Chapter 2. Some Geometrical Aspects of Space-Time and Relativity


Rolf Dahm
Beratung für Informationssysteme und Systemintegration, Mainz, Germany

Part of the book: Future Relativity, Gravitation, Cosmology


In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski’s approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein’s 1910 paper on the geometry of the Lorentz group. To include microscopic representations, we discuss few aspects of Wigner’s and Weinberg’s ’boost’ approach to describe ’any spin’ with respect to its reductive Lie algebra and coset theory, and we relate the physical identification to objects in P 5 based on the case (1, 0) ⊕ (0, 1) of the electromagnetic field. So instead of following this – in some aspects – special and misleading ’old’ representation theory, based on 4-vector calculus and tensors, we provide and use an alternative representation based on line geometry which – besides comprising known representation theory – is capable of both describing (classical) projective geometry of 3-space as well as it yields spin matrices and the classical Lie transfer. In addition, this geometry is capable of providing a more general route to known Lie symmetries, especially of the su(2)⊕i su(2) Lie algebra of special relativity, as well as it comprises gauge theories and affine geometry. Thus it serves as foundation for a future understanding of more general representation theory of relativity based, however, on roots known from classical projective geometry and P 5 . As an application, we discuss Lorentz transformations in point space in terms of line and Complex geometry, where we can identify them as a subset of automorphisms of the Plücker-Klein quadric M2 4 of P 5 . In addition, this description provides an identification as a special, but singular parameterization of the tetrahedral Complex, too. As such, we propose to generalize and supersede the usual rep theory of relativity by an embedding into the general geometry of P 5 , and the use of appropriate concepts of projective and algebraic geometry in Plücker’s sense by switching geometrical base elements and using transfer principles.

Keywords: relativity, unification, quantum field theory, Dirac theory, Lie algebras, Lie groups, geometry, projective geometry, line geometry, line Complex, Complex geometry, congruences, null systems


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