CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Subextremal RN surface gravity has no uniform positive lower bound

Statement refuted

Fix a mass length $m > 0$ . For each length parameter $q_{e} \in R$ with $0 < q_{e}^{2} < m^{2}$ , take the electric RN family with mass parameter $M = m$ and magnetic parameter $q_{m} = 0$ , and define $Q^{2} : = q_{e}^{2}$ , $r_{\pm} : = m \pm \sqrt{m^{2} - Q^{2}}$ , and $κ_{+} (q_{e}) : = (r_{+} - r_{-}) / (2 r_{+}^{2})$ . The false claim is that there exists a constant $c > 0$ , with units of inverse length, such that $κ_{+} (q_{e}) \geq c$ for every such $q_{e}$ .

Statement references

RN roots, Killing horizons and surface gravity

Facts & Assumptions

Given: A fixed $m > 0$ and the subextremal electric RN family with $q_{m} = 0$ and $0 < q_{e}^{2} < m^{2}$ .

[F1]

Dyonic Reissner-Nordström metric and field domains: $Q^{2} = q_{e}^{2} + q_{m}^{2}$ and the static RN metric uses $f (r) = 1 - 2 m / r + Q^{2} / r^{2}$ .

[F2]

RN roots, Killing horizons and surface gravity: when $m > 0$ and $Q^{2} > 0$ , the outer surface gravity is $κ_{+} = (r_{+} - r_{-}) / (2 r_{+}^{2})$ with $r_{\pm} = m \pm \sqrt{m^{2} - Q^{2}}$ .

[F3]

Explicit rational and real power regimes: for $a > 0$ , $\sqrt{a}$ is the unique positive real solution of $s^{2} = a$ .

[F4]

Dedekind-complete ordered real field: $R$ is a complete ordered field.

[F5]

For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ : in any complete ordered field, for every $ϵ > 0$ there is a natural $N \geq 1$ with $1 / N < ϵ$ .

[A1]

The fixed mass parameter satisfies $m > 0$ ; all mass and charge parameters have length units and $κ_{+}$ has inverse-length units.

Counterexample

1.1givenA1F1F2F3algebra

For each integer $n \geq 2$ , choose $q_{e, n} = m \sqrt{1 - n^{- 2}}$ and $q_{m, n} = 0$ . By [F3], $Q_{n}^{2} = q_{e, n}^{2} = m^{2} (1 - n^{- 2})$ , so $0 < Q_{n}^{2} < m^{2}$ and $\sqrt{m^{2} - Q_{n}^{2}} = m / n$ . Applying [F2] gives $r_{+, n} = m (1 + 1 / n)$ , $r_{-, n} = m (1 - 1 / n)$ , and $κ_{+, n} = \frac{r_{+, n} - r_{-, n}}{2 r_{+, n}^{2}} = \frac{n}{m (n + 1)^{2}}, 0 < κ_{+, n} < \frac{1}{m n} .$

2.1givenA1F4F5step 1.1algebra∎

Let $c > 0$ be any proposed inverse-length lower bound. Since $m c > 0$ , [F4]–[F5] allow a natural $N \geq 1$ with $1 / N < m c$ . For every integer $n \geq 2 N$ , $1 / n \leq 1 / (2 N) < m c$ , so step 1.1 gives $0 < κ_{+, n} < 1 / (m n) < c$ . Thus the sequence $κ_{+, n}$ tends to zero through subextremal RN data, contradicting any positive uniform lower bound. Every member of the sequence remains subextremal; only its limit reaches $Q^{2} = m^{2}$ .

Remarks

The counterexample concerns the unscaled outer surface gravity as the charge approaches extremality at fixed mass. It makes no claim about the scaled throat coordinates or about uniform estimates for waves.

Depends on

Dependency tree · 2 levels

Sources

Max Weissenbacher, Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes, Arch. Rational Mech. Anal. 248, 118 (2024)