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Wave energy and black-hole perturbation interfaces — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Wave energy and black-hole perturbation interfaces — model 1

Physical scope

A conditional scalar mode-trace calculation of the stationary Kerr horizon flux sign; it proves no global scattering or amplification result.

Example

Let $0 < ∣ a ∣ < M$ and let $Ω_{H} \neq 0$ be the Kerr horizon angular velocity. Choose an integer $m$ with $m Ω_{H} > 0$ and set $ω = m Ω_{H} / 2$ . Conditional on a nonzero smooth scalar horizon mode trace $u$ on an open mode domain $P$ with these parameters, and on a smooth real scalar field $ψ$ defined on a neighborhood of $P$ in the regular extension with $ψ ∣_{P} = Re u$ , at each point whose full stationary integral-curve segment of length $2 π / ω$ remains in $P$ its shifted horizon frequency is $\tilde{ω} = - m Ω_{H} / 2 < 0$ and its time-averaged stationary horizon flux density is nonpositive, is zero at trace nodes, and is strictly negative wherever the trace is nonzero. On every compact smooth spacelike cut patch $D$ whose full stationary integral-curve segments remain in the mode domain and which meets its nonzero set, its integrated averaged flux is $F_{D} = - (m Ω_{H})^{2} \int_{D} ∣ u ∣^{2} d μ_{D} / 8 < 0$ . This is the local mode-trace sign calculation in the superradiant frequency range. It does not construct a global wave solution or establish a scattering amplification factor.

Example references

Kerr horizon mode and shifted horizon frequency

Time-averaged stationary flux of a Kerr horizon mode

Facts & Assumptions

Given: The Kerr parameters and a nonzero smooth mode trace with $m Ω_{H} > 0$ and $ω = m Ω_{H} / 2$ .

[F1]

Kerr horizon mode and shifted horizon frequency: $\tilde{ω} = ω - m Ω_{H}$ for a smooth joint mode of $T$ and $Φ$ .

[F2]

Time-averaged stationary flux of a Kerr horizon mode: for a nonzero smooth mode trace with $ω > 0$ , the averaged stationary horizon flux density is $\frac{1}{2} ω \tilde{ω} ∣ u ∣^{2}$ .

[A1]

${A C}_{ω}$ is the only choice principle inherited through the Kerr horizon definitions (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

A nonzero smooth horizon mode trace with these parameters is assumed for this example, together with a smooth real scalar field $ψ$ on a neighborhood of its mode domain whose restriction is $Re u$ . For each compact cut patch $D$ used below, its mode domain contains the full stationary integral-curve segment of length $2 π / ω$ from every point of $D$ ; this item does not establish the trace's existence as a global solution of $□_{g} ψ = 0$ .

Verification

1.1givenA1A2F1algebra

The chosen frequency satisfies $ω = m Ω_{H} / 2 > 0$ and $\tilde{ω} = ω - m Ω_{H} = - m Ω_{H} / 2 < 0$ .

2.1givenA2F2step 1.1algebra

The horizon-flux identity [F2] gives $⟨ Q (χ, T) ⟩_{T} = \frac{1}{2} ω \tilde{ω} ∣ u ∣^{2} = - (m Ω_{H})^{2} ∣ u ∣^{2} / 8$ . This is nonpositive at every point, zero at nodes and strictly negative where $u \neq 0$ ; being a nonzero mode does not exclude nodes.

3.1givenA2F2step 2.1algebra∎

For a compact smooth spacelike cut patch $D$ whose stationary integral-curve segments remain in the mode domain and which meets the nonzero set, [F2] proves $\int_{D} ∣ u ∣^{2} d μ_{D} > 0$ and finiteness of that integral. Integrating step 2.1 therefore gives $F_{D} = - (m Ω_{H})^{2} \int_{D} ∣ u ∣^{2} d μ_{D} / 8 < 0$ . At any nonzero mode point whose full period integral-curve segment remains in the mode domain, such a sufficiently small patch is supplied by the local cut construction in [F2]. The conclusion remains conditional on the given trace, without a global wave or scattering existence claim.

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

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 — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Wave energy and black-hole perturbation interfaces — model 1

Example

Example references

Facts & Assumptions

Verification

Wave energy and black-hole perturbation interfaces — model 2

Example

Example references

Facts & Assumptions

Verification

Sources