Interacting vector fields in relativity without relativity

Abstract

Barbour, Foster andÓ Murchadha have recently developed a new framework, called here the 3-space approach, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric-matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -and more -without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang-Mills fields minimally coupled to gravity.

Relativity without relativity

“…Since our approach exploits 3-geometry to the maximal extent possible but nothing else, we can say that Maxwellian theory is uniquely inherent in Riemannian 3-geometries. We have already noted that, within our scheme, the attempt to couple a single 3-vector field to scalar fields leads to the standard U(1) gauge coupling [12], and the attempt to let a collection of 3-vector fields interact among themselves leads to Yang-Mills gauge theory [13].…”

“…Thus we obtain classical gauge theory. A paper just completed by Edward Anderson and one of us [13] shows that in the BSW framework 3-vector fields can interact among themselves only as Yang-Mills fields minimally coupled to gravity.…”

The Definition of Mach’s Principle

“…Equally striking is the fact that the simplest theories of a single 3-vector field interacting with Machian geometry and of a set of 3-vector fields interacting with geometry and among themselves turn out to be Maxwell and Yang-Mills fields, respectively. These results, which show that all the well-established modern dynamical theories can be derived from Machian principles, are obtained in [36,39]. It should be said that some of the uniqueness claims made in [36] relied on unjustified tacit assumptions that ruled out more complicated possibilities, as is noted in the careful study of [40,41].…”

The Fundamental Importance of Discourse in Theoretical Physics

“…And indeed, this condition appears to be satisfied by the Yorke scaling and gravitational degrees of freedo m arguments of Barbour et al cited earlier. [5], [6], [7], [8], [10] However, Clarke also argues that because the Fried mann solutions contain a Cauchy surface, at least a part of our universe must be (geodesically) incomp lete. At this point Clarke makes a fundamental discourse error, but one which is easily fixed, and once fixed, allo ws his theoretical exposition to continue uninterrupted (p. 12):…”