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Using the method of first integrals, we find an exact solution for the relativistic motion of a charge in orthogonal and uniform electric and magnetic fields with respect to laboratory time and for any value of the dimensionless governing parameter equal to the ratio of the magnetic field strength to the electric field strength.

General Relativity rewritten in conformal units identifies conformal intervals with the real observational distances. This identification gives a base to explain all epochs of the Universe evolution including Ia supernova luminosity long distance-redshift relation by the dominance of the Casimir vacuum energy of all physical fields. A set of arguments is discussed in favor that SNe Ia data in the conformal units can be an evidence of the conformal twistor structure of the space-time as a nonlinear realization of the affine group, like the nonlinear realization of chiral symmetry and phenomenological Lagrangian is evidence of the quark structure of hadrons.

Strong gravitation plane waves are represented in terms of the Maurer-Cartan spin connection coefficients in cosmological background. It was shown that the diffeo-invariance of spin connection coefficients leaves only one degree of freedom of the strong gravitation plane waves in contrast to the metric approach, where gravitation waves have two degrees of freedom like photons in QED. The Hilbert action of gravitation waves in terms of spin connection coefficients takes the form of a bilinear field theory.

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type la supernova luminosity distances-redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that, the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of “squeezed oscillator”. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.

The General Relativity formulated with the aid of the spin connection coefficients is considered in the finite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution of the General Relativity describes the vacuum creation of the matter in the empty Universe at the electroweak epoch and the dilaton vacuum energy plays a role of the dark energy., Comment: 9 pages, 1 figure, submitted to PLB

General Relativity and Standard Model are formulated in terms of scale-invariant variables where the initial data are integrals of motion. In this case, the Hubble law can be explained by a cosmological evolution of particle masses. Supernovae type Ia data and the CMB energy budget in the model are in agreement with the dominance of a scalar field kinetic energy density and an intensive cosmological creation of primordialW, Z, and Higgs bosons from vacuum. Some arguments are discussed testifying to that two-photon processes of primordial particle annihilation and decays form three peaks in the CMB power spectrum, and their values and positions l = 220, 546, 800 are in agreement with the QED coupling constant,Weinberg’s angle, and the Higgs particle mass of about 118 GeV.

General Relativity and Standard Model are considered as a theory of dynamical scale symmetry with definite initial data compatible with the accepted Higgs mechanism. In this theory the Early Universe behaves like a factory of electroweak bosons and Higgs scalars, and it gives a possibility to identify three peaks in the Cosmic Microwave Background power spectrum with the contributions of photonic decays and annihilation processes of primordial Higgs, W, and Z bosons in agreement with the QED coupling constant, Weinberg's angle, and Higgs' particle mass of about 118 GeV., Comment: version to appear in Yadernaya Fizika

We show that relativistic theories of strings and of gravity that are invariant with respect to general covariance transformations can be formulated in terms of physical coordinates with nonzero evolution Hamiltonians by identifying these coordinates with some dynamic variables.

We derive equations relating the Fermi-Walker and the congruent Weyl transports. Using these equations, we show that a non-Abelian gauge field can result in the Thomas precession of a gyroscope. We find solutions to the equations for such a non-Abelian gauge field.

The connection between the Fermi–Walker transport and the Weyl non-Abelian gauge field is established. A theoretical possibility of detecting the Weyl gauge field caused by the Thomas precession of a gyroscope is discussed.

The aim of the present paper is to find a spinor current—a source—in the Weyl non-Abelian gauge theory whose distinguishing feature is that it involves no abstract gauge space. It is shown that the desired spinor representation of the Weyl gauge group can be constructed in the space of antisymmetric tensor fields in the form of a 16-component quantity for which a gauge-invariant Lagrangian is established. The relationship between the Weyl non-Abelian gauge potential and the Cartan torsion field, and the question of where the interactions in question could manifest are discussed.

Boundary equations for the relativistic string with masses at ends are formulated in terms of geometrical invariants of world trajectories of masses at the string ends. In the three-dimensional Minkowski space $E^1_2$, there are two invariants of that sort, the curvature $K$ and torsion $\kappa$. For these equations of motion with periodic $\kappa_i(\tau+n l)=\kappa(\tau)$, constants of motion are obtained., Comment: 8 pages, REVTEX, no figures and tables

It is shown that the congruent transport introduced by Weyl in 1921 determines a non-Abelian gauge field. The simplest gaugeinvariant equations of this field are proposed. Its connection with torsion is discussed.

It is shown that the congruent transference introduced by Weyl in 1921 defines a non-Abelian gauge field. The simplest gauge-invariant equations are proposed for this field. Its relation with the Riemann–Cartan geometry is also discussed.

The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer-Cartan forms. The dominance of the Casimir vacuum energy of physical fields provides a good description of the type Ia supernova luminosity distance--redshift relation. Introducing the uncertainty principle at the Planck's epoch within our model, we obtain the hierarchy of the Universe energy scales, which is supported by the observational data. We found that the invariance of the Maurer-Cartan forms with respect to the general coordinate transformation yields a single-component strong gravitational waves. The Hamiltonian dynamics of the model describes the effect of an intensive vacuum creation of gravitons and the minimal coupling scalar (Higgs) bosons in the Early Universe., 37 pages, version submitted to Gen. Rel. Grav

Dynamic equations in the theory of a relativistic string with point masses at the ends are formulated only in terms of geometric invariants of the world trajectories of the massive ends of the string (curvature ka and torsion kappa a of the trajectories). These characteristics allow the authors to reproduce the string world surface up to its position in Minkowski space E1. The torsions kappa a( tau ), a=1, 2 obey a system of second-order differential equations with delay describing the retardation effects of the interaction of masses through the string, the ka being constants. A new particular solution to these equations is found that corresponds to periodic torsions.

The Hamiltonian formulation of general relativity (GR) is considered in finite space-time and a specific frame of reference given by the diffeo-invariant components of the Fock simplex in terms of the Dirac -- ADM variables. The evolution parameter and energy invariant with respect to the time-coordinate transformations are constructed by the separation of the cosmological scale factor and its identification with the spatial averaging of the metric determinant, so that the dimension of the kinemetric group of diffeomorphisms coincides with the dimension of a set of variables whose velocities are removed by the Gauss-type constraints in accordance with the second N\"other theorem. This coincidence allows us to solve the energy constraint, fulfil Dirac's Hamiltonian reduction, and to describe the potential perturbations in terms of the Lichnerowicz scale-invariant variables distinguished by the absence of the time derivatives of the spatial metric determinant. It was shown that the Hamiltonian version of the cosmological perturbation theory acquires attributes of the theory of superfluid liquid, and it leads to the generalization of the Schwarzschild solution. The astrophysical application of this approach to GR is considered under supposition that the Dirac -- ADM Hamiltonian frame is identified with that of the Cosmic Microwave Background radiation distinguished by its dipole component in the frame of an Earth observer., Comment: 30 pages, 3 figures

A version of the cosmological perturbation theory in general relativity (GR) is developed, where the cosmological scale factor is identified with spatial averaging of the metric determinant logarithm and the cosmic evolution acquires the pattern of a superfluid motion: the absence of "friction-type" interaction, the London-type wave function, and the Bogoliubov condensation of quantum universes. This identification keeps the number of variables of GR and leads to a new type of potential perturbations. A set of arguments is given in favor of that this "superfluid" version of GR is in agreement with the observational data., 8 pages, "Long Communication" at the XXVIII Spanish Relativity Meeting "A Century of Relativity Physics" Oviedo, Spain, 6-10 Sept. 2005; http://fisi24.ciencias.uniovi.es/ere05.html; To be published in "AIP Conference Proceedings"

The version of the cosmological perturbation theory based on exact resolution of energy constraint is developed in accordance with the diffeomorphisms of general relativity in the Dirac Hamiltonian approach. Such exact resolution gives one a possibility to fulfil the Hamiltonian reduction and to explain the ``CMBR primordial power spectrum'' and other topical problems of modern cosmology by quantization of the energy constraint and quantum cosmological origin of matter., 11 pages, 1 figure. Invited talk at the V International Conference on Non-Accelerator New Physics(Dubna, Russia, June 20--25, 2005)

The Hamiltonian approach to cosmological perturbations in general relativity in finite space-time is developed, where a cosmological scale factor is identified with spatial averaging the metric determinant logarithm. This identification preserves the number of variables and leads to a cosmological perturbation theory with the scalar potential perturbations in contrast to the kinetic perturbations in the Lifshitz version which are responsible for the ``primordial power spectrum'' of CMB in the inflationary model. The Hamiltonian approach enables to explain this ``spectrum'' in terms of scale-invariant variables and to consider other topical problem of modern cosmology in the context of quantum cosmological creation of both universes and particles from the stable Bogoliubov vacuum., Comment: 10 pages, text was changed according to the version accepted for publication in Physics Letters B

A spinor current-source is found in the Weyl non-Abelian gauge theory which does not contain the abstract gauge space. It is shown that the searched spinor representation can be constructed in the space of external differential forms and it is a 16-component quantity for which a gauge-invariant Lagrangian is determined. The connexion between the Weyl non-Abelian gauge potential and the Cartan torsion field and the problem of a possible manifestation of the considered interactions are considered., Comment: 16 pages, latex, no figures

Using the methods of differential geometry it is shown that the Born-Infeld scalar field in two-dimensional space-time and the relativistic string in three dimensions are described by the same non-linear Liouville equation utt-uxx=R eu. This equation admits soliton solutions which may be stable or unstable, and there are periodical solutions among the stable ones. In the quantum case the solitons can be interpreted as massive particles either stable or unstable with respect to the stability of the corresponding classical solution. The periodical soliton generates a series of resonances which have the equidistant mass spectrum. This result appears to be well suited to the theory of the closed relativistic string. In four dimensions the relativistic string is described by the same Liouville equation, but for the complex-valued function u.

General solutions for the system of nonlinear equations in the second order partial derivatives with two independent variables are obtained. They determine the basic differential forms of the two-dimensional minimal surface embedded inton-dimensional pseudo-Euclidean space.

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation ϕττ − ϕσσ = Hshϕ. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.

A differential-geometric formulation of the dynamics of a relativistic string with masses at its ends is considered in the Minkowski space E/sub 2//sup 1/. The surface swept out by the string is described by differential forms and is bounded by two curves - the worldlines of its massive ends. These curves have a constant geodesic curvature, and their torsion is determined only up to an arbitrary function on the interval (0, 2/pi/). Equations are obtained that determine the world surface of the string as a function of the curvature and torsion of the trajectories of its massive ends. For the choice of the constant torsions for which the mass points move along helices, the surface of the relativistic string is a helicoid.

Exact classical solutions of the equations of motion for the infinite relativistic string loaded with a point-like mass are obtained in terms of the Fourier integrals. The gauge when the time evolution parameter τ is proportional to the proper time of a point-like particle is used. The gauge conditions in this case are the second-class constraints. So in the quantum case the commutators of the dynamical variables have to be defined by the Dirac brackets.

The central part of our work was the proof that the dynamics on the physical submanifold of the phase space determined by the constraint equations and the gauge conditions is Hamiltonian. It is this that is the central problem in the general investigation of the dynamics in the phase space of systems with degenerate Lagrangians [11]. In [1] and in the monographs [2–4] the Hamiltonicity of the dynamics on the physical subspace was assumed implicitly. In constrast, we have given a consistent derivation of the corresponding Hamiltonian equations (3.15) and (3.16).

The relativistic string model is investigated in a space-time of a constant curvature (de Sitter universe). The fundamental differential quadratic forms of the world surface of the string are considered as the dynamical variables. The coefficients of these forms obey two nonlinear equations $$\varphi _{,11} - \varphi _{,22} = e^\varphi \cos \theta + Ke^{ - \varphi } ,\theta _{,11} - \theta _{,22} = e^\varphi \sin \theta .$$ The Lax representation for this system is obtained.