Physics theoremStatement: AI-adaptedProof: AI-generated

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Energy-condition inputs for the transformed geometry

PDF p. 6: §1.4, dominant energy; PDF p. 13: §3.1, nonnegative Jacobi potential; PDF p. 25: §4, Raychaudhuri area comparison; PDF p. 26: §4, Raychaudhuri area comparison; PDF p. 27: §4, Raychaudhuri focusing

Statement

Adopt the regular stationary vacuum construction with the original exterior section maximal, $T ∣_{S} \equiv̸ 0$ , and mass normalized to one. Let $(g_{q}, F_{q})$ be its Harrison family for $q \in [0, 1]$ , on the exterior $E$ and the supplied open neighborhood $O$ of the bifurcation sphere $S$ . Assume the Axiom of Choice. All contractions and causal character below use $g_{q}$ .

Use the original smooth future timelike field $N$ to orient each transformed metric by $N_{q}^{o r} = A_{q}^{- 1} N$ , where $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ is the Harrison tangent map for this same family. This field is smooth and $g_{q}$ -timelike on $E \cup O$ , and agrees with $N$ at $q = 1$ . Future timelike vectors below satisfy $g_{q} (v, N_{q}^{o r}) < 0$ ; future causal vectors use the corresponding closed cone.

Define $Q_{a b}^{(q)} = (F_{q})_{a c} (F_{q})_{b}^{c} - \frac{1}{4} (g_{q})_{a b} (F_{q})_{c d} (F_{q})^{c d}$ . For every future timelike vector $v$ , the vector $- (Q^{(q)})^{a}_{b} v^{b}$ is future causal or zero. Thus the Maxwell tensor has the dominant energy condition, in particular for $g_{0}$ . For every unit timelike vector $n$ and every nonzero null vector $L$ , $Ric (g_{q}) (n, n) \geq 0$ and $Ric (g_{q}) (L, L) \geq 0$ .

Consequently, on any supplied smooth spacelike maximal candidate section for $g_{q}$ , with future unit normal $n_{q}$ , induced positive-definite metric $h_{q}$ and second fundamental form $k_{q}$ , the potential $∣ k_{q} ∣_{h_{q}}^{2} + Ric (g_{q}) (n_{q}, n_{q})$ is nonnegative. This applies in particular for $q \in [δ, 1]$ when $δ > 0$ and the indicated section exists. The null inequality is the null-convergence input used in the article's Raychaudhuri area-comparison and focusing arguments; it does not by itself assert those arguments' additional hypotheses or conclusions.

Facts & Assumptions

[A1]

The physical model and regular stationary vacuum hypotheses, original maximal section and mass-one normalization are those of Stationary vacuum model and asymptotic normalization, Regular non-degenerate stationary vacuum spacetime, Spacelike sections, maximality and lapse–shift data and Mass-one convention and Harrison denominator. The Axiom of Choice is assumed. The Axiom of Choice

[F1]

Under these hypotheses, the Harrison construction gives the smooth Lorentzian metrics and smooth spacetime two-forms satisfying the source-free Einstein–Maxwell equations on $E \cup O$ for each $q \in [0, 1]$ . Its regular Harrison data have positive denominators on this common domain, and $(g_{1}, F_{1}) = (g, 0)$ . Proposition 2.1: Einstein–Maxwell equations and regularity

[F2]

A source-free geometric electrovacuum datum has a time-oriented Lorentzian metric of signature $(-, +, +, +)$ , a real two-form, and the equation $Ric = 2 Q$ with the displayed Maxwell tensor. Geometric electrovacuum equations and charge convention

[F3]

For such a datum, Maxwell algebra gives the dominant energy condition for $Q$ , nonnegative Ricci contraction on unit timelike vectors, and nonnegative Ricci contraction on nonzero null vectors. Maxwell algebra supplies timelike and null convergence

[F4]

A spacelike section has positive-definite induced metric, a future unit normal and a second fundamental form. Maximal means that its mean curvature vanishes. Spacelike sections, maximality and lapse–shift data

[F5]

For these same regular Harrison data on $D_{q} > 0$ , the algebraic supplier defines $A_{q} v = v + D_{q}^{- 1} W_{q} (v) T$ and proves $g_{q} = (D_{q} / 4) A_{q}^{*} g$ and smooth invertibility of $A_{q}$ . For the original smooth timelike field $N$ , it proves that $A_{q}^{- 1} N$ is a smooth transformed time orientation, with the future-cone correspondence and $A_{1} = I$ for the single-potential family. Invertibility, Lorentz signature and stationary norm

Proof

Given: The construction and candidate-section hypotheses in the Statement.

Goal: Prove the stated energy and convergence inequalities and the nonnegativity of the potential.

1.1A1F1F5algebra

Fix $q \in [0, 1]$ and use the regular positive-denominator Harrison data on $E \cup O$ from [F1]. They satisfy the hypotheses of [F5] for the same $g_{q}$ and $W_{q}$ . In particular $N_{q}^{o r} = A_{q}^{- 1} N$ is a smooth field and $g_{q} (N_{q}^{o r}, N_{q}^{o r}) = (D_{q} / 4) g (N, N) < 0$ . Also $g_{q} (v, N_{q}^{o r}) = (D_{q} / 4) g (A_{q} v, N)$ , so this field defines precisely the future convention in the Statement. At $q = 1$ , $A_{1} = I$ gives the original time orientation. Thus no assumption that the original field $N$ itself stays timelike for $g_{q}$ is needed.

2.1A1F1F2F3step 1.1

Keep the original manifold orientation and use the time-orientation field constructed in step 1.1. The equations and spacetime regularity in [F1] now make $(E \cup O, g_{q}, N_{q}^{o r}, o, F_{q})$ a source-free geometric electrovacuum datum in [F2], with every index operation taken using $g_{q}$ . Applying [F3] gives the future-causal or zero energy current and the timelike and null Ricci inequalities at each point. The argument includes $q = 0$ and requires no limiting passage.

3.1F4step 2.1algebra∎

On a supplied candidate section, its induced metric is positive definite. In an $h_{q}$ -orthonormal tangent frame, $∣ k_{q} ∣_{h_{q}}^{2}$ is a sum of squares and hence is nonnegative. Its future unit normal is timelike, so the Ricci term is nonnegative by step 2.1. Their sum is therefore nonnegative, including throughout the stated positive-parameter range whenever the section exists. The same step supplies null convergence wherever the article additionally invokes a valid Raychaudhuri argument.

Source notes

The article citation attributes the uses described above. The proof uses the completed construction, the locally proved Harrison pullback time orientation, and the Maxwell-algebra supplier; it does not derive the Raychaudhuri equation, area comparison, focusing, or existence of the candidate sections.

Depends on

Used by · 6

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Sources

The uniqueness of smooth stationary black holes · PDF pp. 6, 13, 25–27; §§1.4, 3.1 and 4