Scalar charge bounds for extremal black hole formation

Berend Schneider

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arXiv · 2608.23355

Scalar charge bounds for extremal black hole formation

Berend Schneider

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Abstract

I prove that a collapsing charged scalar field with scalar charge $\mathfrak{e}$ that forms an exactly extremal Reissner--Nordström black hole à la Kehle--Unger with radius $r_+$ must satisfy $|\mathfrak{e}|r_+ > \frac{1}{3}$. In contrast, I also show that no such bound exists for subextremal collapse: a scalar field with an arbitrary $\mathfrak{e} \ne 0$ can form a subextremal black hole with an arbitrary (subextremal) charge-to-mass ratio. Complementing the extremal bound, I prove that a Schwarzschild black hole can become extremal if $|\mathfrak{e}|r_+ > \sqrt{\frac{3 + \sqrt{33}}{3}}$. The fact that $|\mathfrak{e}|r_+$ can be taken to be of order unity could be considered evidence that the third law of black hole mechanics in vacuum is false.

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Berend Schneider. 2026-08-24. Scalar charge bounds for extremal black hole formation. https://arxiv.org/abs/2608.23355

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