  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. The presence of the null
cone adds new elements to this analogy, particularly regarding the smooth
foliation of the asymptotic boundary.
