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Asymptotic time-translation flux of a charged scalar mode

Statement

Use the fixed subextremal electric Reissner–Nordström exterior, gauge, charged derivative, real normalized spherical harmonic $Y$ , and mode conventions of Massless charged-scalar radial scattering data. Let $ψ = A_{i n} e^{- i ω t} Y (Ω) u (r_{*}) / r$ be any admitted real-frequency scattering mode with $ω > 0$ and $A_{i n} \neq 0$ , and let $R$ be its reflection coefficient. Form $T_{a b} [ψ; g, A, e]$ by the scalar stress-tensor definition. With $ξ = \partial_{t}$ , define the outward radial flux through the sphere of radius $r$ by

$P (r) : = - \int_{S^{2}} r^{2} T^{r}_{t} [ψ; g, A, e] d ω_{S^{2}} .$

Let $ψ_{i n} : = A_{i n} e^{- i ω t} Y e^{- i Θ} / r$ and $ψ_{r e f} : = A_{i n} e^{- i ω t} Y R e^{i Θ} / r$ be the two leading asymptotic components. Define their nonnegative directional flux magnitudes by

$P_{i n} : = \lim_{r \to \infty} \int_{S^{2}} r^{2} T^{r}_{t} [ψ_{i n}; g, A, e] d ω_{S^{2}}, P_{r e f} : = - \lim_{r \to \infty} \int_{S^{2}} r^{2} T^{r}_{t} [ψ_{r e f}; g, A, e] d ω_{S^{2}} .$

Then

$P_{i n} = 2 ω^{2} ∣ A_{i n} ∣^{2}, P_{r e f} = 2 ω^{2} ∣ A_{i n} ∣^{2} ∣ R ∣^{2}, \frac{P_{r e f}}{P_{i n}} = ∣ R ∣^{2} .$

This is a conditional tensor calculation for the specified mode asymptotics; it asserts neither that such a mode exists nor that its flux has been observed.

Statement references

Massless charged-scalar radial scattering data

Complex scalar stress tensor with a supplied potential

Facts & Assumptions

Given: The exact static exterior and gauge of the scattering definition, a real normalized harmonic, an admitted mode with its stated two-term asymptotics and differentiated asymptotics at infinity, and the tensor assignment in the stress-tensor definition.

[F1]

Massless charged-scalar radial scattering data: $A_{t} = - Q / r$ , $D_{t} = \partial_{t} + i e Q / r$ , $D_{r} = \partial_{r}$ , $g^{r r} = f$ , $\partial_{r_{*}} = f \partial_{r}$ , and $Θ^{'} (r_{*}) = ω - e Q f / r \to ω$ ; also $\int_{S^{2}} Y^{2} d ω_{S^{2}} = 1$ .

[F2]

Complex scalar stress tensor with a supplied potential: $T_{a b} = \overline{D_{a} ψ} D_{b} ψ + \overline{D_{b} ψ} D_{a} ψ - g_{a b} g^{c d} \overline{D_{c} ψ} D_{d} ψ$ .

[A1]

The separated mode has the normalized asymptotics $u = e^{- i Θ} + R e^{i Θ} + o (1)$ and $u^{'} = - i Θ^{'} e^{- i Θ} + i Θ^{'} R e^{i Θ} + o (1)$ as $r \to \infty$ , with all parameters real and $ω > 0$ .

Proof

1.1givenF1A1algebra

Since $A_{t} = - Q / r$ and $ω > 0$ , the time derivative satisfies $D_{t} ψ = - i ω ψ + o (r^{- 1})$ . Using $D^{r} = f \partial_{r} = \partial_{r_{*}}$ and $\partial_{r_{*}} (u / r) = u^{'} / r - f u / r^{2}$ , the differentiated asymptotics and $Θ^{'} \to ω$ give $ψ = \frac{A_{i n} e^{- i ω t} Y}{r} (e^{- i Θ} + R e^{i Θ}) + o (r^{- 1}), D^{r} ψ = \frac{A_{i n} e^{- i ω t} Y}{r} (- i ω e^{- i Θ} + i ω R e^{i Θ}) + o (r^{- 1}) .$

2.1givenF1F2A1step 1.1algebra

Since $g^{r}_{t} = 0$ , [F2] gives $T^{r}_{t} = 2 Re (\overline{D^{r} ψ} D_{t} ψ)$ . Substitution of step 1.1 gives the diagonal terms $ω^{2} (1 - ∣ R ∣^{2})$ ; the cross terms are $z - \overline{z}$ for $z = ω^{2} R e^{2 i Θ}$ and have zero real part. Integrating $Y^{2}$ over the sphere and taking $r \to \infty$ therefore gives the net outward flux $P (r) \to 2 ω^{2} ∣ A_{i n} ∣^{2} (∣ R ∣^{2} - 1)$ .

3.1givenF1F2A1step 1.1algebra∎

For the incoming component, $D^{r} ψ_{i n} = - i ω ψ_{i n} + o (r^{- 1})$ and $D_{t} ψ_{i n} = - i ω ψ_{i n} + o (r^{- 1})$ , so $T^{r}_{t} [ψ_{i n}] = 2 ω^{2} ∣ ψ_{i n} ∣^{2} + o (r^{- 2})$ ; its positive inward magnitude is $2 ω^{2} ∣ A_{i n} ∣^{2}$ after sphere integration. For the reflected component, $D^{r} ψ_{r e f} = i ω ψ_{r e f} + o (r^{- 1})$ while $D_{t} ψ_{r e f} = - i ω ψ_{r e f} + o (r^{- 1})$ , so its nonnegative outward magnitude is $2 ω^{2} ∣ A_{i n} ∣^{2} ∣ R ∣^{2}$ . Their ratio is $∣ R ∣^{2}$ because $ω > 0$ and $A_{i n} \neq 0$ .

Remarks

The Coulomb phase in the scattering definition is retained; only its derivative limit $Θ^{'} \to ω$ enters the asymptotic stress calculation. The result concerns the prescribed geometric normalization and supplies no scattering-solution existence claim.

Asymptotic time-translation flux of a charged scalar mode

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources