Charged Black Holes and Multidimensional Gravity

A multidimensional generalization of the Reissner-Nordström solution of general relativity is obtained for the case of n Ricci-flat internal spaces. A two-parameter family of black-hole solutions for an arbitrary dimensionality D is selected. Nontrivial black holes with D > 4 are shown to exist only with a nonzero electric charge. Observational consequences are discussed, in particular, a violation of Coulomb's law.     

Charged black holes and additional dimensions

A multidimensional generalization is obtained for the Reissner-Nordstrom solution of general relativity theory for the case of n Ricci-plane internal spaces with inclusion of the dilaton field. A two-parameter family of solutions that describe black holes of arbitrary dimensionality D is identified. It is shown that nontrivial black holes with dimensionality D > 4 exist only for nonzero electric charge. The observational consequences, particularly a violation of Coulomb's law, are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 24–28, July, 1991.     

Decomposition of the space of covariant two-tensors on R3

We show that the decomposition of the space of covariant two-tensors onR ³ is true in weighted Hölderian spaces, as in weighted Sobolev spaces, in the general case, that is without supposing the metric near the flat metric. M. Cantor proved, first, that a splitting of two-covariant tensor fields onR ⁿ in weighted Sobolev spaces was true. We apply this result to solve the problem of constraints, in general relativity; we show that this problem admits a solution in the most general case.     

Kaluza-Klein cosmology: Phenomenology and exact solutions with three-component matter

We explain the general fact that Friedmann models in Kaluza-Klein cosmologies, in which ordinary space-time is supplemented by internal factor spaces, are equivalent to the motion of tensorial-mass particle in a scalar field. We find exact solutions for an important class of three-component matter in the case of one internal space of dimensiond. The three components in question are the curvature of the internal space, the Zeldovich matter, and dust of the protoradiation type. The method includes one-component solutions for all the different models of compactification discussed so far.     

Conformal flatness, non-Abelian Kaluza-Klein reduction and quaternions

The non-Abelian Kaluza-Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group Sp(d/4). Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza-Klein spaces.     

Kaluza-Klein Cosmology: Friedmann Models with Phenomenological Matter

We reduce the FRIEDMANN models in generalized KALUZA -KLEIN cosmologies, in which the ordinary space-time is supplemented by internal factor spaces, to the motion of a tensorial mass particle in a scalar field. Some general properties of these models as well as exact solutions for the case of one internal space are discussed for different mixtures of phenomenological matter components.     

Path integration on Darboux spaces

In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D _I-D _IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.     

Supersymmetries and Constants of Motion in Taub-NUT Spinning Space

We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT spinning space. A simple exact solution, corresponding to trajectories lying on a cone, is given.     

Integration of the einstein-dirac equations for steckel spaces of type (3.1)

We consider the problem of classification of Steckel spaces satisfying the system of self-consistent Einstein-Dirac equations with a cosmological term for the case when the spaces allow for isotropic complete sets of integrals of motion of type (3.1). The exact solution obtained contains four arbitrary functions of one variable. Tomsk State Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 3–9, February, 1997.     

Integrable multicomponent perfect fluid multidimensional cosmology. I

Multidimensional cosmological models with a space-time consisting ofn (n 2) Einstein spaces are investigated for a special class of multicomponent perfect fluid as a matter source. The dynamical behaviour of the universe is described. In the case of static internal spaces the external space evolves like a Friedmann universe with changing effective equation of state. Some of the models considered are integrable and classical as well as quantum solutions are found. Some of them represent wormholes. Quantum wormholes have a discrete spectrum.     

How Model Sets Can Be Determined by Their Two-point and Three-point Correlations

We show that real model sets with real internal spaces are determined, up to translation and changes of density 0, by their 2- and 3-point correlations. We also show that there exist pairs of real (even 1D) aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose 2- and 3-point correlations are identical but which are not related by either translation or inversion of their windows. All these examples are pure point diffractive. Placed in the context of ergodic uniformly discrete point processes, the result is that real point processes of model sets based on real internal windows are determined by their second and third moments.     

Perturbative effective theory in an oscillator basis?

The effective interaction problem in nuclear physics is believed to be highly nonperturbative, requiring extended high-momentum spaces for accurate solution. We trace this to difficulties that arise at both short and long distances when the included space is defined in terms of a basis of harmonic oscillator Slater determinants. We show, in the simplest case of the deuteron, that both difficulties can be circumvented, yielding highly perturbative results in the potential even for modest (approximately 4 variant Planck's over 2pi omega) included spaces.     

SU(3) ⊗ SU(2) ⊗ U(1) from D = 11 supergravity

In this paper we show that D = 11 supergravity admits an infinite discrete class of solutions having the phenomenological group SU(3) ? SU(2) ? U(1) as a symmetry of the internal space M₇. These solutions lead, in dimensional reduction, to SU(3) ? SU(2) ? U(1) gauge fields.In general all these spaces produce a complete breaking of supersymmetry except in one case where N = 2 supersymmetry survives. The parameter which classifies the solutions is a rational number q/p which describes the embedding of the stability subgroup SU(2) ? U(1) ? U(1) of M₇ in SU(3) ? SU(2) ? U(1). For all q/p ≠ 1 the holonomy group is SO(7) and all supersymmetries are broken. For q/p = 1 the holonomy group is SU(3) and two supersymmetries survive. In this last case we can also find a solution with internal photon curl $\text{F}{}_{\mathrm{\alpha \beta \gamma \delta }}\text{≠ 0}$. It breaks all sypersymmetries.     

Non-static electromagnetic fields in general relativity. IV

An investigation is made of Rainich's ‘already unified field theory’ from the standpoint of continuous groups of motions, and a solution is obtained for the case where the electromagnetic field is non-static and the space-time admits a three-parameter continuous group of motions whose minimum invariant varieties are three-dimensional spaces. There exists a divergence-free electromagnetic field for values of t at which the metric is not singular.     

A four-dimensional \Lambda CDM-type cosmological model induced from higher dimensions using a kinematical constraint

A class of cosmological solutions of higher dimensional Einstein field equations with the energy-momentum tensor of a homogeneous, isotropic fluid as the source are considered with an anisotropic metric that includes the direct sum of a 3-dimensional (physical, flat) external space metric and an $n$ -dimensional (compact, flat) internal space metric. A simple kinematical constraint is postulated that correlates the expansion rates of the external and internal spaces in terms of a real parameter $\lambda $ . A specific solution for which both the external and internal spaces expand at different rates is given analytically for $n=3$ . Assuming that the internal dimensions were at Planck length scales when the external space starts with a Big Bang ( $t=0$ ), they expand only 1.49 times and stay at Planck length scales even in the present age of the universe (13.7 Gyr). The effective four dimensional universe would exhibit a behavior consistent with our current understanding of the observed universe. It would start in a stiff fluid dominated phase and evolve through radiation dominated and pressureless matter dominated phases, eventually going into a de Sitter phase at late times.     

Einsteins Feldtheorie mit Fernparallelismus und Diracs Elektrodynamik. III (Statische kugelsymmetrische Lösung der genäherten Feldgleichungen)

Einstein's Field Theory with Teleparallelism and Dirac's Elektrodynamics. III (Static, Spherically Symmetric Solution of the Approximated Field Equations) The static, spherially symmetric solution of the equation of gravitation, considered by Treder [1] and Kreisel [2] and connecting the electromagnetic potential by the Dirac-gauge with the time-like tetrad, is given for the case of small contributions of the space-like tetrads, which corresponds to Einstein-Dirac's electrodynamics. The threedimensional spaces of the solution are two-bladed. The two blades are connected by a wormhole. The solution has a zero of the determinant the metric in the second blade at r = ∞. Therefore the manifold is geodesically incomplete.     

Accelerating cosmologies from spacelike branes

We point out that the recently proposed model of a flat four-dimensional universe with accelerated expansion in string or M theory is a special case of time-dependent solutions that the author found under the name of spacelike (S) branes. We also show that similar accelerating models can be obtained from S branes if the internal space is chosen to be hyperbolic or flat spaces.     

Physical vacuum properties and internal space dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization, however, is not unique. The case of 7-dimensional Riemannian space of signature 7(−) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of the Lie algebra E ₈. Considerations are presented, from which it follows that the least-dimen-si-on space bearing on physics is the Riemannian 11-dimensional space of signature 1(−)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density. Alexander Vasil'evich Pushkin was born on 13 April 1947 in St. Petersburg and died on 17 August 2004 in Sarov.     

$\mathcal{PT}$ Invariant Complex E8 Root Spaces

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E ₈-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser-Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.