Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner
Yen Chin Ong
TL;DR
The paper investigates whether the Kasner sequence, originating from the arc-to-chord ratio of certain complex curves, manifests in black hole physics through the charge-to-mass ratio $Q/M$ for Reissner-Nordström and the rotation-to-mass ratio $a/M$ for Kerr black holes. By pairing RN area quantization with the Kasner ratio, it shows that the even subsequence $\frac{2\sqrt{l+1}}{l+2}$ emerges naturally as admissible $Q/M$ values, with the first term matching the quantum dominance bound $\frac{2\sqrt{2}}{3}$, while the Kerr case aligns differently, linking $a/M$ to classical geometric constraints such as ISCO/ergosphere relations and embedding conditions. The work further connects these ratios to horizon-radius relations, inversion symmetry, and a broader web of relationships to random graphs, Ramanujan graphs, and complex interpolation problems, suggesting a deeper mathematical structure behind quantum aspects of black holes. Overall, the paper proposes a revised Kasner-based conjecture: the even Kasner subsequence corresponds to quantized RN black-hole parameters, while odd terms are only approachable, inviting further quantum-gravity derivations and cross-disciplinary links to graph theory and analytic function 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-Nordström and Kerr black holes, respectively. In this work, in the Reissner-Nordström 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.