Homogeneity, Isotropy, and Determinism Force a Quadratic Spacetime Interval: A Derivation of Relativity Without Light

Abstract

We show that a few physical principles -- 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. Formalizing these as axioms about an "invariant interval" function $D:\mathbb{R}^n\to\mathbb{R}$ ($n\geq 3$), we find that smoothness and homogeneity force $D$ to be homogeneous of degree $p > 0$; determinism -- that an inertial worldline be uniquely fixed by its initial point and direction -- makes its geodesics straight lines; and isotropy -- that the isometry group act transitively on each level set, with the stabilizer of a reference direction reversing every transverse direction -- forces $D(v) = C\,(v^T S v)^{p/2}$ for a nondegenerate symmetric matrix $S$ and $p > 0$, with $p = 2$ (so that $D$ is exactly quadratic) when $S$ is indefinite. Thus the only admissible invariant intervals are powers of nondegenerate quadratic forms. The signature of $S$ is otherwise free: the definite case is Euclidean geometry and the indefinite case includes both Minkowski and ultrahyperbolic geometries, the two cases distinguished by the absence or presence of a null cone.