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Specified time Killing stationary-limit set and geodesic constants

Statement

Let $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ in the selected ingoing KN model. For the specified stationary field $T = \partial_{v}$ , $g (T, T) = - \frac{r^{2} - 2 M r + q^{2} + a^{2} \cos^{2} θ}{Σ}, Σ = r^{2} + a^{2} \cos^{2} θ .$ Define its ergoregion by $g (T, T) > 0$ and its stationary-limit set by $g (T, T) = 0$ , using the specified terminology in Causal solutions, Killing horizons, escape and metric extensions. At each angle where the discriminant is nonnegative, the stationary-limit radii are $r_{E \pm} (θ) = M \pm \sqrt{M^{2} - q^{2} - a^{2} \cos^{2} θ} .$ Only positive values of these algebraic roots belong to the selected $r > 0$ model; in particular a zero lower root is excluded. If $a \neq 0$ , then $g (T, T) = a^{2} \sin^{2} θ / Σ_{H} > 0$ at every nonpolar point of the outer horizon $r_{+} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ , and $r_{E +} (θ) > r_{+}$ . The outer stationary-limit set meets the outer horizon on the rotation axis. For each affinely parameterized geodesic, $g (T, \dot{γ})$ and $g (Ψ, \dot{γ})$ are constant, where $Ψ = X \partial_{Y} - Y \partial_{X}$ generates rotations. These statements refer to the specified fields and model.

Facts & Assumptions

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

[F1]

Regular ingoing KN metric, field and sphere axes: the metric is smooth on the full sphere product, its coefficients depend on $r, θ$ but not $v$ , and it is invariant under the global rotation field $Ψ$ . Its $v v$ coefficient is $g_{v v} = - 1 + (2 M r - q^{2}) / Σ$ .

[F2]

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor: a smooth field is Killing if its metric Lie derivative vanishes; along an affine geodesic, the pairing of a Killing field with the tangent is constant. Equivalently, the Killing equation is $\nabla_{a} K_{b} + \nabla_{b} K_{a} = 0$ .

[F3]

Explicit rational and real power regimes: nonnegative real square roots are unique and $\sqrt{0} = 0$ .

[L1]

Causal solutions, Killing horizons, escape and metric extensions, Definition (c): a specified stationary Killing field is timelike on a nonempty open region; its ergoregion is its positive-norm set, its stationary-limit set is its zero-norm set, and its ergosurface is the relative boundary of the former. These notions do not identify a null horizon.

[A1]

$Ψ$ is the smooth generator of rotations on $S^{2}$ and equals $\partial_{ψ}$ away from the axes. The angles are dimensionless and $M, a, q, r$ have units of length.

Proof

1.1L1F1F2F3A1algebra

The $v v$ metric coefficient in [F1] is $- 1 + \frac{2 M r - q^{2}}{Σ} = - \frac{r^{2} - 2 M r + q^{2} + a^{2} \cos^{2} θ}{Σ},$ which proves the stated norm. Since $Σ > 0$ , its zero set is the quadratic equation in the numerator; the quadratic formula and [F3] give the stated two real algebraic roots exactly when the discriminant is nonnegative. Retaining only $r > 0$ gives the stationary-limit set in the selected model. A zero lower root occurs exactly when $q = 0$ and $a \cos θ = 0$ , and is outside this model. The metric is independent of $v$ and invariant under rotations, so $L_{T} g = L_{Ψ} g = 0$ and [F2] makes both fields Killing. Moreover $g (T, T) \to - 1$ as $r \to \infty$ uniformly in angle, so $T$ is timelike on a nonempty large-radius open region and is stationary in the sense of [L1].

2.1F1F3step 1.1A1algebra

Write $r_{+} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ and $Σ_{H} = r_{+}^{2} + a^{2} \cos^{2} θ$ . Put $D = M^{2} - a^{2} - q^{2}$ , $u = \sqrt{D + a^{2} \sin^{2} θ}$ and $w = \sqrt{D}$ . For $a \neq 0$ and $\sin θ \neq 0$ , $(u - w) (u + w) = a^{2} \sin^{2} θ > 0$ and $u + w > 0$ , hence $u > w$ . Thus the outer stationary-limit radius $r_{E +} = M + u$ strictly exceeds $r_{+} = M + w$ . At the horizon, $2 M r_{+} = r_{+}^{2} + a^{2} + q^{2}$ , and substitution in step 1.1 yields $g (T, T) = a^{2} \sin^{2} θ / Σ_{H}$ . This is positive off the axis and zero on the axis, proving the stated local distinction.

3.1F2step 1.1algebra∎

Let $γ$ be an affine geodesic and let $K$ be either Killing field from step 1.1. Metric compatibility and the product rule give $\frac{d}{d λ} g (K, \dot{γ}) = g (\nabla_{\dot{γ}} K, \dot{γ}) + g (K, \nabla_{\dot{γ}} \dot{γ}) .$ The second term is zero by the affine geodesic equation. The first is zero because the Killing equation in [F2] says the symmetric part of $\nabla K$ vanishes, while $\dot{γ} \otimes \dot{γ}$ is symmetric. The derivative therefore vanishes and both contractions are constant on the geodesic interval.

Remarks

The current terminology carrier supplies the stated ergoregion and stationary-limit definitions, with zero-radius algebraic roots excluded from this model. The formulas and Killing identities are proved for the specified fields; no event-horizon classification or observation is asserted.

Specified time Killing stationary-limit set and geodesic constants

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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