Axiomatization of Minkowski 3-space for synthetic treatment of the hyperbolic case

Develop an axiomatization of Minkowski 3-space M equipped with the indefinite quadratic form x^2 + y^2 − z^2 that is adequate to carry out a synthetic proof of the diagonal Pythagorean theorem for the hyperboloid model of the hyperbolic plane, analogous to the Euclidean-sphere embedding approach that uses circle geometry and disk areas.

Background

The paper provides a geometric, embedding-based proof of a Pythagorean theorem formulation that holds in Euclidean, spherical, and hyperbolic geometries. The spherical case leverages the sphere embedded in Euclidean 3-space and the bread-crust theorem to define areas of circular disks. The hyperbolic case uses the hyperboloid embedded in Minkowski 3-space and analogous geometric properties.

In discussing axiomatic foundations, the author notes that Hilbert’s axioms do not handle areas of circles, which prevents a purely axiomatic proof in that framework. For the spherical case, the embedding and bread-crust theorem provide a synthetic route to area. The author suggests that a similar synthetic treatment for the hyperbolic case would be possible given a suitable axiomatization of Minkowski 3-space, but explicitly states that they do not know of such an axiomatization.