PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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RN roots, Killing horizons and surface gravity

Statement

Let $M > 0$ , $Q^{2} = q_{e}^{2} + q_{m}^{2} > 0$ , and use $f (r) = 1 - 2 M / r + Q^{2} / r^{2}$ on the selected ingoing RN model. If $M^{2} \geq Q^{2}$ , the positive roots $r_{\pm} = M \pm \sqrt{M^{2} - Q^{2}}$ are null constant- $r$ hypersurfaces. For the Killing field $ξ = \partial_{v}$ normalized by the displayed coordinate for which $g (ξ, ξ) \to - 1$ as $r \to \infty$ , each is a Killing horizon in the local sense of Causal solutions, Killing horizons, escape and metric extensions, with signed surface gravity $κ_{\pm} = \frac{f^{'} (r_{\pm})}{2} = \frac{r_{\pm} - r_{\mp}}{2 r_{\pm}^{2}} .$ At extremality $M^{2} = Q^{2}$ , the coincident root is $r_{H} = M$ and $κ_{H} = 0$ . If $M^{2} < Q^{2}$ , $f$ has no positive root and no constant- $r$ null hypersurface of this form. These are local properties of the selected metric; they do not classify event horizons in a maximal spacetime.

Facts & Assumptions

Given: $M > 0$ , $Q^{2} > 0$ , the selected ingoing RN metric, and the exact current meanings of null hypersurface, Killing horizon and surface gravity.

[F1]

Selected ingoing RN extension and time orientation: $g = - f d v^{2} + 2 d v d r + r^{2} γ$ , $g^{- 1} (d r, d r) = g^{r r} = f$ , and $ξ = \partial_{v}$ is a global product field.

[F2]

Explicit rational and real power regimes: the square root of a nonnegative real is its unique nonnegative real root, with $\sqrt{0} = 0$ . This current batch-2 definition is used only for the exact root convention.

[F3]

Killing fields of a Lorentzian metric: a smooth vector field $ξ$ is Killing when $L_{ξ} g = 0$ .

[L1]

Causal solutions, Killing horizons, escape and metric extensions, Definition (b): a level with nonzero conormal is null exactly when that conormal has zero metric square. A Killing horizon has a nonempty dense nonzero-generator portion with tangent null normal Killing field, and its signed surface gravity is fixed by the ambient covector equation $d (g (ξ, ξ)) = - 2 κ_{H} ξ^{♭}$ .

[A1]

All parameters are real and square roots are nonnegative. The parameter and radius units are metres; $κ$ has units of inverse length.

Proof

1.1givenF2A1algebra

Multiplying $f$ by $r^{2}$ gives $r^{2} f = r^{2} - 2 M r + Q^{2} .$ If $M^{2} \geq Q^{2}$ , the quadratic formula gives the stated roots. Because $0 < Q^{2} \leq M^{2}$ and $M > 0$ , the nonnegative root $s = \sqrt{M^{2} - Q^{2}}$ satisfies $s < M$ : otherwise $s^{2} \geq M^{2}$ , contradicting $s^{2} = M^{2} - Q^{2} < M^{2}$ . Thus both $M + s$ and $M - s$ are positive. If $M^{2} < Q^{2}$ , then $r^{2} f = (r - M)^{2} + (Q^{2} - M^{2}) > 0$ for every $r > 0$ , so there is no positive root.

2.1F1F3L1step 1.1algebra

The inverse $v, r$ block in [F1] gives $g^{- 1} (d r, d r) = g^{r r} = f$ . Thus a positive root gives a null level hypersurface. The metric coefficients are independent of $v$ , hence $L_{ξ} g = 0$ and $ξ$ is Killing by [F3]. At a root, $ξ^{♭} = - f d v + d r = d r$ and $d r (ξ) = 0$ , so $ξ$ is null, tangent and normal to that level. The product field $\partial_{v}$ is nonzero everywhere on the level, so its nonzero-generator part is the whole level, nonempty and dense. This verifies the Killing-horizon conditions in [L1].

3.1L1step 2.1algebra

On the root set, $ξ^{♭} = d r$ and $d (g (ξ, ξ)) = d (- f) = - f^{'} (r_{H}) d r .$ Comparing with the normalization in [L1] gives $κ_{H} = f^{'} (r_{H}) / 2$ . Factoring $f = (r - r_{+}) (r - r_{-}) / r^{2}$ and differentiating at either root gives $f^{'} (r_{\pm}) = \frac{r_{\pm} - r_{\mp}}{r_{\pm}^{2}},$ which proves the stated formula, including its negative sign at the inner simple root. If $M^{2} = Q^{2}$ , then $r_{+} = r_{-} = M$ and the formula gives zero.

4.1F1L1step 1.1algebra∎

If $M^{2} < Q^{2}$ , step 1.1 gives $f > 0$ for every $r > 0$ . Therefore $g^{- 1} (d r, d r) > 0$ and no positive constant- $r$ level is null. This rules out only the radial-root horizons under discussion, not other global horizon notions.

Remarks

The horizon and signed surface-gravity terms use the current exact vocabulary Definition. The proof establishes their conditions for this selected model, without an event-horizon identification or an independent acceptance claim.

RN roots, Killing horizons and surface gravity

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources