Chemical algebra. VII: ImproperG-weighted metrics of non-compact groups: Lorentz group in the Minkowski space

Abstract

The principles and the generalized equation of chemical algebra is extended to a Minkowskian substrateE endowed with its improper non-definite-positive metric, where the non-compact 6-parameter groupG of the Lorentz transformations operates. Given a map μ_u,u(g) = μ(gu)m(g) onG, a “line element”ds ² is formulated at each point marked by a vectoru. Assuming “μ = 1” and “m(g) :≠ 0 ⇒g is a pure Lorentz transformation (without a spatial rotation)”, the isotropic hypothesis (m depends on a single parameter out of three inG) is first studied. In general,ds ² does not define a Riemannian manifold unless one additional condition onm is imposed. Several relationships are established which are useful for the calculation of the metric tensor and the curvature tensor.