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Smooth Maxwell tensor formula without division by the stationary norm

Statement

Assume the family and orientation in Regular Lorentzian metric from the normalized rank-one map. Let $F = F + i ⋆_{g} F$ satisfy $i_{T} F = \frac{1}{2} d σ$ and $d σ (T) = 0$ . Put $H_{q} = 1 + q - (1 - q) σ$ , $D_{q} = ∣ H_{q} ∣^{2}$ , and $Z_{q} = - \sqrt{1 - q^{2}} (\overline{H_{q}} / H_{q}) (F + (2 D_{q})^{- 1} W_{q} \land d σ)$ . Then $Z_{q}$ is self-dual for $g_{q}$ and equals $F_{q} + i ⋆_{g_{q}} F_{q}$ for the real two-form $F_{q} = Re Z_{q}$ . It is spacetime smooth, continuous in $q \in [0, 1]$ , and $F_{1} = 0$ . Moreover $i_{4 T / D_{q} (p)} Z_{q} = d Φ_{q}$ where $Φ_{q} = - \sqrt{1 - q^{2}} (1 + σ) / H_{q}$ .

Facts & Assumptions

[F1]

The metric, metric index contraction and stationary scalar conventions use Stationary Lorentzian data and tensor conventions.

[F2]

Self-duality and fiberwise pullback use Complex self-dual forms and metric-dependent contraction.

[F3]

The orientation-preserving map and metric are those of Regular Lorentzian metric from the normalized rank-one map.

[F4]

The smooth family $W_{q}$ and the contraction normalization $W_{q} (T) = D_{q} (p) - D_{q}$ are supplied by Lifting a normalized stationary primitive to the Harrison parameter.

[F5]

The Hodge star is the unique smooth map characterized on real two-forms by $α \land ⋆_{g} β = ⟨ α, β ⟩_{g} {v o l}_{g}$ , where $⟨ α, β ⟩_{g} = \frac{1}{2} α_{a b} β^{a b}$ ; it satisfies $⋆_{g}^{2} = - id$ . These are mathematical conclusions of Lorentzian exterior calculus and Hodge star.

[F6]

Interior product of a smooth form by a smooth vector field is the pointwise operation of Interior product of a form by a vector field: $(i_{X} ω)_{x} (v_{2}, \dots, v_{k}) = ω_{x} (X_{x}, v_{2}, \dots, v_{k})$ for $k \geq 1$ . On the complex forms of [F2], extend it by $i_{X} (U + i V) = i_{X} U + i i_{X} V$ .

[F7]

The exterior derivative and its local coordinate formula and graded product rule are supplied by Lorentzian exterior calculus and Hodge star. For a smooth real function $u$ , $(d u) (X) = X (u)$ . For a smooth complex function $u + i v$ with real $u, v$ , use the componentwise extension $d (u + i v) = d u + i d v$ ; the product rule extends by complex bilinearity. All derivatives here are in spacetime at fixed $q$ .

Proof

Given: The hypotheses in the Statement and the Facts above.

Goal: Establish the stated conclusions.

1.1givenF1F2F3F5F6

Expanding the alternating form gives $A_{q}^{*} F = F + D_{q}^{- 1} W_{q} \land i_{T} F = F + (2 D_{q})^{- 1} W_{q} \land d σ$ . The defining metric-and-volume identity for the Hodge star is preserved by an orientation-preserving invertible linear map; in dimension four a positive conformal factor cancels on two-forms. Consequently $A_{q}^{*} F$ is self-dual for $(D_{q} / 4) A_{q}^{*} g$ . Complex scalar multiplication preserves this eigenspace. Writing $Z_{q} = U + i V$ and comparing real parts in $⋆_{g_{q}} Z_{q} = - i Z_{q}$ gives $V = ⋆_{g_{q}} U$ .

2.1givenF1F3F4F6F7algebrastep 1.1∎

Put $C = D_{q} (p)$ . Using $W_{q} (T) = C - D_{q}$ from [F4] and $d σ (T) = 0$ , contraction with $T$ gives $i_{T} (A_{q}^{*} F) = \frac{1}{2} (1 + W_{q} (T) / D_{q}) d σ = C d σ / (2 D_{q})$ . Multiplication by the displayed factor and $4 / C$ yields $- 2 \sqrt{1 - q^{2}} d σ / H_{q}^{2}$ , using $D_{q} = H_{q} {\overline{H}}_{q}$ . Using the scalar exterior derivative and its componentwise complex extension in [F7], differentiating $Φ_{q}$ gives exactly this expression. Positivity of $D_{q}$ removes every spacetime singular denominator, and the only endpoint factor $\sqrt{1 - q^{2}}$ is continuous and vanishes at one. This proves the regularity and endpoint claims, without asserting differentiability in $q$ at one.

Source notes

This conditional calculation follows §3.2 (3.24)–(3.25) of the cited independent source. No existence of the stationary primitive or Einstein–Maxwell solution is inferred from this calculation alone.

Depends on

Used by · 1

Regular Harrison Einstein–Maxwell family

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equation (3.5); Section 3.2, equations (3.12), (3.19)–(3.25) and the paragraphs immediately before (3.24) and after (3.25).

The regular electromagnetic tensor is the displayed complex multiple of the rank-one pullback of the original self-dual form. Its contraction with the normalized stationary field is the derivative of the displayed Harrison potential; it is spacetime smooth and parameter continuous with zero field at q=1.

Hypotheses: The source uses stationary vacuum data and a smooth normalized primitive. This item assumes the smooth family, positive denominators, orientation, contraction normalization and scalar identities conditionally; no existence theorem or field equation is imported.

Reading limits: Consulted the named HTML formulas and adjacent explanations. The local algebra is checked using the declared mathematical suppliers; no Einstein–Maxwell, primitive-existence, flux or horizon result is claimed to have been reconstructed.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equation (3.5); Section 3.2, equations (3.12), (3.19)–(3.25) and the paragraphs immediately before (3.24) and after (3.25).