The five-dimensional angle metric of spacetime is defined, and its connection with the conformal (angle-preserving) group C of transformations of spacetime explained. This is an application to physics of the "sphere geometry" developed in the last century by Liouville, F. Klein, Möbius et al. The extra degree of freedom λ plays several observable roles in solutions of the field equations of the theory (which are uniquely fixed by C-invariance and gauge-invariance under the assumed internal symmetries). In the solution for a gauge boson with arbitrarily moving point source, λ appears as a microscopic "parameter" which enforces a nonzero minimum time lag in causal signal propagation. We show how this enables a nonsingular self-interaction to be defined in classical particle motion having the correct properties. There is the correct radiation-reaction term, but unphysical features of the four-dimensional theory: third order motion equations, runaway solutions, infinite "electromagnetic" mass, etc. are avoided. In free field wave function solutions λ is seen to be conjugate to mass (just as r is to p and t is to E) and provides a mass operator.