The field equations governing the gravitational field of a uniformly rotating axially symmetric source are reformulated in terms of a simple variational principle. The new formalism affords a concise unified derivation of the solutions discovered by Weyl and Papapetrou, and permits a simple derivation of the Kerr metric in terms of prolate spheroidal coordinates. More complex solutions are identified by applying perturbation theory.
The coupled Einstein-Maxwell field equations are reformulated in terms of a pair of complex functions which have especially simple forms in the case of known axially symmetric stationary solutions. The formalism affords, in particular, a simple derivation of a solution previously guessed by Newman et al.
For axial symmetric stationary vacuum fields the Einstein equations reduce to a system derivable from a simple Lagrangian. An investigation of its form invariance leads to a method to construct from known solutions generalized solutions with one additional parameter. The method is applied to Weyl's class and to Kerr metric.
Contains an investigation of spaces with a two parameter Abelian isometry group in which the Hamilton-Jacobi equation for the geodesics is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. The spaces satisfying the stronger condition that the corresponding Schrödinger equation is separable are isolated in a canonical form for which Einstein's vacuum equations and the source-free Einstein-Maxwell equations (with or without a Lambda term) can be solved explicitly. A fairly extensive family of new solutions is obtained, including the previously known solutions of deSitter, Kasner (1925), Taub-NUT (1951), and Kerr (1963) as special cases.
Methods are discussed with which one may derive theorems which allow one to generate new solutions of the Einstein-Maxwell equations from old ones. The old solutions used to generate new ones must admit at least one nonnull Killing vector and may be required to satisfy other conditions, depending on the theorem derived. Examples of derivable theorems are shown; these theorems are used in turn to show how generation of new solutions is accomplished. Examples of the latter are shown such as generation of Brill or electrified NUT space from the Schwarzschild solution, generation of a new twisted Melvin universe from flat space, and generation of a new generalization of the Ozsvath-Schucking metric. Possible physical interpretations, uses, and extensions of this type of theorem are discussed.
Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n-dimensional pseudo-Riemannian manifold (a group is said to be orthogonally transitive if its surface of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n-p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n-2)-parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the cirulcation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space-time (n=4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source-free Maxwell-Einstein equations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure-vacuum case).
General methods are developed for constructing stationary solutions of the Einstein-Maxwell equations from known Einstein-Maxwell fields or vacuum fields. These methods are based on the investigation of an abstract Riemannian V4 characteristic of the Einstein-Maxwell equations. The application to Kerr's solution demonstrates the practical use of this procedure.
Multipole moments are defined for static, asymptotically flat, source-free solutions of Einstein's equations. The definition is completely coordinate independent. We take one of the 3-surfaces V, orthogonal to the timelike Killing vector, and add to it a single point Lambda at infinity. The resulting space inherits a conformal structure from V. The multipole moments of the solution emerge as a collection of totally symmetric, trace-free tensors, P, Pa, Pab, ... at Lambda. These tensors are obtained as certain combinations of the derivatives of the norm of the timelike Killing vector. (For static space-times, this norm plays the role of a "Newtonian gravitational potential.") The formalism is shown to yield the usual multiple moments for a solution of Laplace's equation in flat space, the dependence of these moments on the choice of origin being reflected in the conformal behavior of the P's. As an example, the moments of the Weyl solutions are discussed.
A method is described for constructing, from any source-free solution of Einstein's equations which possesses a Killing vector, a one-parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space-times having a Killing vector.
A scheme is introduced which yields, beginning with any source-free solution of Einstein's equations with two commuting Killing fields for which a certain pair of constants vanish (e.g., the exterior field of a rotating star), a family of new exact solutions. To obtain a new solution, one must specify an arbitrary curve (up to parametrization) in a certain three-dimensional vector space. Thus, a single solution of Einstein's equation generates a family of new solutions involving two arbitrary functions of one variable. These transformations on exact solutions form a non-Abelian group. The extensive algebraic structure thereby induced on such solutions is studied.
The Einstein-Maxwell field equations in the presence of one Killing vector are shown to possess covariance under an eight-parameter group of linear substitutions in the field variables. This internal symmetry group is isomorphic to SL(2,1). Three of the degrees of freedom correspond to gauge transformations, but the remaining ones allow us to generate a five-parameter family of solutions given a single solution.
The Einstein equations for stationary axially symmetric gravitational fields are written in several extremely simple forms. Using a tensor generalization of the Ernst potential, we give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations and gage transformations, and (iii) the infinite parameter group K of Geroch which combines both. We then show how the same thing can be done to the Einstein-Maxwell equations. The englarged internal group H' now includes the Harrison transformations, and is isomorphic to SU(2,1). The enlarged group K' contains even more parameters, and generates even more potentials and conservation laws.
From Einstein-Maxwell fields which are stationary and axially symmetric, we show how to construct an infinite hierarchy of potentials. The potentials form a representation of K', the infinite-parameter symmetry group of the Einstein-Maxwell equations. For flat space, the hierarchy is calculated explicitly.
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 d parameter. The final form of the metric depends on two ordinary differential equations of the second order. When d is not an integer, these equations define unfamiliar transcendental functions for which rapidly converging series expansions of several types are available. When d is an integer, the solutions are polynomials or rational functions of spheroidal coordinates and define the discrete Tomimatsu-Sato series, for which those authors give the cases d = 1,2,3,4. One of two equations is solved explicitly for the case d = 5 and efficient algorithms are presented which make it possible to perform such calculations by hand. The metric and Ernst potentials assume simple functional forms on the symmetry axis. Actually, this three-parameter family of asymptotically flat solutions is shown to be contained in a family of unphysical solutions with six non-trivial parameters, one of which is the familiar NUT parameter.
In a previous paper, the author 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, we consider two interesting special cases of the previous family, which must be constructed by a limiting process. These will be interpreted as a `rotating Curzon metric' and a `generalised extreme Kerr metric.' In addition, approximate forms for the original metrics are given for the cases of slow rotation and small deformation.
We further investigate the infinite hierarchy of potentials and nonlinear symmetry transformations given in a previous paper. We outline a general method of explicitly calculating the potentials for a given spacetime. We show that some of the transformations can be exponentiated to finite values of their parameters, making them available as a means of generating new solutions. In particular, we show how our transformations may be used to generate all static solutions (including Schwarzschild), starting from nothing but flat space.
For stationary, axially symmetric vacuum metrics, the authors give a series of transformations b(k) which automatically preserve asymptotic flatness. They show how to generate the Kerr metric from the Schwarzschild, using b(0). They also show, using b (k), that the Tomimatsu-Sato (TS) class of metrics must be larger than previously realized, and for d=2 there is a five-parameter TS metric. As an example, they present a two-parameter metric from this family which they claim to be a new, physically realistic, asymptotically flat, rotating vacuum solution.
We give a series of transformations b k, k = 0,1,... which may be used to generate new stationary axially symmetric vacuum solutions from ones already known. These transformations have the important property of preserving asymptotic flatness. As one example of their use, we show how to generate the Kerr metric from Schwarzschild. As a second example, we generate a new five-parameter vacuum solution which contains the d=2 Tomimatsu-Sato solution as a special case.
This paper shows that a new class of axially symmetric static electrovac solutions and the Kerr solution are obtainable from the van Stockum metric. The new class contains an infinite set of asymptotically flat solutions (in closed form), each of which involves an arbitrary set of parameters. The parameters have to be interpreted as functions of mass m, charge e, and higher electric multipole moments a i of the particle. The case e = a i = 0 leads to the Darmois metric. Well-known and new examples are given.
We present transformation formulas which facilitate the determination of the metrics, electromagnetic fields, connections and Weyl tensors of those electrovac spacetimes which result when a given solution of the Einstein-Maxwell equations with an isometry is subjected to the transformations of the Kinnersley group. Several applications of our calculational procedures are given as illustrations, and a number of general theorems are presented. In particular, we infer that when we apply such techniques to the only known solution of Petrov type N with twisting principal null rays, the new solutions which result will be algebraically general.
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 e2 g = -g11 g44, 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, e2 g takes a much simpler functional form than either the metric coefficients g44, g34 and g33, or the Ernst potentials, E and x. Three methods are given for the construction of the full metric from e2 g. A duality principle is invoked to provide a very similar field equation for the metric coefficient e2 g -2 u = - g11.
The techniques of the preceding paper are applied to several cases where the g equation may be solved by separation of variables in the form g = g1(r) + g2(t), where g1(r) is either zero or a very simple function and g2(t) 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, j and t, are derived. This problem reduces to finding separable solutions of the dual of the g-equation of the form e2 g - 2 u = R(r,t) [ f(r) + g(t) ] in four special co-ordinate systems, (r,t), where R(r,t) is a prescribed simple function. A comparison is made with the canonical Schrödinger separable metric forms of Carter.
A linear eigenvalue problem in the spirit of Lax is constructed for the stationary, axially symmetric Einstein equations. It is conjectured that this entails the complete integrability of the system.
A Bäcklund transformation for the Ernst equation arising in general relativity in connection with several physical problems is derived, using the pseudo-potential method of Wahlquist and Estabrook (1975). A prolongation structure is also constructed, using a method of writing the equations in terms of differential forms, and an equation in the spirit of Lax (1968) is constructed, somewhat different from that given by Maison (1978). Possible uses of the Bäcklund transformation to generate new solutions are mentioned.
This paper proves the theorem: `All stationary axially symmetric solutions of Einstein's field equations are available by solving a partial differential equation which involves derivatives up to the fourth order of a single real function.' This differential equation is discussed and some solutions are found.
A technique is given for generating from any one-parameter family of stationary, axisymmetric vacuum solutions of Einstein's equation with a certain dependence on the parameter, a two-parameter family. The technique is applicable to almost all known families of stationary axisymmetric solutions. Some new solutions are also presented.
A generation theorem for solutions of Einstein equations is presented. It consists mainly of algebraic steps. With its aid, one obtains from an "old" solution (e.g., from the Minkowski space) "new" solutions with an arbitrary number of constants. The method of repeated application of potential and coordinate transformations considered by Geroch is included.
A linear eigenvalue problem in the spirit of Lax is constructed for the nonlinear differential equations describing stationary, axially symmetric Einstein spaces. In suitable variables these equations yield a generalization of the well-known sine-Gordon equation. The similarity of the system to the nonlinear σ ...
For every nonnull Killing vector K of any given electrovac, there exists a group of transformations HK of the gravitational and electromagnetic potentials of Ernst. This is the gorup which is a nonlinear representation of SU(2,1) and was developed by Kinnersley on the basis of work by Ehlers, Harrison, and Geroch. For every K of Minkowski space (MS), we compute the set HK(MS) of all electrovacs derived from MS by noniterative application of HK; the results include appropriate null tetrads, the connection forms, the conform tensors, and (in the discussion) the group of all motions of every member of every HK(MS). Each conform tensor is type Npp (plane gravitational wave) or type D, and the principal null vector(s) are also eigenvectors of the Maxwell field. Except for those K which represent infinitesimal rotations about a timelike 2-surface of MS followed by null translations in that 2-surface, each K has a corresponding MS Killing vector L such that the G2 generated by K and L has nonnull surfaces of transitivity and is invertible. The discussion covers properites of the principal null rays and the Maxwell fields, Killing tensors of the results (one of the Npp families admits an irreducible Killing tensor of Segré characteristic [(11)(11)], and the precise conditions under which a Killing vector of an electrovac is also an MS Killing vector. Also, some deductions are made concerning the Petrov class and principal null ray properties of the second generation electrovacs which would result from further applications of SU(2,1). Those points of MS which are possible singularities of electrovacs in HK(MS) are classified. The conditions under which an electrovac in HK(MS) has all of R4 (except for curvature singularities) as its domain are found; in particular, such an extension to R4 exists whenever the one-parameter group generated by K has no fixed points or whenever one restricts HK to the Ehlers or Harrison transformations.
The authors rewrite Einstein's equations for stationary axially symmetric gravitational fields, using a pair of noncanonical intrinsically-defined coordinates. They show that both field equations of the Ernst formulation of this problem can be solved identically, by means of a new superpotential K. One more field equation remains to be satisfied, however. It can be expressed as a single fourth-order equation for K, or as a pair of coupled second-order equations. The approach works equally well for the wave metrics one can get from the stationary case via complex coordinate transformations. The method is illustrated by using it to derive a new class of wave solutions.
We analyze various forms of duality symmetry transformations occurring in the theory of axially symmetric gravitational fields and their relation to standard Bäcklund transformations. Appropriately interpreted, duality rotations provide genuine Bäcklund transformations for the fundamental SL(2,R) invariants associated with the metric, consisting of elementary algebraic substitutions; thereby the construction of the metric reduces to the solution of linear equations.
This paper shows that the field equations and the hierarchy of potentials for static electrovac fields can be formulated in close analogy to the stationary vacuum ones. A list of transformations, some of them previously unknown, will be given for the latter case.
A method is presented for generating stationary, axisymmetric, asymptotically flat, vacuum solutions of Einstein's equation with an arbitrary number of mass and angular-momentum parameters. As an example, a two-parameter asymptotically flat solution, generalizaing extreme Kerr, is generated from flat space. It is conjectured that the method, applied to the general static metric, can be used to generate all stationary, axisymmetric, asymptotically flat metrics.
A new series of transformations is presented for generating stationary axially symmetric asymptotically flat vacuum solutions of Einstein's equations. The application requires only algebraic manipulations to be performed. Several examples are given of new stationary axisymmetric solutions obtained in this way. It is conjectured that the transformations, applied to the general Weyl metric, can be used to generate systematically all stationary metrics with axial symmetry.
The construction of spinning mass solutions of Einstein's vacuum field equations, which can be obtained by applying Kinnersley-Chitre transformations to known soutions, is facilitated by our discovery of a linear integral equation of the Cauchy type, the solution of which yields directly the generating function F(t) of the Kinnersley-Chitre hierarchy of potentials associated with the transformed spacetime metric.
Our previously presented integral equation formulation of the Kinnersley-Chitre transformation theory is generalized to the case of electrovac-to-electrovac transformations. The solution of the integral equation for a case in which the kernel has a finite number of simple poles is obtained. In particular, we show that when the transformation correponding to one simple pole is applied to Minkowski space, one obtains the Ehlers transform of the extreme charged Kerr-NUT (Newman-Unti-Tambourino) space. We also find the general solution corresponding to a confluence of two simple poles.
Several recently developed techniques for generating asymptotically flat solutions of the stationary axisymmetric vacuum gravitational field equations are compared. First, the author shows how the generalized Tomimatsu-Sato (1972, 1973) solutions of Cosgrove (1977, 1979) may be derived by means of an asymptotic flatness preserving transformation group, independently discovered by Neugebauer (1979) as a Bäcklund transformation. Second, he points out that two successive Bäcklund transformations of Harrison preserve asymptotic flatness while introducing four new parameters and are closely related to the double rank-zero transformations of Hoenselaers, Kinnersley and Xanthopoulos (1979). Third, a powerful commutation theorem of Neugebauer is used to show how multiple Harrison (1978) transformations generating asymptotically flat fields with an arbitrary number of extra multipoles can be constructed by algebraic steps alone when the first transformation is known. Finally, the author calculates explicitly the double Harrison transforms of the Weyl solutions and the generalized Tomimatsu-Sato solutions.
This paper shows that a new class of axially symmetric static electrovacuum/magnetovacuum solutions is obtainable from Weyl's class of static vacuum solutions. The new class contains an infinite set of asymptotically flat solutions (in closed form) each of which involves an arbitrary set (d,ai) of parameters. These parameters have to be interpreted as functions of mass m, charge e, and higher electric/magnetic multipole moments ai of the particle. The case d = 0, ai = 0 leads to the Darmois solution and the case d = 0, ai = 0 leads to the results of [1]. The case d = 0, e = ai =0 leads to the Schwarzschild solution, the case d = 0, ai = 0, e = 0 leads to the Reissner-Nordström solution. To get more general examples is a lengthy but straightforward exercise.
This paper shows that new classes of axially symmetric static electrovacuum solutions are obtainable from the static Weyl's class of vacuum solutions by means of a generation technique. (From paragraph one.)
The author plans on a continued search for a solution of this current problem. (From last paragraph.)
The aim of this paper is to show that the axially symmetric stationary Einstein fields are minimal surfaces in a characteristic Riemannian 4-space (not to be confused with space-time). (From paragraph one.)
We investigate the precise interrelationships between several recently developed solution-generating techniques capable of generating asymptotically flat gravitational solutions with arbitrary multipole parameters. The transformations we study in detail here are the Lie groups Q and Qbar of Cosgrove, the Hoenselaers-Kinnersley-Xanthopoulos (HKX) transformations and their SL(2) tensor generalizations, the Neugebauer-Kramer discrete mapping, the Neugebauer Bäcklund transformations I1 and I2, the Harrison Bäcklund transformations, and the Belinsky-Zakharov (BZ) one- and two-soliton transformations. Two particular results, among many reported here, are that the BZ soliton transformations are esentially equivalent to Harrison transformations and that the generalized HKX transformation may be deduced as a confluent double soliton transformation. Explicit algebraic expressions are given for the transforms of the Kinnersley-Chitre generating functions under all of the above transformations. In less detail, we also study the Kinnersley-Chitre b transformations, the non-null HKX transformations, and the Hilbert problems proposed independently by Belinsky and Zakharov, and Hauser and Ernst. In conclusion, we describe the nature of the exact solutions constructible in a finite number of steps with the available methods.
A simple recursion formula is presented for calculating stationary axisymmetric (asymptotically flat) Einstein fields with any number of constants. It generates Kerr (Schwarzschild) particles from the vacuum (Minkowski space) and stationary asymptotically flat solutions from static ones.
The Ernst function of an axially symmetric stationary asymptotically flat spacetime involving an arbitrary harmonic function and an arbitrary number of constants is presented and discussed.
Presents a new exact solution of Einstein's field equations: the nonlinear superposition of two Kerr-NUT solutions. The Tomimatsu-Sato d = 2 solution is contained as a limiting case.
A Bäcklund transformation for the Ernst equation of general relativity, published earlier by this author, is used to derive a new large family of vacuum metrics with two commuting Killing vectors from the family of Weyl or Einstein-Rosen metrics. Thus, any solution of the axially symmetric Laplace or wave equation yields a new solution of the Ernst equation. Asymptotically flat Weyl metrics yield new asymptotically flat metrics. The solutions are nonstationary and may exhibit solitonlike behavior.
A homogeneous Hilbert (Riemann) problem (HHP) is introduced for carrying our the Kinnersley-Chitre transformations of the set V of all axially symmetric stationary vacuum spacetimes, and the spacetimes which are like the axially symmetric stationary ones except that both Killing vectors are spacelike. A proof, which is independent of the Kinnersley-Chitre formalism, establishes that the HHP transforms the potential (for certain closed self-dual 2 forms) F0(x,t) of any given member of V into the potential F(x,t) of another member of V. Two illustrative examples involving the Minkowski space F0(x,t) are given. The representation used for the Geroch group K, the singularities and gauge of the potentials, and possible applications of the HHP are discussed.
The homogeneous Hilbert problem which we recently formulated for Kinnersley-Chitre transformations of vacuum spacetimes is here generalized to handle transformations of electrovac spacetimes. This provides in particular a simple derivation of our previously published integral.
The method of Bäcklund transformations for stationary axisymmetric vacuum gravitational fields is applied to the Minkowski space. Two successive steps lead just to the Kerr-NUT solution.
Presented is a Bäcklund transformation of the Einstein-Maxwell equations which is a generalization of Neugebauer's I2 transformation.
The precise interrelations between the Bäcklund transformations, the inverse scattering method and an infinite number of divergence-free currents are obtained for the Ernst equation.
The authors obtained a Bäcklund transformation of the axially symmetric stationary Einstein-Maxwell equation which is a generalization of Neugebauer's I1 transformation (1979) in the case of the Ernst equations. This transformation is shown to generate the generalized Ehlers transformation and the Harrison transformation (1978).
Presented is a systematic approach to the transformation theories for the Ernst equation from the viewpoint of the Bäcklund transformation. It is explicitly shown that the method of Clairin gives a simple derivation of various transformations such as transformations found by Ehlers, Neugebauer, and Harrison.
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 C4 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 thes 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.
The algorithms are studied which construct stationary axisymmetric asymptotically flat spacetimes. The multipole moment structure of the resulting spacetimes is determined. The static metric and the transformations which generate any stationary metric are obtained.
A study is made on the connection among stationary equations for the classical continuous isotropic Heisenberg model (CCIHM), the Ernst equation for the axisymmetric gravitational field (ASGF) and the Yang equations for the self-dual SU(2) gauge field (SDGF). This is done by reducing the Yang equations to a four-dimensional (4d) version of the Ernst equation and by parmetrizing the Ernst potential in terms of variables q and j analogous to two angles of rotation of spins in the CCIHM. Both of the Ernst equation and the Yang equations are thereby reduced to a pair of equations for q and j quite similar to those for the CCIHM, the latter reducing to the former by taking q rightarrow i q. This shows one-to-one correspondence of solutions to the field equations between the CCIHM and the SDGF or the ASGF, though spatial dimensionality, symmetry and boundary conditions of physical significance may be different for these different cases. By paying attention to particular solutions for which j satisfies the Laplace equation, three types of solutions are obtained for the SDGF by using analogy with hydrodynamics, one of which is also applicable to the ASGF. For the first type new multi-vortex or multi-instanton solutions similar to vortex solutions for the two-dimensional (2d) CCIHM are obtained for the 4d SDGF. The second type is axially symmetric solutions for the three-dimensional (3d) SDGF and the ASGF written in terms of the Painlevé transcendents of the third kind. The third type is uniform-flow solutions for the 3d SDGF given as solutions to the 2d static sine-Gordon equation.
The Ernst equation for gravitational fields with a two-parameter isometry group is formulated as a vanishing-curvature condition on an SU(2) or SU(1,1) bundle, both in the elliptic and hyperbolic cases. Bäcklund transformations are introduced as a special case of gauge transformations, and strong Bäcklund transformations are obtained in that context.
The equation zmn = 2 zm zn/(z+z^*) for the complex variable z is equivalent to the field equation for a SO(2,1)-invariant nonlinear s-model in two dimensions. A connection on a SU(2) bundle is given, such that the requirement of the vanishing curvature of the connection implies that equation. Bäcklund transformations, considered as gauge transformations of the connection, are explicitly obtained.
In the literature, various systems of linear eigenvalue equations from which the Einstein-Maxwell equations for stationary axisymmetric exterior fields follow as the integrability conditions were derived. In the present paper, these linear systems are shown to be equivalent; the explicit transformations mapping one form to another are given.
The Einstein-Maxwell equations for stationary axisymmetric exterior fields are shown to be the integrability conditions of a set of linear eigenvalue equations for pseudopotentials. The prolongation structure in the spirit of Wahlquist and Estabrook (1975) of the Einstein-Maxwell equations contains the SU(2,1) Lie algebra. A new mapping of known solutions to other solutions has been found.
Soliton physics has made considerable progress in solving nonlinear problems. The authors relate the soliton concept to the stationary axisymmetric vacuum fields in general relativity. They 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 field. The functional transformations form an infinite-parameter group which contains the Kinnersley-Geroch group as a subgroup.
It is shown that Harrison's (1978, 1980) Bäcklund transformation for the Ernst equation of general relativity is a two-parameter subset (not subgroup) of the infinite dimensional Geroch group K. The author exhibits the specific matrix u(t) appearing in the Hauser-Ernst representation of K for vacuum spacetimes which gives the Harrison transformation. Harrison transformations are found to be associated with quadratic branch points of u(t) in the complex t plane. The coalescence of two such branch points to form a simple pole exhibits in a simple way the known factorization of the (null generalized) HKX transformation into two Harrison transformations. He also shows how finite (i.e., already exponentiated) transformations in the B group and nonnull groups of Kinnersley and Chitre (1977, 1978) can be constructed out of Harrison and/or HKX transformations. Similar considerations can be applied to electrovac spacetimes to provide hitherto unknown Bäcklund transformations. As an example, he constructs a six-parameter transformation which reduces to the double Harrison transformation when restricted to vacuum and which generates Kerr-Newman-NUT space from flat space.
The Einstein equations for gravitational fields in a vacuum simplify considerably when two commuting isometries exist. In the case of two space-like Killing fields (e.g., cylindrical symmetry), and assuming orthogonal transitivity, the author reduces the equations.
An Einstein spacetime E admitting a two-parameter Abelian group G2 of isometries is described covariantly on a two-dimensional manifold S. From the two independent Killing vector fields xa (a=1,2) that generate the group G2 the authors obtain a 2 × 2 matrix. The authors assume that both Killing vectors are spacelike, (or) that one of the Killing vectors is timelike.
A new parametric infinitesimal transformation to preserve the equation of motion and linearization formulation in Kinnersley-Chitre-Hauser-Ernst formulation of gravity is proposed. Based on the transformation the G × C(t) hidden symmetry algebraic structure is explicitly shown.
The author makes a quantitative comparison between the pure nonsoliton part of the inverse scattering method of Belinskii and Zakharov (1978) (BZ) and the homogeneous Hilbert problem of Hauser and Ernst (1979) (HE), these being two independent representations of an infinite dimensional subgroup K of the Geroch group K of invariance transformations for spacetimes with two commuting Killing vectors. An explicit formula for the BZ representing matrix function G0(l) in terms of the HE representing matrix function u(t) is derived. It is shown how certain solution generating techniques (e.g., Harrison's Bäcklund transformation, HKX transformation, generation of Weyl solution from flat space, generation of n-Kerr-NUT solution from n-Schwarzschild) can be derived directly from the BZ formalism, including the soliton part in some cases, thereby bringing our understanding of the BZ formalism up to the level of the more fully developed HE formalism. A technical point which needed to be resolved along the way was how to analytically continue the complex matrix potential F(t) across a quadratic branch cut onto the second Riemann sheet. Finally, the author considers how the subgroup K subset K represented by the BZ and HE formalisms can be enlarged either by simple limiting transitions or by relaxing boundary conditions.
The equations for a gravitational field in the presence of a two-parameter Abelian group of isometries (e.g., stationary axisymmetric fields or cylindrical waves) are shown to be equivalent to the construction of a connection (with vanishing curvature) on a principal bundle with an appropriate structure group. Examples of analogous constructions for other equations are mentioned. The gauge transformations leaving invariant the functional form of the connection transform solutions into solutions and provide a method for finding Bäcklund transformations, examples of which are given.
An exterior differential system (linear) is presented that is `integrable' in the sense of Cartan's theory if and only if Einstein's vacuum field equations hold. The analogy is emphasized with the twistor programme and with the theory of Hamiltonian systems admitting a Lax Pair sometimes called `completely integrable.' Supersymmetry seems again relevant for realistic systems.
We will briefly review the prolongation structure theory for nonlinear systems, such as nonlinear evolution equations and soliton equations, and its applications to the solution generating techniques for the gravitational fields with two commuting Killing vectors.
Based on the theory of Wahlquist and Estabrook, we have presented a set of new fundamental equations of prolongation structures for nonlinear systems and a criterion of completeness of conservation laws of the systems by means of the theory of nonlinear realization of connection due to Lu and Guo. We have also proposed a general requirement for auto-Bäcklund transformation from a point of view of prolongation structure theory and whereupon we have shown that the problem of searching for a kind of auto-Bäcklund transformations can be reduced to a principal homogeneous Hilbert problem.
By means of the prolongation structure theory for nonlinear systems, we have dealt with the various solution generating techniques for gravitational fields with two commuting Killing vectors, especially the group theoretic technique due to Kinnersley and Chitre, Hauser and Ernst, and others. We have also introduced the principal inverse scattering problems with respect to the principal pseudopotentials, i.e., the W-potentials, for both vacuum and electrovac cases. Finally, we will show the solutions of n-fold Kerr family and n-fold charged Kerr family by solving the corresponding principal homogeneous Hilbert problem.
This work is devoted to the interaction of sourceless gravitational and neutrino fields within the scope of general relativity. The stress is on the properties of these fields when an Abelian group of motions G 2 on V2 is present.
We obtain explicit expressions for infinitesimal regular Riemann-Hilbert transforms. Using them, the group theoretical aspects of infinitesimal RH transforms are discussed with an eye at the comparison with the hidden symmetry transformations proposed by us before. We find that the RH transforms have very rich group structure; e.g., in the 2-d principal chiral models, their group contains two Kac-Moody subalgebras. But not all of them are nontrivial hidden symmetries of the theory.
Vector Bäcklund transformations which relate solutions of the vacuum Einstein equations having two commuting Killing fields are introduced. Such transformations generalize those found by Pohlmeyer in connection with the nonlinear s-model. A simple algebraic superposition principle, which permits the combination of Bäcklund transforms in order to get new solutions, is given. The superposition preserves the asymptotic flatness condition, and the whole scheme is manifestly O(2,1)-invariant.
Is comment on above paper by Chinea.
A new family of exact solutions of the stationary axially symmetric vacuum Einstein equations is presented. The internal symmetries, SL(2,R) rotation, and duality of parametrization, are combined to construct a Bäcklund transformation. For the special ansatz of the field equations, the Bäcklund transformation can be integrated and the hierarchy of ansatz is generated, recursively. The Riemann-Hilbert problem is also discussed for the inverse scattering formulas generating nonlocal symmetries.
By generalizing our previous treatment of hidden symmetry in two-dimensional chiral models, a simple and explicit approach is proposed to the Geroch group for the vacuum Einstein equations with the metric tensor depending only on two variables ("two-dimensional reduced gravity"). An infinite number of infinitesimal transformations for the metric tensor preserving the equations of motion are symmarized by an explicit parametric transformation and the commutators among them are further identified to be those of the well-known Geroch group.
Using our new, simplified approach the infinite dimensional invariance group (the Geroch group) for vacuum Einstein (or Einstein-Maxwell) equations with metrics depending on only two coordinates is enlarged with the number of generators doubled. Its Lie algebra is extended from the half Kac-Moody algebra SL(2,R) × R[t] to the full one SL(2,R) × R[t,t-1] (without central charge), when the group is realized by nontrivial transformations on the metric tensor.
The three known Bäcklund transformations for the Ernst equation are derived using a modification of the Wahlquist-Estabrook prolongation procedure. The modification requires that the equation to be studied be cast into a set of differential forms and their exterior derivatives, such that all coefficients are constant (a "CC ideal"). Analysis of the resulting equations produces 16 solutions composed of the three basic transformations combined with identity and other essentially trivial transformations. The group structure of the transformations is discussed. A Bäcklund transformation (already known) for the Ernst-Maxwell equations can be found by the same method. Promising generalizations are mentioned.
After discovering an infinite group of transformations for steady-state, axisymmetric gravitational fields, Geroch (1972) formulated the hypothesis that an arbitrary, asymptotically flat space can be obtained from Minkowski space by means of a suitable transformation from the group. The author offers a proof that arbitrary free gravitational, electromagnetic, and neutrino fields in the general theory of relativity wtih Abelian groups of motions G2 on V2 can be obtained from a Minkowski space (generalized to the case of the presence of neutrinos) by means of a displacement along the orbit of some infinite dimensional, extended group L¥ of transformations.
We present two new alternative representations for the action of an arbitrary number N of HKX transformations on an arbitrary Weyl solution. The resulting new class of Ernst potentials depends on 2N real parameters. Applications of the derived representations are shown.
The author describes a new approach to the problem of understanding stationary axisymmetric solutions of Einstein's vacuum equations, different from the `Bäcklund transformation' approach which has recently been extensively developed. It translates the problem into one of complex geometry, using the machinery of twistor theory. This, in turn, leads to a procedure which, in principle, generates all solutions. Some explicit examples are presented.
The application of the inverse scattering method to the Einstein-Maxwell equations for stationary axisymmetric exterior fields leads the authors to an explicit formula for the Ernst and electromagnetic potentials of new exact solutions generated from an arbitrary seed solution.
The infinite parameter group acting on stationary, axisymmetric solutions of Einstein's vacuum field equations found by Geroch can be considered as a Banach Lie Group.
A survey of the method inverse scattering transform is given and a differential geometric interpretation of the inverse scattering equations is presented. Einstein-Maxwell field equations for space-times admitting nonnull commuting two Killing vector fields are integrated by giving the 2N-soliton construction. One soliton constructions of the gravitational field and of the self-dual Yang-Mills field equations are shown to be equivalent to the recently found Bäcklund transformations.
The author describes a new method of integrating the system of Einstein-Maxwell equations for solutions with an abelian group G2 of motions on V2. His method permits one to write down the general solution of this system in terms of values of the Ernst potentials E and F on the axis of symmetry E(0,z)=e(z), F(0,z)=f(z), in the form of ratios of the determinants constructed from the Fourier coefficients ek and fk of the functions e(z + i r cos q) = s ek cos k q, f(z + i r cos q) = s fk cos k q. It is assumed that the arbitrary functions e(z) and f(z) admit holomorphic (not necessarily single-sheet) continuation into the complex domain.
The Einstein equations describing gravitational fields in vacuum are written as a compact exterior system of spinor-valued forms. A second system of equations is given, such that their integrability conditions are satisfied by virtue of the Einstein equations. This suggests the possibility of integrating the field equations by means of an inverse type procedure.
The Einstein-Maxwell fields j(r,z), G(r,z), W(r,z) and a (r,z) of a uniformly rotating axially symmetric charged source describe minimal surfaces in a six-dimensional Riemannian space with the line element
dS2 = - 2 da dW + 1/2 WF-2 |dG + x0 j¯ dj|2 - x0 WF-1 |dj|2 - 2 x0 W e2 a F-1 p(F,c) dW2.
The minimal surface formalism affords interesting reformulations of the field equations governing the rotating body problem and a purely geometric characterisation of the electromagnetic and gravitational field inside and outside the source.
Following Estabrook and Wahlquist, Einstein's vacuum field equations are represented by a differential ideal form as its restriction on space-time. Prolongating this ideal on a higher dimensional manifold a Bäcklund transformation is found, including the generalized Kerr-Schild transformation as a special case.
We formulate stationary axially symmetric (SAS) Einstein-Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight-dimensional flat manifold and formulate SAS Einstein-Maxwell fields as a s-model. We then give, in a coordinate free way, a Belinskii-Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov-Mikhailov and Mikhailov-Yarimchuk.
An explicit determinantal formula for the N-fold Bäcklund transform of any pair of AKNS fields q(x,t), r(x,t) is derived. In particular this formula includes all N-soliton solutions of nonlinear evolution equations which belong to the AKNS class.
The newly developed transformations, which allow to generate solutions of Einstein's vacuum equations from given ones, are meanwhile very powerful tools for constructing new solutions. We prove the equivalence of Kramer-Neugebauer and HKX transformations and construct the double Kerr solution by means of four HKX transformations. We analyse the resulting solution in detail and present an analytic continuation of the parameters to pass from underextreme to hyperextreme constituents. We pose the conditions for asymptotic flatness, for existence of an axis between the massive objects and for balance caused by the repulsing interaction of the angular momenta. We define mass and angular momentum of the single constituents and compute explicitly these quantities and additionally the total mass and the total angular momentum of the solution. The distance and force between the massive objects are defined and given in a suitable approximation. We prove that two black holes--constituents possessing an horizon--cannot be in balance. The above mentioned conditions are solved for equal hyperextreme constituents which balance due to spin-spin repulsion. We give the distance of balance. Finally we compare rotating and nonrotating two mass systems of equal masses and equal distance between them and estimate the change of force between the masses caused by the spin-spin repulsion.
A set of `extended' regularity conditions is discussed which have to be satisfied on the rotation axis if the latter is assumed to be also an axis of symmetry. For a wide class of energy-momentum tensors these conditions can hold at the origin of the Weyl canonical coordinate. For static and cylindrically symmetric space-times the conditions can be derived from the regularity of the Riemann tetrad coefficients on the axis. For stationary space-times, however, the extended conditions do not necessarily hold, even when 'elementary flatness' is satisfied and when there are no curvature singularities on the axis. The authors generalise the result by Davis and Caplan (1971) for cylindrically symmetric stationary Einstein-Maxwell fields, by proving that only Minkowski space-time and a particular magnetostatic solution possess a regular axis of rotation. Further, several sets of solutions for neutral and charged rigidly and differentially rotating dust are discussed. In most cases only very narrow classes of solutions satisfy the extended requirements.
The authors show that solutions univalent in the r-z plane form only a subclass of the variety of all solutions, which are generally multivalent solutions and which can have a nontrivial topology. The first step in this direction involves the use of the orbits of the Kinnersley-Chitre group, meromorphic on compact Riemannian surfaces which are homeomorphic to the spheres with N handles. In this case the data on the symmetry axis may be given by branches of algebraic functions. The viewpoint according to which solutions with irrational orbits of L¥ lead to solutions in which a massless spinorial field is present is not true.
The Einstein equations in vacuum are written as a closed differential ideal of matrix-valued differential forms with constant coefficients. Several properties of the resulting equations, such as the existence of an associated integrability system and of conserved matrix currents, as well as the treatment of specific Petrov types, are briefly considered. We also show that the field equations can be expressed as a deformation (depending on a complex parameter lambda) of the Bianchi identity for an adequate Poincaré or de Sitter connection.
We extend here to the case of N Abelian gauge fields interacting with gravitation an earlier proof that essentially all stationary axisymmetric vacuum fields can be obtained from the Minkowski space solution by means of Kinnersley-Chitre transformations.
The Cartan differential ideal is constructed for the vacuum field equations with cosmological constant and for the electrovacuum field equations. Two different classes of prolongations of these ideals are given. Some Bäcklund transformations are proposed, and the Painlevé property of the vacuum field equations is discussed.
The use of differential forms has played an important role in the formulation of differential geometry, and in connection with that, in the formulation of general relativity. Previously, this was primarily limited to finding the Riemann tensor as coefficients of the curvature 2-form, but it has been shown how to write the general vacuum field equations in terms of 3-forms. These treatments assume a null tetrad basis, but the author presents a more general approach to the problem.
The Ernst potential of a stationary axisymmetric asymptotically flat vacuum solution determines uniquely formal multipoles, which are very closely related to the multipole moments invariantly defined by Geroch (1970) and Hansen (1974).
A new approach to the solution of certain differential equations, the double complex function method, is developed, combining ordinary complex numbers and hyperbolic complex numbers. This method is applied to the theory of stationary axisymmetric Einstein equations in general relativity. A family of exact double solutions, double transformation groups, and n-soliton double solutions are obtained.
The new methods are presented for solving Einstein-Maxwell equations for the exterior fields of rotating charged masses with axial symmetry, having at least one regular point on the symmetry axis. They include the scalar singular integral equation and Gelfand-Levitan-Marchencho integral equation. The role of scattering coefficients are played by Ernst data on the axis. The connection between Riemann surfaces of Ernst data on the axis and space time topology is discussed as well.
In this work is presented a method of obtaining solutions of the Einstein-Maxwell system in complete form, when on an arbitrary static background are generated arbitrary stationary solitons. In particular, it is demonstrated that the single sheet solutions that were found earlier form only a subclass of the set of all solutions which, generally speaking, exhibit multiple sheets and nontrivial topology.
The author derives in explicit form functional algebras and groups, which are consistent with certain partial differential equations describing nonlinear wave processes in a continuous medium and the behavior of free classic fields. These automorphic groups make it possible to find arbitrary solutions of these equations using a shift with respect to a one-parameter subgroup that converts one solution of these equations into another.
Einstein's equations in the Newman-Penrose formalism for vacuum, vacuum with cosmological constant, and electrovacuum fields are expressed as Cartan ideals. Two different prolongations of these ideals are obtained. These two types of prolonged ideals generalize previous prolongations for vacuum fields to vacuum with cosmological constant and electrovacuum fields. Some Bäcklund transformations are obtained for vacuum, vacuum with cosmological constant, and electrovacuum fields. These Bäcklund transformations include the generalized Kerr-Schild (GKS) transformation, and a two-parameter generalization of the GKS transformation. GKS transformations are studied in detail. Expressions for the transformation of Newman-Penrose quantities are given and algebraic properties are discussed. It is shown that the GKS transformation cannot give algebraically general and asymptotically flat vacuum and electrovacuum space-time metrics.
The Painlevé property of the vacuum Einstein field equations is investigated. It is observed that the field equations possess this property when spacetime admits commuting, nonnull two Killing vector fields.
Two types of linear prolongations of the vacuum gravitational field equations are given. First one is shown to produce nontrivial Bäcklund transformations and the second one is given in a form that resembles the prolongation of the self-dual Yang-Mills and of the s-model field equations.
We present here a simplified and corrected proof of the conjecture due to Geroch that essentially all stationary axially symmetric metrics can be derived from the Minkowski space metric using Kinnersley-Chitre transformations.
By comparing the differential equations possessing both the Painlevé property and prolongation structure, it is conjectured that if a partial differential equation passes the Painlevé test, it admits a nontrivial prolongation structure. The Painlevé proverty of the vacuum gravitational field equations is studied. It is also conjectured that the vacuum Einstein field equations pass the Painlevé test if the space-time geometry admits non-null two Killing vector fields.
It is shown that the `hidden symmetries' of the reduced Einstein equations investigated by Kinnersley and others have a simple description in terms of Ward's twistor construction of stationary axisymmetric vacuum spacetimes.
The set of stationary, axially symmetric solutions of Eintein's vacuum field equations is acted on by some infinite dimensional group (Geroch). A precise definition of this group is given as the central extension of a group of holomorphic functions in SL(2). This group acts in the nonlinear way known from s-models on functions with values in SL(2) which are solutions of a system of linear differential equations and at the same time parametrize an infinite dimensional coset space. This implementation is shown to be directly related to the `inverse scattering method' known for `completely integrable systems'.
A new formulation is given of the Einstein field equations for vacuum gravitational fields with a single non-null Killing vector. In the new method a set of partial differential equations is derived for the components gab of the metric of the quotient 3-space associated with these spacetimes. The remaining field variables are then obtained by integrating a total Riccati system and a straightforward set of total differential equations. In this way the integration of the Einstein equations is split into two disjoint problems and is thereby simplified. Moreover, the new formulation also leads to the possibility of obtaining a Bäcklund map for the one Killing vector case.
By reducing the Ward correspondence, we show that there is a correspondence between stationary axisymmetric solutions of the vacuum Einstein equations and a class of holomorphic vector bundles over a reduced twistor space, which is a compact one-dimensional, but non-Hausdorff, complex manifold. We show that the solutions generated by Ward's ansätze correspond to bundles which have a simple behaviour on the `real axis' in the reduced space. We identify the Geroch group (Kinnersley and Chitre's group K) with a subgroup of the loop group of GL(2,C) and we describe its orbits. We also identify some of the subgroups which preserve asymptotic flatness.
A new and large class of exact solutions of the stationary axisymmetric Einstein equation, which are expressed in terms of the Riemann theta function, is constructed. The properties of the constructed "finite-gap" solutions differ significantly from those of the well-known finite-gap solutions (for example, of the Korteweg-de Vries equation and the nonlinear Schrödinger equation). In particular, the dependence on the dynamical variables in the final expressions is given by a trajectory on a manifold of moduli of algebraic curves, and not on the Jacobi manifold of a given curve. In a degenerate case the constructed solutions include all the main known solutions that can be expressed in terms of elementary functions.
The Belinsky-Zakharov inverse scattering method is extended to a double form, followed by a generation of some new solutions of the double-complex Ernst equation. Therefore, a physical double n-soliton solution corresponding to a pair of gravitation soliton solutions with physical signature can be generated for any positive integer n. In the case of the Weyl type background solution the concrete expression of a double-soliton solution is given, thus a sequence of known solutions and a sequence of new solutions, i.e., the Letelier sequence and its dual sequence are obtained. Consequently, by means of an element of the symmetry gorup of the BZ equation, the author obtains an infinite network of soliton solutions and revises the Nakamura chain of solutions.
Clifford-algebra valued differential forms are shown to provide a natural formalism for the description of gravitational fields in vacuum. The field equations and their integrability conditions are combined into a simple pair of Clifford-algebra valued exterior equations. Conserved currents and de Sitter algebras appear in a natural way. The explicit relation with a recent formulation in terms of spinor forms is given.
For the Ernst equation, a hierarchy of ansatz that generates determinantal solutions of the Ernst equation is proposed. The ansatz is described explicitly in the inverse scattering formalism and it is show that the corresponding exact solutions are determinantal solutions that have been constructed by Kyriakopoulos [Phys. Rev. D 30, 1158 (1984)] and Vein [Class. Quantum Gravit. 2, 899 (1985)].
We find a linearization of the Ernst equation by means of universal Grassmann manifold (UGM) techniques. All local analytic solutions defined at the origin are obtained by solving an initial value problem for a linear differential equation on a UGM. We give an explicit formula which represents solutions of the Ernst equation. By using this formula, we generate several special solutions.
In this paper, it is the intent to apply the Riemann-Hilbert transformation developed by Hauser and Ernst (1980) in providing a new representation of the Virasoro group. It is found that the Geroch group that acts on the solution space of the Einstein field equations is extended to the semidirect product of the Virasoro and Kac-Moody groups; also, the relationship between the infinitesimal transformation given previously (1987, 1988) and the infinitesimal Riemann-Hilbert transformation is pointed out. Finally, it is shown that the well-known Neugebauer Bäcklund transformation can be derived from the Riemann-Hilbert transformation.
A soliton geometry is introduced on manifolds with arbitrary dimensions. The usual soliton connection 1-form defined by Crampin et al. is recovered when the soldering form is a 0-form. It is shown that Einstein's vacuum field equations admit a soliton connection and a soldering 1-form. An associated linear equation with a spectral parameter of Einstein's vacuum field equations are found and some properties of this equation are explored. An example of a Bäcklund transformation is also given.
Application of the method of the inverse problem to the Einstein equations is carried to the construction of new explicit solutions of the Einstein-Maxwell system in terms of Riemann theta functions. In the vacuum case formulas are obtained both for solutions of the Ernst equations and for coefficients of the metric in the formalism of Belinskii-Zakharov U-V pairs. Due to the fact that the U-V pairs used here have a variable spectral parameter, the dynamics is given by a trajectory in a variety of moduli of algebraic curves, and explicit introduction into the solutions of functional parameters turns out to be possible. Among the solutions constructed is a large subclass of new, asymptotically flat solutions. Degeneration of these solutions affords new formulas for multisoliton solutions describing a system of several black holes.
Recently it has been shown that the methods of algebraic geometry first used for finding periodic and almost periodic solutions of KdV, HSh, SG and other equations may be successfully applied to study the solutions of nonlinear equations with a variable spectral parameter in associated zero-curvature representation. In this work this treatment is extended to the case of the self-duality equation. It seems to be the first example of a four-dimensional non-linear equation solvable by the method of finite-gap integration. Two broad classes of finite-gap solutions for each --- SU(2) and SU(1,1) gauge groups are constructed in terms of multidimensional theta-functions. The dynamics of the solutions is given by the movement of the hyperelliptic curve with moving branch points and a divisor of the poles in the moduli space of algebraic curves. In the general case our solutions have no periodicity property. We show how one-instanton solution and 5N-parametric t'Hooft family of instantons may be obtained by the degeneration of general formulae.
We apply the infinitesimal Hou-Li transformations d~(u) to an arbitrary Weyl solution, and find that they do not preserve asymptotic flatness. However, we find a closely related set of new transformations d(u) that do have this desirable property. We determine the action of these new transformations on the multipole moments of the solution.
By using the double complex function method, two regular double Riemann-Hilbert problems are established; therefore, two double Kac-Moody algebras are given. These structures show that in two-dimensional reduced gravity, in fact, there is more exquisite hidden symmetry than common nonlinear systems.
The development of a Riemann-Hilbert (RH) problem approach to the representation of the Virasoro group introduced in our earlier paper (1989) is continued. The Rouche theorem is employed to prove that there exists a set of scalar functions in the complex tau plane, in terms of which the structure of the Virasoro group can be decided. The general form of the RH problem will also be presented and the uniqueness of its solution will be proven. For future use, a standard form of a matrix Fredholm equation equivalent to our RH problem will be deduced. This may help to establish in a future paper the existence of a solution of the RH problem.
The stationary axially symmetric vacuum Einstein equations possess an extremely large group of symmetry transformations. We give a unified treatment of all of the currently known transformations, by considering their action on the boundary values of a general solution upon the symmetry axis. We use these results to construct two infinite families of transformations B(u) and D(u) which preserve asymptotic flatness. Attempting to understand these transformations leads to the definition of a new gauge, the HKX gauge, in which they take their simplest form.
The physical properties of algebraic geometric solutions of stationary axisymmetric vacuum Einstein's equations are discussed. It appears that these solutions describe an interaction of a few localized rotating string-like objects on an arbitrary static background. If such an object collapses to a point, then it produces a Kerr-NUT black hole.
It is shown how Kac-Moody algebras arise from symmetry reductions of a four dimensional generally covariant field theory, via a 3+1 canonical decomposition. This observation may be of significance in the quantization of certain midi-superspace reductions of general relativity.
By using the double inverse scattering method, the problem of how to find new soliton solutions of the stationary axisymmetric vacuum field equations is studied. It is found that, for some kinds of seed solutions, the scattering wave functions can be directly obtained. Some examples of applications are given. Similar results in the cylindrically symmetric case are also discussed.
The method developed by one of the authors for the construction of exact solutions of the Einstein-Maxwell equations by setting the behaviour of the Ernst potentials on the symmetry axis is discussed in full detail. Some new results regarding the solution of the integral equations and the construction of the 3 x 3 matrix potential H are presented. Two new examples of the application of the method are considered, one of which is the stationary vacuum solution for a rotating mass, and the second one is the 4-parameter metric which can be used for the description of the exterior field of a charged magnetised spinning source.
Einstein-Maxwell fields are studied for which the complex functional relationship characterizing most of the well known important solutions such as the Kerr-Newman black hole does not hold. There is a family of spacetimes in this general class which is govered by the ordinary third-order differential equation
{ [(1 + s2) F
+ 1/F]ss
[(1 + s2)
F - 1/F]
3 }s
= Q [(1 + s2)
F + 1/ F]ss [(1 + s2)
F - 1/F]
for the potential Phi. For certain values of the constant Q, these fields exhibit undulations as the cylindrical radius varies.
In this paper we present a more transparent version of an earlier construction of genus g algebraic-geometric solutions of the Einstein equation. For one of the metric coefficients we obtain a new expression that allows us to construct the coefficient in terms of derivatives of the function y(l) (solution of associative linear system). Finally, we proceed with an analysis of the two simplest genus 1 (elliptic) solutions.
The Ernst equation is formulated on an arbitrary Riemann surface. Analytically, the problem reduces to finding solutions of the ordinary Ernst equation which are periodic along the symmetry axis. The family of (punctured) Riemann surfaces admitting a non-trivial Ernst field constitutes a `partially discretized' subspace of the usual moduli space. The method allows us to construct new exact solutions of Einstein's equations in vacuo with non-trivial topology, such that different `universes', each of which may have several black holes on the symmetry axis, are connected through necks bounded by cosmic strings. We show how the extra topological degrees of freedom may lead to an extension of the Geroch group, and discuss possible applications to string theory.
The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints.
On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature' equation, and then employing techniques used in the study of two dimensional integrable models. Two infinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed.
It is also shown that the sl(2,R) observables can be associated with a solution generating technique which is linked to that given by Geroch (1971).
We present a method for generating exact solutions of Einstein equations in vacuum using harmonic maps, when the spacetime possesses two commuting Killing vectors. This method consists in writing the axisymmetric stationary Einstein equations in vacuum as a harmonic map which belongs to the group SL(2,R), and decomposing it in its harmonic "submaps". This method provides a natural classification of the solutions in classes (Weyl's class, Lewis' class, etc.).
We express the complex potential E and the metrical fields w and g of all stationary axisymmetric vacuum spacetimes that result from the application of two successive quadruple-Neugebauer (or two double-Harrison) transformations to Minkowski space in terms of data specified on the symmetry axis, which are in turn easily expressed in terms of multipole moments. Moreover, we suggest how, in future papers we shall apply our approach to do the same thing for those vacuum solutions that arise from the application of more than two successive transformations, and for those electrovac solutions that have axis data similar to that of the vacuum solutions of the Neugebauer family.
Generalizing a method presented in an earlier paper, we express the complex potentials E and F of all stationary axisymmetric electrovac spacetimes that correspond to axis data of the form E(z,0) = (U - W)/(U + W), F(z,0) = V/(U + W), where U = z2 + U1 + U2, V = V1 z + V2, W = W1 z + W2, in terms of the complex parameters U1, V1, W1, U2, V2, and W2, that are directly associated with the various multipole moments.
We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that the both quantities are based on the "same" complexification. In the two-Killing-vector reduction, we demonstrate Ehlers' and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated with these symmetries are explicitly obtained. We show that the gl(2,R) loop algebra constructed previously in the loop representation is not the Lie algebra of the Geroch group itself. We also point out that the recent argument on the equivalence to a chiral model is based on a gauge-choice which cannot be achieved generically.
It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by reformulating the Ernst equation and its associated linear system in terms of a nonautonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the tau function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a "two-time" Hamiltonian approach, and which opens new avenues for the quantization of such systems.
A new class of exact solutions to the axisymmetric and stationary vacuum Einstein equations containing n arbitrary complex parameters and one arbitrary real solution of the axisymmetric three-dimensional Laplace equation is presented. The solutions are related to Jacobi's inversion problem for hyperelliptic Abelian integrals.
Nagatomo's universal Grassmann manifold scheme (1989) 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.
The full set of equations describing the gravitational and electromagnetic fields of a rotating, magnetized, charged relativistic disk is formulated as a set of integral linear equations.
We discuss a class of solutions to the Ernst equation (the stationary axisymmetric Einstein equations) obtained as solutions of a generalized scalar Riemann-Hilbert problem on a hyperelliptic Riemann surface. The singular structure of these solutions is studied for arbitrary genus of the Riemann surface. A subclass is given for which the Ernst potential is everywhere regular besides at a contour that can be identified with the surface of a body of revolution. It turns out that the recently discussed rigidly rotating dust disk belongs to this class.
Within a class of algebro-geometric solutions to the Ernst equation we identify a physically interesting subclass: The solutions are regular except at a closed surface, asymptotically flat, and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance, a relativistic star or galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters that can possibly be used to solve certain boundary value problems for the Ernst equation such as the rigidly rotating dust disk. The solutions can have ergoregions, a Minkowskian limit, and an ultrarelativistic limit where the metric approaches the extreme Kerr solution.
Riemann-Hilbert problems are an important solution technique for completely integrable differential equations. They are used to introduce a free function in the solutions which can be used at least in principle to solve initial or boundary value problems. But even if the initial or boundary data can be translated into a Riemann-Hilbert problem, it is in general impossible to obtain explicit solutions. In the case of the Ernst eqution, however, this is possible for a large class because the matrix problem can be shown to be gauge equivalent to a scalar one on a hyperelliptic Riemann surface that can be solved in terms of theta functions. As an example we discuss the rigidly rotating dust disk.
Starting with the Weyl solution that is known to be valid for any real value of the deformation parameter d, the solution of the Ernst equation in an axially symmetric gravitational field caused by a rotating source is given in a perturbation expansion of the variables l (=q2/ p2) and b (=y2-1) for a real value of d. The proof of the validity of the solution is given to first order in l and all orders in b.
A solution of the Ernst equation in an axially symmetric gravitational field caused by a rotating source is given in a compact form using the prolate spheroidal coordinates for an arbitrary real value of the deformation parameter d. The proof of the validity of the solution is given analytically for integer d, and to lowest orders in l (=q2/ p2) for real d. Numerical results confirming the validity of the solution are thoroughly given to higher order in l for real d.
We discuss the relationship between Schlesinger system and stationary axisymmetric Einstein's equation on the level of algebro-geometric solutions. In particular, we calculate all metric coefficients corresponding to solutions of Ernst equation in terms of theta-functions constructed in [20,21,25].
We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Meinel and Neugebauer can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a previous Letter and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance, a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit, and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulas for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest nonstatic solutions (which are of genus 2) to which the rigidly rotating dust disk belongs.
Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The soluiton of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.
A procedure is described for matching a given stationary axisymmetric perfect fluid solution to a not necessarily asymptotically flat vacuum exterior. Using data on the zero pressure surface, the procedure yields the Ernst potential of the matching vacuum metric on the symmetry axis. From this the full metric can be constructed using a variety of well established procedures.
In addition to the papers listed above, I should like to call attention to Nail R. Sibgatullin's book, Oscillations and Waves in Strong Gravitational and Electromagnetic Fields, Springer-Verlag (1991), especially pp. 133--137, which section is entitled "Proof of the Geroch Hypothesis." This book is an updated and modified translation of a Russian edition of 1984.
• I have a reprint or preprint of the papers that are so marked. In some other cases I found the author's abstract by looking up Physics Abstracts and the journal in which the paper was published. In very few cases, where the author had supplied no abstract, I had to write my own by excerpting phrases from the paper itself.
Supported in part by grants PHY-93-07762, PHY-96-01043 and PHY-98-00091 from the National Science Foundation to FJE Enterprises.