We construct a general approach to the decomposition of the tangent bundle of pseudo-Riemannian manifolds into direct sums of subbundles, and the associated decomposition of geometric objects. An invariant structure Hr defined as a set of r projection operators is used to induce decomposition of the geometric objects into those of the corresponding subbundles. We define the main geometric objects characterizing decomposition. Invariant nonholonomic generalizations of the Gauss–Codazzi–Ricci’s relations have been obtained. All the known types of decomposition (used in the theory of frames of reference, in the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory of stationary spaces, and so on) follow from the present work as special cases when fixing a basis and dimensions of subbundles, and parametrization of a basis of decomposition. Various methods of decomposition have been applied here for the unified multidimensional Kaluza–Klein theory and for relativistic configurations of a pe...

Split structures in general relativity and the Kaluza–Klein theories

A version of the five-dimensional unified theory of gravitation, electromagnetism, and scalar field is developed. It is shown that in this theory the main features of Kaluza's five-dimensional theory and the Weyl one, based on non-Riemannian geometry and on conformal mapping, are combined. Some reasons are pointed out for choosing the physical 4-metric to be conformal (with the factorϕ2=−G55) to the 4-metric obtained by 1+4 splitting of the initial five-dimensional manifold. It is shown that the electrical charge and current appear in the geometrical theory if the condition of cylindrical symmetry in the fifth coordinate is substituted by the condition of quasicylindrical symmetry (i.e., the physical 4-metric and the vector potential of electromagnetic field remain independent of the fifth coordinate, while the scalar field depends on it). Two kinds of the most important exact solutions of the 15 field equations are considered. They are (1) static spherically symmetrical solutions and (2) homogeneous isotropic cosmological models.

The Unified Field Theory, Combining Kaluza's Five-Dimensional and Weyl's Conformal Theories

The general-relativistic Ohm’s law for a two-component plasma which includes the gravitomagnetic force terms even in the case of quasi-neutrality has been derived. The equations that describe the electromagnetic processes in a plasma surrounding a neutron star are obtained by using the general relativistic form of Maxwell equations in a geometry of slow rotating gravitational object. In addition to the general-relativistic effect first discussed by Khanna and Camenzind (Astron. Astrophys. 307:665, 1996) we predict a mechanism of the generation of azimuthal current under the general relativistic effect of dragging of inertial frames on radial current in a plasma around neutron star. The azimuthal current being proportional to the angular velocity ω of the dragging of inertial frames can give valuable contribution on the evolution of the stellar magnetic field if ω exceeds 2.7×1017(n/σ) s−1 (n is the number density of the charged particles, σ is the conductivity of plasma). Thus in general relativity a rotating neutron star, embedded in plasma, can in principle generate axial-symmetric magnetic fields even in axisymmetry. However, classical Cowling’s antidynamo theorem, according to which a stationary axial-symmetric magnetic field can not be sustained against ohmic diffusion, has to be hold in the general-relativistic case for the typical plasma being responsible for the rotating neutron star.

Ohm’s law for plasma in general relativity and Cowling’s theorem

This paper presents a coordinate-dependent 3+ 1 decomposition of the general relativity field equations in terms of a scalar potentialψ≡c2[(−g44)1/2−1], a vector potentialAi≡cg4i/(−g44)1/2, and the three-space metricγij≡gij−g4ig4j/g44. The equations are exact and the form of the decomposed equations is valid in any coordinate system.

Coordinate-dependent 3 + 1 formulation of the general relativity field equations

Shirokov's effect is reexamined and analyzed with the help of a monadic method for defining reference systems. It is shown that the difference in the periods of oscillations is of the same nature as the phenomenon of the displacement of Mercury's perihelion. This effect is calculated in an analogous manner in Kerr's metric.

Small oscillations of test bodies in circular orbits in the Schwarzschild and Kerr metrics (Shirokov's effect)

In this note the general monad method is systematically represented, and it is shown how it may be reduced to its two basic special gauges. The last section deals with two kinds of canonical formalism, “coordinate” and “referential” ones, based on the kinemetric gauge.

V. I. Antonov

Papers

Monad method and canonical formalism of general relativity