The emergence of a Kaluza-Klein microgenometry from the invariants of optimally Euclidean Lorentzian spaces

“The predominantly inductive methods appropriate to the youth of science are giving place to tentative deduction. Such a theoretical structure needs to be very thoroughly elaborated before it can lead to conclusions which can be compared with experience. Here too the observed fact is undoubtedly the supreme arbiter; but it cannot pronounce sentence until the wide chasm separating the axioms from their verifiable consequences has been bridged by some intense, hard thinking. …there is no other way to the goal.” A. Einstein

Abstract

It is shown that relativistic spacetimes can be viewed as Finslerian spaces endowed with a positive definite distance (ω⁰, mod ωⁱ) rather than as pariah, pseudo-Riemannian spaces. Since the pursuit of better implementations of “Euclidicity in the small” advocates absolute parallelism, teleparallel nonlinear Euclidean (i.e., Finslerian) connections are scrutinized.

The fact that (ω^μ, ω₀ ⁱ) is the set of horizontal fundamental 1-forms in the Finslerian fibration implies that it can be used in principle for obtainingcompatible new structures. If the connection is teleparallel, a Kaluza-Klein space (KKS) indeed emerges from (ω^μ, ω₀ ⁱ), endowed ab initio with intertwined tangent and cotangent Clifford algebras. A deeper level of Kähler calculus, i.e., the language of Dirac equations, thus emerges. This makes the existance of an intimate relationship between classical differential geometry and quantum theory become ever more plausible. The issue of a geometric canonical Dirac equation is also raised.