Chemical algebra. VII: ImproperG-weighted metrics of non-compact groups: Lorentz group in the Minkowski space |
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Authors: | Remi Chauvin |
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Institution: | (1) Laboratoire de Chimie de Coordination du C. N. R. S., Unité 8241, liée par conventions à l'Université Paul Sabatier et à l'Institut National Polytechnique, 205 Route de Narbonne, 31077 Toulouse Cedex, France |
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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 u,u(g) = (gu)m(g) onG, a line element ds
2 is formulated at each point marked by a vectoru. Assuming ![ldquo](/content/gx2478352811p138/xxlarge8220.gif) = 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. |
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