ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Wave energy and black-hole perturbation interfaces — model 2

Physical scope

The photon-sphere obstruction to a uniform unweighted local-energy bound for real test waves on the fixed Schwarzschild exterior, using locally proved whole-annulus concentration.

Example

Fix $s > 0$ and $0 < δ < s / 2$ on the Schwarzschild exterior. The photon sphere at $r = 3 s / 2$ obstructs a nondegenerate integrated local energy estimate with no trapping loss on the entire annulus $K_{δ}$ . Precisely, there is no finite constant $C$ , independent of the smooth real finite-energy solution $ψ$ of $□_{g} ψ = 0$ , such that

$\int_{0}^{\infty} E_{T, δ} [ψ] (x^{0}) d x^{0} \leq C E_{T} [ψ] (0) .$

Here $T = \partial_{0}$ , and the total and annular energies are those of the scalar test-wave tensor on the fixed background. The obstruction already holds among waves with compact smooth initial data. It is a mathematical statement about test waves, not a gravitational or nonlinear black-hole stability result.

Example references

Nondegenerate integrated local energy estimate near the Schwarzschild photon sphere

Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2

Exact Schwarzschild waves concentrated in a photon-sphere annulus

Facts & Assumptions

Given: The Schwarzschild exterior with $s > 0$ , the compact trapping neighborhood $K_{δ}$ around $r = 3 s / 2$ , and the stationary energies in the Example.

[F1]

Nondegenerate integrated local energy estimate near the Schwarzschild photon sphere: the proposed estimate is $\int_{0}^{\infty} E_{T, δ} [ψ] (x^{0}) d x^{0} \leq C E_{T} [ψ] (0)$ with one constant $C$ for all smooth finite-energy solutions.

[F2]

Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2: the future null circular geodesic has $r = 3 s / 2$ and $x^{0} (τ) = x_{0}^{0} + e_{0} τ / f (3 s / 2)$ with $e_{0} / f (3 s / 2) > 0$ , so it crosses each exterior static time slice.

[F3]

Dominant energy inequality for the real scalar test tensor: $Q (n, T) \geq 0$ wherever the static slice normal $n$ and $T$ are future causal.

[A1]

${A C}_{ω}$ is the only choice principle in the smooth Schwarzschild and tangent-bundle conventions (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F4]

The locally proved concentration lemma gives, for every $H > 0$ and $0 < ϵ < 1$ , a real smooth exact wave with compact smooth initial data, $E_{T} (0) = 1$ , and $E_{T, δ} (x^{0}) \geq 1 - ϵ$ on every slice $0 \leq x^{0} \leq H$ . It uses the same full radial annulus and Countable Choice, with no additional geometric or choice hypothesis. Exact Schwarzschild waves concentrated in a photon-sphere annulus

Verification

1.1givenA1F2algebra

By [F2], the null geodesic has constant radius $r = 3 s / 2$ , and $x^{0} (τ)$ increases strictly through every real value. Because $δ > 0$ , each static slice segment of this orbit lies inside $K_{δ}$ , the integration region defining $E_{T, δ}$ .

2.1givenF3F4step 1.1algebra

Use [F4] with $ϵ = 1 / 2$ . For every finite $T > 0$ it supplies a wave $ψ_{T}$ with unit initial stationary energy and $E_{T, δ} [ψ_{T}] (x^{0}) \geq 1 / 2$ on $[0, T]$ . By [F3], the local energy density is nonnegative on the whole Schwarzschild annulus, so $\int_{0}^{\infty} E_{T, δ} [ψ_{T}] (x^{0}) d x^{0} \geq \int_{0}^{T} \frac{1}{2} d x^{0} = T / 2$ .

3.1F1F4step 2.1algebra∎

If a uniform estimate constant $C < \infty$ existed, choose $T > 2 C$ . Step 2.1 gives a left side greater than $C$ , while the proposed estimate and $E_{T} [ψ_{T}] (0) = 1$ bound it by $C$ , a contradiction. Thus the nondegenerate no-loss estimate fails for the exact real wave class in the Example.

Wave energy and black-hole perturbation interfaces — model 2

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources