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KN Killing generators, surface gravity, area and potential

Statement

Let $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ in the selected ingoing KN model. For either $r_{H} = r_{\pm} = M \pm \sqrt{M^{2} - a^{2} - q^{2}}$ set $R_{H} = r_{H}^{2} + a^{2}$ , $Σ_{H} = r_{H}^{2} + a^{2} \cos^{2} θ$ , and $χ_{H} = \partial_{v} + Ω_{H} Ψ, Ω_{H} = \frac{a}{R_{H}},$ where $Ψ = X \partial_{Y} - Y \partial_{X}$ is the smooth rotation field on $S^{2}$ . The surface $r = r_{H}$ is a null hypersurface and $χ_{H}$ is its future null normal and Killing generator. Under the specified normalization $d (g (χ_{H}, χ_{H})) = - 2 κ_{H} χ_{H}^{♭}$ in Causal solutions, Killing horizons, escape and metric extensions, $κ_{H} = \frac{Δ^{'} (r_{H})}{2 R_{H}} = \frac{r_{H} - r_{o t h e r}}{2 R_{H}}, Area (S_{H}) = 4 π R_{H}, - A_{i n} (χ_{H}) = \frac{q r_{H}}{R_{H}} .$ At the double root $κ_{H} = 0$ . These are local geometric calculations on the specified model; they assert no maximality or uniqueness.

Facts & Assumptions

Given: The exact ingoing KN metric and potential with $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ .

[F1]

Regular ingoing KN metric, field and sphere axes: the product metric and potential are smooth on $R_{v} \times (0, \infty)_{r} \times S^{2}$ , with $Σ = r^{2} + a^{2} \cos^{2} θ > 0$ , and the future timelike field is $N = \partial_{v} - (1 + f_{0}^{2}) \partial_{r} / 2$ . On a polar chart, $g_{v v} = \frac{- Δ + a^{2} \sin^{2} θ}{Σ}, g_{v ψ} = - \frac{a (R - Δ) \sin^{2} θ}{Σ}, g_{ψ ψ} = \frac{(R^{2} - a^{2} Δ \sin^{2} θ) \sin^{2} θ}{Σ},$ $g_{v r} = 1$ , $g_{r ψ} = - a \sin^{2} θ$ , and $g_{θ θ} = Σ$ ; $A_{i n} = - q r Σ^{- 1} (d v - a \sin^{2} θ d ψ)$ .

[F2]

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor: a smooth field with zero metric Lie derivative is Killing; along an affinely parameterized geodesic, its metric pairing with the tangent is constant.

[F3]

Classification of future and past causal vectors by a timelike field: a nonzero null vector is future-directed when its pairing with the selected future timelike field is negative.

[F4]

Explicit rational and real power regimes: each nonnegative real has a unique nonnegative square root, with $\sqrt{0} = 0$ .

[L1]

Causal solutions, Killing horizons, escape and metric extensions, Definition (b)-(c): the null conormal criterion, nonempty dense regular Killing-generator portion, ambient signed normalization $d (g (χ, χ)) = - 2 κ_{H} χ^{♭}$ , and intrinsic positive induced area density apply to the specified horizon and section.

[A1]

$Ψ$ is the global smooth generator of rotations of the round sphere; on polar charts it equals $\partial_{ψ}$ . The section $S_{H}$ is the sphere at fixed $v, r_{H}$ . Parameters and radial coordinates have length units; angles are dimensionless, and $Ω_{H}, κ_{H}$ have inverse-length units.

Proof

1.1F1F2F4A1algebra

The quadratic factors as $Δ = (r - r_{+}) (r - r_{-})$ , with $D = M^{2} - a^{2} - q^{2} \geq 0$ . Its nonnegative root $s = \sqrt{D}$ satisfies $s < M$ : if $s \geq M$ , then $s^{2} \geq M^{2}$ , contrary to $s^{2} = M^{2} - (a^{2} + q^{2}) < M^{2}$ . Hence both roots $M \pm s$ are positive. The metric coefficients are independent of $v$ and invariant under rotations, so [F2] makes $\partial_{v}$ , $Ψ$ , and their constant linear combination $χ_{H}$ Killing.

2.1F1F3L1step 1.1A1algebra

In a polar chart, substitute $Ω_{H} = a / R_{H}$ into the metric at $r = r_{H}$ . The result is $χ_{H}^{♭} = \frac{Σ_{H}}{R_{H}} d r .$ The multiplier is positive. Since $d r (χ_{H}) = 0$ , this identity gives $g (χ_{H}, χ_{H}) = 0$ and shows $χ_{H}$ is tangent and normal to the level $r = r_{H}$ . It also gives $g^{- 1} (d r, d r) = (R_{H} / Σ_{H})^{2} g (χ_{H}, χ_{H}) = 0$ , so the level is null. Moreover $d r (N) = - (1 + f_{0}^{2}) / 2 < 0$ , whence $g (χ_{H}, N) = (Σ_{H} / R_{H}) d r (N) < 0$ ; [F3] makes the generator future. The generator has $v$ component one everywhere, so its nonzero-generator part is the whole horizon, nonempty and dense. This verifies the Killing-horizon conditions in [L1]. The formula is smooth at the axes because it is an identity of global tensors agreeing on the dense polar charts.

3.1F1L1step 2.1algebra

Direct contraction on a polar chart gives $g (χ_{H}, χ_{H}) = - \frac{Δ}{Σ} {(1 - a \sin^{2} θ Ω_{H})}^{2} + \frac{\sin^{2} θ}{Σ} (R Ω_{H} - a)^{2} .$ At $r_{H}$ the second squared factor vanishes, as does its first radial derivative. Also $1 - a^{2} \sin^{2} θ / R_{H} = Σ_{H} / R_{H}$ . Therefore $d (g (χ_{H}, χ_{H})) = - \frac{Δ^{'} (r_{H}) Σ_{H}}{R_{H}^{2}} d r on r = r_{H} .$ Comparing with [L1] and step 2.1 yields $κ_{H} = Δ^{'} (r_{H}) / (2 R_{H})$ . Differentiating the factorization of $Δ$ gives $Δ^{'} (r_{H}) = r_{H} - r_{o t h e r}$ , including zero at the double root.

4.1F1L1step 3.1algebra∎

The metric induced on the fixed- $v, r_{H}$ sphere is $Σ_{H} d θ^{2} + \frac{R_{H}^{2}}{Σ_{H}} \sin^{2} θ d ψ^{2} .$ Its area density is $R_{H} \sin θ$ , so integration over $0 \leq θ \leq π$ , $0 \leq ψ < 2 π$ gives $4 π R_{H}$ . Contracting the potential in [F1] gives $- A_{i n} (χ_{H}) = \frac{q r_{H}}{Σ_{H}} (1 - \frac{a^{2} \sin^{2} θ}{R_{H}}) = \frac{q r_{H}}{R_{H}},$ which is independent of $θ$ .

Remarks

The current vocabulary supplies the exact Killing-horizon, signed surface-gravity and area definitions. The coordinate calculations above verify those interfaces for the displayed smooth metric, including both roots and the double-root limit.

KN Killing generators, surface gravity, area and potential

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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