Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as ‘curvature/signature (in)dependent trigonometry’ and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an ‘absolute trigonometry’, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton–Hooke and galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are ‘universality’ and ‘(self-)duality’: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical (riemannian) spaces also has a version for the remaining six ‘spacetimes’; in most cases these equations are new. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry; this can be successfully applied for other rank-one spaces as well (e.g. the complex type, as the quantum space of states), and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.