Harrison Transformation Research Articles

Charged dual string vacua from interacting rotating black holes via discrete and nonlinear symmetries

Using the stationary formulation of the toroidally compactified heterotic string theory in terms of a pair of matrix Ernst potentials we consider the four-dimensional truncation of this theory with no U(1) vector fields excited. Imposing one timelike Killing vector permits us to express the stationary effective action as a model in which gravity is coupled to a matrix Ernst potential which, under certain parametrization, allows us to interpret the matter sector of this theory as a double Ernst system. We generate a web of string vacua which are related to each other via a set of discrete symmetries of the effective action (some of them involve S-duality transformations and possess non-perturbative character). Some physical implications of these discrete symmetries are analysed and we find that, in some particular cases, they relate rotating black holes coupled to a dilaton with no Kalb–Ramond field, static black holes with non-trivial dilaton and antisymmetric tensor fields, and rotating and static naked singularities. Further, by applying a nonlinear symmetry, namely, the so-called normalized Harrison transformation, on the seed field configurations corresponding to these neutral backgrounds, we recover the U(1)n Abelian vector sector of the four-dimensional action of the heterotic string, charging in this way the double Ernst system which corresponds to each one of the neutral string vacua, i.e., the stationary and the static black holes and the naked singularities.

On explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations describing dust disks

AbstractWe review explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro‐vacuum. The Einstein‐Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three‐dimensional gravity where the matter is given by a SU(2,1)/S[U(1,1)× U(1)] nonlinear sigma model. The SU(2,1) invariance of the stationary Einstein‐Maxwell equations can be used to construct solutions for the electro‐vacuum from solutions to the pure vacuum case via a so‐called Harrison transformation. It is shown that the corresponding solutions will always have a non‐vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro‐vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter‐rotating dust disks. To obtain algebro‐geometric solutions with vanishing total charge which are of astrophysical relevance, three‐sheeted surfaces have to be considered. The matter in the disk is discussed following Bičák et al. We review the ‘cut and glue’ technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter‐rotating dust moving on geodesics. In electro‐vacuum the condition of geodesic movement is replaced by electro‐geodesic movement. As an example we discuss a class of Harrison‐transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro‐dust can be given.

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. The Harrison transformed hyperelliptic solutions are discussed.

Harrison transformation of hyperelliptic solutions and charged dust disks

We use a Harrison transformation on solutions to the stationary axisymmetric Einstein equations to generate solutions of the Einstein-Maxwell equations. The case of hyperelliptic solutions to the Ernst equation is studied in detail. Analytic expressions for the metric and the multipole moments are obtained. As an example we consider the transformation of a family of counter-rotating dust disks. The resulting solutions can be interpreted as disks with currents and matter with a purely azimuthal pressure or as two streams of freely moving charged particles. We discuss interesting limiting cases as the extreme limit where the charge becomes identical to the mass, and the ultrarelativistic limit where the central redshift diverges.

Generating branes via sigma models

Starting with the D-dimensional Einstein-dilaton-antisymmetric form equations and assuming a block-diagonal form of a metric we derive a $(D-d)$-dimensional $\sigma$-model with the target space $SL(d,R)/SO(d) \times SL(2,R)/SO(2) \times R$ or its non-compact form. Various solution-generating techniques are developed and applied to construct some known and some new $p$-brane solutions. It is shown that the Harrison transformation belonging to the $SL(2,R)$ subgroup generates black $p$-branes from the seed Schwarzschild solution. A fluxbrane generalizing the Bonnor-Melvin-Gibbons-Maeda solution is constructed as well as a non-linear superposition of the fluxbrane and a spherical black hole. A new simple way to endow branes with additional internal structure such as plane waves is suggested. Applying the harmonic maps technique we generate new solutions with a non-trivial shell structure in the transverse space (`matrioshka' $p$-branes). It is shown that the $p$-brane intersection rules have a simple geometric interpretation as conditions ensuring the symmetric space property of the target space. Finally, a Bonnor-type symmetry is used to construct a new magnetic 6-brane with a dipole moment in the ten-dimensional IIA theory.

Generation of colliding gravitational-electromagnetic waves by means of a Harrison transformation

Generation of colliding gravitational-electromagnetic waves by means of a Harrison transformation

Magnetic generalizations of the Carter metrics

Magnetic generalizations of all Carter metrics—Carter A, B(+), B(−), and D solutions—are obtained by applying a Harrison transformation to the Ernst potentials determined from a linear combination of two Killing vectors. The most general magnetized Carter-A metric contains, among other solutions, the magnetized Kerr–Newman metrics and, modulo limiting transitions, the magnetized branches B(+), B(−), and D.

Colliding-wave generalizations of the Nutku-Halil metric in the Einstein-Maxwell theory.

A class of nondiagonal cylindrically symmetric algebraically general solutions of the Einstein-Maxwell equations is given. These solutions are interpretable as gravitational colliding waves supporting an electromagnetic field; they are generalizations of the Nutku-Halil solution. The new metric, derived by a Harrison transformation, is endowed with four free parameters.

The most general magnetized Kerr–Newman metric

The complete magnetic generalization of the Kerr–Newman (KN) metric obtainable by applying a Harrison transformation to the Ernst potentials determined for an arbitrary linear combination of the two Killing vectors of the seed KN metric is derived.

Magnetized diagonal Weyl metrics

A generalization of the diagonal Weyl metrics, which is obtained by a Harrison transformation over the Ernst potential ε, is presented. The Ernst potential is defined for a linear combination of the Killing vectors ∂τ and ∂σ. The cases which correspond to Kasner and Voorhees metrics are analyzed in some detail.

Two equal Kerr-Newman sources in stationary equilibrium.

By applying the Harrison transformation to two equal Kerr black holes spinning around the same axis we find the solution corresponding to two equal Kerr-Newman black holes. We succeed in removing all stress singularities along the axis and simultaneously giving the system total positive mass. However, there still persists a singularity off the axis in the plane with respect to which the holes are located symmetrically. We conjecture that two black holes cannot be in stationary equi- librium except for the exceptional case of two nonrotating extreme Reissner-Nordstroem black holes.

On the Prolongation Structure of Ernst Equation

An SL(2R) × R1(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is presented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neuge-bauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbert problem (HHP), would be covered by this condition.

Bäcklund transformations in the Hauser–Ernst formalism for stationary axisymmetric spacetimes

It is shown that Harrison’s Bäcklund transformation for the Ernst equation of general relativity is a two-parameter subset (not subgroup) of the infinite-dimensional Geroch group K. We exhibit 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. We also show how finite (i.e., already exponentiated) transformations in the B group and nonnull groups of Kinnersley and Chitre 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, we construct 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.

Bäcklund transformations of the axially symmetric stationary Einstein-Maxwell equations II

We obtained a Bäcklund transformation of the axially symmetric stationary Einstein-Maxwell equation which is a generalization of Neugebauer's I 1 transformation in the case of the Ernst equations. This transformation is shown to generate the generalized Ehlers transformation and the Harrison transformation.