On the comoving distance as an arc-length in four dimensions

Abstract

The inner product provides a conceptually and algorithmically simple method for calculating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The key to this is that just as a distance between two points `on' the surface of the ordinary 2-sphere ${\cal S}^2$ is simply an arc-length (angle multiplied by radius) in ordinary Euclidean 3-space ${\cal E}^3$, the distance between two points `on' a 3-sphere ${\cal S}^3$ (a 3-hyperboloid ${\cal H}^3$) is simply an `arc-length' in Euclidean 4-space ${\cal E}^4$ (Minkowski 4-space ${\cal M}^4$), i.e. an `hyper-angle' multiplied by the curvature radius of the 3-sphere (3-hyperboloid).