Stationary Axisymmetric Vacuum Research Articles

Solitons in five-dimensional gravity

The application of the generalized Backlund transformation to the projective field theory equations for stationary axisymmetric vacuum fields leads us to an explicit formula for the electric, magnetic, rotational, gravitational, and scalar potentials of new exact solutions generated from an arbitrary seed solution.

The conformal potential of the stationary axisymmetric vacuum

AbstractIn a stationary axisymmetric vacuum gravitational field, the conformal structure of the 3‐space is determined by the symmetric, trace free and divergence‐less tensor Yir. Using the Killing vector Ki of the axisymmetry, the conformal potential U can be defined by U,i = εijkKjYkrKr. Conversely, the tensor Yik is given algebraically in terms of the gradient U,i of the conformal potential. An attempt is made here to re‐formulate the field equations Rλμ = 0 in terms of the conformal potential. Introducing the Ernst potential as a complex coordinate, the cylindrical radius can be eliminated from the field equations.

The use of generating techniques for space-times with two non-null commuting Killing vectors in vacuum and stiff perfect fluid cosmological models

The SL(2) formalism of Kinnersley and Chitre (1978) for generating techniques in stationary axisymmetric vacuum space-times is adapted to work for space-times with two spacelike commuting Killing vectors. The Kinnersley-Chitre formalism is known to be explicitly related to the inverse scattering transformation of Belinskii and Zakharov (1979) which has already been applied in both types of above mentioned space-times. Using the hyperbolic versions of various Kinnersley-Chitre transformations the interrelation between various vacuum cosmological solutions is obtained. For each solution the Kinnersley-Chitre generating function FAB(t) and corresponding Belinskii-Zakharov eigenfunction psi ( lambda ) is given. These vacuum solutions are then converted to scalar field/stiff perfect fluid solutions using the algorithm of Wainwright, Ince and Marshman (1979).

A new class of asymptotically flat stationary axisymmetric vacuum gravitational fields

We present the full four-dimensional metric for a stationary axisymmetric solution of the vacuum Einstein equations related to the Ernst potential which we obtain by application of anarbitrary number of rank-zero HKX transformations to the general static Weyl solution. The metric function ω is determined in a completely algebraically way. We perform a suitable Ehlers transformation to ensure asymptotic flatness and give the expression for the total mass of this asymptotically flat solution by analyzing the behavior of the solution on the positive z axis.

Proof of a Geroch conjecture

We prove that any given stationary axisymmetric vacuum space-time (SAV) can be generated from Minkowski space by at least one Kinnersley–Chitre transformation, i.e., by at least one member of the Geroch group K, provided that the metric tensor and the Killing vectors are C3 in a domain which covers at least one point of the axis at which one of the Killing vectors characterizing the space-time is timelike. We find that the set of all Kinnersley–Chitre transformations which map any given SAV into another given SAV is uniquely determined by the initial and final values of the Ernst potential on the axis. An explicit formula for these K-C transformations in terms of the initial and final axis values is given; this formula generalizes an analogous one which Xanthopoulos found for the asymptotically flat SAV’s.

Soliton concept in general relativity

Soliton physics has made considerable progress in solving nonlinear problems. This paper is meant to relate the soliten concept to the stationary axisymmetric vacuum fields in general relativity. We present a functional transformation which, working as a nonlinear creation operator, generates gravitational fields of isolated sources. When applied to flat space-time (“gravitational vacuum”) this operation leads to a nonlinear superposition of an arbitrary number of Kerr particles. This superposition also includes the Tomimatsu-Sato fields. The functional transformations form an infinite-parameter group which contains the Kinnersley-Geroch group as a subgroup.

A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. II. Separable solutions

For pt.I see ibid., vol.11, no.12 (1978). The techniques of the preceding paper are applied to several cases where the gamma equation may be solved by separation of variables in the form, gamma = gamma 1( rho )+ gamma 2( tau ), where gamma 1( rho ) is either zero or a very simple function and gamma 2( tau ) satisfies an ordinary differential equation of the fourth order. Among the exact solutions constructed are the full six-parameter family of generalised Tomimatsu-Sato solutions, the rotating Curzon solution, the Kinnersley-Kelley solution and a class of solutions recently found by Ernst. Two new classes of solutions are presented as well as several new particular solutions expressible in closed form. All stationary axisymmetric vacuum metrics with a non-trivial second-rank Killing tensor whose components do not depend on the ignorable co-ordinates, phi and t, are derived. This problem reduces to finding separable solutions of the dual of the gamma equation of the form, e2 gamma -2u=R( rho , tau )(f( rho )+g( tau )), in four special co-ordinate systems, ( rho , tau ), where R( rho , tau ) is a prescribed simple function. A comparison is made with the canonical Schrodinger separable metric forms of Carter.

A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. I. General theory

A new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented. The basic variable is e2γ = -g11g44 and satisfies a field equation of the fourth differential order which may be interpreted as the condition that a certain 2-space has constant curvature, K = -1. The principal motivation is that for many known solutions and all known asymptotically flat (non-static) solutions, e2γ takes a much simpler functional form than either the metric coefficients, g44, g34 and g33, or the Ernst potentials, E and ξ . Three methods are given for the construction of the full metric from e2γ. A duality principle is invoked to provide a very similar field equation for the metric coefficient, e2γ-2u = -g11.

Limits of the generalised Tomimatsu-Sato gravitational fields

The author has previously presented a new three-parameter family of exact asymptotically flat stationary axisymmetric vacuum solutions of Einstein's equations which contains the solutions of Kerr and Tomimatsu-Sato (TS) as special cases. In this paper, he considers two special cases of the previous family which must be constructed by a limiting process. These are interpreted as a 'rotating Curzon metric' and a 'generalised extreme Kerr metric'. In addition, approximate forms for the original metrics are given for the case of slow rotation and small deformation.

New family of exact stationary axisymmetric gravitational fields generalising the Tomimatsu-Sato solutions

A new three-parameter family of exact solutions of the stationary axisymmetric vacuum Einstein equations, which represent rotating bounded sources, are presented. This family contains the solutions of Kerr and Tomimatsu-Sato as special cases, and may be regarded as a generalisation of the latter to arbitrary continuous delta parameter. The final form of the metric depends on two ordinary differential equations of the second order. When delta is not an integer, these equations define unfamiliar transcendental functions for which rapidly converging series expansions of several types are available. When delta is an integer, the solutions are polynomial or rational functions of spheroidal coordinates and define the discrete Tomimatsu-Sato series, for which those authors give the cases delta =1, 2, 3, 4.

On the stationary axisymmetric Einstein–Maxwell field equations

We show the existence of a formal identity between Einstein’s and Ernst’s stationary axisymmetric gravitational field equations and the Einstein–Maxwell and the Ernst equations for the electrostatic and magnetostatic axisymmetric cases. Our equations are invariant under very simple internal symmetry groups, and one of them appears to be new. We also obtain a method for associating two stationary axisymmetric vacuum solutions with every electrostatic known.

Structure of the Tomimatsu-Sato Gravitational Field

The new Tomimatsu-Sato solution is a stationary axisymmetric vacuum solution of the gravitational field equations which has been proposed as a possible external field for rotating stellar objects. The solution is a Petrov type-I rotating generalization of a static Weyl solution. It has the unusual feature of a naked singularity outside a Killing horizon. In this note, after the asymptotic flatness of the solution is demonstrated, the energy and angular momentum are calculated and found to be $m$ and $ma$, respectively. The locations of the horizons, infinite red-shift surfaces, curvature singularities, and closed timelike lines are given. The Killing horizon is found to be nonregular, and thus cannot be identified as an event horizon, contrary to the claim of Tomimatsu and Sato.