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Regular ingoing KN metric, field and sphere axes

Statement

Let $M, a, q \in R$ have units of metres. Set $Σ = r^{2} + a^{2} \cos^{2} θ, Δ = r^{2} - 2 M r + a^{2} + q^{2}, R = r^{2} + a^{2},$ and let $S^{2} = {(X, Y, Z) \in R^{3} : X^{2} + Y^{2} + Z^{2} = 1}$ have unit round metric $γ$ and outward area form $ω$ . Define the global one-form $β : = X d Y - Y d X$ and the product $M_{i n} : = R_{v} \times (0, \infty)_{r} \times S^{2},$ oriented by $d v \land d r \land ω$ . On this product define $\begin{aligned} g_{i n} = & - (1 - \frac{2 M r - q^{2}}{Σ}) d v^{2} + 2 d v d r - 2 a d r β - 2 a \frac{2 M r - q^{2}}{Σ} d v ⊙ β + Σ γ + a^{2} (1 + \frac{2 M r - q^{2}}{Σ}) β^{2}, \\ A_{i n} = & - \frac{q r}{Σ} (d v - a β), F_{i n} : = d A_{i n}, \end{aligned}$ where $u ⊙ w = (u \otimes w + w \otimes u) / 2$ . The time-orientation field is $f_{0} : = 1 - \frac{2 M r - q^{2}}{Σ}, N : = \partial_{v} - \frac{1 + f_{0}^{2}}{2} \partial_{r} .$ These formulas define smooth Lorentzian metric and field data on the whole sphere for every $r > 0$ , including each positive root of $Δ$ ; no maximal extension, $r = 0$ locus, or global uniqueness is asserted. On the dense pole-free set where $Δ \neq 0$ , they are related to the Boyer-Lindquist data by the advanced coordinate change proved below.

The source-free geometric electrovac equations extend across $Δ = 0$ conditionally on the exact current interface of Fully verified electric Kerr-Newman electrovacuum: its geometric pole-free equations are used on the dense complement of the roots and axes, and smoothness then extends their residuals. That supplier is a current draft theorem and is not claimed as independently accepted here.

Facts & Assumptions

Given: Real parameters $M, a, q$ , $r > 0$ , the unit sphere, the exact Boyer-Lindquist KN datum and its current verification interface.

[F1]

Kerr-Newman metric and potential on the exact Boyer-Lindquist domain: on $Σ > 0$ , $Δ \neq 0$ and $0 < θ < π$ , the metric and potential are the displayed BL tensors with $A = - q r Σ^{- 1} (d x^{0} - a \sin^{2} θ d ϕ)$ ; its well-definedness calculation gives determinant $- Σ^{2}$ in $(x^{0}, r, y = \cos θ, ϕ)$ and signature $(-, +, +, +)$ . The definition asserts no axis or root value.

[F2]

Geometric electrovac equations and flux normalization: the selected equations are $d F = 0$ , $d (⋆_{g} F) = 0$ and $Ric = 2 Q (F)$ , with signature $(-, +, +, +)$ and metric volume form matching the chosen orientation. This definition is not a field-equation proof.

[F3]

Fully verified electric Kerr-Newman electrovacuum: its exact geometric Statement says those equations hold on every pole-free BL domain with $Σ > 0$ , $Δ \neq 0$ and $0 < θ < π$ . The carrier is a draft theorem with an unaccepted proof; only this exact geometric interface is used conditionally.

[F4]

Lorentzian exterior calculus and Hodge star: the Hodge map is the smooth map determined by the metric and oriented metric volume form, with $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ on two-forms.

[F5]

Smoothness and transformation of the inverse-metric components: the inverse of a smooth nondegenerate metric is smooth, so curvature and Hodge residual tensors built from smooth metric and field data are continuous.

[F6]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Time orientation, future and past timelike/null/causal vectors, and the closed cones: a smooth Lorentzian metric with a smooth everywhere timelike field has the time orientation selected by that field.

[A1]

$M, a, q$ are real, $r > 0$ , and $Σ = r^{2} + a^{2} Z^{2} > 0$ . No sign condition on $M$ or $Δ$ is imposed.

Proof

1.1A1algebra

The sphere coordinates $X, Y, Z$ and one-form $β = X d Y - Y d X$ are smooth globally. Since $r > 0$ , $Σ = r^{2} + a^{2} Z^{2} > 0$ ; every coefficient in the displayed product tensors is therefore smooth, and $F_{i n}$ is smooth by exterior differentiation. In polar charts, $β = \sin^{2} θ d ψ$ and $γ = d θ^{2} + \sin^{2} θ d ψ^{2}$ .

1.2F1A1algebra

Away from the axes and roots, put $β = \sin^{2} θ d ψ$ and set $d v = d x^{0} + \frac{R}{Δ} d r, d ψ = d ϕ + \frac{a}{Δ} d r .$ This triangular coordinate change has Jacobian determinant $1$ . Substitution in the product metric gives exactly the BL metric in [F1]. Also $d v - a \sin^{2} θ d ψ = d x^{0} - a \sin^{2} θ d ϕ + \frac{Σ}{Δ} d r .$ Thus $A_{i n}$ differs from the BL potential by the radial one-form $- q r Δ^{- 1} d r$ , whose exterior derivative is zero because its coefficient depends only on $r$ . The two field strengths agree. Finally $d v \land d r \land ω = d x^{0} \land d r \land ω$ , so this change preserves the stated orientation.

2.1F1step 1.1step 1.2A1algebra

On $0 < θ < π$ , the determinant in coordinates $(v, r, θ, ψ)$ is $- Σ^{2} \sin^{2} θ$ : it is the determinant $- Σ^{2}$ from [F1], multiplied by $(d y / d θ)^{2} = \sin^{2} θ$ ; the coordinate change in step 1.2 has Jacobian $1$ . This determinant is nonzero when $Δ \neq 0$ and extends by continuity to $Δ = 0$ , since the product metric coefficients are smooth by step 1.1. At a polar axis $β = 0$ as a covector on the sphere tangent space; the radial block is $(\begin{array}{cc} - f_{0} & 1 \\ 1 & 0 \end{array})$ with determinant $- 1$ , and the angular block is $Σ γ$ , positive definite. So the metric is nondegenerate at the axes too. On a large- $r$ pole-free region it has signature $(-, +, +, +)$ by [F1] and step 1.2. A continuous nondegenerate symmetric form has locally constant signature; the product manifold is connected, so this signature holds everywhere.

3.1F6step 1.1step 2.1algebra

The field $N$ is smooth and global by the product construction. Direct contraction gives $g_{i n} (N, N) = - (f_{0}^{2} + f_{0} + 1) = - ((f_{0} + \frac{1}{2})^{2} + \frac{3}{4}) < 0.$ Thus $N$ is everywhere timelike, and by [F6] it selects the stated smooth time orientation on the whole product.

4.1F2F3F4F5step 1.1step 1.2step 2.1step 3.1algebra∎

By [F3] the Einstein and Maxwell residual tensors vanish on the pole-free BL domains. The coordinate and field identities in step 1.2 carry that zero residual to the ingoing formulas on the dense set $Δ \neq 0$ , $0 < θ < π$ . By steps 1.1, 2.1 and 3.1, the metric and field are smooth and nondegenerate on the full product; [F4-F5] therefore make each residual tensor continuous, including at the axes and roots. A continuous tensor zero on a dense subset is zero on its closure. This proves the stated conditional equation continuation exactly when the interface [F3] is established, without extending beyond the selected ingoing model.

Remarks

Metric, field and orientation regularity are established directly from the displayed product formulas. The electrovac-equation clause remains conditional on the current pole-free KN verification draft; it adds no maximality, uniqueness or empirical claim.

Regular ingoing KN metric, field and sphere axes

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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