Factor structure of the Tomimatsu‐Sato solutions and their recurrence relations

We consider stationary axisymmetric vacuum solutions of Einstein's equations for which the Ernst potential is rational in prolate spheroidal coordinates. Extending an earlier study we show that there are several new expressions which are factorizable. In particular, we concentrate on the Tomimatsu‐Sato solutions and their recurrence relations. Various continuum limits of the recurrence relations will be discussed.     

Axially symmetric solutions generated by the complexification technique

Banerjee and Andrade generated axially symmetric electrovac solutions from the spherically symmetric Reissner-Nordström solution by giving a complex translation along theZ axis. We use the same method for generating new more complicated axially symmetric solutions from known solutions. Several such examples are given.     

Classification of stationary space-times

A systematic approach to the geometric structure of stationary gravitational fields is presented. The algebraic type of the trace-free Ricci tensor together with the propagation properties of the eigenrays in the background 3-space defined by the Killing trajectories serve as a basis for classifying the solutions of the stationary field equations. The eigenrays are the integral curves belonging to the solutions _A of the eigenvalue problemG _A ^B _B=_A,G _A ^B spinor representing the gravitational field in the background space. Many of the already known stationary metrics can be derived in the present scheme but new solutions of the field equations are also obtained. The possible types of the vacuum and electrovac fields are discussed in their connection with the corresponding exact solutions.Work honoured by a Fifth Gravity Research Foundation Award in 1973.Leverhulme Visiting Fellow.     

Electrovac fields with geodesic eigenrays

The stationary states of electrovac fields for which the geodesic eigenrays of both the Maxwell and gravitational field coincide are investigated. The fact that the Kerr-Newman solutions belong to this class lends physical interest to the fields considered here. Particular attention is devoted to fields with shearing eigenrays since principal null congruences do not coincide with eigenrays in this case, and so earlier approaches to the problem fail. By the generalisation of a theorem on the corresponding vacuum case, it is proven that no spherical solutions exist in the shearing class, with the exception of the fields with ²¦G_O¦²-¦H_O¦²=O.The metrics with O, admitted by the theorem, can either be calculated from the corresponding vacuum solutions by a relatively simple procedure, or, if not, we list them explicitly in this paper.     

Event horizons in static electrovac space-times

The following theorem is established. Among all static, asymptotically flat electrovac fields with closed, simply-connected equipotential surfacesg _{0 0}=const.. the only ones which have regular event horizonsg _{0 0}=0 are the Reissner-Nordström family of spherisymmetric solutions withmG ^1/2|e|/c. In the special case where the gravitational coupling of the electromagnetic energy density is neglected (G=0) all solutions are computed explicitly, thus extending an earlier result ofGinzburg for a magnetic dipole inSchwarzschild's space-time. Possible implications for gravitational collapse are briefly discussed.On leave of absence from the Mathematics Department, University of Alberta, Edmonton, Canada.     

Axially symmetric electrovac solutions

Two classes of electrovac solutions are obtained in oblate spheroidal coordinates, which are the electromagnetic analogs of Zipoy's monopole and dipole solutions. The asymptotic behavior of the solutions is studied to gain some insight into the nature of the source of the gravitational and electromagnetic fields. A similar stationary solution of the pure gravitational field is found to belong to Papapetrou's class.     

Stationary axially symmetric electrovac solutions in the general scalar-tensor theory

A method is presented which reduces the Bergmann-Wagoner-Nordtvedt field equations for a stationary axisymmetric electrovac space-time, to the Einstein-Maxwell equations. In this formalism the solution generation technique of Singh and Rai for Brans-Dicke theory yields a particular class of solutions, for which the conformal scalar field depends upon the radial coordinate only. As an application of the method, new cylindrically symmetric and nonstatic scalar-Maxwell solutions are obtained for null and non-null electromagnetic fields.     

Inverted black holes and anisotropic collapse

The structure of static vacuum and electrovacuum fields with the most simple symmetry is discussed within the framework of the general theory of relativity. The general electrovacuum fields with plane and pseudospherical symmetries and the special case of fields with cylindrical symmetry contain horizons with the R and T regions interchanged by comparison with the Schwarzschild solution (the inverted black hole structure); the horizons disappear in the presence of a massless scalar field and for small deviations from plane symmetry. The physical consequences of this structure for the simplest models of anisotropic collapse are briefly discussed. A simple method is described for constructing the Penrose diagrams for a given metric, and examples are given of the construction of new diagrams for the electrovacuum solutions with a term.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 32–38, June, 1979.     

Gravitation and Electrodynamics over SO(3,3)

In a series of papers, an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in space-time. The extra dimensions are given a physical meaning as coordinatized matter. The inhomogeneous energy-momentum relations for the interacting fields in space-time are automatically generated by the simple homogeneous relations in 6-d. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variational principle. The Einstein–Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.     

Reciprocity and equivalence in reciprocal and non-reciprocal media through reflection transformations of the current distributions

The electromagnetic wave fields generated by arbitrary electric and equivalent magnetic current distributions are expressed by means of a Maxwell operator in anisotropic, gyrotropic or bianisotropic media. Provided that the constitutive tensorK(r), (which relates the wave-field vectorsD andB toE andH), has in each case the appropriate symmetry in its spatial variation, Lorentz-type reciprocity relations are obtained connecting the given current distributions and their wave fields with a transformed (reflected) set of current distributions and their fields. Reflections are with respect to a plane, a line or a point, depending on the symmetry structure of the constitutive tensor. Modified Lorentz reciprocity appears as a special case of the reflection transformations. A related set of reflection transformations yields equivalence (rather than reciprocity) relationships, in which mirrored current distributions generate mirrored wave fields. Various applications are discussed.     

Space-time compactification induced by nonlinear sigma models,gauge fields and submersions

The nonlinear sigma model coupled to gravity in (4+K) dimensions is used to trigger the compactification of space-time. A very general class of solutions is given by submersions from the extradimensional space onto the space in which the scalar fields take values. The existence of some vertical Killing vectors on the extra space can produce massless gauge bosons.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Solutions of the Field Equations and Possible Meaning of the Hermitian Theory of Relativity

A procedure is stated, which allows to build solutions of the Hermitian theory of relativity from known solutions of the general theory of relativity. Solutions depending on three co-ordinates, built from Minkowski metric, as well as Hermitian generalizations of the Weyl-Levi Civita solution are shown. They suggest that the imaginary part of the fundamental tensor may encompass fields of different physical behaviour, like the electromagnetic field and a field responsible for forces which do not depend on the distance between charges which cannot exist as individuals. In the generalizations of the Weyl-Levi Civita solution these fields appear to be decoupled from gravitation in a peculiar way.     

Black holes in static electrovac space-times

The theorem of Israel which characterizes the Reissner-Nordström solutions as the only well behaved asymptotically flat electrovac spaces with a simple regular horizon is extended by weakening the assumptions. Critical points of the gravitational potential are not a priori excluded and the topology of the eguipotential surfaces is not restricted. The regularity of the horizon is formulated in terms of bounds for certain geometrical quantities and the assumption of existence, in some extension, of a bifurcation surface for the horizons is not made. The possibilities of non-static or non-conservative electromagnetic fields in a static space-time are discussed and excluded by physical arguments.     

Point Interactions in One Dimension and Holonomic Quantum Fields

We introduce and study a family of quantum fields, associated to δ-interactions in one dimension. These fields are analogous to holonomic quantum fields of Sato et al. in Holonomic quantum fields I–V (Publ. RIMS, Kyoto University, 14: 223–267, 1978; 15: 201–278, 1979; 15: 577–629, 1979; 15: 871-972, 1979; 16: 531–584, 1979). Corresponding field operators belong to an infinite-dimensional representation of the group in the Fock space of ordinary harmonic oscillator. We compute form factors of such fields and their correlation functions, which are related to the determinants of Schroedinger operators with a finite number of point interactions. It is also shown that these determinants coincide with tau functions, obtained through the trivialization of the det^*-bundle over a Grassmannian associated to a family of Schroedinger operators.     

Induced geometry and confinement

A classical, Poincaré invariant dynamical system is developed which contains, besides the natural metric _v , an induced metricg _v that is generated by a real scalar dynamical field. It is shown that scalar fields whose dynamics are governed by the induced metric can be consistently introduced. Also, point particles which follow timelike quasi-geodesic trajectories can be introduced. The reaction forces acting ong _v due to the presence of these fields and particles are computed. A discussion of causality and geometrical confinement is given.     

The Ernst equation in general relativity

Stationary axisymmetric gravitational fields are governed by the Ernst equation. Its internalSU(1, 1) symmetry gives rise to a linear problem and to Bäcklund transformations which map known solutions into new ones. The main features of the solution generating techniques are outlined, the generalization to Einstein-Maxwell fields is discussed and some applications are given.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.This paper summarizes some common work with G. Neugebauer. I would like to thank him for many stimulating discussions.     

The spontaneous generation of magnetic fields at high temperature in a supersymmetric theory

The spontaneous generation of magnetic and chromomagnetic fields at high temperature in the minimal supersymmetric standard model is investigated. The consistent effective potential including the one-loop and the daisy diagrams of all bosons and fermions is calculated and the magnetization of the vacuum is observed. The mixing of the generated fields due to the quark and s-quark loop diagrams and the role of superpartners are studied in detail. It is found that the contribution of these diagrams increases the magnetic and chromomagnetic field strengths as compared with the case of a separate generation of fields. The magnetized vacuum state is found to be stable due to the magnetic masses of gauge fields included in the daisy diagrams. Applications of the results obtained are discussed. A comparison with the standard model case is given. Received: 18 October 2002 / Revised version: 23 December 2002 / Published online: 3 March 2003 RID="a" ID="a" e-mail: vadimdi@yahoo.com RID="b" ID="b" e-mail: skalozub@ff.dsu.dp.ua     

3-Dimensional “relativity” for axisymmetric stationary space-times

The equivalence of the axisymmetric stationary vacuum gravitational field problem to a 3-dimensional relativity theory in the presence of a certain scalar matter field is shown. An invariant classification can be achieved with respect to the algebraic structure of the 3-dimensional trace-free Ricci-tensor. The extension of these results to electrovac spaces is also discussed.