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Fixed-coordinate curve is spacelike in the rotating ergoregion

Physical scope

Conditional demonstration that a fixed-angular-coordinate stationary curve is spacelike in a chosen exterior region of the rotating KN model. It supplies no energy-extraction argument, maximum-extension result, or observation and does not rule out timelike trajectories with angular motion.

Statement

Take the zero-magnetic-flux KN model with SI mass, angular-momentum and electric-charge inputs $m_{*} > 0, J, Q_{e}$ , mapped by SI and geometric solution parameter identifications to lengths $M, a, q$ . Assume $M > 0$ , $a \neq 0$ , and $M^{2} \geq a^{2} + q^{2} > 0$ . Use the selected ingoing product and its future orientation from Regular ingoing KN metric, field and sphere axes, and write $N$ for that product's designated smooth future timelike field. Let $r_{+} = M + \sqrt{M^{2} - a^{2} - q^{2}}, r_{E +} = M + \sqrt{M^{2} - q^{2}} .$ At the equator, choose any $r$ with $r_{+} < r < r_{E +}$ and fixed azimuth $ψ_{0}$ , and set $ω_{0} = (\cos ψ_{0}, \sin ψ_{0}, 0) \in S^{2}$ . The candidate fixed-coordinate curve is the smooth map $γ : R \to M_{i n}, v \mapsto (v, r, ω_{0}) .$ Its tangent is $T = \partial_{v} \neq 0$ and, by Specified time Killing stationary-limit set and geodesic constants, $g (T, T) > 0$ , so it is a regular spacelike curve. The exact definition Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold makes a future timelike worldline a $C^{1}$ map $η : I \to M_{i n}$ on an interval $I$ with nonempty interior whose tangent satisfies $g (η^{'}, η^{'}) < 0$ and $g (η^{'}, N) < 0$ at every parameter value; its tangent is therefore nonzero. Thus $γ$ is not a worldline of a massive observer and cannot represent one at rest in these stationary coordinates. For ideal freely falling massive test bodies, Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework separately adopts a future timelike geodesic worldline. The ergoregion here means the positive-norm set of the specified stationary Killing field, as defined in Causal solutions, Killing horizons, escape and metric extensions. Curves with angular motion require a separate norm calculation; no energy-extraction or empirical claim follows.

Facts & Assumptions

Given: The pure-electric KN SI inputs, $M > 0$ , $a \neq 0$ , $M^{2} \geq a^{2} + q^{2} > 0$ , the selected ingoing model, and the adopted massive-test-body interpretation.

[F1]

SI and geometric solution parameter identifications: $M = G_{N} m_{*} / c^{2}$ , $a = J / (m_{*} c)$ , $q_{e} = \sqrt{G_{N} / (4 π ε_{0} c^{4})} Q_{e}$ , and $ε_{0} μ_{0} c^{2} = 1$ ; zero magnetic flux makes the KN seed parameter $q = q_{e}$ .

[F2]

Verified rotating electric KN physical model: on the stated electric KN domains with $Δ \neq 0$ , the SI-rescaled datum is a conditional source-free equation-based model solution; its field-equation proof is draft and not independently accepted.

[F3]

Regular ingoing KN metric, field and sphere axes: $M_{i n} = R_{v} \times (0, \infty)_{r} \times S^{2}$ carries the smooth selected ingoing metric and future timelike field $N = \partial_{v} - \frac{1}{2} (1 + f_{0}^{2}) \partial_{r}$ , where $f_{0} = 1 - (2 M r - q^{2}) / Σ$ ; $T = \partial_{v}$ is defined on the full product. Its field-equation continuation remains conditional on the current verification draft.

[L1]

Specified time Killing stationary-limit set and geodesic constants: $g (T, T) = - (r^{2} - 2 M r + q^{2} + a^{2} \cos^{2} θ) / Σ$ ; at the equator the outer stationary-limit radius is $r_{E +} = M + \sqrt{M^{2} - q^{2}}$ , it strictly exceeds $r_{+}$ when $a \neq 0$ , and the norm is positive between the roots. Its proof remains draft and is used conditionally.

[F4]

Causal solutions, Killing horizons, escape and metric extensions: for a specified stationary Killing field, the ergoregion is its positive-norm set; this definition alone does not identify a null horizon.

[A1]

Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework adopts a future timelike geodesic worldline for an ideal freely falling massive test body; it does not derive that physical rule from the KN equations. The exact mathematical worldline condition from Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold is a $C^{1}$ map on an interval with nonempty interior whose tangent satisfies $g (η^{'}, η^{'}) < 0$ and $g (η^{'}, N) < 0$ for the selected future timelike field $N$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the inherited geometric metric convention uses only ${A C}_{ω}$ . The finite choice of a radius between two explicit real roots uses no family selection or full AC.

Proof

1.1givenF1F3algebra

From $a \neq 0$ and $M^{2} \geq a^{2} + q^{2}$ , we have $M^{2} - q^{2} \geq a^{2} > 0$ . Therefore $r_{E +} - r_{+} = \sqrt{M^{2} - q^{2}} - \sqrt{M^{2} - a^{2} - q^{2}} > 0,$ since the first radicand exceeds the second and both square roots are nonnegative. Both radii are positive, so the stated open interval is nonempty. Retain the arbitrary radius $r$ of the Statement with $r_{+} < r < r_{E +}$ . In particular $r > 0$ , which places every point of $γ$ in the selected product.

2.1L1step 1.1algebra

On the equator, [L1] gives $g (T, T) = - (r^{2} - 2 M r + q^{2}) / r^{2}$ . The numerator factors as $(r - r_{E -}) (r - r_{E +})$ , where $r_{E -} = M - \sqrt{M^{2} - q^{2}} < M \leq r_{+}$ ; the chosen radius satisfies $r_{E -} < r < r_{E +}$ , so the numerator is strictly negative. Thus $g (T, T) > 0$ . The fixed-coordinate curve has derivative $T$ and is spacelike at every point.

3.1A1step 2.1

By the future timelike worldline condition in [A1], a curve representing a massive observer must have a future timelike tangent. Since the curve in step 2.1 has a spacelike tangent, it cannot be such an observer at rest in the chosen stationary coordinates. This conclusion concerns only tangent $T$ ; an angularly moving curve has a different tangent and needs its own causal-norm check.

4.1F2F3F4A1step 3.1A2∎

The result uses the conditional KN solution and ergoregion geometry only as specified in [F2]-[L1]. It establishes no extraction process, horizon maximality, observation, or measured property.

Fixed-coordinate curve is spacelike in the rotating ergoregion

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources