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Outer KN escape boundary on the selected ingoing model

Statement

Let $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ in the selected ingoing KN model. Put $r_{+} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ . Let $E$ be the set of points from which a future piecewise- $C^{1}$ causal path reaches arbitrarily large $r$ , using the specified definition Causal solutions, Killing horizons, escape and metric extensions. Then $E = {p : r (p) > r_{+}}, \partial_{M_{i n}} E = {p : r (p) = r_{+}} .$ This result concerns the explicit escape observable on the selected ingoing manifold, including the extremal case. It does not classify all maximal KN extensions, the inner Cauchy horizon, collapse, or cosmic censorship.

Facts & Assumptions

Given: The selected ingoing KN metric and orientation, with the parameter range in the Statement.

[F1]

Regular ingoing KN metric, field and sphere axes: on the full sphere product, $Σ = r^{2} + a^{2} \cos^{2} θ > 0$ and $N = \partial_{v} - (1 + f_{0}^{2}) \partial_{r} / 2$ is a smooth future timelike field. In polar coordinates the nonzero metric components are $g_{v v} = \frac{- Δ + a^{2} \sin^{2} θ}{Σ}, g_{v r} = 1, g_{v ψ} = - \frac{a (R - Δ) \sin^{2} θ}{Σ}, g_{r ψ} = - a \sin^{2} θ, g_{ψ ψ} = \frac{(R^{2} - a^{2} Δ \sin^{2} θ) \sin^{2} θ}{Σ}, g_{θ θ} = Σ,$ where $R = r^{2} + a^{2}$ and $Δ = (r - r_{+}) (r - r_{-})$ .

[F2]

KN Killing generators, surface gravity, area and potential: under the stated hypotheses $r_{\pm} = M \pm \sqrt{M^{2} - a^{2} - q^{2}}$ are positive and the outer root is $r_{+}$ .

[F3]

Smooth causal-cone collar barrier with screen directions: on a collar of $r = r_{+}$ , for future causal $Z$ and $τ = v - r$ one has $d τ (Z) > 0$ for $Z \neq 0$ , $d r (Z) \leq C ∣ r - r_{+} ∣ d τ (Z)$ , and $d r (Z) \leq 0$ on the root. Its proof uses the exact KN metric, including all angular directions. For the zero vector $d τ (Z) = 0$ , so along a future causal curve piece $τ^{'} \geq 0$ .

[F4]

Classification of future and past causal vectors by a timelike field: for a nonzero causal vector, future-directedness is equivalent to negative pairing with a selected future timelike field.

[F5]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion makes a continuous real function on a compact interval Riemann integrable; The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive then makes the derivative of its integral function equal to its integrand. The exponential function is smooth and $(\exp)^{'} = \exp$ and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ give the integrating-factor derivative along each smooth curve piece.

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : if a real function $H$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then $H (b) - H (a) = H^{'} (c) (b - a)$ for some $c \in (a, b)$ . Consequently a nonnegative derivative makes $H$ nondecreasing, and a nonpositive derivative makes it nonincreasing, on each closed $C^{1}$ curve piece.

[L1]

Causal solutions, Killing horizons, escape and metric extensions, Definition (d): the escape set uses continuous future piecewise- $C^{1}$ causal paths, with finite $C^{1}$ subdivisions on compact parameter segments and nonzero future causal piece tangents. A single such path must reach arbitrarily large values of the specified radius; the escape boundary is relative to the selected model, without a conformal completion.

[A1]

Square roots are nonnegative real roots, as in Explicit rational and real power regimes. In KN let $Ψ = X \partial_{Y} - Y \partial_{X}$ be the smooth sphere rotation field; it coincides with $\partial_{ψ}$ off the axes.

Proof

1.1F1F2F4A1algebra

On a polar chart put $ℓ = - \partial_{r}, n = \frac{R}{Σ} \partial_{v} + \frac{Δ}{2 Σ} \partial_{r} + \frac{a}{Σ} \partial_{ψ} .$ Substitution of [F1]'s six metric entries gives $g (ℓ, ℓ) = g (n, n) = 0$ , $g (ℓ, n) = - 1$ , and $g (n, N) = - \frac{1}{2} (\frac{Δ}{Σ} + 1 + f_{0}^{2}) .$ The same components give $g (n, \partial_{r}) = 1$ because $R - a^{2} \sin^{2} θ = Σ$ , and direct contraction gives $g (n, \partial_{v}) = - Δ / (2 Σ)$ . Since $N = \partial_{v} - (1 + f_{0}^{2}) \partial_{r} / 2$ , these yield the displayed $g (n, N)$ . For the null identity, after multiplying by $Σ^{3}$ the numerator is $R^{2} (- Δ + a^{2} s^{2}) + R Δ Σ - 2 a^{2} R (R - Δ) s^{2} - a^{2} Δ s^{2} Σ + a^{2} (R^{2} - a^{2} Δ s^{2}) s^{2} = 0,$ where $s^{2} = \sin^{2} θ$ and $Σ = R - a^{2} s^{2}$ ; the pairing $g (ℓ, n) = - 1$ follows from the $r v$ and $r ψ$ terms. The vector $n$ extends smoothly across the polar axes as $(R / Σ) \partial_{v} + (Δ / (2 Σ)) \partial_{r} + (a / Σ) Ψ$ . For $r > r_{+}$ , [F2] gives $Δ > 0$ , so $g (n, N) < 0$ ; [F4] makes $n$ future null. Its integral curves satisfy $d v / d s = R / Σ > 0$ , $d r / d v = Δ / (2 R)$ and $d ψ / d v = a / R$ .

1.2F3F5F6L1algebra

For a path starting below $r_{+}$ , apply [F3] on its final collar segment before a proposed first root contact. With $w = r_{+} - r \geq 0$ , $w^{'} \geq - C w τ^{'}$ , so [F5] and the chain rule make $(e^{C τ} w)^{'} \geq 0$ on each $C^{1}$ piece. This product is continuous on each closed piece and differentiable in its interior, so [F6] makes it nondecreasing there. Continuity patches those inequalities across the finite partition. This contradicts $w$ reaching zero from its positive value at the starting point of that collar segment: use the initial point if already in the collar, or the last crossing of a fixed smaller collar lower boundary otherwise. If a path starts at the root and has a positive excursion, a component of ${r > r_{+}}$ has a left endpoint with $r = r_{+}$ . Restrict to an initial portion inside the collar, stopping at the first crossing of $r_{+} + δ / 2$ if necessary, where $δ$ is a collar width from [F3]. On that portion $h = e^{- C τ} (r - r_{+})$ has derivative $h^{'} \leq 0$ by [F3] and [F5]. It is continuous on each closed piece and differentiable in its interior, so [F6] on the finitely many pieces makes $h$ nonincreasing. Continuity from its zero left-end value contradicts positivity. Therefore no point with $r \leq r_{+}$ can escape.

2.1F2F5L1step 1.1A1algebra

Fix any $r_{0} > r_{+}$ and initial $v_{0}, ω_{0}$ . For $r \geq r_{0}$ , set $v (r) = v_{0} + \int_{r_{0}}^{r} \frac{2 R (s)}{Δ (s)} d s, α (r) = \int_{r_{0}}^{r} \frac{2 a}{Δ (s)} d s .$ The integrands are smooth on each finite interval $[r_{0}, r]$ and [F5] gives their derivatives. Rotate $ω_{0}$ about the axis by $α (r)$ to obtain a smooth sphere path, including if it passes through a pole. The resulting curve has tangent $γ^{'} (r) = \frac{2 R}{Δ} \partial_{v} + \partial_{r} + \frac{2 a}{Δ} Ψ = \frac{2 Σ}{Δ} n .$ Its coefficient is positive because $Σ, Δ > 0$ , so step 1.1 shows it is future null. Reparameterizing by $s = r - r_{0} \in [0, \infty)$ gives the path domain in [L1]. It exists for every finite $r \geq r_{0}$ and reaches arbitrarily large radii. Thus every point with $r > r_{+}$ belongs to $E$ .

3.1L1step 1.2step 2.1algebra∎

Step 2.1 includes every point with $r > r_{+}$ in $E$ , while the earlier collar argument excludes every point with $r \leq r_{+}$ . The boundary of this open radial half-space in $R_{v} \times (0, \infty)_{r} \times S^{2}$ is exactly $r = r_{+}$ . This proves the stated boundary under the conditional escape interface [L1].

Remarks

The current causal-vocabulary Definition (d) supplies the exact escape observable used here. The cone barrier and outward null paths are proved for the selected metric and orientation, without a maximal-extension argument.

Outer KN escape boundary on the selected ingoing model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources