Nonlocal integral conservation laws and tetradic currents in the Einstein-Cartan theory. II. Problem of the gravitational energy and tetradic currents

Einstein-Cartan gravitational field equations are formed as Maxwell equations: the codifferential of the tetrad differential is equal to the conserved tetradic current. This yields local integral conservation laws for any tetrad field and, in particular, allows solving the old problem of the gravitational energy in the general relativity and Einstein-Cartan theories.     

Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice

Using the notion of a general conical defect, the Regge Calculus is generalized by allowing for dislocations on the simplicial lattice in addition to the usual disclinations. Since disclinations and dislocations correspond to curvature and torsion singularities, respectively, the method we propose provides a natural way of discretizing gravitational theories with torsion degrees of freedom like the Einstein-Cartan theory. A discrete version of the Einstein-Cartan action is given and field equations are derived, demanding stationarity of the action with respect to the discrete variables of the theory.     

Field equations and contracted Bianchi identities in the generalized Einstein-Cartan theory

The generalized Palatini variational principle for the gravitational field interacting with a matter field is considered. The contracted Bianchi identities are proved and the reduction of the generalized Einstein-Cartan equations to the set of 6 dynamical equations and 4 constraints is presented.     

The Brans-Dicke scalar field in Einstein-Cartan theory

In the framework of Einstein-Cartan (EC) theory, the Brans-Dicke (BD) theory is considered and it is found that a scalar field nonminimally coupled to the gravitational field gives rise to torsion, even though the scalar field has zero spin. The metric equations stay the same if the coupling constant is rescaled, but the equations of motion of a test particle, derived from the conservation equations, differ from those of the usual BD theory without torsion. The gravitational red-shift value differs considerably from the usual prediction of general theory of relativity (GTR), and rules out the possibility of a torsion version of BD theory for<6.     

New Approach to the U(2,2)-Symmetry in Spinor and Gravitation Theory

Generally-relativistic Dirac equation is reinterpreted as an approximation to the more fundamental model based on the quadruplet of Klein-Gordon particles with the internal hermitian metric of the neutral signature (++−−). Compensating field of the local group U(2,2) of internal symmetries splits under the GL(2, C)-reduction into gravitational co-tetrad, Einstein-Cartan-Weyl affine connection, and an auxiliary co-tetrad-like object. Equation for the matter field shows under the SL(2,C)-reduction a reasonable correspondence with the generally relativistic Dirac theory. Masses of fermions may be dynamically generated by non vanishing “vacuum” values of geometrodynamical quantities. There is also a convincing correspondence between Yang-Mills equations for the U(2,2)-gauge field and the Einstein-Cartan theory, or more general metric-affine theories of gravitation. In our approach the tetrad field is neither a reference frame nor a compensating field of translations in originally Minkowskian space, but a part of the U(2,2)-gauge multiplet.     

On the effect of spin on the gravitational field

Using a solution of Trautman's recently formulated Einstein-Cartan equations, it is shown that if an isolated body is embedded in an expanding universe consisting of a dust of spinning particles, then the local gravitational field of the body is influenced by spin, even when the cosmological constant is neglected.     

Inflation in Einstein-Cartan theory with energy-momentum tensor with spin

Generalized, or “power-law”, inflation is shown to necessarily exist for a simple, anisotropic, (Bianchi type I) cosmology in the Einstein-Cartan gravitational theory with the Ray-Smalley improved energy-momentum tensor with spin. Formal solution of the EC field equations with the fluid equations of motion explicitly shows inflation caused by the RS spin angular kinetic energy density.     

Topological and physical effects of rotation and spin in the general relativistic theory of gravitation

Interrelations of the intrinsic momentum (spin), rotation of material distributions, and intrinsic momentum of the gravitational field are investigated in the context of the general relativistic theory of gravitation involving the general relativity theory (GRT) and the Einstein-Cartan theory. It is demonstrated that the spin density vector of the gravitational field s _g ⁱ is equal to the rotor of the tetrad reference point ωⁱ=ɛ^iklm e _k ^(a) e_(a)l,m/2 to within the factor 1/κ (s _g ⁱ =ω/κc). It is demonstrated that the vector s _g ⁱ is proportional to the spin density vector of the gravitating field sⁱ (ω)=jc(Ψγⁱγ₅Ψ)/2 as well as the pseudovector of space-time torsion Qⁱ in the Einstein-Cartan theory, which in both cases induces a cubic nonlinearity of the spinor field. An expression for the energy-momentum density tensor of the eddy gravitational field is derived. It is also demonstrated that the free eddy gravitational field with polarized spin can form “mole holes.” An ideal fast-rotating self-gravitating fluid can cause a similar effect. The corresponding exact solutions of joint systems of the Einstein and rotating ideal fluid equations are presented. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 57–60, October, 2007.     

Strong gravity with torsion and the Cabibbo angle

The classical field equations of the strong gravity theory are modified by introducing spin and torsion so as to resemble the field equations of the Einstein-Cartan theory. The effects of the hadronic spin upon the strong metric are evaluated using a semiclassical approximation, and it is pointed out that the torsion contributions are not negligible. A possible interpretation of the Cabibbo angle as a strong gravity correction of the weak semileptonic Lagrangian is suggested.     

Brans-Dicke引力理论的弱场近似

Brans-Dicke引力理论是重要的修正引力理论之一,其对于研究星体运动、宇宙演化以及解释宇宙现象等起着重要的作用.寻找Brans-Dicke引力理论场方程的解对于Brans-Dicke引力理论的研究和发展具有重要意义.但由于Brans-Dicke引力理论场方程本身的高度非线性性,使得一般情况下精确求解非常困难,特殊情况下也只能求得部分精确解.幸运的是大多数情况下引力场比较弱,且在低速的条件下求场方程的近似解相对容易.本文基于弱场低速条件,详细地求解了Brans-Dicke引力理论的弱场低速近似解.首先基于弱场条件,将标量场和度规写为一阶微扰展开的形式;然后将标量场和度规代入相应的场方程得到相应的线性场方程,通过选取特定的规范条件进一步简化线性场方程;最后求解出简化的线性场方程的低速近似解.本文的求解方法可以为Brans-Dicke引力理论的教学和研究提供有益的参考.     

Free vector field in a homogeneous space with torsion

The effect of torsion of space-time, induced by the spin of a vector field, upon the physical properties of the field is studied within the framework of Einstein-Cartan gravitational theory. It is shown that the torsion leads to the appearance of nonlinearities in the vector field, as a consequence of which soliton solutions are possible under certain conditions. A cosmological model without singularities is constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 26–29, September, 1979.     

Mini-bangs in cosmology and astrophysics

The ideas originally proposed to discuss continuous creation of matter are reconsidered in the context of the big bang cosmological models. It is shown that singularity-free big bang models are possible under the modified field equations of general relativity. However, the case is made out that matter creation takes place in several mini-bangs at different epochs rather than in one big bang. The implications of this idea for high energy astrophysics and for gravitational radiation are discussed.     

The path of a sound signal in Einstein and Einstein-Cartan theory of gravitation

The path equation of a sound signal in general relativity is studied, It is shown that the sound path deflects towards stronger gravitational fields and is expected to defect less in a medium with polarization than in one without it, using the Einstein-Cartan theory of gravitation.     

Lorentz invariance of gravitational Lagrangians in the space of reference frames

The recently proposed theories of gravitation in the space of reference framesS are based on a Lagrangian invariant with respect to the homogeneous Lorentz group. However, in theories of this kind, the Lorentz invariance is not a necessary consequence of some physical principles, as in the theories formulated in space-time, but rather a purely esthetic request. In the present paper, we give a systematic method for the construction of gravitational theories in the spaceS, without assuming a priori the Lorentz invariance of the Lagrangian. The Einstein-Cartan equations of gravitation are obtained requiring only that the Lagrangian is invariant under proper rotations and has particular transformation properties under space reflections and space-time dilatations     

A spin-coefficient approach to Weyssenhoff fluids in Einstein-Cartan theory

The generalized Newman-Penrose formalism is used to analyze semiclassical aligned spin fluids satisfying the Weyssenhoff restriction in the framework of Einstein-Cartan theory. Some general properties are derived and the formalism is then used to obtain two classes of exact solution. One has a flat metric, but the fluid has in general nonzero acceleration, expansion, and shear. It is characterized by two arbitrary constants and two functions of two variables satisfying one partial differential equation. In the other class the fluid has nonzero acceleration and vorticity, and the free gravitational field is of typeD. It is characterized by three arbitrary constants and an arbitrary function of two spacelike coordinates.     

Extension of the York field decomposition to general gravitationally coupled fields

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.     

NULL GEODESIC EQUATION EQUIVALENT TO THE GEOMETRIC OPTICS EQUATION

In this paper, we proved that the null geodesic equations in general isotropic metric are rigorously equivalent to the differential equations bf light rays in a medium of flat 3-dimension.,31 space, which are the fundamental equations of geometrical optics, when the effect bf gravitational field tb light ray is taken as that of a variable "refractive index"。Our results provide the basic way to strictly describe the characteristics of some gravitational fields in terms of the Euclidean framework. It may be useful in the computation of modern astrometry and gravitational Lens effects. Finally, as a special example,the deflection of Iight by gravitational field is discussed by means of the formula of pure classical geometrical optics.     

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.