A spherically symmetric entity with the Weyl-Dirac geometry holding in its interior is investigated. The structure is determined by the presence of the Dirac gauge function, which creates a mass density. Two models are obtained, one that can describe a cosmic body, the other an elementary particle.

Classical models of elementary particles, regarded as the ultimate constituents of the known particles, are considered in the framework of general relativity. The work is based on that of López, using the Kerr-Newman solution of the Einstein field equations.

It is suggested that the dark matter of the universe is due to the presence of a scalar field described by the gauge function introduced by Dirac in his modification of the Weyl geometry. The behavior of such dark matter is investigated.

The Weyl geometry, as modified by Dirac, can lead to the presence of a field consisting of bosons of spin 1 and finite mass. It was proposed earlier that these bosons, called weylons, form the bulk of the dark matter in the universe. The development of this Weylian dark matter is investigated from the time of its creation until the present, and an acceptable cosmological behavior is obtained. One finds that this dark matter was unimportant in the early stages of the universe but became important at the time of galaxy formation and may have played a role in this process.

An elementary particle is described as a spherically symmetric solution of the Klein-Gordon equation and the Einstein equations of general relativity. It is found that it has a mass of the order of the Planck mass. If one assumes that the motion of its center of mass is determined by the Dirac equations, then it has a spin of 1/2.

The energy of the universe, including the energy of the matter and that of the gravitational field, is investigated with the help of the Einstein gravitational pseudo-tensor. It is found that the total energy vanishes.

From the equations of general relativity for the radius of a closed homogeneous isotropic universe a Schrödinger equation for a particle is obtained. In the case of a universe filled with pressureless matter (dust) the equation is like that for thes states of a hydrogenlike atom. The miniuniverses obtained in this way have quantized masses of the order of the Planck mass.

Recently a homogeneous cosmological model free from singularities was proposed, based on the general relativity theory. It described a closed universe (k = +1), initially filled with prematter, characterized by a density ρ equal to the Planck density and a pressureP = −ρ, and undergoing oscillations. In the present work the case of a similar, but spatially flat, universe (k = 0) is investigated. In this case there is an initial geometric singularity (the scale factorR = 0), but not a physical one, since the initial density is finite. This universe begins its existence at a timet = −∞ and, after going through the prematter and radiation-dominated eras, reaches the matter-dominated state and continues to expand indefinitely.

Weyl proposed a geometry that differed from Riemannian geometry, which underlies general relativity, in that it contained a vector that could be interpreted as describing the electromagnetic field. Dirac modified this geometry to remove certain difficulties and based it on a variational principle which gave satisfactory field equations for gravitation and electromagnetism. However, by changing the value of a parameter appearing in his variational principle one gets, instead of electromagnetism, a field of massive particles of spin 1, which can be assumed to interact with ordinary matter only through gravitation. It is suggested that these bosons, called weylons, provide most of the dark matter in the universe.

It is shown that all spherically symmetric distributions of prematter in the framework of general relativity are static. These results provide a justification for the models of elementary particles proposed previously.

The rest-frame of the universe determines a universal, or absolute time, that given by a clock at rest in it. The question is raised whether one can have a satisfactory universal time in general relativity if a gravitational field is present, i.e., whether there are coordinates such that the coordinate time is the time given everywhere by a clock at rest and they provide the correct description of our everyday experience. Several attempts are made to find such coordinates, but the results are unsatisfactory. The question is still open, but it may be that there is no significance to such a universal time in general relativity, because to have a clock at rest in a gravitational field requires nongravitational forces.

Elementary particles, regarded as the constituents of quarks and leptons, are described classically in the framework of the general relativity theory. There are neutral particles and particles having charges±1/3e. They are taken to be spherically symmetric and to have mass density, pressure, and (if charged) charge density. They are characterized by an equation of state P=−ρ suggested by earlier work on cosmology. The neutral particle has a very simple structure. In the case of the charged particle there is one outstanding model described by a simple analytic solution of the field equations.

The possibility of an elementary particle in the framework of classical bimetric general relativity is explored further. A model is considered which is filled with a pressureless primal fluid having a fixed ratio of charge density to mass density. This ratio is assumed to be0, ±ε 0 , where ε 0 is a universal constant <0.5. If the particle charge is assumed to be ±1/3e, the mass is a fraction of the Planck mass, the fraction being greater than0.0285.

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with Nv vector dimensions and Ns spinor dimensions, where Nv=2k and Ns=2k, k⩾3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with Nv−4 vector and Ns−4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.

Annals of the New York Academy of SciencesVolume 470, Issue 1 p. 378-378 A Compact Object in the Bimetric Theory AMOS HARPAZ, AMOS HARPAZ Department of Physics, Technion-Israel Institute of Technology Haifa, Israel Also the University of Haifa, School of Education of the Kibbutz Movement, Oranim.Search for more papers by this authorNATHAN ROSEN, NATHAN ROSEN Department of Physics, Technion-Israel Institute of Technology Haifa, IsraelSearch for more papers by this author AMOS HARPAZ, AMOS HARPAZ Department of Physics, Technion-Israel Institute of Technology Haifa, Israel Also the University of Haifa, School of Education of the Kibbutz Movement, Oranim.Search for more papers by this authorNATHAN ROSEN, NATHAN ROSEN Department of Physics, Technion-Israel Institute of Technology Haifa, IsraelSearch for more papers by this author First published: May 1986 https://doi.org/10.1111/j.1749-6632.1986.tb48003.x AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume470, Issue1Twelfth Texas Symposium On Relaivistic AstrophysicsMay 1986Pages 378-378 RelatedInformation