CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Equatorial rotating ergosurface differs from horizon

Statement refuted

The outer stationary-limit set of the specified asymptotic time-translation field $T = \partial_{v}$ is always the same hypersurface as the outer KN Killing horizon. The example below refutes this universal identification in the selected ingoing model.

Facts & Assumptions

Given: The selected ingoing KN model with $M = 2 m$ , $a = 1 m$ , $q = 0 m$ , evaluated at the equator $θ = π / 2$ .

[F1]

Specified time Killing stationary-limit set and geodesic constants: the current A-page proposition computes $g (T, T)$ and the equatorial stationary-limit radius $r_{E} = M + \sqrt{M^{2} - q^{2}}$ ; its terminology is fixed by the current vocabulary Definition.

[F2]

KN Killing generators, surface gravity, area and potential: the current A-page proposition gives the outer root $r_{H} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ and its null Killing generator; its proof verifies the current exact horizon vocabulary.

[F3]

Explicit rational and real power regimes: the nonnegative square roots in this example are uniquely defined.

[A1]

The concrete choice satisfies $a \neq 0$ and $M^{2} - a^{2} - q^{2} = 3 m^{2} > 0$ , so both displayed outer-root expressions are real; the equatorial angle is $θ = π / 2$ .

Counterexample

1.1F1F3A1algebra

At $θ = π / 2$ , the norm in [F1] vanishes when $r^{2} - 2 M r + q^{2} = 0$ . Since $M^{2} - q^{2} = (M^{2} - a^{2} - q^{2}) + a^{2} > 0$ , its outer root is the real value $r_{E} = M + \sqrt{M^{2} - q^{2}}$ .

2.1F2F3A1step 1.1algebra

The outer horizon radius in [F2] is $r_{H} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ . Put $u = \sqrt{M^{2} - q^{2}}$ and $w = \sqrt{M^{2} - a^{2} - q^{2}}$ . Then $u, w \geq 0$ and $(u - w) (u + w) = a^{2} > 0$ . Since $u + w > 0$ , we have $u > w$ , hence $r_{E} > r_{H}$ . Therefore these two level sets are distinct at the equator, refuting the universal claim.

3.1F1F2A1algebra∎

At the equatorial horizon, [F1] gives $g (T, T) = a^{2} / r_{H}^{2} > 0$ , whereas [F2] gives a null horizon generator. Thus the stationary field itself is spacelike there and is not the horizon generator. On the rotation axis $\sin θ = 0$ the displayed outer radii coincide; this is the boundary case and does not restore the universal claim. The deductions remain conditional on [F1-F2]'s stated interfaces.

Remarks

This is a mathematical comparison of two level sets for specified fields. It asserts no observation and no global event-horizon classification.

Equatorial rotating ergosurface differs from horizon

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources