On Lorentz-Minkowski geometry in real inner product spaces

Abstract

Let A" be a real inner product space of finite or infinite dimension ^2, and let ρ φ Ο be a fixed real number. The following results will be presented in this note.A. A surjective mapping σ : X -» X preserving Lorentz-Minkowski distances 0 and Q in one direction must be a Lorentz transformation.B. The causal automorphisms of X, dim X ^ 3, are exactly the products δ λ, where λ is an orthochronous Lorentz transformation and δ a dilatation jc -> αχ, R 9 a > 0.C. If Q > 0, there exist A" and an injective σ : X -> X preserving Lorentz-Minkowski distance ρ, such that σ is not a Lorentz transformation. This result can be extended, mutatis mutandis, to Euclidean and Hyperbolic Geometry. It could be possible that Theorem 1 also holds true in the infinite-dimensional case provided that ρ < 0. However, a proof, if it exists, is not yet known. Result C shows that Theorem 1 cannot be extended to the infinite-dimensional case if ρ > 0, not even in the injective case.

Möbius transformations and the configuration space of a Hilbert snake

“…All these results can be generalized to the context of a Hilbert space (cf. [3] and [11] for instance). Therefore the group M(S H ) of Möbius transformations of the unit sphere S H of a Hilbert space H is also isomorphic to some subgroup SO0(H, 1) of the group O(H, 1) of linear Lorentz transformations of a Lorentz structure on H = R ⊕ H (for more details see Subsection 2.2).…”

“…On the other hand, on H, we consider the hyperbolic distance δ characterized by (cf [3]) cosh δ(x, y) = 1 + |x| 2 1 + |y| 2 − < x, y > and δ(x, y) ≥ 0. …”

“…If H is finite dimensional, each isometry is a bijection, but it is no longer true in general, if H is infinite dimensional (see [3]). …”

“…On the other hand, we consider the hyperboloid H1 = {u = (s, x) ∈ H : |u| The link between the Lorentz transformations of H and the hyperbolic transformations of H is given by the following result (cf [3]). …”

“…The following result is classical in the finite dimensional case and in the infinite dimensional case it is more or less included in [3] or [11] Proposition 2.2.1. Let A ∈ O(H, 1), there exists, v ∈ H with v = 0, such that A has the following decomposition:…”