Rotating 5D-Kaluza-Klein Space-Times from Invariant Transformations

Using invariant transformations of the five-dimensional Kaluza-Klein (KK) field equations, we find a series of formulae to derive axially symmetric stationary exact solutions of the KK theory starting from static ones. The procedure presented in this work allows to derive new exact solutions up to very simple integrations. Among other results, we find exact rotating solutions containing magnetic monopoles, dipoles, quadrupoles, etc., coupled to scalar and to gravitational multipole fields.     

Static solutions of the Brans-Dicke equations

It is shown that, given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection. Spherically and axially symmetric fields are considered explicitly. It is demonstrated how some solutions of the Brans-Dicke equations may be obtained without having to solve any field equations explicitly at all.     

Odd-Soliton Solutions to the Einstein Equations

Using the Inverse Scattering Method (ISM) of Belinskii and Zakharov a new odd-soliton solutions to the Einstein's field equations for an axially symmetric space-time in general relativity are obtained in the determinant form and shown to include Weyl's half-integral delta static solutions in a special case.     

Linearization procedure for cylindrically and axially symmetric space-times

An investigation of the vacuum Einstein gravitational field equations for cylindrically and axially symmetric space-times is presented which leads to an equivalent differential system involving a simple nonlinearity only. The case when this equivalent system is linear is analyzed in detail and two methods for generating solutions of the Einstein vacuum equations are set up. As a result, in the axially symmetric case the linearity of the equivalent system characterizes completely the Kramer-Neugebauer transforms of Papapetrou line elements. Accordingly, Weyl solutions are shown to generate exhaustively both Lewis and van Stockum solutions. Analogous results are obtained also in the cylindrically symmetric case.     

The axially symmetric non-rotating vacuum solutions of Rosen's equations

It is shown that all axially symmetric non-rotating solutions of Rosen's field equations can be expressed in terms of two harmonic functions as well as that the total energy of Rosen's metric isMc ².     

Axially symmetric stationary electrovac solutions

Perjes and Israel and Wilson have given independently a new class of solutions of the sourcefree Einstein-Maxwell equations, which can be interpreted as the external gravitational and electromagnetic fields of a spinning source with unit specific charge. Starting from Zipoy's solutions in oblate and prolate spheroidal coordinates for the source-free gravitational field we generate some axially symmetric stationary solutions of the source-free Einstein-Maxwell equations by using Perjes' method. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are studied in order to gain some insight into the nature of the source. The solution in prolate spheroidal coordinates is found to contain closed timelike lines.     

Axially symmetric solutions of einstein equations

Two exact axially symmetric solutions of the gravitational field equations, which depend on a number of arbitrary real constants, are derived.     

Anisotropic Bianchi-I universe with phantom field and cosmological constant

We study an anisotropic Bianchi-I universe in the presence of a phantom field and a cosmological constant. Cosmological solutions are obtained when the kinetic energy of the phantom field is of the order of anisotropy and dominates over the potential energy of the field. The anisotropy of the universe decreases and the universe transits to an isotropic flat FRW universe accommodating the present acceleration. A class of new cosmological solutions is obtained for an anisotropic universe in case an initial anisotropy exists which is bigger than the value determined by the parameter of the kinetic part of the field. Later, an autonomous system of equations for an axially symmetric Bianchi-I universe with phantom field in an exponential potential is studied. We discuss the stability of the cosmological solutions.      

One-soliton solutions from Laplace’s seed

One-soliton solutions of axially symmetric vacuum Einstein field equations are presented in this paper. Two sets of Laplace’s solutions are used as seed and it is shown that the derived solutions reduce to some already known solutions when the constants are properly adjusted. An analysis of the solutions in terms of the Ernst potential is also presented. It is found that the solutions do not reduce to the Euclidean form at spatial infinity. However, in the static limit, Weyl solutions are obtained for half integral ∂-values.     

Analytical properties and exact solutions of static plasma equilibrium systems in three dimensions

For static reductions of isotropic and anisotropic magnetohydrodynamics plasma equilibrium models, a complete classification of admitted point symmetries and conservation laws up to first order is presented. It is shown that the symmetry algebra for the isotropic equations is finite-dimensional, whereas anisotropic equations admit infinite symmetries depending on a free function defined on the set of magnetic surfaces. A direct transformation is established between isotropic and anisotropic equations, which provides an efficient way of constructing new exact anisotropic solutions. In particular, axially and helically symmetric anisotropic plasma equilibria arise from classical Grad-Shafranov and JFKO equations.     

On axially symmetric solutions to the Higgs-Kibble-Yang-Mills field equations

The field equations for axially symmetric generalizations of the 't Hooft-Polyakov magnetic monopole are written out in a preferred coordinate system. It is argued that no soliton solutions exist when the product 2eg of the electric chargee with the magnetic chargeg is an even integer.Dedicated to Achille Papapetrou on the occasion of his retirement.     

Axially symmetric solutions in general scalar-tensor theory

We generalise Ernst's derivation of the axially symmetric solutions of Einstein's field equations to the general scalar-tensor theory proposed by Nordtvedt. The solution of the Nordtvedt theory differs by a conformai transformation from the Brans-Dicke solution. The Kerr-like solution of the Nordtvedt theory is obtained as an example.     

Bäcklund transformations for chiral field equations

We give a general procedure to obtain Bäcklund transformations for nonlinear partial differential equations derived as compatibility conditions between some given generalized Lax pair of operators. We apply this technique to obtain a set of Bäcklund transformations for three-dimensional and axially symmetric two-dimensional chiral field equations.     

Non-existence of stationary,axially symmetric,asymptotically flat solutions of the Einstein equations for dust

It is shown that the Einstein equations for stationary, axially symmetric distributions of dust do not admit regular, asymptotically flat solutions, provided the mass density is strictly positive everywhere.     

Rotating,charged, and uniformly accelerating mass in general relativity

A new general class of solutions of the Einstein-Maxwell equations is presented. It depends on seven arbitrary parameters that group in a natural way into three complex parameters m + in, a + ib, e + ig, and the cosmological constant λ. They correspond to mass, NUT parameter, angular momentum per unit mass, acceleration, and electric and magnetic charge. The metric is in general stationary and axially symmetric. These solutions are of type D and contain as special cases all known solutions of type D belonging to this class. The known solutions are recovered by performing limiting transitions. An appropriate limit of our solutions describes an electromagnetic field in flat spacetime. We investigate the properties of that field. Its singular region corresponds in general to two circles moving with uniform acceleration in the positive and negative directions along the axis of symmetry. One can easily extend our solutions to the complex domain. Then it turns out that the metric can be written in a double Kerr-Schild form.     

Two-Soliton Solutions of Axially Symmetric Metrics

Stationary axially symmetric solutions of Einstein's field equations generated by the soliton technique are presented in this paper with Laplace's solution as seed. The solutions are asymptotically flat and the Schwarzschild, Kerr and Kerr-NUT metrics are contained in it. The constructed solutions possess an event horizon. The surface area of the event horizon is evaluated. Finally, the solutions presented in this paper are compared with the solutions of Manko. A simple transformation technique is discussed by which one can directly obtain the solutions of Gutsunaev and Manko simply by adjusting a parameter related to the Inverse Scattering Method.     

Periastron shift in Weyl class spacetimes

The periastron position advance for geodesic motion in axially symmetric solutions of the Einstein field equations belonging to the Weyl class of vacuum solutions is investigated. Explicit examples corresponding to either static solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or stationary solutions (single rotating Chazy-Curzon and Kerr black hole) are discussed. The results are then applied to the case of S2-SgrA^* binary system of which the periastron position advance will be soon measured with a great accuracy.     

A new construction technique for exact axially symmetric static electrovac solutions of the Einstein-Maxwell equations

Coordinates adapted to the full electrostatic symmetries of a charged axially symmetric static system are constructed and applied to the coupled Einstein-Maxwell equations. A new solution is presented and the framework for general solutions is developed. The Weyl solutions g₀₀(Φ) are readily extracted from the formalism.     

The axially symmetric flow-through problem for the Navier-Stokes equations in variables “vorticity-stream function”

The axially symmetric steady-state motion of a viscous incompressible fluid in a region of the type of a spherical layer is considered. The boundary problem for the Navier-Stokes equations is formulated in terms of the stream function and vorticity. The solvability of this problem was previously established provided that the fluid flow rate through each layer boundary is sufficiently small. The main result of this study is in proving the solvability of the axially symmetric flow-through problem without limitations on the magnitude of the flow rate. The proof is based on the a priori Dirichlet estimate of the flow velocity field. The asymptotic properties of the solution near the symmetry axis are established conditionally. The proof procedure admits the generalization to the case of axially symmetric problems in multiply-connected domains.