On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

No black holes: A gravitational gauge theory possibility

The most general lowest order lagrangian that can be formed from gauge-derived vierbein invariants is constrained by the hypothesis that the speed of light as measured by conventional rods and clocks of atomic constitution is independent of direction in a gravitational field. It is shown that the standard weak field observational tests of general relativity serve to eliminate all possible combinations of parameters in this constrained lagrangian except two. One parameter choice gives the isotropic Schwarzchild black hole metric of the general theory of relativity. The other allowable choice leads to an exponential metric of the class proposed by Yilmaz, corresponding in strong fields to large red shifts without black hole formation.     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Sphaleron glueballs in NBI theory with symmetrized trace

We derive a closed expression for the SU(2) Born–Infeld action with the symmetrized trace for static spherically symmetric purely magnetic configurations. The lagrangian is obtained in terms of elementary functions. Using it, we investigate glueball solutions to the flat space NBI theory and their self-gravitating counterparts. Such solutions, found previously in the NBI model with the “square root–ordinary trace” lagrangian, are shown to persist in the theory with the symmetrized trace lagrangian as well. Although the symmetrized trace NBI equations differ substantially from those of the theory with the ordinary trace, a qualitative picture of glueballs remains essentially the same. Gravity further reduces the difference between solutions in these two models, and, for sufficiently large values of the effective gravitational coupling, solutions tends to the same limiting form. The black holes in the NBI theory with the symmetrized trace are also discussed.     

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.     

Exact theory of the (Einstein) gravitational field in an arbitrary background space-time

The Lagrangian based theory of the gravitational field and its sources at the arbitrary background space-time is developed. The equations of motion and the energy-momentum tensor of the gravitational field are derived by applying the variational principle. The gauge symmetries of the theory and the associated conservation laws are investigated. Some properties of the energymomentum tensor of the gravitational field are described in detail and the examples of its application are given. The desire to have the total energymomentum tensor as a source for the linear part of the gravitational field leads to the universal coupling of gravity with other fields (as well as to the self-interaction) and finally to the Einstein theory.     

Defining Euler-Lagrange fields in terms of almost tangent structures

It is shown that a differentiable manifold with an almost tangent structure provides a suitably general setting for lagrangian dynamics, in analogy to the way that a symplectic manifold provides a suitably general setting for hamiltonian dynamics; and it is shown how a closed 1-form on a manifold with almost tangent structure determines a vector field, its Euler-Lagrange field, whose integral curves are solutions of the Euler-Lagrange equations for a suitable lagrangian function.     

Kähler-Dirac ghosts for self-dual fields

We present the generalization to spacetime dimension D=4n+2 of the Lorentz covariant quadratic lagrangian for pairs of (anti)self-dual fields previously obtained by the authors in D=2. In the process BRST quantizing this lagrangian a first-order quadratic lagrangian for ghost (anti)self-dual fields is found which, after gauge fixing, can be written in terms of bispinors and it turns out to be a Kähler-Dirac lagrangian. The coupling to gravity is straightforward and the gravitational anomaly due to (anti)self-dual fields is obtained directly from an action principle.     

Weak gravitational field in Finsler–Randers space and Raychaudhuri equation

The linearized form of the metric of a Finsler–Randers space is studied in relation to the equations of motion, the deviation of geodesics and the generalized Raychaudhuri equation are given for a weak gravitational field. This equation is also derived in the framework of a tangent bundle. By using Cartan or Berwald-like connections we get some types “gravito-electromagnetic” curvature. In addition we investigate the conditions under which a definite Lagrangian in a Randers space leads to Einstein field equations under the presence of electromagnetic field. Finally, some applications of the weak field in a generalized Finsler spacetime for gravitational waves are given.     

General relativity in Newtonian form

It is shown how the use of coordinates where time is measured with clocks moving radially in a spherically symmetric gravitational field leads to general relativistic dynamical expressions that are exactly identical to corresponding expressions in Newtonian theory. The general formalism is developed for the case where the stress-energy tensor is that of a perfect fluid. Expressions like the Newtonian inverse square gravitational law, the Newtonian equation of continuity for fluid flow, Newtonian conservation of energy, etc., follow quite naturally from the fully-fledged exact general relativistic equations. Specific examples involving cosmology and gravitational collapse are given.     

The variant of post-Newtonian mechanics with generalized fractional derivatives

In this article, we investigate mathematically the variant of post-Newtonian mechanics using generalized fractional derivatives. The relativistic-covariant generalization of the classical equations for gravitational field is studied. The equations (i) match the weak Newtonian limit on the moderate scales and (ii) deliver a potential higher than Newtonian on certain large-distance characteristic scales. The perturbation of the gravitational field results in the tiny secular perihelion shift and exhibits some unusual effects on large scales. The general representation of the solution for the fractional wave equation is given in the form of retarded potentials. The solutions for the Riesz wave equation and classical wave equation are clearly distinctive in an important sense. The hypothetical gravitational Riesz wave demonstrates the space diffusion of the wave at the scales of metric constant. The diffusion leads to the blur of the peak and disruption of the sharp wave front. This contrasts with the solution of the D'Alembert classical wave equation, which obeys the Huygens principle and does not diffuse.     

Solitons in five-dimensional gravity

The application of the generalized Bäcklund transformation to the projective field theory equations for stationary axisymmetric vacuum fields leads us to an explicit formula for the electric, magnetic, rotational, gravitational, and scalar potentials of new exact solutions generated from an arbitrary seed solution.     

Scalar-tensor theories and the principle of equivalence

We investigate the restrictions on scalar-tensor theories of gravitation implied by the assumptions: (i) the field equations are derivable from an action principle, (ii) units of mass length and time are defined by atomic standards, and (iii) the principle of equivalence holds whenever gravitational self-energy can be neglected. We show that in all these theories the presence of gravitational energy in a system leads to violations of the principle of equivalence.     

Quantum constraint algebra for a closed bosonic string in a gravitational and dilaton background

We derive the quantum constraint algebra for a closed bosonic string moving in a gravitational and dilaton background to first order in '. The hamiltonian approach is used to directly compute the quantum constraint commutators and calculate the c-and q-number anomalies that arise at the quantum level. The requirement that the algebra preserves the conformal invariance leads to the known background field equations.     

Quantization effects in the plasma universe

It is suggested that a unification of the morphology of the solar system, anomalous intrinsic red shifts of quasars and galaxies, the structure of the hydrogen atom, the Einstein equations of general relativity, and Maxwell's equations can be accomplished by a basic consideration of the minimum-action states of cosmic and/or virtual vacuum field plasmas. A formalism of planetary formation theory leads naturally to a generalization which describes relativistic gravitational field theory in terms of a `pregeometry'. A virtual plasma associated with the vacuum state is postulated. It is demonstrated that the relaxed state of the virtual plasma underlies Einstein's field equation and predicts the proper form for the effective gravitational potential generated by the Schwarzschild solution of those equations. A further extension of the theory demonstrates that it also predicts the structure of the hydrogen atom described in terms of the Schrodinger equation of quantum mechanics. These concepts are applied in an attempt to explain the quantized anomalous red shifts in related galaxies as observed by H. Arp and J.H. Sulentic (1985). A possible unified field theory is suggested based on the above-mentioned concepts     

Generalised cosmological friedmann equations without gravitational singularity

Within the framework of the gauge approach to gravitation, including terms in the lagrangian quadratic in the curvature and torsion tensors, a generalised Friedman equation for a homogeneous isotropic cosmology is obtained. This equation avoids the gravitational singularity with infinite mass density.     

Gravitational Faraday Effect Produced by a Ring Laser

Using the linearized Einstein gravitational field equations and the Maxwell field equations it is shown that the plane of polarization of an electromagnetic wave is rotated by the gravitational field created by the electromagnetic radiation of a ring laser. It is further shown that this gravitational Faraday effect shares many of the properties of the standard electromagnetic Faraday effect. An experimental arrangement is then suggested for the observation of this gravitational Faraday effect induced by the ring laser.     

Existence of dual mass: Linear theory

The field equations of general relativity are symmetrized in a manner similar to Dirac's symmetrization of electromagnetism. This symmetrization allows us to predict the existence of a gravitational, magnetic-like mass which we call dual mass. Time-independent solutions for both rotating and nonrotating sources are constructed for these generalized equations. The gravitational field produced by the dual mass source is compared with the gravitational field that follows from the linearized NUT and Kerr-NUT metrics, and an identification of the NUT parameter with a dual mass monopole is made.     

A smooth oscillating cosmological solution

A metric is given, determined by the field equations derived from the Lovelock lagrangian in a Kaluza—Klein theory of gravity, whose projection onto the observable external space yields a smooth oscillating cosmological solution.