# Thirty years of studies of integrable reductions of Einstein's field equations

@article{Alekseev2010ThirtyYO, title={Thirty years of studies of integrable reductions of Einstein's field equations}, author={George A. Alekseev}, journal={arXiv: General Relativity and Quantum Cosmology}, year={2010} }

More than thirty years passed since the first discoveries of various aspects of integrability of the symmetry reduced vacuum Einstein equations and electrovacuum Einstein - Maxwell equations were made and gave rise to constructions of powerful solution generating methods for these equations. In the subsequent papers, the inverse scattering approach and soliton generating techniques, B\"acklund and symmetry transformations, formulations of auxiliary Riemann-Hilbert or homogeneous Hilbert…

## 25 Citations

### Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group G2

- Mathematics, Physics
- 2016

We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group G2 with nonnull orbits and…

### Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group G2

- Mathematics, PhysicsProceedings of the Steklov Institute of Mathematics
- 2016

We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group G2 with nonnull orbits and…

### The non-autonomous chiral model and the Ernst equation of General Relativity in the bidifferential calculus framework

- Mathematics
- 2011

The non-autonomous chiral model equation for an m m matrix function on a two-dimensional space appears in particular in general relativity, where for m = 2 a certain reduction of it determines…

### Travelling waves in expanding spatially homogeneous space–times

- Physics, Mathematics
- 2015

Some classes of the so-called ‘travelling wave’ solutions of Einstein and Einstein–Maxwell equations in general relativity and of dynamical equations for massless bosonic fields in string gravity in…

### Geroch group for Einstein spaces and holographic integrability

- Physics, Mathematics
- 2015

We review how Geroch's reduction method is extended from Ricci-flat to Einstein spacetimes. The Ehlers-Geroch SL(2,R) group is still present in the three-dimensional sigma-model that captures the…

### Dark energy in some integrable and nonintegrable FRW cosmological models

- Physics
- 2011

One of the greatest challenges in cosmology today is to determine the nature of dark energy, the sourse of the observed present acceleration of the Universe. Besides the vacuum energy, various dark…

### Weyl metrics and Wiener-Hopf factorization

- Mathematics
- 2019

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that…

### Weyl metrics and Wiener-Hopf factorization

- MathematicsJournal of High Energy Physics
- 2020

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that…

### Conformal mechanics of homogeneously curved FRW cosmology: Möbius Invariance, CVH algebra and relation to AdS2

- Physics, Mathematics
- 2021

We show that the homogeneously curved Friedmann-Robertson-Walker (FRW) cosmology filled with a massless scalar field enjoys a Noether symmetry under the 1d conformal group SL(2,R). Concretely, the…

### The Geroch group in Einstein spaces

- Physics, Mathematics
- 2014

Gerochʼs solution-generating method is extended to the case of Einstein spaces, which possess a Killing vector and are thus asymptotically (locally) (anti) de Sitter. This includes the reduction to a…

## References

SHOWING 1-10 OF 83 REFERENCES

### Monodromy transform approach to solution of the Ernst equations in General Relativity

- Mathematics
- 1999

The approach, referred to as "monodromy transform", provides some general base for solution of all known integrable space - time symmetry reductions of Einstein equations for the case of pure vacuum…

### Gravitational solitons and monodromy transform approach to solution of integrable reductions of Einstein equations

- Mathematics, Physics
- 2001

### A New Integral Equation Form of Integrable Reductions of the Einstein Equations

- Mathematics
- 2001

We further develop the monodromy transformation method for analyzing hyperbolic and elliptic integrable reductions of the Einstein equations. The compatibility conditions for alternative…

### Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein’s field equations

- Mathematics
- 2005

We show that for the fields depending on only two of the four space-time coordinates, the spaces of local solutions of various integrable reductions of Einstein’s field equations are the subspaces of…

### Integrability of Generalized (Matrix) Ernst Equations in String Theory

- Mathematics
- 2005

We elucidate the integrability structures of the matrix generalizations of the Ernst equation for Hermitian or complex symmetric (d×d)-matrix Ernst potentials. These equations arise in string theory…

### Stationary gravitational solitons with axial symmetry

- Physics
- 2010

In a previous paper we have shown1 that in the case in which the metric tensor depends on only two variables the gravitational equations form a system which i s integrable by the method of the…

### Integrability of the symmetry reduced bosonic dynamics and soliton generating transformations in the low energy heterotic string effective theory

- Physics
- 2009

Integrable structure of the symmetry reduced dynamics of massless bosonic sector of the heterotic string effective action is presented. For string background equations that govern in the space-time…

### Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II. Production of the black ring solution

- Mathematics
- 2006

We study vacuum solutions of five-dimensional Einstein equations generated by the inverse scattering method. We reproduce the black ring solution which was found by Emparan and Reall by taking the…

### Infinite hierarchies of exact solutions of the einstein and einstein-maxwell equations for interacting waves and inhomogeneous cosmologies

- Mathematics, PhysicsPhysical review letters
- 2000

This work presents infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations as represented by their Ernst potentials, which are constructed using the so-called "monodromy transform" approach and the new method for the solution of the linear singular integral equation form of the reduced Einstein equations.

### Integrability of five-dimensional minimal supergravity and charged rotating black holes

- Physics
- 2010

We explore the integrability of five-dimensional minimal supergravity in the presence of three commuting Killing vectors. We argue that to see the integrability structure of the theory one…