Projective relativity

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

The synthesis of general relativity and quantum theory is embodied in the gauge principle that emerges as a natural feature of projective relativity and explains the unihcation of the electromagnetic and gravitational helds. A brief introduction to the concept of gauge invariance is provided in a second Appendix. 

Bazin M. Schiffer, (1965) Introduction to General Relativity, McGraw-Hill, NY. 

Bergmann, P.G. (1976) Introduction to the theory of relativity, Dover edition, NY, p.268. 

Gell-Mann, M. (1979) in D. Huff O. Prewett, The Nature of the Physical Universe 1976 Nobel Conference, 1979, New York, p.29. 

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In other studies, solanidane-induced teratogenicity was proposed to depend upon C-22 configuration, c/Fig, (10), and the teratogenicity of epimeric solanidanes was suggested to correlate with the direction in which the unshared pair of electrons on the nitrogen atom projects relative to the steroidal plane [S2]. Earlier structure - terata relations of plant steroidal alkaloids have been summarized [S3]. 

Attempts to accommodate the electromagnetic field in addition to the gravitational lead to a further generalization of the Riemannian geometry, for instance to the five-dimensional Kaluza metric or projective relativity of Veblen. 

Although Einstein must have been familiar with the concepts of projective relativity, developed by his colleagues Oswald Veblen, Banesh Hoffmann and... 

The purpose of projective relativity is to derive the equivalent of Einstein s field equations in homogeneous projective coordinates, which requires definition of projective scalars, vectors, displacements, connections and tensors in projective space. Such procedures are described in detail in the monograph. 

Applied to the potentials of the electromagnetic field the coordinate system is determined only to within an additive gradient, which is the well-known property of the vector potential of the Maxwell field. In common practice it is necessary to assume the gauge invariance, which appears naturally in projective relativity. 

In projective relativity the field equations contain, in addition to the gravitational and electromagnetic fields, also the relativistic wave equation of Schrodinger and, as shown by Hoffinann (1931), are consistent with Dirac s equation, although the correct projective form of the spin operator had clearly not been found. The problem of spin orientation presumably relates to the appearance of the extra term, beyond the four electromagnetic and ten gravitational potentials, in the field equations. It correlates with the time asymmetry of the magnetic field and spin. 

The isotropic lines of a projective identity mapping define the local complex Minkowski space of special relativity directly. By taking the circular points at infinity into account the global projective space of general projective relativity is obtained. No other topology reveals the transition from special to general relativity as such a simple consequence of curved space-time. 

Independent of our investigation Einstein and Mayer (Bibl. 1930,3) have published a "unified field theory", which led to essentially the same results as ours (compare Chap. VIII). Furthermore there recently appeared several works by J.A. Schouten and D. von Dantzig (Bibl. 1931, 7 1932, 3, 4, 8, 9 1933, 1) in which the projective relativity theory was treated in different forms. It is remarkable that - in a mathematical sense - all these theories seem to converge together. One can therefore hope to find a final solution of the unification problem in this way. We say more about the limits of this solution in chap. VII. 

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