On defining the spin moment in the tetrad theory of gravitation

A general definition of the spin moment is presented in the tetrad formulation of the relativistic theory of gravitation; it is based on the conditions for the invariance of the corresponding action integral relative to infinitesimal tetrad transformations (the so-called tetrad spin moment) and infinitesimal coordinate transformations (the so-called coordinate spin moment). It is shown that the tetrad formulation of the general theory of relativity (TFGTR) and the tetrad theory of gravitation (TTG) in a space of absolute parallelism lead to fundamentally different definitions of spin, since in the Riemannian geometry of the TFGTR only the coordinate spin moment is physically meaningful, whereas in the space of absolute parallelism of the TTG only the tetrad spin moment has essential significance. It is also indicated that the Pellegrini-Plebanski theory (PPT) leads to an unsatisfactory hybrid definition of spin in the form of the coordinate spin moment of the gravitational and boson fields and the tetrad spin moment of the gravitational and fermion fields, the gravitational field entering into these spin moments of the PPT with opposite signs.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 68–71, May, 1976.     

Møller's tetrad theory of gravitation as a special case of Poincaré gauge theory—A coincidence?

Møller's tetrad theory of gravitation is included in Poincaié gauge theory for a special choice of parameters. In both theories the conservation theorems are derived from the field equations. They have the same form as in Einstein's theory. We compare the invariance properties of the field equations and discuss questions concerning the interpretation and measurability of the tetrad coefficients.     

Reissner–Nordström anti-de Sitter black holes and energy

We show that the tetrad field whose metric gives the Reissner–Nordström anti-de Sitter black holes gives the correct value of energy in Møller tetrad theory of gravitation.     

Reference systems and gravitation

It is shown how the formalism of the tetrad theory of gravitation used by Treder (1967a, b, 1970) follows from the more general fibre bundle formalism. This is of interest in the study of the relations between tetrad theories and the general theory of relativity. In particular, the breaking of the principle of general relativity and the interpretation of tetrad fields as reference systems are considered in greater detail.     

Weak equivalence principle and gauge theory of the tetrad gravitational field

The tetrad theory of gravitation corresponding to the Treder formulation of the weak equivalence principle is incompatible with the customary method for constructing a gauge theory for a tetrad gravitational field. In this formulation, the Lagrangian of the nongravitating mass is a direct covariant generalization of the partially relativistic expression to a Riemannian space-time V₄. This incompatibility is at odds with the resutt found in the tetrad formulation of the general theory of relativity derived from the requirement of localization of the Poincaré group.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 18–21, April, 1978.     

Theory of a gravitational gauge field with localization of the de Sitter group

A de Sitter-invariant gauge theory is formulated for the case where a 40-component de Sitter A-field is present. It is shown that the theory coincides with the Poincare-invariant gauge theory in a space with torsion with a cosmological term. Two other versions of a de Sitter-invariant theory are also discussed: the first is a metric theory of gravitation in a Riemann space; the second is a de Sitter-invariant generalization of the tetrad theory of gravitation in a space of absolute parallelism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 50–53, November, 1986.     

Relativity and equivalence principles in a gauge theory of gravitation

Conclusion The principal difficulty that has obstructed the formulation of gauge gravitation for more than twenty years now is the fact that an Einstein gravitational field represents a metric or a tetradic field, while gauge fields are connections on fiber bundles.The popular approach to the resolution of this problem lies in attempts to represent tetrad fields as gauge fields of the translation subgroup within the framework of the gauge theory of the Poincaré group, but the existing set of variants of the latter theory indicate that it is a long way from completion.Our approach [2, 3] insists that in a gauge theory, apart from gauge fields, the situation of spontaneous breaking of symmetry can also admit Goldstone and Higgs fields, under which is subsumed the metric (tetrad) gravitational field by virtue of the fact that, as we have shown above, the equivalence principle is included in the gauge theory of gravitation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 79–82, June, 1981.     

On Treder-Type Tetrad Theories of Gravitation I. Equivalence Principle and Invariance Properties

The Principle of relativity and the equivalence principle are the most important foundation of any theory of gravitation. We can formulate these principles by the help of the LORENTZ and the EINSTEIN groups. If we start with an action functional, the demand of invariance of this action with respect to these groups makes possible to get detailed conclusions about the general structure of theories of gravitation. EINSTEIN'S idea, to interpret gravitation as deformation of the local inertial systems of the special theory of relativity, leads to bi-tetrad theories, which we call TREDER-type tetrad theories. In this theories a sufficient number of gauge parameters is introduced in order to ensure the invariance of the action functional without limitations for the field variables.     

Variational formalism and field equations of tetrad gravitation theory in the Weyl-Cartan space

A variational formalism of tetrad gravitation theory is developed in the Weyl-Cartan space with independent variations in the tetrad coefficients, metric tensor components, and affine connectivity coefficients that considers the Weyl condition imposed on the nonmetricity based on the method of undetermined Lagrange multipliers. The gravitational field equations are derived for the Lagrangian comprising all possible quadratic convolutions of curvature, torsion, and nonmetricity tensors in addition to the linear component. Differential identities are obtained for the general gravitational Lagrangian in the Weyl-Cartan space. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 56–59, June, 2006.     

Determination of the Lagrangian of the tetrad gravitational field

By requiring correspondence with Newtonian gravitational theory and the Lorentz covariant theory of nongravitational matter and by establishing the simplest possible form of the linear approximation of the field equations, the gravitational Lagrangian of the tetrad theory of gravitation is determined uniquely. It contains two characteristic constants: Einstein's gravitational constant and the specific dimensionless “teleparallel” constant ω ≈ 1.     

Scalar-tetrad theories of gravity

A general theory of gravitation is constructed using a tetrad and a scalar field. The resulting theory, called a scalar-tetrad theory, does not contain Einstein's or the Brans-Dicke theories as special cases. However, there is a range of scalar-tetrad theories with the same post-Newtonian limit as Einstein's theory. Two particular models are interesting because of their simplicity.     

Tetrad fields and metric tensor in the gauge theory of gravitation

The most relevant geometrical aspects of the gauge theory of gravitation are considered. A global definition of the tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle. It is finally shown how to construct the fundamental geometrical objects on space-time, starting from B.     

Matrix theory of gravitation

A new classical theory of gravitation within the framework of general relativity is presented. It is based on a matrix formulation of four-dimensional Riemann-spaces and uses no artificial fields or adjustable parameters. The geometrical stress-energy tensor is derived from a matrix-trace Lagrangian, which is not equivalent to the curvature scalar R. To enable a direct comparison with the Einstein-theory a tetrad formalism is utilized, which shows similarities to teleparallel gravitation theories, but uses complex tetrads. Matrix theory might solve a 27-year-old, fundamental problem of those theories (Sect. 4.1). For the standard test cases (PPN scheme, Schwarz schild-solution) no differences to the Einstein-theory are found. However, the matrix theory exhibits novel, interesting vacuum solutions.     

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⁾     

Feldgleichungen der TREDERschen Gravitationstheorie,die aus einem Gravitationstheorem herleitbar sind. II

In the frame work of non-linear generalizations of TREDER 's tetrad theory of gravitation considered in part I. a pure bimetric gravitation theory results for the LAGRANG ian Ω⁽¹⁾_F with ω₂ = 1. The discussion of the post-NEWTON ian approximation given in I. has demonstrated that must be: ω₂ = ?1 ? 2ω₁. - However, a LAGRANG ian with ω₁ = ? ω₂ = ?1 is identical with GUPTA 's post-NEWTON ian approximation for EINSTEIN 's general relativistic LAGRANG ian. Therefore, for ω₁ = ? ω₂ = ? 1 the EINSTEIN effects are resulting evidently and the question discussed in I. the tetrad formalism becomes non-important.     

On Mø11er's tetrad theory of gravitation cosmology

We use a particular tetrad field to describe homogeneous isotropic cosmologies in the background of Møller's theory of gravitation. We prove that the universe can be closed or open for any possible value of the density of energy. The relation between the apparent magnitudem and the number of galaxies with a magnitude greater thanm is proved to be different from that of general relativity.     

Localization of the Einstein group and a nonsingular cosmological model of space-time with torsion

We consider an Einstein-invariant gauge theory of gravitation (EGT), obtained by localizing the group of motions of a homogeneous static Einstein Universe. Taking into account the cosmological term, we find exact solutions of EGT are as nonsingular homogeneous Isotropic cosmological models with both the metric and the torsion regular. It is shown that EGT satisfies the principle of correspondence with Newton's theory of gravity and with the tetrad theory of gravity in the space of absolute parallelism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fisika, No. 10, pp. 13–17, October, 1986.     

Double duality and hidden gauge freedom in the poincaré gauge theory of gravitation

We investigate the double duality ansatz of the Poincaré gauge theory of gravitation. It is shown that many known exact solutions belong, as special cases, to larger families of solutions. These families of solutions include several arbitrary functions and can be generated by a transformation which is a Lorentzian rotation of the connection with fixed tetrad. Several new spherically symmetric and wavelike exact solution are presented.     

Regular Charged Solutions in Mφller＇s Tetrad Theory of Gravitation

We find the most general tetrads which give a regular charged spacetime in tetrad theory of gravitation. The metric is a static one and it includes the Schwarzschild and Fteissner Nordstrom black holes. The energy content contained in a sphere of radius R is calculated using the superpotential given by Mφller in the context of Weitzenbock spacetime.