Spaces with Torsion from Embedding, and the Special Role of Autoparallel Trajectories

As a contribution to the ongoing discussion of trajectories of spinless particles in spaces with torsion we show that the geometry of such spaces can be induced by embedding their curves in a euclidean space without torsion. Technically speaking, we define the tangent (velocity) space of the embedded space imposing non-holonomic constraints upon the tangent space of the embedding space. Parallel transport in the embedded space is determined as an induced parallel transport on the surface of constraints. Gauss’ principle of least constraint is used to show that autoparallels realize a constrained motion that has a minimal deviation from the free, unconstrained motion, this being a mathematical expression of the principle of inertia. In contrast, geodesics play no special role in the constrained dynamics, making them less likely candidates for particle trajectories. 1. On an affine manifold equipped with a metric, there exist two preferred connections compatible with the metric [1]. One is the Riemann connection defined only by the metric. In a coordinate basis in the tangent space of the manifold, the coefficients of this connection are Christoffel symbols Γ̄μνκ = gκλΓ̄νκ λ = 1 2 (gμν,κ + gμκ,ν − gνκ,μ) . (1) Here gμν are components of the metric tensor, and indices after a comma stand for the corresponding derivatives, Tμν...,λκ... = ∂λ∂κ · · ·Tμν.... By definition, the covariant derivative formed with this Riemann connection satisfies the metricity condition D̄μgνκ = ∂μgνκ − Γ̄νμ gλκ − Γ̄κμ gνλ = 0 . (2) Apart from Γ̄μν , there exists also a Cartan connection Γμν . It satisfies the same compatibility condition with the metric: Dμgνκ = ∂μgνκ − Γνμ gλκ − Γκμ gνλ = 0 . (3) 1 Email: [email protected]; [email protected]; URL: http://www.physik.fuberlin.de/ ̃kleinert. Phone/Fax: 0049/30/8383034 on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia; a DFG fellow.