Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner

TL;DR

This paper explores the appearance of specific ratios in black hole physics, relating them to complex analysis, quantum interpretations, and graph theory, revealing interdisciplinary connections and potential implications for quantum information dynamics.

Contribution

It establishes novel links between black hole ratios, Kasner's conjecture, complex analysis, and graph theory, suggesting new perspectives on black hole entropy and quantum information.

Findings

Ratios relate to black hole charge and rotation parameters.

Connections between Kasner's conjecture and black hole entropy quantization.

Mathematical similarities with complex interpolation and graph theory.

Abstract

The ratios $\sqrt{8/9}=2\sqrt{2}/3\approx 0.9428$ and $\sqrt{3}/2 \approx 0.866$ appear in various contexts of black hole physics, as values of the charge-to-mass ratio $Q/M$ or the rotation parameter $a/M$ for Reissner-Nordstr\"om and Kerr black holes, respectively. In this work, in the Reissner-Nordstr\"om case, I relate these ratios with the quantization of the horizon area, or equivalently of the entropy. Furthermore, these ratios are related to a century-old work of Kasner, in which he conjectured that certain sequences arising from complex analysis may have a quantum interpretation. These numbers also appear in the case of Kerr black holes, but the explanation is not as straightforward. The Kasner ratio may also be relevant for understanding the random matrix and random graph approaches to black hole physics, such as fast scrambling of quantum information, via a bound related to Ramanujan graph. Intriguingly, some other pure mathematical problems in complex analysis, notably complex interpolation in the unit disk, appear to share some mathematical expressions with the black hole problem and thus also involve the Kasner ratio.