Scalar charge bounds for extremal black hole formation

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.

arXiv comment15 pages, 1 figure