An Assessment of Evans’ Unified Field Theory II

Evans attempted to develop a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. In an accompanying paper I, we analyzed this theory and summarized it in nine equations. We now propose a variational principle for a theory that implements some of the ideas that have been (imprecisely) indicated by Evans and show that it yields two field equations. The second field equation is algebraic in the torsion and we can resolve it with respect to the torsion. It turns out that for all physical cases the torsion vanishes and the first field equation, together with Evans’ unified field theory, collapses to an ordinary Einstein equation.     

The Spinning and Curving of Spacetime: The Electromagnetic and Gravitational Fields in the Evans Field Theory

The unification of the gravitational and electromagnetic fields achieved geometrically in the generally covariant unified field theory of Evans implies that electromagnetism is the spinning of spacetime and gravitation is the curving of spacetime. The homogeneous unified field equation of Evans is a balance of spacetime spin and curvature and governs the influence of electromagnetism on gravitation using the first Bianchi identity of differential geometry. The second Bianchi identity of differential geometry is shown to lead to the conservation law of the Evans unified field, and also to a generalization of the Einstein field equation for the unified field. Rigorous mathematical proofs are given in appendices of the four equations of differential geometry which are the cornerstones of the Evans unified field theory: the first and second Maurer-Cartan structure relations and the first and second Bianchi identities. As an example of the theory, the origin of wavenumber and frequency is traced to elements of the torsion tensor of spinning spacetime.     

Semi-Teleparallel Theories of Gravitation

A class of theories of gravitation that naturally incorporates preferred frames of reference is presented. The underlying space-time geometry consists of a partial parallelization of space-time and has properties of Riemann—Cartan as well as teleparallel geometry. Within this geometry, the kinematic quantities of preferred frames are associated with torsion fields. Using a variational method, it is shown in which way action functionals for this geometry can be constructed. For a special action the field equations are derived and the coupling to spinor fields is discussed.     

Einstein: Distant parallelism and electromagnetism

Einstein's approach to unified field theories based on the geometry of distant parallelism is discussed. The simplest theory of this type, describing gravitation and electromagnetism, is investigated. It is found that there is a charge-current density vector associated with the geometry. However, in the static spherically symmetric case no singularity-free solutions for this vector exist.     

A gauge-covariant bimetric tetrad theory of gravitation and electromagnetism

In order to get to a geometrically based theory of gravitation and electromagnetism, a gauge covariant bimetric tetrad space-time is introduced. The Weylian connection vector is derived from the tetrads and it is identified with the electromagnetic potential vector. The formalism is simplified by the use of gauge-invariant quantities. The theory contains a contorsion tensor that is connected with spinning properties of matter. The electromagnetic field may be induced by conventional sources and by spinning matter. In absence of spinning matter, the equations are identical with those of the gauge-covariant bimetric theory.⁽²³⁾     

A variational principle for Newton-Cartan theory

In the framework of a space-time theory of gravitation a variational principle is set up for the gravitational field equations and the equations of motion of matter. The general framework leads to Newton's equations of motion with an unspecified force term and, for irrotational motion, to a restriction on the propagation of the shear tensor along the streamlines of matter. The field equations obtained from the variation are weaker than the standard field equations of Newton-Cartan theory. An application to fluids with shear and bulk viscosity is given.     

Scale-covariant gravitation and electromagnetism

The theory of scale-covariant gravity is extended to include charged matter and electromagnetism at the classical level. The possibility of charge creation exists and the creation rate of charge differs from the creation rate of matter. A variational principle for scale-covariant gravity and electromagnetism coupled to a charged perfect fluid is given.Supported by a grant from the National Research Council of Canada.     

Gravitation and Electrodynamics over SO(3,3)

In a series of papers, an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in space-time. The extra dimensions are given a physical meaning as coordinatized matter. The inhomogeneous energy-momentum relations for the interacting fields in space-time are automatically generated by the simple homogeneous relations in 6-d. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variational principle. The Einstein–Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.     

New energy-momentum tensor in relativistic gravitation theory

Expressions for a new canonical energy-momentum tensor and an internal angular momentum of the gravitational field are derived in the context of bimetric relativistic gravitation theory based on the variational principle. A system of relations for the determining parameters of the gravitational field and matter involving, in particular, the continuity condition for the energy-momentum flux density is formulated on the discontinuity surface. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 52–58, April, 2006.     

Gravitational Energy-Momentum and Conservation of Energy-Momentum in General Relativity

Based on a general variational principle, Einstein-Hilbert action and sound facts from geometry, it is shown that the long existing pseudotensor, non-localizability problem of gravitational energy-momentum is a result of mistaking different geometrical, physical objects as one and the same. It is also pointed out that in a curved spacetime, the sum vector of matter energy-momentum over a finite hyper-surface can not be defined. In curvilinear coordinate systems conservation of matter energy-momentum is not the continuity equations for its components. Conservation of matter energy-momentum is the vanishing of the covariant divergence of its density-flux tensor field. Introducing gravitational energy-momentum to save the law of conservation of energy-momentum is unnecessary and improper. After reasonably defining “change of a particle’s energy-momentum”, we show that gravitational field does not exchange energy-momentum with particles. And it does not exchange energy-momentum with matter fields either. Therefore, the gravitational field does not carry energy-momentum, it is not a force field and gravity is not a natural force.     

Group deformations and relativity

A theory of deformation (homcomorphism but non-isomorphism) of topological groups is developed. In particular, a theory of deformation of subgroups structure is considered. The whole formalism is based on conceptions of holonomicity and relative geometry.A field is postulated to deform the symmetry group of a free physical system. It is shown that the classical fields deform the Poincare group. Thanks to this fact, gravitation appears as space-time curvature (non-holonomicity of the Lorentz subgroup mapping); and electromagnetism reveals itself by space-time torsion (non-holonomicity of the translation subgroup mapping). From physically evident premises it follows that space-time also has a torsion in the rotating and accelerated systems of reference.     

TheB (3) field as a link between gravitation and electromagnetism in the vacuum

The emergence of theB ⁽³⁾ field in vacuo has shown that electromagnetism is non-Abelian and similar in structure to gravitation. In this paper the Christoffel symbol used in general relativity is developed for electromagnetism in curvilinear coordinates: The former becomes describable as the antisymmetric part of the gravitational Ricci tensor. Therefore gravitation and electromagnetism are respectively the symmetric and antisymmetric parts of thesame Ricci tensor within a proportionality factor. Both fields are obtained from the Riemann curvature tensor, both are expressions of curvature in spacetime.     

有限精度函数理论与Einstein-Maxwell方程的光子解

提出了有限精度函数理论,并基于该理论求出了有限精度的Einstein-Maxwell方程的光子解,揭示了经典光子的电磁、引力与时空几何的重要性质. 关键词： 有限精度函数理论 Einstein-Maxwell方程 光子解     

Wesson’s Induced Matter Theory with a Weylian Bulk

The foundations of Wesson’s induced matter theory are analyzed. It is shown that the empty—without matter—5-dimensional bulk must be regarded as a Weylian space rather than as a Riemannian one. Revising the geometry of the bulk, we have assumed that a Weylian connection vector and a gauge function exist in addition to the metric tensor. The framework of a Weyl–Dirac version of Wesson’s theory is elaborated and discussed. In the 4-dimensional hypersurface (brane), one obtains equations describing both fields, the gravitational and the electromagnetic. The result is a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk. In special cases on obtains on the brane the equations of Einstein–Maxwell, or these of the original induced matter theory.     

The R.I. Pimenov unified gravitation and electromagnetism field theory as semi-Riemannian geometry

More than forty years ago R.I. Pimenov introduced a new geometry—semi-Riemannian one—as a set of geometrical objects consistent with a fibering pr: M _n → M _m . He suggested the heuristic principle according to which the physically different quantities (meter, second, Coulomb, etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple-fibered semi-Riemannian geometry is the most appropriate one for the treatment of more than three different physical quantities as unified geometrical field theory. Semi-Euclidean geometry ³ R ₅⁴ with 1-dimensional fiber x ⁵ and 4-dimensional Minkowski space-time as a base is naturally interpreted as classical electrodynamics. Semi-Riemannian geometry ³ V ₅⁴ with the general relativity pseudo-Riemannian space-time ³ V ₄, and 1-dimensional fiber x ⁵, responsible for the electromagnetism, provides the unified field theory of gravitation and electromagnetism. Unlike Kaluza-Klein theories, where the fifth coordinate appears in nondegenerate Riemannian or pseudo-Riemannian geometry, the theory based on semi-Riemannian geometry is free from defects of the former. In particular, scalar field does not arise. The text was submitted by the author in English.     

On Measuring Standards in Weyl’s Geometry

In Weyl’s geometry the nonintegrability problem and difficulties in defining measuring standards are reconsidered. Approaches removing the nonintegrability of length in the interior of atoms are given, so that atoms can serve as measuring standards. The Weyl space becomes a well founded framework for classical theories of electromagnetism and gravitation.