Lorentzian angles and trigonometry including lightlike vectors

TL;DR

This paper introduces a new concept of Lorentzian angles that includes lightlike vectors, with applications in gravitational theories and Lorentzian geometry, and provides a proof of a Lorentzian Gauss-Bonnet theorem.

Contribution

It defines Lorentzian angles for null vectors and demonstrates their relevance in gravitational action and geometric theorems.

Findings

Defined Lorentzian angles for lightlike vectors

Applied angles to Regge-Calculus and gravitational action

Proved a Lorentzian Gauss-Bonnet theorem

Abstract

We define a concept of Lorentzian angle that works even when one or both of the directions involved is null (lightlike). Such angles play a role in Regge-Calculus, in the boundary- and corner- terms for the gravitational action, and in the Lorentzian Gauss-Bonnet theorem (for which we provide a proof).