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 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.

Wave energy and black-hole perturbation interfaces — model 1

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources