General relativity, Lauricella's hypergeometric function $F_D$ and the theory of braids

TL;DR

This paper explores the connection between the solutions of equations in general relativity involving Lauricella's hypergeometric function and the mathematical theory of braids, revealing a topological link.

Contribution

It establishes a fundamental connection between the domain of Lauricella's hypergeometric function in relativity and the theory of braids, highlighting a novel topological aspect.

Findings

Topological properties of the domain ${\

Connection between general relativity solutions and braid theory

Abstract

The exact (closed form) solutions of the equations of motion in the theory of general relativity that describe motion of test particle and photon in Kerr and Kerr-(anti) de Sitter spacetimes all involve the multivariable hypergeometric function of Lauricella $F_D$: Kraniotis [Class. Quantum Grav. {\bf 21} 2004, 4743; Class. Quantum Grav. {\bf 22} 2005, 4391; Class. Quantum Grav. {\bf 24} 2007, 1775]. The domain of variables ${\cal D}_n$ of the corresponding function depends on the first integrals of motion associated with the isometries of the Kerr-(anti) de Sitter metric and Carter's constant $Q$ as well as on the cosmological constant $\Lambda$ and the Kerr (rotation) parameter. In this work we discuss the topological properties of the domain ${\cal D}_n$ and in particular its fundamental connection with the theory of braids. An intrinsic relationship of general relativity with the pure braids is established.