Stationary Axisymmetric Gravitational Fields Research Articles

Constants of motion in stationary axisymmetric gravitational fields

The motion of test particles in stationary axisymmetric gravitational fields is generally nonintegrable unless a nontrivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr-de Sitter spacetime is the only example known to date. Proposed astrophysical tests of the black-hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for nontrivial constants of motion polynomial in the momenta and obtain two theorems. First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr-de Sitter spacetime, the mass moments of this class satisfy a "no-hair" recursion relation $M_{2l+2}=a^2 M_{2l}$, and the constant is Noether-related to a second-order Killing-St\"ackel tensor. Second, solving a new set of quartic integrability conditions, we establish nonexistence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized "no-hair" recursion relation $M_{2l+4}=(a^2+b^2)M_{2l+2}-a^2b^2 M_{2l}$. The full set of quartic integrability conditions, however, cannot be satisfied nontrivially by any stationary axisymmetric vacuum potential.

Double Superpotential and Generation of New Solutions of Stationary Axisymmetric Gravitational Fields

In this paper COX’s superpotential is extended to double complex form firstly, then how to get hyperbolic complex superpotential from the known ordinary complex superpotential is discussed, finally, new solutions of stationary axisymmetric gravitational fields are shown with a few specific cases.

On the generalized Tomimatsu-Sato solutions

We consider stationary axisymmetric vacuum gravitational fields. We use the factor structure of rational solutions to derive two second order equations for two functions. From the solutions of those equations the Ernst potential can be found by quadrature. Non-rational solutions generalize the Tomimatsu-Sato solutions.

Maximal acceleration is non-rotating

In a stationary axisymmetric spacetime, the angular velocity of a stationary observer whose acceleration vector is Fermi-Walker transported is also the angular velocity that locally extremizes the magnitude of the acceleration of such an observer. The converse is also true if the spacetime is symmetric under reversing both t and together. Thus a congruence of non-rotating acceleration worldlines (NAW) is equivalent to a stationary congruence accelerating locally extremely (SCALE). These congruences are defined completely locally, unlike the case of zero angular momentum observers (ZAMOs), which requires knowledge around a symmetry axis. The SCALE subcase of a stationary congruence accelerating maximally (SCAM) is made up of stationary worldlines that may be considered to be locally most nearly at rest in a stationary axisymmetric gravitational field. Formulae for the angular velocity and other properties of the SCALEs are given explicitly on a generalization of an equatorial plane, infinitesimally near a symmetry axis, and in a slowly rotating gravitational field, including the far-field limit, where the SCAM is shown to be counter-rotating relative to infinity. These formulae are evaluated in particular detail for the Kerr-Newman metric. Various other congruences are also defined, such as a stationary congruence rotating at minimum (SCRAM), and stationary worldlines accelerating radially maximally (SWARM), both of which coincide with a SCAM on an equatorial plane of reflection symmetry. Applications are also made to the gravitational fields of maximally rotating stars, the Sun and the Solar System.

Linear double universal Grassmann manifold method for the stationary axisymmetric vacuum gravitational field equations

Nagatomo's universal Grassmann manifold scheme is extended to a double form, which is used to find the exact solutions of the stationary axisymmetric vacuum gravitational field equations. Some new results are given.

Double harmonic mappings of Riemannian manifolds and its applications to stationary axisymmetric gravitational fields

Harmonic mappings of Riemannian manifolds are discussed by a double complex function method, and the double-complex Ernst equation and the related Backlund transformations are naturally derived. Further, the Ernst solution and its dual solution are obtained by two different methods, respectively. Therefore, the results obtained by A. Eris are extended to a double form.

New exact solutions of Einstein's field equations: Gravitational force can also be repulsive!

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are 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 Backlund 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.

Stationary axisymmetric electrovacuum fields in the generalized theory of relativity

The problem of stationary axisymmetric gravitational fields is formulated in the framework of a generalized theory of gravitation. It is shown that this problem can be solved if the analogous solution in Einstein's theory of gravitation is known. A theorem is formulated on the basis of which it is possible, given the stationary vacuum solution to the problem, to find the metric in the magnetostatic case.

Shell sources of stationary axisymmetric gravitational fields

A number of solutions for material shell sources of stationary axisymmetric gravitational fields are presented. Explicit solutions are found for shells lying on equipotential hypersurfaces (gtt = const) and generating static “monopole” fields in prolate and oblate spheroidal coordinates (Zipoy-Voorhees fields). Numerical solutions are found for shells lying on hypersurfaces of constantgϕ/gϕϕ and generating Kerr- and Tomimatsu-Sato (δ = 2) fields. The shells have minimum areas allowed by the energy conditions of Hawking and Ellis.

Stationary axisymmetric gravitational fields in the bimetric theory

Stationary axisymmetric fields are considered in the framework of Rosen's bimetric theory of gravitation. Analytic solutions in the approximation quadratic in the angular velocity are found outside the mass distribution. A method is proposed for finding the metric within the mass distribution and determining the internal structure and the integrated parameters of rotating objects in which the state of the matter is described by a single-parameter equation.

Stationary axisymmetric fields in Generalized Theory of Gravitation

The problem of stationary axisymmetric gravitational fields is formulated within the framework of Generalized Theory of Gravitation. It is shown that solutions of the problem mentioned above may be found, if analogous solutions in General Relativity are obtained. As an illustration a Kerr-like solution is offered. A generation theorem for finding magnetostatic solution from stationary vacuum solutions is proposed.

Stationary axisymmetric post-Minkowskian metrics.

We obtain an approximate stationary axisymmetric gravitational field up to terms going as ${R}^{\mathrm{\ensuremath{-}}4}$ (${R}^{2}$=${x}^{2}$+${y}^{2}$+${z}^{2}$). It may be the exterior metric to a spacelike bounded distribution of matter. We check the reliability of our solution by comparing it with the Weyl metric. Sources with different multipole structures are found, depending on the system of harmonic coordinates we use to write down the metric.

The rankN HKX transformations: New stationary axisymmetric gravitational fields

The previous formulation ofN rank 0 HKX transformations by means of determinants is revisited. The new expressions for the metric functions permit the limits which yield the rankN transformations. As an instance we give the Ernst potential for such transformations acting on flat space.

Slow rotation in the bimetric theory of gravitation

Field equations and their exterior solutions in the approximate quadratic in angular velocity are obtained for stationary axisymmetric gravitational fields. The problem of rotating configurations consisting of a degenerate real baryon gas is solved in the framework of bimetric theory, and the results are compared with analogous ones in general relativity. It is shown that the results obtained by Falik and Opher (1979) are incorrect. 15 references.

Prolongation Structure Approach to the Group-Theoretical Technique for Generating Stationary Axisymmetric Space-Times

The group-theoretical technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)× structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem (HHP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.

Projection tensor formalism for stationary axisymmetric gravitational fields

A two-dimensional projection tensor formalism for treating spacetimes with two commuting Killing vectors is presented and its use is illustrated by two examples---an alternative proof of a sufficient condition for such a spacetime to be orthogonally transitive and an application to the Rainich-Misner-Wheeler geometrization of the electromagnetic field to derive a result previously obtained by Michalski and Wainright. Finally, the admissible Petrov types are investigated and it is proved, in particular, that all vacuum fields with two commuting orthogonally transitive Killing vectors ${{\ensuremath{\xi}}_{0}}^{\ensuremath{\mu}}$ and ${{\ensuremath{\xi}}_{1}}^{\ensuremath{\mu}}$ satisfying $\mathrm{det}({\ensuremath{\xi}}_{a}^{\ensuremath{\mu}}{\ensuremath{\xi}}_{b\ensuremath{\mu}})\ensuremath{\ne}\mathrm{const}$, except the van Stockum metric (which is in general type II), are either type I or type D.

The Kerr metric and stationary axisymmetric gravitational fields

A treatment of Einstein’s equations governing vacuum gravitational fields which are stationary and axisymmetric is shown to divide itself into three parts: a part essentially concerned with a choice of gauge (which can be chosen to ensure the occurrence of an event horizon exactly as in the Kerr metric); a part concerned with two of the basic metric functions which in two combinations satisfy a complex equation (Ernst’s equation) and in one combination satisfies a symmetric pair of real equations; and a third part which completes the solution in terms of a single ordinary differential equation of the first order. The treatment along these lines reveals many of the inner relations which characterize the general solutions, provides a derivation of the Kerr metric which is direct and verifiable at all stages, and opens an avenue towards the generation of explicit classes of exact solutions (an example of which is given).

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.