Ehlers Transformation Research Articles

Symmetries of heterotic string theory

We study the symmetries of the two-dimensional Heterotic string theory by following the approach of Kinnersley et al. for the study of stationary axially-symmetric Einstein-Maxwell equations. We identify the finite-dimensional symmetries which are the analogues of the groups G′ and H′ for the Einstein-Maxwell equations. We also give the constructions for the infinite number of conserved currents and the affine o ̂ (8,24) symmetry algebra in this formulation. The generalized Ehlers and Harrison transformations are identified and a parallel between the infinite-dimensional symmetry algebra for the heterotic string case with s ̂ l(3, R) that arise in the case of Einstein-Maxwell equations is pointed out.

Ehlers transformations and string effective action

We explicitly obtain the generalization of the Ehlers transformation for stationary axisymmetric Einstein equations to string theory. This is accomplished by finding the twist potential corresponding to the moduli fields in the effective two dimensional theory. Twist potential and symmetric moduli are shown to transform under an O( d, d) which is a manifest symmetry of the equations of motion. The non-trivial action of this O( d, d) is given by the Ehlers transformation and belongs to the set O(d)×O(d) O(d) .

Generalized magnetic universe solutions.

In this paper we apply the techniques which have been developed over the last few decades for generating nontrivially new solutions of the Einstein-Maxwell equations from seed solutions for simple spacetimes. The simple seed spacetime which we choose is the "magnetic universe" to which we apply the Ehlers transformation. Three interesting non-singular metrics are generated. Two of these may be described as "rotating magnetic universes" and the third as an "evolving magnetic universe." Each is causally complete - in that all timelike and lightlike geodesics do not end in a finite time or affine parameter. We also give the electromagnetic field in each case. For the two rotating stationary cases we give the projection with respect to a stationary observer of the electromagnetic field into electric and magnetic components.

Nondiagonal seed universes and a network of double gravitational soliton universes

By using the double Ehlers transformations andγ transformations, new non-diagonal seed solutions are obtained. From these seed solutions we obtain a network of double gravitational soliton solutions. The double gravitational inverse scattering method is used to give some concrete examples of new solutions.

A family of electrovac colliding wave solutions of Einstein’s equations

Beginning with any colliding wave solution of the vacuum Einstein equations, a corresponding electrified colliding wave solution can be generated through the use of a transformation due to Harrison [J. Math. Phys. 9, 1744 (1968)]. The method, long employed in the context of stationary axisymmetric fields, is equally applicable to colliding wave solutions. Here it is applied to a large family of vacuum metrics derived by applying a generalized Ehlers transformation to solutions published recently by Ernst, García, and Hauser (EGH) [J. Math. Phys. 28, 2155, 2951 (1987); 29, 681 (1988)]. Those EGH solutions were themselves a generalization of solutions first derived by Ferrari, Ibañez, and Bruni [Phys. Rev. D 36, 1053 (1987)]. Among the electrovac solutions that are obtained is a charged version of the Nutku–Halil [Phys. Rev. Lett. 39, 1379 (1977)] metric that possesses an arbitrary complex charge parameter.

On generating new solutions of the Ernst equation from old solutions

If E is an arbitrary generic solution of the Ernst equation and E its complex conjugate the most general expression E'=G(E, E, rho , z), is determined which is also a solution of this equation. If E is complex it is shown that G cannot depend on E and E simultaneously, and that the most general E' is the Ehlers transformation. Also it is proven that, if E is real, the most general E' is E'=(Ec+ic1ew)/(ic2Ec+ew), where c, c1 and c2 are arbitrary real constants and w is an arbitrary real solution of the Laplace equation in the axially symmetric case. In addition two other expressions are given for E'. The three expressions can also be derived from the Ehlers transformation.

Colliding gravitational plane waves with noncollinear polarization. I

An Ehlers transformation on the Ernst potential for the Nutku–Halil solution [Phys. Rev. Lett. 39, 1379 (1977)] provides a new solution of the Einstein field equations describing colliding gravitational plane waves with noncollinear polarization, the first of an infinite sequence of solutions that can be generated using techniques described in this paper.

On colliding waves that develop time–like singularities: a new class of solutions of the Einstein–Maxwell equations

Solutions of the Einstein–Maxwell equations are found that provide generalizations of a solution discovered by Bell and Szekeres, which represents the collision of impulsive gravitational waves coupled with electromagnetic shock-waves in a conformally flat space-time. Starting with the Bell–Szekeres solution in a form more general than their original one (though equivalent to it) and applying to it a so-called Ehlers transformation, we obtain a new family of Petrov type-D space-times in which horizons form and subsequently two-dimensional time-like singularities develop. A second solution provides a generalization of the Bell–Szekeres solution in the same way as the axisymmetric distorted static black-hole solutions provide a generalization of the Schwarzschild solution. This second solution also forms a horizon but the time-like singularity that develops is three-dimensional. The mathematical theory that is developed seems specially adapted to the solution of these and related problems.

On the inverse-scattering method generation of gravitational waves and other new solution-generating algorithms

The possibility of generation of a new exact solution to the vacuum Einstein equations using the inverse-scattering method together with other simple new solution-generating algorithms like the hyperbolic version of the Lewis and the Ehlers transformations is examined. The relation of the hyperbolic version of the Papapetrou solution with the solutions generated by using the inverse-scattering method is studied. Also a relation of the Papapetrou solution with the Bianchi II vacuum cosmological model is established. The integration of the potential that appears in the Ernst type of formulation of the Einstein equations is explicitly performed for an-soliton solution. Then-soliton solution is cast in a determinantal form. And the singular behaviour of a particular class ofn-soliton solutions is analysed.

A new class of asymptotically flat stationary axisymmetric vacuum gravitational fields

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

A new class of bipolar vacuum gravitational fields

We present the Ernst potential and the full four-dimensional metric for a stationary axisymmetric solution of the vacuum Einstein equations, which we obtain by application of two rank-zero H. K. X. transformations (Hoenselaers, Kinnersley & Xanthopolous J. math. Phys . 20, 2530 (1979)) to the general static Weyl solution. A suitable Ehlers transformation ensures asymptotic flatness.

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.

SU(2,1) generation of electrovacs from Minkowski space

For every nonnull Killing vector K of any given electrovac, there exists a group of tranformations ℋK of the gravitational and electromagnetic potentials of Ernst. This is the group 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 properties 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 Segre 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.