Fermion Absorption Cross Section and Topology of Spherically Symmetric Black Holes

TL;DR

This paper establishes a topological relation between the low-energy fermion absorption cross section of spherically symmetric black holes and their Euler characteristic, linking geometry, topology, and black hole physics.

Contribution

It introduces a novel relation connecting fermion absorption cross section to the Euler characteristic and employs topological methods to analyze black hole absorption properties.

Findings

Absorption cross section is related to the Euler characteristic of the black hole.

Topological numbers like Hopf indices and Brouwer degrees characterize absorption features.

The absorption cross section density is determined by singularities of the timelike Killing vector.

Abstract

In 1997, Liberati and Pollifrone in Phys. Rev. D56 (1997) 6458 (hep-th/9708014) achieved a new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by $S=\chi A/8$. In this work we present a relation between the low-energy absorption cross section for minimally coupled fermions and the Euler characteristic of (3+1)-dimensional spherically symmetric black holes, i.e. $\sigma =\chi g_h^{-1}A$. Based on the relation, using the Gauss--Bonnet--Chern theorem and the $\phi$-mapping method, an absorption cross section density is introduced to describe the topology of the absorption cross section. It is shown that the absorption cross section and its density are determined by the singularities of the timelike Killing vector field of the spacetime and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally.