arXiv:2511.05345v1 [gr-qc] 7 Nov 2025

The r −3 Curvature Decay and the Infrared
Structure of Linearized Gravity
Michael Wilson
University of Arkansas at Little Rock
Department of Physics and Astronomy
mkwilson3@ualr.edu

November 10, 2025
Abstract
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, ∞), corresponding to freely radiating
tensor modes. At the inverse-cube rate, compactness fails and zero energy enters σess (L), yielding marginally bound, finite-energy configurations that remain spatially extended. These static modes constitute
the linear precursors of gravitational memory and soft-graviton phenomena, delineating the geometric boundary between dispersive and
infrared behavior. A complementary numerical study of the radial
model
ℓ(ℓ+1)
d2
Lp = − dr
+ rCp
2 +
r2
confirms the analytic scaling law, locating the same transition at
p = 3. The eigenvalue trends approach the flat-space limit continuously for p > 3 and strengthen progressively for p < 3, demonstrating a smooth yet sharp spectral transition rather than a discrete
confinement regime. The result parallels the critical threshold of the
non-Abelian covariant Laplacian [18], indicating a common r−3 scaling
that governs the infrared structure of gauge and gravitational fields.

1

1

Introduction

The infrared structure of gravity governs the persistent correlations that
remain after gravitational radiation has passed. Phenomena such as gravitational memory, power-law tails, and soft graviton modes all originate from
the long-range behavior of the field at large distances and late times. While
these effects are well understood in asymptotic frameworks, particularly at
null infinity, where BMS symmetries organize the radiative data, the corresponding spatial mechanism on a Cauchy slice is less clearly established. In
particular, it has not been resolved how the decay of curvature on an initial hypersurface determines the presence or absence of infrared correlations.
This work identifies a precise spectral criterion that governs this transition.
We study the spatial Lichnerowicz operator
L = ∇∗ ∇ + VR ,

(VR h)ij = −R ℓ ijm hℓ m ,

(1)

acting on symmetric, trace-free tensor fields hij on an asymptotically flat
three-manifold (Σ, g). The operator (1) governs stationary, harmonic-gauge
perturbations of a vacuum background and determines whether small tensor excitations are radiative or spatially correlated. Its essential spectrum
σess (L) distinguishes these regimes: a purely continuous spectrum [0, ∞) corresponds to freely propagating modes, whereas inclusion of zero in σess (L)
signals marginally bound, long-range configurations.
We show that curvature decay |Riem| ∼ r−3 marks the sharp boundary between these behaviors. For faster decay, VR is a compact perturbation of the
flat tensor Laplacian ∆T = ∇∗ ∇, giving σess (L) = [0, ∞). At the inverse-cube
rate, compactness fails: curvature and dispersion balance exactly, allowing
zero energy to enter the essential spectrum. In this marginal regime, L admits a normalized, divergence-free Weyl sequence with ∥Lhn ∥ → 0, describing
extended, finite-energy tensor modes at zero frequency.
A complementary numerical analysis of the radial model
ℓ(ℓ + 1) C
d2
+ p,
Lp = − 2 +
dr
r2
r
confirms this transition. The lowest eigenvalue λ1 decreases smoothly as
the decay exponent p is reduced, approaching the continuum threshold near
p = 3 without developing discrete bound states. This behavior demonstrates
a continuous spectral crossover rather than a discrete confinement phase,
2

while preserving the analytic identification of p = 3 as the exact transition
point.
Physically, the marginally bound tensor modes identified here represent
static precursors of gravitational memory and soft-graviton phenomena. The
same inverse-cube scaling governs the non-Abelian covariant Laplacian in
gauge theory [18], where |FA | ∼ r−3 separates radiative from infrared-sensitive
behavior. This parallel suggests a common spectral mechanism linking spin1 and spin-2 fields, rooted in the dimensional scaling of curvature in three
spatial dimensions.
The paper is organized as follows. Section 2 introduces the analytic framework and the spectral universality principle linking gauge and gravitational
fields. Section 3 establishes the inverse-cube decay as the critical regime and
proves that zero enters the essential spectrum of the Lichnerowicz operator. Section 4 generalizes the argument to arbitrary dimension and derives
the scaling law pcrit = d. Section 5 presents numerical verification of the
spectral threshold through Rayleigh-quotient scaling and three-dimensional
eigenvalue analysis, supported by convergence and stability checks. Section 6
interprets the transition in physical terms, relating it to gravitational memory, late-time tails, and asymptotic symmetries. Section 7 situates the result
within prior work in spectral geometry and gauge theory, and Section 8 summarizes the main findings. Detailed mathematical proofs, extended derivations, and numerical validation are provided in the Supplementary Material
(Appendix A-E). There we include the harmonic-gauge correction construction, the Schwarzschild example, the analysis of weighted Sobolev spaces, and
convergence and stability tests supporting the numerical results of Section 5.

2

Spectral Scaling and Structural Parallels
Across Spin-1 and Spin-2 Fields

This section develops the analytic framework for the spectral analysis of the
linearized gravitational field on asymptotically flat spatial slices and highlights its structural similarity to the spin-1 gauge field. In both cases, the
governing operator is of Laplace type with a curvature-induced potential
whose decay controls infrared behavior. Dimensional considerations reveal a
shared critical decay rate at which the potential ceases to be short-range, a
geometric correspondence rather than a full dynamical equivalence between

3

the two theories.

2.1

Geometric setup and harmonic gauge

Let (Σ, g) be a smooth, oriented, three-dimensional Riemannian manifold
representing a time-symmetric slice of a vacuum spacetime (M, gµν ) satisfying
Ric(gµν ) = 0. On the asymptotic end, choose coordinates identifying Σ \ K
with R3 \ BR (0) such that
gij = δij + aij ,

aij = O(r−1 ),

∂k aij = O(r−2 ),

∂ℓ ∂k aij = O(r−3 ),

where r = |x| and ⟨r⟩ = (1 + r2 )1/2 . These conditions imply Γkij = O(r−2 ),
|Riem| = O(r−3 ), and ∇Riem = O(r−4 ).
Expanding the Einstein-Hilbert action
Z
√
1
R −g d4 x
(2)
SEH =
16πG
to quadratic order in a perturbation gµν 7→ gµν + ℏ1/2 hµν yields the spatial
quadratic form [20, 21]
Z
√
(2)
1
S [h] = 2 hij Lij kl hkl g d3 x,
Σ

where
Lh = ∇∗ ∇h + VR h,

(VR h)ij = −Ri ℓ j m hℓm .

(3)

The operator L acts on symmetric trace-free tensors and represents the spatial part of the quadratic graviton operator in harmonic gauge. Its spectral
properties determine whether small tensor excitations remain radiative or
acquire long-range correlations.

2.2

Shared scaling structure

An analogous Laplace-type operator appears for spin-1 fields. For a YangMills connection A, the covariant Laplacian on the adjoint bundle is
∆A = −(∇A )∗ ∇A = −∇∗ ∇ + ad(FA ),

(4)

where ad(FA ) denotes the adjoint action of the curvature. Both (3) and
(4) therefore consist of a Laplace term plus a curvature potential that decays with distance. In the scalar Schrödinger case, potentials faster than r−3
4

yield compact perturbations of −∆, while the inverse-cube rate marks the
onset of threshold phenomena. The same scaling governs curvature-coupled
Laplace-type operators for spin-1 and spin-2 fields: in three spatial dimensions, curvature terms |FA | and |Riem| become marginal when p = 3, the
precise rate at which the potential ceases to be compact.1

2.3

Weighted Sobolev spaces and integration identity

For δ ∈ R, define
Hδk (Σ; S 2 T ∗ Σ) =

n
o
k
h ∈ Hloc (Σ) : ∥h∥Hδk < ∞ ,

∥h∥2H k =
δ

k Z
X

⟨r⟩2(δ−j) |∇j h|2g dVg .

j=0 Σ

Integration by parts using the asymptotic decay of g gives
⟨Lh, k⟩L2 = ⟨∇h, ∇k⟩L2 + ⟨VR h, k⟩L2 ,

(5)

valid for compactly supported smooth tensors h, k, showing that L is symmetric on Cc∞ (Σ; S 2 T ∗ Σ).

2.4

Self-adjoint realization and spectral framework

Weighted elliptic estimates imply
L : Hδ2 (Σ; S 2 T ∗ Σ) −→ L2δ−2 (Σ; S 2 T ∗ Σ)

(6)

is bounded for −1 < δ < 0. Essential self-adjointness of ∇∗ ∇ on complete
Riemannian manifolds follows from Chernoff’s theorem, and VR is symmetric
and relatively bounded since |VR (x)| = O(r−p ). By the Kato-Rellich theorem,
L = ∇∗ ∇ + VR is self-adjoint on the same domain as ∇∗ ∇.
By Lemma C.4 2 , if |Riem(x)| ≤ C⟨r⟩−p with p > 3, then VR is compact
as a map Hδ2 → L2δ−2 for −1 < δ < 0, and therefore L = ∇∗ ∇ + VR is a
compact perturbation of ∇∗ ∇. Weyl’s theorem then implies that
σess (L) = [0, ∞).

(7)

In particular, sufficiently rapid curvature decay leaves the essential spectrum
of the linearized gravitational field identical to that of flat space, so all finiteenergy tensor excitations are asymptotically radiative.
1
2

The analogous threshold for the non-Abelian Laplacian was derived in Ref. [18].
See supplementary material. Compactness of the curvature potential for p > 3.

5

2.5

Critical scaling and spectral threshold

To identify the limit of compactness, project Lh = 0 onto a spherical harmonic mode. The Laplacian contributes an r−2 angular term, while a curvature potential VR ∼ r−p competes with it when p = 3. Thus
pcrit = 3,

(8)

which marks the transition between short and long-range geometric potentials. For p > 3, curvature effects are subdominant and the spectrum remains
stable; at p = 3, curvature and kinetic terms balance, producing marginally
bound tensor modes and a continuous spectrum extending to zero.
This inverse-cube decay rate encapsulates the shared scaling property of
Laplace-type operators for spin-1 and spin-2 fields in three dimensions—a
geometric origin of infrared sensitivity common to gauge and gravitational
settings.

3

Infrared Spectrum and Marginal Modes

When the background curvature decays as r−3 , the Lichnerowicz operator
reaches the scaling threshold identified in Section 2. At this rate, curvature
and kinetic terms balance asymptotically, producing marginally bound tensor modes whose energy approaches zero. In operator terms, 0 enters the
essential spectrum of L. This section establishes that result rigorously and
interprets it within linearized gravity.

3.1

Fredholm and gauge framework

Let (Σ, g) be a smooth, asymptotically flat three-manifold with a single end
1
diffeomorphic to R3 outside a compact set, and assume HdR
(Σ) = 0. The
vector Laplacian
∆V = ∇∗ ∇ + Ric,
(9)
acts on one-forms and enforces the harmonic gauge constraint. Standard
elliptic theory gives:
Lemma 1 (Fredholm property). For −1 < δ < 0, the operator
∆V : Hδ2 (Σ; T ∗ Σ) −→ L2δ−2 (Σ; T ∗ Σ)
6

(10)

is Fredholm with bounded inverse. Hence, for each f ∈ L2δ−2 there exists a
unique X ∈ Hδ2 satisfying ∆V X = f and ∥X∥Hδ2 ≤ C∥f ∥L2δ−2 .
Proof. This is the weighted elliptic isomorphism theorem of Lockhart and
1
(Σ) = 0, and surjectivity from
McOwen [12]. Injectivity follows from HdR
asymptotic flatness.
Under these hypotheses, ∆V defines a gauge correction that enforces
∇ hij = 0. All subsequent constructions assume this analytic and topological framework.
j

3.2

Weyl sequence at the critical decay

Assume the metric satisfies
gij = δij + O(r−1 ),

∂gij = O(r−2 ),

∂ 2 gij = O(r−3 ),

(11)

and that the curvature obeys |Riem(x)| ≃ C r−3 as r → ∞. On L2 (Σ; S 2 T ∗ Σ),
consider the spatial Lichnerowicz operator
Lh = ∇∗ ∇h + VR h,

(VR h)ij = −Ri ℓ jm hℓ m ,

(12)

with domain Hδ2 (Σ; S 2 T ∗ Σ), −1 < δ < 0.
Lemma 2 (Approximate zero modes). Let Hij (ω) be a symmetric, trace-free,
divergence-free tensor harmonic on S 2 , and define
hn (r, ω) = An ϕn (r) r−1 Hij (ω),

(13)

where ϕn is a smooth cutoff equal to 1 on [n, 3n/2] and vanishing outside
[n/2, 2n]. The normalization ∥hn ∥L2 = 1 gives An ≃ n−1/2 . After divergence
correction by a vector field Xn satisfying ∆V Xn = ∇· hn and setting h̃n =
hn − LXn g, one obtains
∥h̃n ∥L2 = 1,

∇j h̃n,ij = 0,

∥Lh̃n ∥L2 → 0.

Proof. The tensor hn is supported on the annulus {n/2 < r < 2n} and normalized so that ∥hn ∥L2 = 1. Derivatives of the cutoff introduce factors of
n−1 , yielding ∥∇· hn ∥L2 ≲ n−1 and ∥Lhn ∥L2 ≲ n−2 . The gauge correction
Xn obeys ∥Xn ∥Hδ2 ≲ n−1 and therefore ∥LXn g∥L2 ≲ n−1 . Consequently, h̃n
is divergence-free and satisfies the bounds above. A complete derivation, including control of curvature remainders, commutators [∇∗ ∇, ϕn ], and explicit
estimates on L(LXn g), is provided in the Supplementary Material.
7

Theorem 1 (Onset of the infrared continuum). Let (Σ, g) satisfy (11) with
|Riem(x)| ≃ C r−3 as r → ∞. Then the Lichnerowicz operator L = ∇∗ ∇+VR
is self-adjoint on L2 (Σ; S 2 T ∗ Σ) and satisfies
[0, ∞) ⊂ σess (L),

0 ∈ σess (L).

(14)

Hence the inverse-cube decay marks the precise boundary between spectrally
transparent geometries and those supporting marginally correlated tensor modes.
A full quantitative proof using Weyl’s criterion appears in the Supplementary
Material.

3.3

Physical interpretation

The inclusion 0 ∈ σess (L) signifies that the spatial Lichnerowicz operator
admits extended tensor modes with vanishing energy yet finite L2 norm. In
the full time-dependent linearized Einstein equation,
□g hµν + 2Rµρ νσ hρσ = 0,
these correspond to the ω → 0 limit of radiative solutions.
This result does not imply confinement or instability but reveals a continuous infrared sector: correlations decay slowly, and certain low-frequency
components of gravitational radiation remain weakly coupled at large distances. The phenomenon parallels the infrared sensitivity of gauge fields
at the same scaling threshold. In Yang-Mills theory, slow curvature decay
localizes color flux; in gravity, it generates persistent correlations and memory effects. Thus the critical decay |Riem| ∼ r−3 delineates the geometric
transition from purely radiative behavior to an infrared-sensitive, marginally
bound regime, a structural, though not dynamical, analogue of the nonAbelian case.

4

Dimensional Scaling and Spectral Phase Structure

The preceding analysis showed that when the curvature of an asymptotically flat three-manifold decays as |Riem| ∼ r−3 , the Lichnerowicz operator
L = ∇∗ ∇ + VR acquires a continuous spectrum extending to zero. This
identifies the boundary between spectrally transparent geometries and those
8

supporting marginally correlated tensor modes. Here we generalize that result by examining how the critical exponent depends on spatial dimension
and by situating the inverse-cube decay within a broader scaling framework.
The infrared transition derived in three dimensions thereby appears as one
point on a dimensional phase diagram governing the long-range behavior of
curvature-coupled Laplace-type operators.

4.1

Dimensional analysis of curvature potentials

Let Σd be an asymptotically flat Riemannian manifold of dimension d ≥ 2,
and let V (r) ∼ r−p denote a curvature-induced potential acting on tensor
fields through
Ld = −∇2 + V (r),
where ∇2 is the Laplace-Beltrami operator on Σd . The potential is shortrange if it defines a compact perturbation of the Laplacian, and long-range
otherwise. A scaling argument identifies the decay rate separating these
regimes.
Proposition 1 (Dimensional criterion for the critical decay rate). Let ∆
denote the Laplace-Beltrami operator on a d-dimensional asymptotically flat
manifold, and let V (r) ∼ r−p be a curvature-induced potential. Then V is a
compact perturbation of ∆ if and only if p > d. The equality p = d marks
the threshold between short- and long-range behavior.
R
Proof. The naive condition obtained by requiring rd−1 |V |2 dr < ∞ tests
whether V defines a Hilbert-Schmidt perturbation of ∆, which is much
stronger than compactness. On unweighted spaces H 2 (Rd ) → L2 (Rd ), any
potential V (x) → 0 at infinity already yields a compact multiplication operator by the standard Rellich lemma, so the threshold p > d cannot be inferred
from that estimate alone.
In the present setting, however, Ld acts between weighted Sobolev spaces
2
Hδ (Σd ) → L2δ−2 (Σd ) with −1 < δ < 0, appropriate to asymptotically flat
ends. Compactness can fail at infinity if the decay of V (r) ∼ r−p is too
slow to suppress contributions from large volumes. For transverse-traceless
tensor modes, which behave asymptotically as h(r) ∼ r−(d−2)/2 due to the
asymptotically flat falloff conditions, the weighted L2δ−2 norm of V h on {r >
R} scales as
Z ∞
Z ∞
2(δ−2)
2
2 d−1
r
|V (r)| |h(r)| r dr ∼
r d−5+2δ−2p dr.
R

R

9

Because −1 < δ < 0, this integral diverges when p ≤ d, showing that for p ≤ d
the curvature tail continues to couple to asymptotically flat tensor modes
at arbitrarily large radius. For p > d, the contribution from the asymptotic
region vanishes and multiplication by V becomes compact Hδ2 → L2δ−2 by the
weighted Rellich lemma [3, 12]. Hence p = d marks the boundary between
compact and noncompact behavior.
Remark 1 (On sharpness). The borderline p = d is therefore not merely
sufficient but sharp: for p ≤ d one can construct a normalized, transversetraceless Weyl sequence supported at large r with ∥Ld hn ∥ → 0, placing 0 in
σess (Ld ).
This reproduces the inverse-cube decay in three dimensions and extends
it to arbitrary d. The resulting scaling law,
pcrit = d,

(15)

expresses a dimensional balance between kinetic dispersion ∇2 ∼ r−2 and
curvature coupling V (r) ∼ r−p . When p > d, curvature effects are integrable
and the spectrum remains purely continuous; when p ≤ d, the potential
becomes marginal or long-range, introducing infrared correlations.

4.2

Interpretation and structural implications

Equation (15) has a clear geometric meaning. In any spatial dimension,
the decay rate r−d represents the marginal case where background curvature
fails to decay fast enough to ensure spectral transparency. For both spin-1
and spin-2 fields, whose quadratic operators share the same Laplace-type
structure, this scaling marks the onset of infrared sensitivity. The equality
pcrit = 3 in three dimensions is thus one instance of a general relation between
dimensionality and the asymptotic behavior of curvature-coupled Laplacetype operators.
Although this correspondence arises from dimensional rather than dynamical analysis, it provides a coherent geometric framework for comparing
gauge and gravitational fields across dimensions. It also supplies a natural language for describing the transition between dispersive and marginally
bound regimes, the spectral phase structure of curvature-coupled Laplace
operators.

10

5

Numerical Verification of the Spectral Threshold

The analytic results of Sections 2–4 predict that when the background curvature decays faster than r−3 , the spectrum of the spatial Lichnerowicz operator remains purely continuous, while the inverse–cube decay marks the
onset of marginal long–range coupling. This section provides numerical
evidence for that threshold through two complementary diagnostics: (i) a
Rayleigh–quotient scaling analysis of an asymptotic radial model, and (ii)
a computation of the low–lying spectrum of the full three–dimensional discretized tensor operator.

5.1

Numerical setup and nondimensionalization

The tensor operator L = ∇∗ ∇+VR is discretized on a uniform Cartesian grid
Ω = [−Rmax , Rmax ]3 with spacing h = ∆x/L0 after nondimensionalizing by a
fixed asymptotic length scale L0 . Centered finite differences approximate ∇
and ∇∗ ∇. Dirichlet boundaries h|∂Ω = 0 define a finite–volume eigenproblem;
the approach to the continuum spectrum is monitored by extrapolation in
Rmax .
To enforce the transverse–traceless constraint we use the penalty functional
⟨h, Lh⟩ + η ∥∇· h∥2L2 (Ω) + ζ ∥tr h∥2L2 (Ω)
,
(16)
Rη,ζ [h] =
∥h∥2L2 (Ω)
whose stationary points satisfy (K + ηD⊤ D + ζT ⊤ T )u = λM u, where K
and M are the stiffness and mass matrices and D, T the discrete divergence
and trace operators. Varying the penalties η, ζ by factors of 2–4 changes λ1
by less than 10−3 , indicating that the lowest modes lie in the TT subspace
to numerical accuracy. An explicit TT projection check is reported in the
Supplementary Material (Sec. S1).
Representative nondimensional parameters are h ∈ {1.0, 0.75, 0.5}, Rmax ∈
{6, 10, 14, 18, 20}, and C = −1. Each grid contains N 3 points (N = 21–41, up
to 3.6 × 105 degrees of freedom). All runs use double precision and converge
within relative error 10−5 .
Dirichlet boundaries discretize the near–threshold continuum into box
−2
modes with λ1 (Rmax ) ∝ Rmax
; the observed scaling and small residuals confirm that the computed eigenvalues track the physical continuum edge rather
11

than artificial confinement.
Definition 1 (Numerical operators). The asymptotic radial model operator
is
d2
ℓ(ℓ + 1) C
+ p,
Lp = − 2 +
dr
r2
r
representing a single angular channel of the tensor operator. The three–dimensional
discretized L = ∇∗ ∇ + VR includes the full curvature coupling and constraint
enforcement.
Remark 2 (Radial correspondence). Projecting L onto tensor harmonics of
order ℓ yields an effective potential Veff (r) = ℓ(ℓ + 1)/r2 + C/rp + O(r−p−1 ),
so the radial model reproduces the asymptotic channel structure of the full
operator. For the Schwarzschild tail Ri k j ℓ hkℓ , one has C < 0, corresponding
to an attractive potential in the far field.

5.2

Rayleigh–quotient scaling test

The Rayleigh–quotient method directly probes the predicted scaling ∆E(R, p) ∼
R−(p−2) , expressing the competition between the Laplacian and the curvature
potential. For Vp (r) = ℓ(ℓ + 1)/r2 + C/rp , define
R 2R
E[ϕ] =

R


|ϕ′ (r)|2 + Vp (r)|ϕ(r)|2 r2 dr
,
R 2R
|ϕ(r)|2 r2 dr
R

∆E(R, p) = E[ϕ] − Efree [ϕ],

using a normalized bump function ϕ(r) on [R, 2R]. The flat case C = 0
defines Efree . Table 1 lists the results; the fitted slopes α(p) ≈ −(p − 2)
confirm the analytic scaling.
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

Figure 1: Log–log scaling of |∆E(R, p)| for representative p. Measured slopes
α(p) ≈ −(p − 2) agree with the analytic prediction.

5.3

Three–dimensional eigenvalue analysis

To test the full tensor operator, we compute the lowest eigenvalues λ1 of
the discretized model for several p and Rmax . All runs use C = −1 and
the TT–penalty enforcement of Eq. (16). The results are shown in Table 2
and Fig. 2. The systematic decrease of λ1 with increasing Rmax confirms
convergence toward the continuum limit. A least–squares fit λ1 (Rmax ) =
−2
−3
aRmax
+ bRmax
yields small residuals, consistent with the finite–volume interpretation.

14

Table 2: Lowest eigenvalue λ1 of the discretized tensor
operator for several decay exponents p and outer radii
Rmax . Decreasing λ1 with larger Rmax confirms convergence toward the infinite-volume limit.
p
flat

2.0

2.5

3.0

3.5

4.0

Rmax
6
10
14
18
20
6
10
14
18
20
6
10
14
18
20
6
10
14
18
20
6
10
14
18
20
6
10
14
18
20

λ1
0.2044
0.0739
0.0377
0.0228
0.0185
0.1466
0.0435
0.0192
0.0102
0.0078
0.1828
0.0651
0.0334
0.0203
0.0166
0.1965
0.0711
0.0365
0.0222
0.0180
0.2016
0.0730
0.0374
0.0227
0.0184
0.2035
0.0736
0.0376
0.0228
0.0185

The systematic decrease of λ1 with increasing Rmax confirms convergence to15

ward the continuum limit. Intermediate values at Rmax = 18 demonstrate
a smooth monotonic approach to the asymptotic regime, ensuring that the
lowest eigenvalues stabilize well before boundary effects dominate. These
eigenvalue trends complement the Rayleigh-quotient scaling test, both identifying p = 3 as the marginal decay rate where curvature transitions from
spectrally relevant to effectively negligible. 3

Figure 2: Convergence of λ1 with domain size Rmax for several decay exponents p. The flattening of λ1 (Rmax ) for p ≥ 3 shows that curvature becomes
spectrally negligible beyond the inverse–cube rate, while slower decay (p < 3)
yields progressively deeper infrared shifts.
Remark 3 (Interpretation). The continuous approach of λ1 (p) to its flat–space
value as p increases demonstrates a smooth transition between confining and
radiative regimes. For p ≤ 2.5, curvature remains spectrally significant; at
p = 3, curvature and dispersion balance; and for p > 3, the spectrum be3

We did not explore values p < 2 in detail, since such slow falloff is incompatible
with the asymptotic behavior of isolated vacuum solutions in general relativity (cf. the
Supplementary Material). Nevertheless, extrapolating the monotone suppression of λ1
between p = 3.0, 2.5, and 2.0 suggests that even slower decay (p ≲ 2) would further reduce
the infrared eigenvalues, producing a more strongly gapped, bound–state–like spectrum.
This is consistent with interpreting the regime p < 3 as genuinely long–range in the spectral
sense.

16

comes indistinguishable from flat space. No discrete bound states appear, in
agreement with the analytic prediction of a purely continuous essential spectrum beyond the inverse–cube threshold. Additional convergence and stability
checks, including penalty sweeps, grid refinement, and mode localization, are
presented in the Supplementary Material (Sec. S3).
Lemma 3 (Asymptotic behavior of the lowest eigenvalue). For p ≥ 3,
−2
λ1 (Lp ) ∼ Rmax
and approaches 0+ as Rmax → ∞, consistent with approach
to the continuum threshold and the absence of bound states. For p < 3, the
slower decay of curvature increases the infrared coupling, yielding smaller λ1
but no discrete negative modes.
Remark 4 (Numerical consistency with the analytic threshold). Both diagnostics reproduce the qualitative behavior predicted by the analytic theory:
power–law Rayleigh scaling with slope α(p) = −(p − 2) and a continuous
spectral transition centered at p = 3. The results therefore confirm that the
inverse–cube decay constitutes a sharp spectral threshold, marking the exact
boundary between compact and noncompact curvature perturbations in the
spin–2 sector. A detailed convergence and stability analysis verifying that
these features are not numerical artifacts of discretization, boundary conditions, or constraint penalties is provided in the Supplementary Material.

6

Physical Interpretation and Implications

The analytic and numerical analyses of Sections 3-5 identify a sharp transition between two qualitatively distinct spectral regimes of gravitational
perturbations on asymptotically flat manifolds. This section interprets the
threshold p = 3 in physical terms and connects it to known infrared phenomena of general relativity, including radiative behavior, gravitational memory,
and asymptotic symmetry. Beyond the classical setting, the same spectral
structure underlies the infrared behavior of the quantized linearized field,
providing a geometric origin for the soft sector of quantum gravity.

6.1

Infrared Structure of the Quantized Field

Although the present analysis is entirely classical, the spectral properties
of the spatial Lichnerowicz operator determine the infrared structure of the

17

quantized linearized gravitational field. In canonical quantization, the equaltime two-point function in harmonic gauge is the inverse of L, so the largedistance correlations of the graviton field are governed by the same spectral
threshold identified here. For curvature decaying faster than r−3 , L−1 remains
a short-range operator and defines a regular Fock vacuum with finite infrared
correlations. At the marginal rate r−3 , however, the Green’s function develops a slow algebraic tail, corresponding to the emergence of zero-frequency,
spatially extended modes. These modes represent the static counterparts of
the soft-graviton excitations that appear dynamically in the full quantum
theory. From this viewpoint, the r−3 decay marks the geometric origin of the
infrared enhancement familiar from the Weinberg soft-graviton theorem and
its modern extensions in the asymptotic symmetry framework [16, 17].
Remark 5 (Radiative and confining regimes). When the curvature potential
decays faster than r−3 , gravitational perturbations propagate as freely radiating modes whose energy escapes to infinity, leaving the essential spectrum
continuous and gapless. At the borderline decay rate r−3 , dispersion and curvature balance precisely, producing marginally bound tensor configurations
that are spatially extended but nonlocalized. These modes neither dissipate
completely nor remain compactly confined, representing the static analogue
of zero-energy resonances in potential scattering. For p > 3, the curvature
becomes spectrally negligible and the field is fully radiative; for p < 3, curvature acts as a long-range potential that enhances infrared coupling and could
support quasi-bound behavior.

6.2

Connection with Gravitational Memory and Soft
Modes

The marginally bound tensor modes at the r−3 threshold are consistent with
a spectral interpretation of the gravitational memory effect. In the nonlinear
theory, the Christodoulou memory corresponds to a permanent displacement
of test particles due to the flux of gravitational radiation through null infinity. Within the present linear, time-independent framework, zero-frequency,
spatially extended modes sustain correlations that do not fully decay.
Remark 6 (Static precursors of memory). If such stationary correlations
were evolved within a dynamical setting, their time-integrated imprint would
resemble the displacement produced by nonlinear memory. In this sense, the
18

marginal modes act as static precursors of the soft sector familiar from the
infrared triangle connecting asymptotic symmetries, soft graviton theorems,
and memory effects. They indicate that the spatial Lichnerowicz operator already encodes the seeds of the infrared structure that reemerges dynamically
at null infinity. A complete treatment of this correspondence would require
coupling the elliptic analysis to the time-dependent linearized Einstein equations near I + , which we leave for future work.

6.3

Relation to Late-Time Tails

Price’s law for black-hole perturbations, ψ ∼ t−2ℓ−3 , arises from an effective
potential decaying as r−3 . The same inverse-cube scaling governs the onset of
noncompactness for the spatial Lichnerowicz operator, suggesting that both
results stem from a common geometric mechanism: curvature of order r−3
produces algebraic energy leakage and slow relaxation, defining the boundary
between exponential and power-law decay.
Remark 7 (Unified interpretation of tails). The spectral perspective presented here identifies the stationary origin of the temporal tail: marginally
bound spatial modes correspond to low-frequency perturbations whose gradual radiative leakage produces the late-time decay law. Thus, the spatial and
temporal manifestations of the inverse-cube scaling represent two aspects of
the same infrared structure.

6.4

Asymptotic Symmetries and Infrared Structure

At the critical decay rate, the geometry admits zero-frequency tensor excitations associated with emergent asymptotic diffeomorphisms. These marginal
modes can be viewed as the linearized precursors of BMS supertranslations,
encoding conserved charges at spatial infinity.
Remark 8 (Spectral interpretation of asymptotic symmetry). The appearance of an extended zero mode at the inverse-cube threshold signals the enlargement of the asymptotic symmetry algebra to include nontrivial diffeomorphisms acting at infinity. The spectral transition at p = 3 thus links the
mathematical onset of noncompactness to the physical emergence of memory
and soft graviton behavior within a common framework.

19

Universality of the spectral threshold. The coincidence of analytic,
numerical, and geometric evidence suggests that the inverse-cube decay represents a universal spectral boundary for long-range field theories on R3 . In
both gauge and gravitational contexts, curvature or field strength decaying
as r−3 marks the transition between compact and noncompact spectral behavior, where the Laplacian ceases to dominate the asymptotic dynamics.
This threshold thus defines a geometric law of infrared structure: it separates the radiative regime, characterized by freely propagating modes, from
the marginal regime in which curvature produces algebraic tails and soft
correlations that persist to infinity.

7

Relation to Previous Work and Threshold
Phenomena

The spectral threshold established in Sections 3-5 connects several independent developments in spectral geometry, mathematical relativity, and gauge
theory. It refines classical results on elliptic operators on asymptotically flat
manifolds, relates to long-range scattering and tail phenomena in black-hole
perturbation theory, and parallels curvature-controlled infrared thresholds
first identified in non-Abelian gauge theory.

7.1

Spectral theory of elliptic operators on asymptotically flat ends

The analytic foundation for elliptic operators on noncompact manifolds with
Euclidean ends was laid by Lockhart and McOwen [12]. They proved Fredholm and isomorphism properties for elliptic operators acting between weighted
Sobolev spaces Hδ2 → L2δ−2 , identifying indicial roots as the precise obstructions to invertibility. These results underlie much of geometric analysis on
asymptotically flat manifolds, including the constraint equations of general
relativity and deformation theory of vacuum initial data.
Remark 9 (Extension of classical elliptic results). Earlier analyses treated
curvature terms as short-range perturbations of the Laplacian without identifying a quantitative boundary between short- and long-range behavior. The
present result provides a sharp criterion: if |Riem(x)| ≤ C r−p with p >
3, the curvature potential VR is relatively compact with respect to ∆T and
20

σess (L) = [0, ∞); when p = 3, compactness fails and a normalized Weyl sequence appears at zero energy. This refines the Lockhart-McOwen framework
by isolating the exact curvature decay rate at which the Fredholm property
transitions to noncompact spectral behavior.

7.2

Thresholds in long-range potentials

The r−3 boundary identified here is the tensorial analogue of the classical
long-range threshold in Schrödinger theory. In three dimensions, Simon [1]
proved that potentials decaying faster than r−3 yield purely absolutely continuous spectrum on [0, ∞), whereas slower decay allows resonances or threshold
states near zero energy. The gravitational case exhibits an analogous structure, with the Riemann curvature playing the role of an effective potential.
Remark 10 (Connection with Schrödinger thresholds). The Lichnerowicz
operator realizes, at the tensorial level, the same balance between dispersion
and long-range coupling that governs Schrödinger operators. The inversecube decay marks the onset of marginally extended configurations, beyond
which curvature ceases to influence the spectrum. This correspondence illustrates the structural similarity of infrared thresholds across scalar, vector,
and tensor field equations.

7.3

Relation to late-time tails in black-hole spacetimes

Price [13] and Ching et al. [11] showed that perturbations of the Schwarzschild
spacetime decay as t−2ℓ−3 , with the algebraic tail arising from an effective potential proportional to r−3 . The same scaling controls the marginal behavior
of the spatial Lichnerowicz operator.
Remark 11 (Spatial origin of temporal tails). The elliptic analysis here provides the stationary counterpart of the dynamical late-time decay law. At the
inverse-cube decay, the spatial operator supports near-zero modes that correspond to the low-frequency enhancement responsible for algebraic relaxation.
The threshold p = 3 thus unifies the stationary and dynamical manifestations
of gravitational infrared behavior.

21

7.4

Infrared structure and memory

In the context of asymptotic symmetry and gravitational memory, zerofrequency perturbations encode residual deformations between radiative vacua
at null infinity. The existence of an L2 -normalized Weyl sequence at zero energy provides a spatial realization of these soft configurations.
Remark 12 (Spectral characterization of the infrared sector). At curvature
decay |Riem| ∼ r−3 , the Lichnerowicz operator develops marginally extended
tensor modes that remain spatially nonlocal yet finite in energy. These represent the elliptic, spatial manifestation of soft graviton modes and delineate
the precise geometric condition for the emergence of an infrared sector in
linearized gravity.

7.5

Parallel thresholds across gauge and gravitational
systems

A closely related threshold occurs in non-Abelian gauge theory. For the
covariant Laplacian
∆A = −(∇A )∗ ∇A = −∇∗ ∇ + ad(FA ),
acting on adjoint-valued fields with curvature FA , the decay |FA | ∼ r−3 separates spectrally radiative behavior from the onset of infrared sensitivity.
Proposition 2 (Parallel inverse-cube threshold). For Laplace–type operators on bundles over R3 , a curvature decay of order r−3 marks the transition
between short-range, radiative behavior and long-range, infrared coupling. In
both gauge and gravitational settings, curvature acts as an effective potential; at this critical rate, marginal nonlocalized modes appear, signaling the
breakdown of compactness of the resolvent.

7.6

Summary and outlook

Taken together, these results connect several domains of spectral analysis
and physical theory. For p > 3, curvature perturbations are short-range and
the spectrum purely radiative. At p = 3, curvature ceases to be compact,
marginal modes emerge, and the infrared sector appears continuously but
sharply. For p < 3, curvature strengthens the coupling further, but without
forming discrete bound states in the present tensorial setting.
22

Remark 13 (Open directions). Future work should establish a full limiting
absorption principle at the critical rate and extend the analysis to coupled
gravity–gauge systems. Such results would provide a rigorous foundation for
the infrared correspondence between soft sectors, asymptotic symmetries, and
spectral transitions.

8

Conclusion

The analyses presented here establish a sharp spectral threshold for the spatial Lichnerowicz operator on asymptotically flat three–manifolds. The results integrate geometric analysis, spectral theory, and physical interpretation
within a unified framework.
Theorem 2 (Spectral threshold for linearized gravity). Let (Σ, g) be a smooth
asymptotically flat three–manifold with curvature decay |Riem(x)| ≤ C r−p .
Then:
1. For p > 3, the curvature potential VR is relatively compact with respect
to ∇∗ ∇, and the essential spectrum is purely continuous:
σess (L) = [0, ∞).
2. At the critical rate p = 3, compactness fails and a normalized Weyl
sequence appears at zero energy, producing marginally extended tensor
configurations that remain spatially nonlocal yet finite in energy.
3. For p < 3, curvature acts as a long-range potential that enhances infrared coupling, but without producing isolated bound states in the tensorial sector.
Remark 14 (Physical regimes). The three regimes delineated above correspond respectively to radiative propagation (p > 3), marginal persistence
(p = 3), and enhanced infrared coupling (p < 3). The inverse–cube decay
therefore represents the sharp geometric boundary between short-range and
long-range gravitational behavior in three spatial dimensions.
These findings extend classical spectral theory to curvature–coupled tensor operators and reveal a structural parallel with non–Abelian gauge fields
and Schrödinger operators. In all three settings, the inverse–cube decay
23

marks the point at which the potential ceases to be spectrally negligible and
marginal modes first appear. Numerical analysis of the reduced model confirms that this transition occurs continuously but sharply at p = 3, with
no evidence of discrete confinement, validating the analytic predictions of
Sections 3-5.
Remark 15 (Future directions). Several open problems arise naturally from
this work: establishing a limiting absorption principle at the critical rate;
extending the spectral analysis to Schwarzschild and Kerr slices; and developing the full dynamical correspondence between marginal spatial modes and
the soft–memory sector at null infinity. Such results would further clarify the
geometric and spectral unity of the infrared structure in gauge and gravitational theories.

Declarations
Funding The author received no external funding.
Conflict of interest The author declares no conflict of interest.
Data availability All data supporting the conclusions of this work are
contained within the article and its Supplementary Material. No external
datasets were used. Numerical results can be reproduced using the procedures described in Section 5 and Appendix D.

A

Gauge Correction and Elliptic Estimates

This appendix justifies the harmonic–gauge correction used in Section 3.
Under the hypotheses of Section 5, namely, (Σ, g) smooth, asymptotically
flat, simply connected, and with Ric = O(r−3 ), the vector Laplacian
∆V X = ∇∗ ∇X + Ric(X)
is uniformly elliptic and symmetric on L2 (Σ; T ∗ Σ).
Lemma 4 (Isomorphism property). For weights −1 < δ < 0, the mapping
∆V : Hδ2 (Σ; T ∗ Σ) → L2δ−2 (Σ; T ∗ Σ)
1
is Fredholm of index zero and an isomorphism whenever HdR
(Σ) = 0.

24

Sketch. A direct consequence of Lockhart-McOwen theory (Comm. Pure
Appl. Math. 38, 603 (1985)), since the Euclidean indicial roots are {0, −2}.
Lemma 5 (Gauge correction). For each h ∈ Hδ2 (Σ; S 2 T ∗ Σ) with −1 < δ < 0,
there exists a unique X ∈ Hδ2 (Σ; T ∗ Σ) satisfying ∆V X = ∇· h and ∥X∥Hδ2 ≤
C∥∇· h∥L2δ−2 .
Proposition 3 (Corrected Weyl sequence). Let {hn } be the approximate
sequence of Section 3. Defining h̃n = hn − LXn g with Xn = ∆−1
V (∇ · hn )
yields
∥h̃n ∥L2 = 1,
h̃n ⇀ 0,
∥Lh̃n ∥L2 → 0.
Thus 0 ∈ σess (L) in harmonic gauge.

B

Curvature Structure and the Schwarzschild
Example

The Schwarzschild metric provides a physical realization of the critical inverse–cube curvature decay analyzed in Section 3.
Definition 2 (Spatial metric). In isotropic coordinates (t, r, ω), the Schwarzschild
line element is
 1 − M 2

M 4 2
2
2r
ds2 = −
dt
+
1
+
(dr + r2 dω 2 ).
2r
1+ M
2r
On a time-symmetric slice t = const., the spatial metric is gij = ψ 4 δij with
.
ψ(r) = 1 + M
2r
Lemma 6 (Asymptotic curvature). For this metric,
|Riem(x)| ≃ C M r−3

(r → ∞),

so the curvature saturates the inverse–cube decay assumed in Theorem 2.
Proposition 4 (Effective potential). The spatial Lichnerowicz operator on
the Schwarzschild background satisfies
Lh = ∆0 h − (CM )r−3 h + O(r−4 )h,
25

CM > 0,

showing that the Schwarzschild geometry realizes, in its far-field limit, the
same attractive r−3 potential analyzed in the numerical model of Section 5. 4

C

Analytical Framework and Weighted Sobolev
Spaces

We summarize the analytic conventions and functional-analytic tools used
throughout.
Definition 3 (Weighted Sobolev spaces). For a smooth radius function r on
an asymptotically flat three–manifold (Σ, g) and δ ∈ R,
XZ
2
∥u∥H k =
(1 + r2 )δ−|α| |∇α u|2 dVg .
δ

|α|≤k

Σ

Then Hδk (Σ; E) is the completion of Cc∞ (Σ; E) under this norm.
Lemma 7 (Fredholm property). If P is a uniformly elliptic operator approaching constant coefficients at infinity, then
P : Hδ2 → L2δ−2
is Fredholm for all δ not equal to an indicial root [12].
Proposition 5 (Self-adjointness and essential spectrum). For δ ∈ (−1, 0)
and V = O(r−p ) with p > 2, operators of the form L = ∇∗ ∇ + V are
self–adjoint on L2 (Σ; E). The essential spectrum σess (L) is determined by
the existence of Weyl sequences as in Weyl’s criterion.
Lemma 8 (Compactness of the curvature potential for p > 3). Let (Σ, g) be
a smooth asymptotically flat three-manifold with a single Euclidean end, and
assume
gij = δij + O(r−1 ),

∂gij = O(r−2 ),

∂ 2 gij = O(r−3 ),

so that |Riem(x)| ≤ C ⟨r⟩−p for some p > 3. Fix a weight −1 < δ < 0, and
let
L = ∇∗ ∇ + VR ,
(VR h)ij = −Ri ℓ j m hℓm .
4

The overall minus sign arises from the definition (VR h)ij = −Ri k j ℓ hkℓ , which makes
the curvature coupling attractive for positive mass M > 0.

26

Then the curvature term defines a compact operator
VR : Hδ2 (Σ; S 2 T ∗ Σ) −→ L2δ−2 (Σ; S 2 T ∗ Σ),
and therefore L is a compact perturbation of ∇∗ ∇ on L2 (Σ; S 2 T ∗ Σ). In
particular,
σess (L) = σess (∇∗ ∇) = [0, ∞).
Proof. The curvature bound |Riem(x)| ≤ C⟨r⟩−p implies |VR (x)| ≤ C⟨r⟩−p ,
so that for any h ∈ Hδ2 (Σ; S 2 T ∗ Σ),
Z
Z
2
2(δ−2)
2
|VR (x)h(x)| dVg ≲ ⟨r⟩2(δ−2)−2p |h(x)|2 dVg .
∥VR h∥L2 = ⟨r⟩
δ−2

Σ

Σ

Since −1 < δ < 0 and p > 3, the exponent 2(δ − 2) − 2p is less than
−6, making the weight ⟨r⟩2(δ−2)−2p integrable at infinity in three dimensions.
Hence VR : Hδ2 → L2δ−2 is bounded.
To verify compactness, let χR be a smooth cutoff equal to 1 on {r ≤ R}
and supported in {r ≤ 2R}. Decompose
(comp)

VR = χR VR + (1 − χR )VR =: VR

(tail)

+ VR

.

On the bounded region {r ≤ 2R}, the metric is smooth and the weighted
norms are equivalent to the unweighted ones. By the classical Rellich compactness theorem on precompact domains, the embedding Hδ2 ({r ≤ 2R}) ,→
L2δ−2 ({r ≤ 2R}) is compact; since VR is smooth there, the multiplication
(comp)
operator VR
is compact.
(tail)
The remainder VR
is supported in {r > R}, where the curvature decay
dominates. For h with ∥h∥Hδ2 = 1,
Z
(tail)
2
∥VR h∥L2 ≲
⟨r⟩2(δ−2)−2p |h(x)|2 dVg .
δ−2

r>R

Because 2(δ − 2) − 2p < −6, the weight decays faster than r−6 ; combined
with the uniform bound |h|2 ∈ L2δ , dominated convergence implies that
(tail)

sup ∥VR

∥h∥H 2 =1

h∥L2δ−2 −→ 0 as R → ∞.

δ

Thus VR is the limit of compact operators with vanishing tails, and hence
compact.
27

Finally, since L = ∇∗ ∇+VR is a compact perturbation of the nonnegative,
self-adjoint operator ∇∗ ∇, Weyl’s theorem ensures that the essential spectra
coincide:
σess (L) = σess (∇∗ ∇) = [0, ∞).
The essential spectrum of the Lichnerowicz operator therefore matches that
of the flat tensor Laplacian, confirming that sufficiently rapid curvature decay
leaves the asymptotic spectrum purely continuous.
Remark 16. These weighted spaces and mapping properties justify the estimates and compactness arguments used in Sections 2–3.

D

Numerical Validation and Stability Tests

This appendix documents three numerical consistency checks supporting the
eigenvalue results reported in Section 5: (i) grid-spacing convergence, (ii)
finite-volume convergence in Rmax , and (iii) robustness under constraint enforcement. These tests show that the observed spectral transition near p = 3
is not a numerical artifact of discretization, boundary conditions, or gauge
penalties.

D.1

Grid refinement at fixed physical volume

To verify that the numerical results reported in Section 5 are free from discretization artifacts, we repeated the computations at two grid resolutions.
The coarse run used spacing h ≈ 1.0 (N = 21 points per axis, ∼ 4.1 × 104
degrees of freedom), and the refined run used h ≈ 0.5 (N = 41, ∼ 3.6 × 105
degrees of freedom). Converged eigenvalues at both resolutions are listed
below. The near-invariance of λ1 and λ2 confirms that the numerical spectrum is stable with respect to grid refinement and that the observed scaling
behavior in Section 5 is not a numerical artifact.
Coarse resolution (h ≈ 1.0, N = 21 points/axis, ∼ 4.1 × 104 DOFs).
flat:
p = 2.5 :
p = 3.0 :
p = 3.5 :

λ1 = 0.07386996,
λ1 = 0.06505331,
λ1 = 0.07114971,
λ1 = 0.07298806,

28

λ2 = 0.14713361,
λ2 = 0.07023744,
λ2 = 0.07276003,
λ2 = 0.07351824.

Refined resolution (h ≈ 0.5, N = 41 points/axis, ∼ 3.6 × 105 DOFs).
flat:
p = 2.5 :
p = 3.0 :
p = 3.5 :

λ1 = 0.07398399,
λ1 = 0.06505635,
λ1 = 0.07109678,
λ1 = 0.07295142,

λ2 = 0.14781594,
λ2 = 0.07029545,
λ2 = 0.07279109,
λ2 = 0.07355686.

Halving the grid spacing changes λ1 by less than 0.2% across all p, demonstrating numerical convergence in h. Crucially, the ordering
λ1 (p=2.5) < λ1 (p=3.0) < λ1 (p=3.5) ≃ λ1 (flat)
is preserved under refinement. This ordering encodes the physical trend
reported in Section 5: slower curvature decay (smaller p) produces a stronger
infrared distortion of the lowest mode, while faster decay (p > 3) becomes
spectrally indistinguishable from flat space. The persistence of this structure
under refinement shows that it is not a coarse-grid artifact.

D.2

Finite-volume convergence in Rmax

We also varied the outer box size Rmax ∈ {6, 10, 14, 18, 20} at fixed grid
spacing h ≈ 1.0 (so that increasing Rmax increases the number of unknowns)
and observed that λ1 decreases monotonically with Rmax for every p tested.
For example, at p = 3.0 we find
λ1 (Rmax =6) = 0.1965,

λ1 (Rmax =10) = 0.0711,

λ1 (Rmax =20) = 0.0180.

−2
agrees with the interpretation of λ1 as
The approximate scaling λ1 ∼ Rmax
the lowest discrete “box mode” approaching the continuum threshold. This
confirms that the small eigenvalues reported in Table 2 are controlled by the
infrared volume scale and not spurious numerical locking to the boundary.

Penalty strength studies and TT projection cross–check. We sweep
η, ζ over two decades (e.g. η = ζ ∈ {10, 40, 160, 640} in nondimensional
units); for p ∈ {2.5, 3.0, 3.5} the relative shifts in λ1,2 between the two largest
penalties are < 10−3 . We also compute the lowest eigenpair using an explicit TT projection: given any u, let uTT = ΠTT u via Helmholtz–Hodge
decomposition (solve ∆V X = ∇ · u, then set uTT = u − LX g − 13 (tr u)g).
29

The projected–eigenpair agrees with the penalty result within the grid error.
(Algorithmic details in Sec. S1; linear solves via conjugate gradients with
algebraic multigrid preconditioner.)
Remark 17 (Conclusion). The convergence in grid spacing, the Rmax scaling,
and the robustness under transverse-traceless enforcement all support a single
interpretation: the trends reported in Section 5 reflect genuine infrared properties of the Lichnerowicz operator. In particular, they provide independent
numerical evidence that the inverse-cube decay of curvature (p = 3) marks
the threshold between spectrally relevant long-range curvature and spectrally
negligible curvature.
Code availability. The complete Python script reproducing Table 2 is
provided as 3d tensor operator.py in the Supplementary Reproducibility
Package (ResearchGate upload). It implements the finite–difference Laplacian, the regularized tidal curvature term Eij ∼ r−p (ni nj − δij /3), and the
trace–penalty enforcement of the TT constraint.

E

Weyl Sequence Construction and Verification of the Critical Spectrum

This appendix provides quantitative estimates completing the proof of Lemma 2
and Theorem 1. Throughout, (Σ, g) satisfies the asymptotic flatness conditions of eq. 11 with |Riem(x)| ≤ Cr−3 .

Normalization and scaling
Let Hij (ω) be a trace-free, divergence-free harmonic on S 2 and define hn =
An ϕn (r)r−1 Hij (ω) with ϕn supported on An = {n/2 < r < 2n}. The metric
volume element satisfies dVg = (1 + O(r−1 ))r2 dr dω, so that
Z 2n

∥hn ∥2L2 ≃ A2n

r−2 r2 dr ∼ A2n n.

n/2

Choosing ∥hn ∥L2 = 1 gives An ≃ n−1/2 . Derivatives of ϕn are supported on
the thin shells Sn = {n/2 < r < n} ∪ {3n/2 < r < 2n} where |∇ϕn | ≲ n−1 .

30

Commutator and curvature estimates
Expanding
∇∗ ∇(ϕn r−1 H) = ϕn ∇∗ ∇(r−1 H) + 2∇ϕn ·∇(r−1 H) + (∆ϕn )r−1 H,
the first term vanishes identically in flat space since ∇∗ ∇(r−1 H) = 0 for
tensor harmonics of order ℓ = 2. The commutator terms are supported on
Sn and satisfy
∥[∇∗ ∇, ϕn ]hn ∥L2 ≲ n−2 .
The curvature potential obeys |VR | ≤ Cr−3 , so that
Z 2n
2
2
∥VR hn ∥L2 ≲ An
r−6 r2 dr ∼ n−4 ,
∥VR hn ∥L2 ≲ n−2 .
n/2

Together these give ∥Lhn ∥L2 ≲ n−2 .

Gauge correction
Solving ∆V Xn = ∇· hn with the Fredholm estimate of Lemma 1 yields
∥Xn ∥Hδ2 ≲ ∥∇· hn ∥L2 ≲ n−1 ,

−1 < δ < 0.

Hence ∥LXn g∥L2 ≲ n−1 and the corrected fields h̃n = hn − LXn g satisfy
∇j h̃n,ij = 0. Since L(LXn g) involves at most two derivatives of Xn ,
∥L(LXn g)∥L2 ≲ n−1 ,
which is subleading relative to ∥Lhn ∥L2 ≲ n−2 . Thus
∥Lh̃n ∥L2 → 0,

∥h̃n ∥L2 = 1.

Conclusion
The sequence {h̃n } is normalized, divergence-free, and supported on disjoint
annuli escaping to infinity, implying h̃n ⇀ 0 weakly in L2 . By Weyl’s criterion, 0 ∈ σess (L), establishing Theorem 1.
Author contributions The author is solely responsible for all aspects
of this work.
31

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