Abstract
The Lorentzian distance formula, conjectured several years ago by Parfionov and Zapatrin, has been recently proved by the second author. In this work we focus on the derivation of an equivalent expression in terms of the geometry of 2-spinors by using a partly original approach due to the first author. Our calculations clearly show the independence of the algebraic distance formula of the observer.
Highlights
The distance between any two points of a Riemannian manifold M can be expressed through Connes’ formula [1, 2]: for any x, y ∈ M d(x, y) = sup f (y) − f (x), ∇f ≤1 (R1)where the supremum is over the continuously differentiable 1-Lipschitz functions
Taking different notations and conventions into account—in particular opposite metric signatures—we can rewrite the Lorentzian distance in the spectral triple formulation as presented by Franco and Eckstein as dLor(x, y) = inf f ∈A
Through the elaboration of the right-hand side, that this equation is equivalent to the Lorentzian distance formula
Summary
We consider the case when dim U = 2 and the 1-dimensional space ∧2U (not U ) is endowed with a positive Hermitian metric. This yields, up to a phase factor, a unique normalized ‘complex symplectic’ tensor ǫ ∈ ∧2U ⋆. Isotropic elements in H are of the form ±u ⊗ uwith u ∈ U , so that there is a natural way of fixing a time orientation in H. can be identified with the space of Dirac spinors, that may be represented as ψ ≡ (u, λ). It should be noticed that we assume no positive Hermitian metric on U Such object is a timelike, future-oriented element h ∈ H∗, so that its assignment is essentially equivalent to fixing an observer. W → M , associated with the choice of a positive Hermitian metric h on W . The Dirac operator ∇/ ≡ −i θaλ γλ∇a ≡ −i γa∇a associated with the spin connection
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