Hyperbolic times in Minkowski space

TL;DR

This paper explores hyperbolic time functions in Minkowski space, highlighting their geometric properties and natural coordinate systems, which are crucial for understanding relativity and wave equations.

Contribution

It provides an expository analysis of hyperbolic time functions, emphasizing their geometric significance and introducing hyperboloidal coordinates as natural in Lorentzian manifolds.

Findings

Hyperbolic time functions relate to asymptotically hyperbolic geometry in relativity.

Hyperboloidal coordinates are natural in Lorentzian manifolds, analogous to spherical coordinates in Riemannian geometry.

The paper offers a geometric perspective on hyperbolic times in Minkowski space.

Abstract

Time functions with asymptotically hyperbolic geometry play an increasingly important role in many areas of relativity, from computing black-hole perturbations to analyzing wave equations. Despite their significance, many of their properties remain underexplored. In this expository article, I discuss hyperbolic time functions by considering the hyperbola as the relativistic analog of a circle in two-dimensional Minkowski space and argue that suitably defined hyperboloidal coordinates are as natural in Lorentzian manifolds as spherical coordinates are in Riemannian manifolds.