Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Rotating model horizons and selected causal escape

Physical scope

Conditional conversion of local horizon geometry and the selected escape boundary of the electric Kerr–Newman solution family to SI parameters. The selected ingoing model does not establish a maximal extension, an event horizon in a conformal completion, a black-hole observation, or empirical validity.

Statement

Let $m_{*} > 0$ , $J, Q_{e} \in R$ be the selected SI mass, angular-momentum and electric-charge inputs, with zero magnetic flux. Use SI and geometric solution parameter identifications to set $M = \frac{G_{N} m_{*}}{c^{2}}, a = \frac{J}{m_{*} c}, q = \sqrt{\frac{G_{N}}{4 π ε_{0} c^{4}}} Q_{e}, α = \sqrt{\frac{4 π G_{N}}{μ_{0} c^{4}}} .$ Assume $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ . In the selected ingoing KN model let $r_{\pm} = M \pm \sqrt{M^{2} - a^{2} - q^{2}}, R_{\pm} = r_{\pm}^{2} + a^{2}, χ_{\pm} = \partial_{v} + \frac{a}{R_{\pm}} Ψ,$ where $Ψ = X \partial_{Y} - Y \partial_{X}$ is the smooth rotation field on the unit sphere and $v$ has length units. For either root $r_{H} = r_{\pm}$ , with $r_{o t h e r}$ the other root, the selected horizon data of KN Killing generators, surface gravity, area and potential give $Area (S_{H}) = 4 π R_{H}, Ω_{H, g e o m} = \frac{a}{R_{H}}, κ_{H, g e o m} = \frac{r_{H} - r_{o t h e r}}{2 R_{H}}, - A_{i n} (χ_{H}) = \frac{q r_{H}}{R_{H}} .$ Thus the area is in m $^{2}$ , $Ω_{H, g e o m}$ and the signed $κ_{H, g e o m}$ are in m $^{- 1}$ , and the associated SI angular rate and signed acceleration convention are $Ω_{H, S I} = c \frac{a}{R_{H}} and κ_{H, a c c, S I} = c^{2} \frac{r_{H} - r_{o t h e r}}{2 R_{H}} .$ With $A_{S I} = A_{i n} / α$ and time-normalized generator $χ_{H, S I} = c χ_{H}$ , the gauge-normalized horizon voltage is $- A_{S I} (χ_{H, S I}) = \frac{c q r_{H}}{α R_{H}};$ This voltage normalization uses the displayed ingoing gauge, whose coordinate limit is specified as follows: for each fixed $v$ and $ω \in S^{2}$ , both coefficient functions in the global one-form expression $A_{i n} = - q r Σ^{- 1} d v + a q r Σ^{- 1} β$ tend to zero as $r \to + \infty$ . At extremality the two roots coincide and the signed surface gravity is zero. Conditional on the current exact draft interface of Outer KN escape boundary on the selected ingoing model and the relative-boundary convention of Causal solutions, Killing horizons, escape and metric extensions, the selected escape set on $M_{i n}$ has relative boundary $r = r_{+}$ . These are properties of the chosen solution family and escape observable, not a maximal-extension or observational claim.

Facts & Assumptions

Given: The selected electric SI inputs, $G_{N}, c, μ_{0} > 0$ , the zero-magnetic-flux KN branch, the parameter inequality in the Statement and the chosen ingoing model.

[F1]

SI and geometric solution parameter identifications: $M = G_{N} m_{*} / c^{2}$ , $a = J / (m_{*} c)$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , $F = α F$ , and $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ ; the coordinates $x^{0} = c t$ and $v$ have length units.

[F2]

Verified rotating electric KN physical model: with zero magnetic flux, the SI-rescaled KN datum on its stated pole-free domains and full-sphere exterior is a conditional solution of the equation-based $Λ = 0$ , $j = 0$ , $T^{m} = 0$ model; the mathematical verification remains a draft premise, not independent proof acceptance.

[F3]

Regular ingoing KN metric, field and sphere axes: the selected product $M_{i n} = R_{v} \times (0, \infty)_{r} \times S^{2}$ carries the smooth ingoing metric and potential $A_{i n} = - q r Σ^{- 1} (d v - a β)$ , with $β = X d Y - Y d X$ and $Σ > 0$ ; its field-equation continuation is conditional on the current KN-verification draft.

[L1]

KN Killing generators, surface gravity, area and potential: for $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ , each root $r_{H}$ has future null Killing generator $χ_{H} = \partial_{v} + (a / R_{H}) Ψ$ , area $4 π R_{H}$ , signed $κ_{H} = Δ^{'} (r_{H}) / (2 R_{H}) = (r_{H} - r_{o t h e r}) / (2 R_{H})$ , and $- A_{i n} (χ_{H}) = q r_{H} / R_{H}$ ; the supplier proof remains draft and is used conditionally.

[L2]

Outer KN escape boundary on the selected ingoing model: on the selected ingoing product and for $M > 0$ , $M^{2} \geq a^{2} + q^{2} > 0$ , the exact current statement gives escape set ${r > r_{+}}$ and its relative boundary ${r = r_{+}}$ for the named path class; its proof is draft and not independently accepted.

[F4]

Causal solutions, Killing horizons, escape and metric extensions: the escape boundary is taken relative to the selected manifold, and the escape set uses continuous future piecewise- $C^{1}$ causal paths with finite $C^{1}$ subdivisions on compact parameter intervals.

[A1]

Geometric time is $t = v / c$ , so the time-normalized generator is $c χ_{H}$ when the angular rate is $c (a / R_{H})$ ; the SI one-form is $A_{S I} = A_{i n} / α$ . The gauge for the voltage is the displayed potential, whose contraction tends to zero at infinity.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric electrovac and intrinsic horizon-area conventions use only ${A C}_{ω}$ , meaning each specified countable family of nonempty sets has a choice function. No new selection is made in these finite parameter conversions, and full AC is not used.

Proof

1.1givenF1F2A2algebra

The exact SI map [F1] gives the displayed real length parameters $M, a, q$ and positive normalization $α$ . Setting magnetic flux to zero makes the KN datum's geometric charge exactly $q = q_{e}$ ; no ADM/Komar identification is involved. The stated inequality gives $D = M^{2} - a^{2} - q^{2} \geq 0$ and $\sqrt{D} < M$ , since $a^{2} + q^{2} > 0$ ; hence both roots are positive.

2.1F2F3L1step 1.1algebra

The factorization $Δ = (r - r_{+}) (r - r_{-})$ identifies these roots on the smooth selected ingoing model [F3]. The area, generator, signed surface gravity and displayed-gauge potential formulas then apply to either root by [L1].

2.2L1step 1.1algebra

When $D = 0$ , $r_{+} = r_{-} = M$ , so the signed difference formula in [L1] gives $κ_{H} = 0$ . When $D > 0$ , it gives positive outer and negative inner values under the displayed sign convention.

2.3L2F3F4step 1.1

Apply [L2] with the exact path class and relative topology in [F4]. Its current statement yields the selected escape boundary $r = r_{+}$ on $M_{i n}$ . This use is conditional on that draft supplier and says nothing about a conformal completion or maximal extension.

3.1F1F3L1A1step 2.1algebra

By [F1] and [A1], changing from length-time generator $χ_{H}$ to the SI time generator multiplies it by $c$ , while the SI potential is $A_{i n} / α$ . Therefore $- A_{S I} (c χ_{H}) = (c / α) (- A_{i n} (χ_{H})) = c q r_{H} / (α R_{H})$ . Also $[a / R_{H}] = m^{- 1}$ , so its SI angular rate is $c a / R_{H}$ ; $[κ_{H}] = m^{- 1}$ , so the stated acceleration convention is $c^{2} κ_{H}$ . The area formula has units of square metres. Finally $Σ = r^{2} + a^{2} \cos^{2} θ \geq r^{2}$ , so $r / Σ \leq 1 / r \to 0$ ; the two coefficient functions $- q r / Σ$ and $a q r / Σ$ of $A_{i n}$ in the $(d v, β)$ expression in the Statement vanish in the specified fixed- $(v, ω)$ limit.

4.1A2step 2.1step 2.2step 2.3step 3.1∎

The horizon formulas in steps 2.1-2.2 and the SI conversions in step 3.1 establish the claimed identifications on the specified model, conditional on the named draft mathematical interfaces. The escape boundary in step 2.3 uses the selected-model path convention. The ${A C}_{ω}$ premise [A2] is inherited as stated; no full choice principle, empirical measurement, or global event-horizon claim has been added.

Rotating model horizons and selected causal escape

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources