Algebraic classification of the gravitational field

We rewrite the tensorial method of Peres [1] for the Petrov classification in terms of the Newman-Penrose formalism [2]. It results an algorithm computationally simpler that the one proposed by DInverno-Russell Clark [3].This revised version was published online in April 2005. The publishing date was inserted.     

Periodic fields in five-dimensional relativity

The concept of periodic fields in the framework of five-dimensional relativity is introduced in a 5-covariant manner. An analysis is given of periodic scalar, vector and bivector fields. The resulting 4-dimensional theories are examined, and it is found that some of them correspond to familiar physical theories in space-time, whereas others require further investigation.     

Some structural features induced by the space-time metrical fluctuation in the theory of gravitational fields

Under the assumption that the so-called space-time fluctuationy(x) in a classical sense, attached to each point of the gravitational field at some microscopic stage, is summarized as the metrical fluctuation in the formg _λκ (x)=g_λκ (x)·exp2σ(y(x)), some new physical aspects induced by the conformal scalarσ(x) (≡σ(y(x))) are found: By introducing the torsionT _κ ^λμ (x) from a general standpoint, the resulting micro-gravitational field is made to have a conformally non-Riemannian structure, where a special form ofT _κ ^λμ (i.e.,T _κ ^λμ =δ _κ ^λ σ_μ?δ _κ ^μ σ_λ(σ_μ=?σ/?x ^μ)) shows some peculiar features. An averaging process with respect toy is taken into account, by which the spatial structure of the corresponding macro-field is shown, in general, to have a somewhat “non”-Riemannian structure due to the contributions of the torsionT _κ ^λμ .     

Quantum space-time and gravitational consequences

Relativistic particle dynamics and basic physical quantities for the general theory of gravity are reconstructed from a quantum space-time point of view. An additional force caused by quantum space-time appears in the equation of particle motion, giving rise to a reformulation of the equivalence principle up to values ofO(L ²), whereL is the fundamental length. It turns out that quantum space-time leads to quantization of gravity, i.e., the metric tensorg _v () becomes operator-valued and is not commutative at different pointsx andy in usual space-time on a large scale, and its commutator depending on the vielbein field (gaugelike graviton field) is proportional toL ² multiplied by a translation-invariant wave function propagated between pointsx andy . In the given scheme, there appears to be an antigravitational effect in the motion of a particle in the gravitational force. This effect depends on the value of particle mass; when a particle is heavy its free-fall time is long compared to that for a light-weight particle. The problem of the change of time scale and the anisotropy of inertia are discussed. From experimental data from testing of the latter effect it follows thatL10^–22cm.     

Phantom scalar fields in five-dimensional Kaluza-Klein theory

The existence of a so-called "phantom" scalar field in some Riemannian spaceV ₄, i.e., a field in which the effective energy momentum tensorT ^(sf) vanishes in the five-dimensional Kaluza-Klein theory, is investigated by means of the integrability conditions for relations of the form _;;=k_,,+bg found in [6]. Phantom fields are found in homogeneous isotropic cosmological models.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–56, February, 1996.     

Another gravitational instability of flat space-time

Witten's demonstration of the instability of the five dimensional Kaluza-Klein ground state can be reduced to four, physical dimensions. One can then conclude that flat space-time at zero temperature with one of the spatial directions wrapped up is unstable. The decay rate is of order exp?L²/16πL²_p where L is the spatial periodicity and L_p is the Planck length. The post tunnelling evolution of the space-time is also discussed.     

Axisymmetric gravitational fields

In this review paper Einstein equations for axisymmetric vacuum fields are introduced in the form given by Lewis. Following Ernst they are reduced to one complex potential equation. Weyl-type, Schwarzschild, Kerr, Tomimatsu-Sato solutions, and their NUT-like generalizations are then discussed.     

Global spinor fields in space-time

This paper is concerned with space-time manifolds that are space- and time-oriented, causal, and possess spinor structures. Five propositions are proven: (1) If a connected, space- and time-oriented manifold is simply con-nected, then it is non-compact; (2) If such a manifold is simply connected, it admits a spinor structure, which, moreover, is unique; (3) If the space-like section of M is compact, then there exists a global system of orthonormal tetrads on M; (4) The necessary and sufficient condition for every space-time M whose space-like section is compact to admit a spinor structure is that M have a global system of orthonormal tetrads; (5) Every space-time M which can be imbedded in R⁶ admits a spinor structure. It is further suggested that in view of the fact that the existence of a spinor structure is related to homotopy properties, space-time manifolds may be classified in terms of their homotopy groups _i (M), i=1,2, 3,4. In a concluding section, some avenues for future research are discussed.Supported by the National Science Foundation.     

Two theorems on flat space-time gravitational theories

The first theorem states that all flat space-time gravitational theories must have a Lagrangian with a first term that is an homogeneous (degree-1) function of the 4-velocityu ⁱ , plus a functional of _ij u ⁱ u ^j . The second theorem states that all gravitational theories, that satisfy the strong equivalence principle have a Lagrangian with a first termg _ij (x)u ⁱ u ^j plus an irrelevant term. In both cases the theories must issue from a unique variational principle. Therefore, under this condition it is impossible to find a flat space-time theory that satisfies the strong equivalence principle.     

Stationary gravitational fields from static fields

In this note it is shown that the method of Misra and Pandey for generating stationary axially symmetric fields from static ones can only give rise to a class of solutions already known.     

Singularities in Weyl gravitational fields

The peculiarities of the scalarS ≡R _ijkl R ^ijkl are exhibited for two axially-symmetric static (Weyl) gravitational fields. By examiningS along curved families of trajectories to the Weyl singularities, examples are found which contradict previous claims by Gautreau and Anderson regarding ‘directional singularities’. Proper circumferences about the Bach and Weyl line-mass singularity are also examined. There is no apparent correlation between the source structure and the behaviour ofS from this analysis.     

Mass-energy in static gravitational fields

The mass-energy equation in static gravitational fields is shown to beE =g ₄₄ mc ², which agrees with the expressionE =m ₀ c ²(dx/ds)ξ_μ for the energy in a gravitational field possessing a timelike Killing vector ξ. For the Schwarzschild field this leads toE _s ?m ₀ c ² + 1/2m ₀ v ² ?km ₀ M/r. For the Reissner-Nordström field an additional term describing the interaction between the mass and the charge is found to be 2πkm ₀ Q ²/c ² r ². In the Kerr-Newman case more terms are found due to the central rotating gravitating mass.     

Neutrino radiation in gravitational fields

A differential equation representing radiation solutions of the general relativistic Weyl equation is derived. Their optical properties and the group of motion of the corresponding energy-momentum tensor are studied. If there exists neutrino radiation the Riemann space must be algebraically special and the propagation of the neutrinos occurs only along one of the principal null directions. Gravitational- and neutrinopp-waves taken together, represent an exact solution of the Weyl-Einstein system of field equations.     

Ghost neutrino fields in flat space-time

The Weyl neutrino equation is integrated in flat space-time assuming that the energy-momentum tensor of the neutrino field vanishes. It is shown that the flux vector of the neutrino field is tangent to a twist-free and shear-free congruence of null geodesics, which is a special Robinson congruence and constitutes a geometrical representation of a null twistor. It is also shown that, conversely, given such a congruence, a ghost neutrino field can be constructed.     

Higher-spin fields in a curved space-time

Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.     

A classification of space-time structures

The paper contains a review of various bundles which may be associated to the bundle of linear frames and used to describe properties of space relevant to physics. Restrictions, extensions, prolongations and reductions are defined in terms of morphisms of principal bundles. It is shown that the holonomic prolongation of a G-structure exist iff the corresponding structure function vanishes. G-connections are related to restrictions of the bundle of second-order frames. It is shown that these restrictions may be used to classify theories of space-time and gravitation. A distinction is made between a projective connection and a geodetic structure. In the framework of the Einstein-Cartan theory, the projective connection of a space-time is compatible with its metric tensor iff the spin density is bivector-valued. As an example, we mention a new theory of gravitation and electromagnetism based on the Weyl-Cartan structure of space-time and on the Yang quadratic Lagrangian.     

Matter fields in the Plebanski-Demianski space-time

We consider the boson (electromagnetic and gravitational) fields and the fermion (electron and neutrino) fields in the Plebanski-Demianski space-time. In the case of the boson field, we observe that the superradiance phenomena occurs, but in the case of the fermion field we find no super-radiance phenomena