` lies in `σ_ess(L)` is the pivot
|
||||
| Weyl sequence | `{h_n}` with `||h_n||=1`, `h_n ⇀ 0`, `||L h_n|| → 0` | A way to prove `0 ∈ σ_ess(L)` | Requires careful gauge correction
|
||||
| Radial model | `L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p` | A tractable “one channel” proxy for scaling | Must not be oversold as the full operator
|
||||
|
||||
---
|
||||
|
||||
## 1. Introduction (what problem is being solved)
|
||||
|
||||
> The paper is trying to turn “infrared structure” from a slogan into a spectral criterion.
|
||||
|
||||
The starting tension is simple:
|
||||
|
||||
- There are well-known long-range gravitational phenomena (memory, tails, soft modes).
|
||||
- Most of the clean theory is phrased at null infinity (asymptotic symmetry frameworks).
|
||||
- It is less clear what a purely spatial (Cauchy slice) mechanism is, and specifically how **curvature decay** controls the existence of persistent low-frequency structure.
|
||||
|
||||
The paper proposes a criterion based on a single operator:
|
||||
|
||||
```text
|
||||
L = ∇*∇ + V_R,
|
||||
|
||||
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
|
||||
```
|
||||
|
||||
It is explicit about the intended interpretation:
|
||||
|
||||
- `σ_ess(L) = [0,∞)` corresponds to freely propagating tensor modes.
|
||||
- `0 ∈ σ_ess(L)` corresponds to marginally bound, spatially extended, finite-energy modes.
|
||||
|
||||
The paper’s central threshold claim (already in the abstract) is:
|
||||
|
||||
```text
|
||||
|Riem| ~ r^-3 is the sharp spectral threshold in 3 spatial dimensions.
|
||||
```
|
||||
|
||||
### How the paper says it will proceed (section map)
|
||||
|
||||
The paper itself says the structure is:
|
||||
|
||||
- Section 2: analytic framework and the scaling principle shared by gauge and gravity operators.
|
||||
- Section 3: the critical decay regime and the “zero enters the essential spectrum” result.
|
||||
- Section 4: general dimension scaling and `p_crit = d`.
|
||||
- Section 5: numerical verification: a radial scaling test plus a full 3D discretized tensor eigenvalue calculation.
|
||||
|
||||
---
|
||||
|
||||
## 2. Spectral scaling and structural parallels across spin‑1 and spin‑2 fields
|
||||
|
||||
> Two different theories can share the same infrared scaling mechanism because the geometry forces it.
|
||||
|
||||
### 2.1 Geometric setup and harmonic gauge (paper definitions)
|
||||
|
||||
The spatial manifold `(Σ, g)` is asymptotically flat. In an asymptotic coordinate chart:
|
||||
|
||||
```text
|
||||
g_{ij} = δ_{ij} + a_{ij}
|
||||
|
||||
a_{ij} = O(r^-1)
|
||||
∂_k a_{ij} = O(r^-2)
|
||||
∂_ℓ ∂_k a_{ij} = O(r^-3)
|
||||
```
|
||||
|
||||
The paper notes that these falloff conditions imply:
|
||||
|
||||
- `Γ^k_{ij} = O(r^-2)`
|
||||
- `|Riem| = O(r^-3)`
|
||||
|
||||
The operator `L` is defined to act on symmetric trace-free tensor fields `h_{ij}`. (The paper works in harmonic gauge; the details are enforced analytically via a vector Laplacian and gauge correction later.)
|
||||
|
||||
### 2.2 Shared scaling structure (spin‑1 analogue)
|
||||
|
||||
The paper draws a structural parallel to Yang–Mills:
|
||||
|
||||
```text
|
||||
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A).
|
||||
```
|
||||
|
||||
Both the spin‑1 and spin‑2 operators have the same schematic form:
|
||||
|
||||
- Laplace term
|
||||
- plus a curvature-induced potential term that decays like `r^-p`
|
||||
|
||||
The claim is not “the theories are the same.” It is “the *spectral scaling* is governed by the same dimensional mechanism.”
|
||||
|
||||
### 2.3 Weighted Sobolev spaces + symmetry identity (what gets used later)
|
||||
|
||||
The analysis is carried out on weighted spaces with weight `δ`:
|
||||
|
||||
```text
|
||||
H^k_δ(Σ; S^2 T*Σ) = { h in H^k_loc : ||h||_{H^k_δ} < ∞ }
|
||||
|
||||
||h||^2_{H^k_δ} = Σ_{j=0..k} ∫_Σ <r>^{2(δ-j)} |∇^j h|^2_g dV_g.
|
||||
```
|
||||
|
||||
Where the shorthand `<r>` is the standard “one plus radius” weight:
|
||||
|
||||
```text
|
||||
<r> = (1 + r^2)^{1/2}.
|
||||
```
|
||||
|
||||
An integration-by-parts identity makes `L` explicitly symmetric on compactly supported smooth tensors:
|
||||
|
||||
```text
|
||||
⟨Lh, k⟩_{L^2} = ⟨∇h, ∇k⟩_{L^2} + ⟨V_R h, k⟩_{L^2}.
|
||||
```
|
||||
|
||||
### 2.4 Self-adjoint realization + essential spectrum in the fast-decay regime
|
||||
|
||||
The paper states (for `-1 < δ < 0`) that the mapping
|
||||
|
||||
```text
|
||||
L : H^2_δ → L^2_{δ-2}
|
||||
```
|
||||
|
||||
is bounded, and that `L = ∇*∇ + V_R` is self-adjoint on the same domain as `∇*∇` under decay assumptions.
|
||||
|
||||
If curvature decays faster than `r^-3` (i.e. `p > 3`), `V_R` is compact in the relevant mapping, and Weyl’s theorem gives:
|
||||
|
||||
```text
|
||||
σ_ess(L) = [0, ∞).
|
||||
```
|
||||
|
||||
### 2.5 Why “3” appears (the core scaling argument)
|
||||
|
||||
The borderline value is obtained by comparing:
|
||||
|
||||
- the `r^-2` angular term induced by the Laplacian after spherical harmonic decomposition,
|
||||
- to a curvature tail `V_R ~ r^-p`.
|
||||
|
||||
The paper summarizes the outcome as:
|
||||
|
||||
```text
|
||||
p_crit = 3 (in three spatial dimensions).
|
||||
```
|
||||
|
||||
This is the first place where the narrative should snap into focus:
|
||||
|
||||
- above 3: curvature tail is spectrally negligible in the weighted setting,
|
||||
- at 3: curvature and dispersion balance,
|
||||
- below 3: curvature becomes long-range in the spectral sense.
|
||||
|
||||
---
|
||||
|
||||
## 3. Infrared spectrum and marginal modes (what “zero in the essential spectrum” means)
|
||||
|
||||
> This is the paper’s most concrete mathematical assertion: it constructs the mechanism that makes `0` appear.
|
||||
|
||||
### 3.1 Fredholm + gauge framework (so the “harmonic gauge” assumption is not hand-waved)
|
||||
|
||||
A key analytic tool is the vector Laplacian acting on one-forms:
|
||||
|
||||
```text
|
||||
Δ_V = ∇*∇ + Ric.
|
||||
```
|
||||
|
||||
The paper states a weighted Fredholm property:
|
||||
|
||||
```text
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
|
||||
is Fredholm and invertible for -1 < δ < 0.
|
||||
```
|
||||
|
||||
Operationally, this gives a well-posed “gauge correction” step: you can solve for a vector field `X` to correct a candidate tensor field so it satisfies the divergence constraint.
|
||||
|
||||
#### Lemma 1 (paper statement; verbatim structure retained)
|
||||
|
||||
For `−1 < δ < 0`:
|
||||
|
||||
```text
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
```
|
||||
|
||||
is Fredholm with bounded inverse. The paper uses this to justify solving `Δ_V X = f` with the estimate `||X||_{H^2_δ} ≤ C ||f||_{L^2_{δ-2}}`.
|
||||
|
||||
### 3.2 Weyl sequence at the critical decay (the construction)
|
||||
|
||||
Assume asymptotic flatness with falloff
|
||||
|
||||
```text
|
||||
g_{ij} = δ_{ij} + O(r^-1)
|
||||
∂g_{ij} = O(r^-2)
|
||||
∂^2 g_{ij} = O(r^-3)
|
||||
```
|
||||
|
||||
and curvature tail
|
||||
|
||||
```text
|
||||
|Riem(x)| ≃ C r^-3 as r → ∞.
|
||||
```
|
||||
|
||||
Then `L` is written again on `L^2(Σ; S^2 T*Σ)`:
|
||||
|
||||
```text
|
||||
L h = ∇*∇ h + V_R h
|
||||
|
||||
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
|
||||
```
|
||||
|
||||
The approximate zero-mode ansatz is (paper definition):
|
||||
|
||||
```text
|
||||
h_n(r, ω) = A_n φ_n(r) r^-1 H_{ij}(ω)
|
||||
|
||||
φ_n(r) = cutoff equal to 1 on [n, 3n/2] and vanishing outside [n/2, 2n]
|
||||
||h_n||_{L^2} = 1 ⇒ A_n ≃ n^-1/2.
|
||||
```
|
||||
|
||||
#### Lemma 2 (what the paper is asserting, operationally)
|
||||
|
||||
The paper’s “approximate zero mode” claim is:
|
||||
|
||||
- If you build `h_n` supported on a large annulus and normalize it (`||h_n||=1`),
|
||||
- then the action of `L` on it becomes small (`||L h_n|| → 0`) as the annulus goes to infinity,
|
||||
- and you can fix the divergence constraint by solving `Δ_V X_n = ∇·h_n` and subtracting a Lie derivative term `L_{X_n} g`.
|
||||
|
||||
The algebraic content is the triple condition:
|
||||
|
||||
```text
|
||||
||\tilde h_n||_{L^2} = 1,
|
||||
∇^j \tilde h_{n,ij} = 0,
|
||||
||L \tilde h_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
The paper’s proof sketch (as extracted) assigns the scale:
|
||||
|
||||
- derivatives of `φ_n` contribute factors `n^-1`,
|
||||
- giving `||∇·h_n||_{L^2} ≲ n^-1` and `||L h_n||_{L^2} ≲ n^-2`,
|
||||
- and then the gauge correction yields `||L_{X_n} g||_{L^2} ≲ n^-1`.
|
||||
|
||||
Then the paper performs a gauge correction:
|
||||
|
||||
- solve `Δ_V X_n = ∇·h_n`
|
||||
- define a corrected sequence `\tilde h_n = h_n - L_{X_n} g`.
|
||||
|
||||
The conclusion is the Weyl sequence criterion:
|
||||
|
||||
```text
|
||||
||\tilde h_n||_{L^2} = 1,
|
||||
∇^j \tilde h_{n,ij} = 0,
|
||||
||L \tilde h_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
This implies:
|
||||
|
||||
```text
|
||||
0 ∈ σ_ess(L)
|
||||
```
|
||||
|
||||
in the critical decay regime.
|
||||
|
||||
#### Theorem 1 (paper statement; what it buys you)
|
||||
|
||||
The paper’s formal conclusion for the critical decay regime is:
|
||||
|
||||
```text
|
||||
[0, ∞) ⊂ σ_ess(L),
|
||||
0 ∈ σ_ess(L).
|
||||
```
|
||||
|
||||
This is the paper’s internal definition of “infrared continuum onset”: `0` is no longer excluded from the essential spectrum when the curvature tail decays like `r^-3`.
|
||||
|
||||
### 3.3 A plain-English interpretation of the Weyl sequence claim (what should stick)
|
||||
|
||||
You can read “`0 ∈ σ_ess(L)`” as:
|
||||
|
||||
- You can build a sequence of increasingly far-out tensor configurations,
|
||||
- each with finite (normalized) `L^2` energy,
|
||||
- whose “operator energy” `||L h||` goes to zero,
|
||||
- meaning the operator behaves like it has “almost-static” extended modes at arbitrarily large radius.
|
||||
|
||||
This is not the same as saying “there is a normalizable bound state.” The paper is describing a marginal, continuum-edge phenomenon, not a discrete eigenvalue below zero.
|
||||
|
||||
---
|
||||
|
||||
## 4. Dimensional scaling and spectral phase structure (generalization)
|
||||
|
||||
> The paper claims the “3” is not accidental; it is `d`.
|
||||
|
||||
The paper introduces a dimensional generalization:
|
||||
|
||||
```text
|
||||
L_d = -∇^2 + V(r), with V(r) ~ r^-p on a d-dimensional asymptotically flat manifold.
|
||||
```
|
||||
|
||||
### Proposition (paper statement)
|
||||
|
||||
The key statement is:
|
||||
|
||||
```text
|
||||
V is a compact perturbation of Δ ⇔ p > d.
|
||||
|
||||
p = d is the threshold between short- and long-range behavior.
|
||||
```
|
||||
|
||||
The paper explains why a naive `L^2` or Hilbert–Schmidt test is too strong, and why the weighted mapping and asymptotic mode scaling matter.
|
||||
|
||||
#### The proof mechanism (as stated in the paper)
|
||||
|
||||
The paper’s proof outline has two key moves:
|
||||
|
||||
1) “`V → 0` at infinity implies compact multiplication on unweighted spaces” is **not** the right test here, because the operator is being controlled on weighted mappings appropriate to asymptotically flat ends.
|
||||
|
||||
2) In the weighted space, compactness fails exactly when the tail contributions fail to vanish at large radius. For the asymptotic tensor mode scaling, the paper states:
|
||||
|
||||
```text
|
||||
h(r) ~ r^{-(d-2)/2}
|
||||
```
|
||||
|
||||
and then estimates the weighted norm of `V h` on `{r > R}` by an integral of the form:
|
||||
|
||||
```text
|
||||
∫_R^∞ r^{d - 5 + 2δ - 2p} dr
|
||||
```
|
||||
|
||||
Because `−1 < δ < 0`, the paper concludes the integral diverges when `p ≤ d`, implying noncompact behavior in the curvature tail.
|
||||
|
||||
If you only keep one operational takeaway from Section 4, it is:
|
||||
|
||||
- the threshold depends on how the asymptotic modes scale in the weighted space,
|
||||
- and that scaling drives the integral that converges or diverges in the tail.
|
||||
|
||||
In 3 dimensions, this reduces back to `p_crit = 3`.
|
||||
|
||||
---
|
||||
|
||||
## 5. Numerical verification of the spectral threshold (what was actually computed)
|
||||
|
||||
> The numerical section is not “just plots.” It has explicit discretization choices and convergence targets.
|
||||
|
||||
### 5.1 Numerical setup and nondimensionalization (paper parameters)
|
||||
|
||||
The full tensor operator `L = ∇*∇ + V_R` is discretized on a uniform Cartesian grid:
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
```
|
||||
|
||||
with spacing `h = Δx / L_0` after nondimensionalization by a fixed length scale `L_0`.
|
||||
|
||||
- Centered finite differences approximate `∇` and `∇*∇`.
|
||||
- Dirichlet boundary conditions impose `h|_{∂Ω} = 0`.
|
||||
- The continuum approach is monitored by extrapolation in `R_max`.
|
||||
|
||||
To enforce transverse–traceless constraints, the paper uses a penalty functional:
|
||||
|
||||
```text
|
||||
⟨h, Lh⟩ + η ||∇·h||^2_{L^2(Ω)} + ζ ( ... )
|
||||
```
|
||||
|
||||
and reports that varying penalties `η, ζ` by factors `2–4` shifts the lowest eigenvalue by less than `10^-3`.
|
||||
|
||||
#### The exact penalty functional (paper equation (16))
|
||||
|
||||
The paper’s explicit form of the penalized Rayleigh quotient is:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇· h||^2_{L^2(Ω)} + ζ ||tr h||^2_{L^2(Ω)} ) / ||h||^2_{L^2(Ω)}.
|
||||
```
|
||||
|
||||
It then states the associated discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u,
|
||||
```
|
||||
|
||||
where `K` and `M` are stiffness and mass matrices and `D, T` are discrete divergence and trace operators.
|
||||
|
||||
Representative nondimensional parameters (paper list):
|
||||
|
||||
- `h ∈ {1.0, 0.75, 0.5}`
|
||||
- `R_max ∈ {6, 10, 14, 18, 20}`
|
||||
- `C = -1`
|
||||
- grid sizes `N = 21–41` (up to `3.6 × 10^5` degrees of freedom)
|
||||
- convergence target: relative error `10^-5`.
|
||||
|
||||
The paper explicitly notes that Dirichlet boundaries discretize a near-threshold continuum into “box modes” with scaling
|
||||
|
||||
```text
|
||||
λ_1(R_max) ∝ R_max^-2.
|
||||
```
|
||||
|
||||
This matters because it is an explicit check that the numerics are tracking continuum-edge behavior rather than a false “confinement.”
|
||||
|
||||
### 5.1 Definition (radial operator vs full 3D operator)
|
||||
|
||||
The paper defines a radial model operator (single angular channel):
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
and describes the full 3D discretized operator as including curvature coupling and constraint enforcement.
|
||||
|
||||
It also states a channel correspondence:
|
||||
|
||||
```text
|
||||
V_eff(r) = ℓ(ℓ+1)/r^2 + C/r^p + O(r^{-p-1}).
|
||||
```
|
||||
|
||||
### 5.2 Rayleigh–quotient scaling test (explicit energy functional)
|
||||
|
||||
The paper uses an energy functional on a normalized bump function `φ(r)` supported on `[R, 2R]`:
|
||||
|
||||
```text
|
||||
E[φ] = ( ∫_R^{2R} ( |φ'(r)|^2 + V_p(r)|φ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |φ(r)|^2 r^2 dr )
|
||||
|
||||
ΔE(R, p) = E[φ] - E_free[φ].
|
||||
```
|
||||
|
||||
It reports a predicted scaling relationship:
|
||||
|
||||
```text
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
and states that fitted slopes `α(p) ≈ -(p-2)` match the analytic scaling.
|
||||
|
||||
### What the numerics are *actually used for*
|
||||
|
||||
The numerics support two claims:
|
||||
|
||||
1) In the radial model, the lowest eigenvalue approaches the continuum threshold near `p = 3`.
|
||||
|
||||
2) In the full 3D discretized tensor operator, the same transition appears without evidence of discrete bound-state formation.
|
||||
|
||||
The important “interpretation constraint” is:
|
||||
|
||||
- the numerics validate a **continuous spectral crossover** centered at `p = 3`, not a dramatic phase change.
|
||||
|
||||
---
|
||||
|
||||
## 6. Physical interpretation and implications (what “infrared structure” means here)
|
||||
|
||||
> The paper is careful: it is still a classical, stationary analysis. It argues the operator already contains the seeds of the soft sector.
|
||||
|
||||
The paper links the operator’s spectrum to a quantized two-point function: in canonical quantization (harmonic gauge), the equal-time two-point function is the inverse of `L`, so large-distance correlations are governed by the same threshold.
|
||||
|
||||
It also states a regime interpretation:
|
||||
|
||||
- for `p > 3`: radiative propagation, fully dispersive
|
||||
- for `p = 3`: marginal persistence (extended but finite-energy)
|
||||
- for `p < 3`: enhanced infrared coupling (without discrete confinement in the tensor sector).
|
||||
|
||||
The paper draws a conceptual link to soft graviton theorems and gravitational memory, but it flags that a complete correspondence would require coupling to the time-dependent linearized Einstein equations near null infinity.
|
||||
|
||||
---
|
||||
|
||||
## 7. Relation to previous work and threshold alignment (why this is not an isolated curiosity)
|
||||
|
||||
> The paper’s “parallel threshold” claim is what turns a one-off scaling observation into a pattern.
|
||||
|
||||
The paper states an analogous threshold in non-Abelian gauge theory:
|
||||
|
||||
```text
|
||||
Δ_A = -(∇_A)* ∇_A = -∇*∇ + ad(F_A),
|
||||
|
||||
|F_A| ~ r^-3 is the analogue threshold.
|
||||
```
|
||||
|
||||
It claims that for Laplace-type operators on bundles over `R^3`, a curvature tail of order `r^-3` separates short-range radiative behavior from long-range infrared coupling.
|
||||
|
||||
It also ties `r^-3` to known scattering thresholds in Schrödinger-type operators.
|
||||
|
||||
---
|
||||
|
||||
## 8. Conclusion (the paper’s end state)
|
||||
|
||||
> If you accept the operator model, then `r^-3` is the boundary between “radiative only” and “infrared sector present” on a spatial slice.
|
||||
|
||||
The paper’s concluding theorem is explicit:
|
||||
|
||||
```text
|
||||
Theorem (paper): Let |Riem(x)| ≤ C r^-p on an asymptotically flat 3-manifold.
|
||||
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling (without isolated bound states in the tensor sector).
|
||||
```
|
||||
|
||||
It adds explicit future directions: limiting absorption principle at critical rate; extension to Schwarzschild and Kerr; dynamical correspondence between spatial modes and soft/memory sector at null infinity.
|
||||
|
||||
---
|
||||
|
||||
## What to do with this (if you are not a gravitational physicist)
|
||||
|
||||
> Treat this as a structural result about a linear operator under asymptotic decay assumptions.
|
||||
|
||||
If you want to use this paper responsibly, the minimum checklist is:
|
||||
|
||||
1) You can restate the operator definitions (`L`, `V_R`, radial `L_p`) without distortion.
|
||||
2) You can state the threshold claim with its scope (“linearized,” “harmonic gauge,” “asymptotically flat,” “spatial slice”).
|
||||
3) You can distinguish “operator has marginal zero-energy Weyl sequence” from “there are bound states” (the paper says no discrete confinement is observed).
|
||||
4) You can keep the generalization `p_crit = d` separate from the physically interpreted 3D statement.
|
||||
|
||||
*If you can’t do those four things, you are not citing the paper. You are citing the vibe.*
|
||||
|
||||
---
|
||||
|
||||
## Appendix A — Formal statements (paper text; structure preserved)
|
||||
|
||||
This appendix exists so you can see the “hard edges” in one place: the operator definitions and the main threshold statements.
|
||||
|
||||
### A.1 Operator definitions (paper equations (1), (9), (12))
|
||||
|
||||
```text
|
||||
L = ∇*∇ + V_R,
|
||||
(V_R h)_{ij} = -R^\ell_{ijm} h_{\ell m}.
|
||||
|
||||
Δ_V = ∇*∇ + Ric.
|
||||
|
||||
L h = ∇*∇ h + V_R h,
|
||||
(V_R h)_{ij} = -R_{i\ell jm} h_{\ell m}.
|
||||
```
|
||||
|
||||
### A.2 Lemma 1 (Fredholm property; as extracted)
|
||||
|
||||
```text
|
||||
For −1 < δ < 0:
|
||||
Δ_V : H^2_δ(Σ; T*Σ) → L^2_{δ-2}(Σ; T*Σ)
|
||||
is Fredholm with bounded inverse.
|
||||
```
|
||||
|
||||
### A.3 Lemma 2 (Approximate zero modes; as extracted)
|
||||
|
||||
```text
|
||||
Let H_{ij}(ω) be symmetric, trace-free, divergence-free on S^2 and define
|
||||
h_n(r, ω) = A_n ϕ_n(r) r^-1 H_{ij}(ω),
|
||||
where ϕ_n = 1 on [n, 3n/2] and 0 outside [n/2, 2n].
|
||||
After divergence correction using Δ_V X_n = ∇·h_n and h̃_n = h_n − L_{X_n} g:
|
||||
||h̃_n||_{L^2} = 1, ∇^j h̃_{n,ij} = 0, ||L h̃_n||_{L^2} → 0.
|
||||
```
|
||||
|
||||
### A.4 Theorem 1 (Onset of the infrared continuum; as extracted)
|
||||
|
||||
```text
|
||||
Let (Σ, g) satisfy the asymptotic flatness conditions (11) with |Riem(x)| ≃ C r^-3.
|
||||
Then:
|
||||
[0, ∞) ⊂ σ_ess(L), and 0 ∈ σ_ess(L).
|
||||
```
|
||||
|
||||
### A.5 Proposition 1 (Dimensional criterion; as extracted)
|
||||
|
||||
```text
|
||||
Let ∆ be the Laplace–Beltrami operator on a d-dimensional asymptotically flat manifold,
|
||||
and V(r) ~ r^-p a curvature-induced potential. Then:
|
||||
V is a compact perturbation of ∆ iff p > d.
|
||||
p = d is the threshold.
|
||||
```
|
||||
|
||||
### A.6 Theorem 2 (Spectral threshold for linearized gravity; paper end state)
|
||||
|
||||
```text
|
||||
Let (Σ, g) be asymptotically flat with |Riem(x)| ≤ C r^-p.
|
||||
1) p > 3 ⇒ σ_ess(L) = [0,∞).
|
||||
2) p = 3 ⇒ compactness fails and a normalized Weyl sequence appears at zero energy.
|
||||
3) p < 3 ⇒ curvature enhances infrared coupling.
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix B — Numerical definitions and the actual computed objects
|
||||
|
||||
This appendix consolidates the “what was computed” layer so the numerics can be read as a check of the operator story, not a vibe.
|
||||
|
||||
### B.1 Domains, parameters, and constraints (paper Section 5.1)
|
||||
|
||||
```text
|
||||
Ω = [-R_max, R_max]^3
|
||||
h ∈ {1.0, 0.75, 0.5}
|
||||
R_max ∈ {6, 10, 14, 18, 20}
|
||||
C = -1
|
||||
Dirichlet boundary: h|_{∂Ω} = 0
|
||||
convergence: relative error 10^-5
|
||||
```
|
||||
|
||||
TT enforcement via the penalty quotient:
|
||||
|
||||
```text
|
||||
R_{η,ζ}[h] = ( ⟨h, Lh⟩ + η ||∇·h||^2 + ζ ||tr h||^2 ) / ||h||^2
|
||||
```
|
||||
|
||||
and the discrete generalized eigenproblem:
|
||||
|
||||
```text
|
||||
(K + η D^⊤ D + ζ T^⊤ T) u = λ M u.
|
||||
```
|
||||
|
||||
### B.2 Radial model operator (paper Definition 1)
|
||||
|
||||
```text
|
||||
L_p = -d^2/dr^2 + ℓ(ℓ+1)/r^2 + C/r^p.
|
||||
```
|
||||
|
||||
### B.3 Rayleigh quotient functional (paper Section 5.2)
|
||||
|
||||
```text
|
||||
E[ϕ] = ( ∫_R^{2R} ( |ϕ'(r)|^2 + V_p(r)|ϕ(r)|^2 ) r^2 dr ) / ( ∫_R^{2R} |ϕ(r)|^2 r^2 dr )
|
||||
ΔE(R, p) = E[ϕ] − E_free[ϕ]
|
||||
ΔE(R, p) ~ R^{-(p-2)}.
|
||||
```
|
||||
|
||||
### B.4 Table 1 (raw extraction block)
|
||||
|
||||
The following is the paper’s Table 1 content as extracted into plain text. It is kept here because it defines the specific numerical values associated with the scaling check.
|
||||
|
||||
```text
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1. The power-law scaling with R follows ∆E ∼
|
||||
R−(p−2) , confirming that p=3 behaves as the marginal
|
||||
case separating decaying from saturating behavior.
|
||||
p
|
||||
2.00
|
||||
|
||||
R
|
||||
∆E(R, p)
|
||||
Efull
|
||||
Efree
|
||||
−2
|
||||
−1
|
||||
10 −2.85 × 10
|
||||
1.50 × 10
|
||||
1.54 × 10−1
|
||||
−2
|
||||
−2
|
||||
20 −2.85 × 10
|
||||
3.74 × 10
|
||||
3.85 × 10−2
|
||||
Continued on next page
|
||||
|
||||
12
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
2.50
|
||||
|
||||
3.00
|
||||
|
||||
3.50
|
||||
|
||||
4.00
|
||||
|
||||
R
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
640
|
||||
10
|
||||
20
|
||||
40
|
||||
80
|
||||
160
|
||||
320
|
||||
|
||||
∆E(R, p)
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−2.85 × 10−2
|
||||
−7.39 × 10−3
|
||||
−5.23 × 10−3
|
||||
−3.69 × 10−3
|
||||
−2.61 × 10−3
|
||||
−1.85 × 10−3
|
||||
−1.31 × 10−3
|
||||
−9.23 × 10−4
|
||||
−1.92 × 10−3
|
||||
−9.62 × 10−4
|
||||
−4.81 × 10−4
|
||||
−2.40 × 10−4
|
||||
−1.20 × 10−4
|
||||
−6.01 × 10−5
|
||||
−3.01 × 10−5
|
||||
−5.02 × 10−4
|
||||
−1.78 × 10−4
|
||||
−6.28 × 10−5
|
||||
−2.22 × 10−5
|
||||
−7.85 × 10−6
|
||||
−2.78 × 10−6
|
||||
−9.81 × 10−7
|
||||
−1.32 × 10−4
|
||||
−3.29 × 10−5
|
||||
−8.23 × 10−6
|
||||
−2.06 × 10−6
|
||||
−5.14 × 10−7
|
||||
−1.29 × 10−7
|
||||
|
||||
13
|
||||
|
||||
Efull
|
||||
Efree
|
||||
−3
|
||||
9.36 × 10
|
||||
9.63 × 10−3
|
||||
−3
|
||||
2.34 × 10
|
||||
2.41 × 10−3
|
||||
5.85 × 10−4 6.02 × 10−4
|
||||
1.46 × 10−4 1.50 × 10−4
|
||||
3.65 × 10−5 3.76 × 10−5
|
||||
1.53 × 10−1 1.54 × 10−1
|
||||
3.83 × 10−2 3.85 × 10−2
|
||||
9.59 × 10−3 9.63 × 10−3
|
||||
2.40 × 10−3 2.41 × 10−3
|
||||
6.01 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
3.76 × 10−5 3.76 × 10−5
|
||||
1.54 × 10−1 1.54 × 10−1
|
||||
3.85 × 10−2 3.85 × 10−2
|
||||
9.63 × 10−3 9.63 × 10−3
|
||||
2.41 × 10−3 2.41 × 10−3
|
||||
6.02 × 10−4 6.02 × 10−4
|
||||
1.50 × 10−4 1.50 × 10−4
|
||||
Continued on next page
|
||||
|
||||
|
||||
Table 1: Rayleigh-quotient energy shift ∆E(R, p) for
|
||||
C = −1 (continued).
|
||||
p
|
||||
|
||||
R
|
||||
640
|
||||
|
||||
∆E(R, p)
|
||||
Efull
|
||||
−8
|
||||
−3.21 × 10
|
||||
3.76 × 10−5
|
||||
|
||||
Efree
|
||||
3.76 × 10−5
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## Appendix C — Verbatim excerpt anchors used in the rewrite
|
||||
|
||||
The following are excerpted from the `pdftotext` extraction and are included to keep wording stable (they are not “new claims” added by this rewrite).
|
||||
|
||||
These are direct excerpts from the `pdftotext` extraction of the paper and are included so the wording is stable.
|
||||
|
||||
- “We identify curvature decay |Riem| ∼ 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 σess (L) = [0, ∞).”
|
||||
- “At the inverse-cube rate, compactness fails and zero energy enters σess (L), yielding marginally bound, finite-energy configurations that remain spatially extended.”
|
||||
- “Proposition 1 (Dimensional criterion for the critical decay rate). … V is a compact perturbation of ∆ if and only if p > d. The equality p = d marks the threshold …”
|
||||
- “Theorem 2 (Spectral threshold for linearized gravity). … For p > 3 … σess(L) = [0, ∞). … At p = 3 … Weyl sequence … For p < 3 … enhances infrared coupling …”
|
||||
</script>
|
||||
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|
||||
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|
||||
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|
||||
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arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.pdf
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|
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|
|||
992e50be5f84477fdbcf72ddea26bec88e6e1b756d948161af1b71bf03fc6742
|
||||
1512
arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.txt
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1512
arxiv/2511.05345/2026-01-01-gm-v3/source/2511.05345.txt
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