Homogeneity, Isotropy, and Determinism Force a Quadratic Spacetime Interval: A Derivation of Relativity Without Light
Deon Nicholas · arXiv preprint (math-ph, gr-qc, math.DG, math.MG) · 2026
We show that smoothness, homogeneity, isotropy, and the determinism of inertial motion force the invariant interval governing the geometry of spacetime to reduce to a quadratic form, without presupposing the existence of light or electromagnetic phenomena. We formalize these principles as axioms about an invariant interval function and prove that smoothness and homogeneity compel it to be homogeneous of degree p > 0; determinism requires geodesics to be straight lines; and isotropy constrains the function to the form D(v) = C(v^T S v)^(p/2) for a nondegenerate symmetric matrix S. When S is indefinite, p must equal 2. Only powers of nondegenerate quadratic forms are admissible invariant intervals, encompassing Euclidean, Minkowski, and ultrahyperbolic geometries.