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Exact charged scattering Wronskian flux identity

Statement

Let $u$ be any scattering solution as defined on the fixed subextremal electric RN exterior, with real $ω > 0$ , real $e, Q$ , unit incoming amplitude, and reflection and transmission coefficients $R, T$ . Then its radial Wronskian flux $j : = Im (\overline{u} \partial_{r_{*}} u)$ is constant and $∣ R ∣^{2} = 1 - \frac{ω - e Q / r_{+}}{ω} ∣ T ∣^{2} .$ Consequently, if $0 < ω < e Q / r_{+}$ and $T \neq 0$ , the reflected radial flux magnitude exceeds the incident radial flux magnitude, since their ratio is $∣ R ∣^{2} > 1$ . This is conditional on an admitted scattering solution and does not assert its existence, a complex-frequency mode, or a nonlinear instability.

Statement references

Massless charged-scalar radial scattering data

Charged radial equation on subextremal electric RN

Facts & Assumptions

Given: The charged radial scattering definition, a smooth solution of its radial equation, and its stated incoming, reflected, and ingoing-horizon asymptotics.

[F1]

Massless charged-scalar radial scattering data: $r_{*}$ tends to $- \infty$ at the outer root and to $+ \infty$ at spatial infinity; the asymptotics are $u = e^{- i Θ} + R e^{i Θ} + o (1)$ with the differentiated phase form at infinity and $u = T e^{- i ω_{H} r_{*}} + o (1)$ at the horizon, where $ω_{H} = ω - e Q / r_{+}$ .

[F2]

Charged radial equation on subextremal electric RN: $u^{''} + K (r) u = 0$ with real $K (r) = (ω - e Q / r)^{2} - f (ℓ (ℓ + 1) / r^{2} + f^{'} / r)$ for every $r > r_{+}$ .

[A1]

$M > 0$ , $0 < ∣ Q ∣ < M$ , $e \in R$ , and $ω > 0$ , so $r_{+} > 0$ and both $ω$ and $ω_{H}$ are real.

Proof

1.1givenF2A1algebra

Differentiate $j = Im (\overline{u} u^{'})$ . The radial equation gives $j^{'} = Im (\overline{u^{'}} u^{'} + \overline{u} u^{''}) = Im (∣ u^{'} ∣^{2} - K ∣ u ∣^{2}) = 0,$ because both terms inside the last imaginary part are real by [F2]. Thus $j$ is constant along the exterior.

2.1givenF1F2A1step 1.1algebra∎

At infinity, $Θ^{'} = ω - e Q f / r \to ω$ . Substitution of the Definition's differentiated asymptotics into $j$ gives $j_{\infty} = ω (∣ R ∣^{2} - 1)$ : the incident and reflected diagonal terms are $- ω$ and $ω ∣ R ∣^{2}$ , while the two cross terms have real sum. At the horizon the asymptotics give $j_{H} = - ω_{H} ∣ T ∣^{2}$ . Equating these limits by step 1.1 and dividing by $ω > 0$ proves the identity. If $0 < ω < e Q / r_{+}$ then $ω_{H} < 0$ ; when $T \neq 0$ the identity yields $∣ R ∣^{2} = 1 + (- ω_{H} / ω) ∣ T ∣^{2} > 1$ . The unit incoming and reflected asymptotic flux magnitudes are $ω$ and $ω ∣ R ∣^{2}$ , so the stated radial-flux comparison follows.

Remarks

The flux here is the conserved radial Wronskian flux of the separated classical test-field equation. The identity is a linear fixed-background scattering calculation. It gives no conclusion about field backreaction, quantum pair production, trapping, exponential growth, or nonlinear stability.

Exact charged scattering Wronskian flux identity

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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