[...]... general properties of the gravitational soliton solutions The case of background solutions with a diagonal metric is discussed in detail A section is devoted to the topological properties of gravitational solitons and we discuss how some features of the sine-Gordon solitons can be translated under some restrictions to the gravitational solitons Some remarkable solutions such as the gravitational analogue... old ones These solutions are generally known as soliton solutions of the gravitational field or gravitational solitons for short, even though they share only some, or none, of the properties that solitons have in other nonlinear contexts x Preface xi Among the soliton solutions generated by the ISM are some of the most relevant in gravitational physics Thus in the stationary axisymmetric case the Kerr... be derived as gravitational solitons Chapter 8 is devoted to the stationary axisymmetric gravitational soliton solutions Now the relevant metric field equations are elliptic rather than hyperbolic, but the ISM of chapter 1 is easily translated to this case We describe in detail how the Schwarzschild and Kerr metrics, and their Kerr–NUT generalizations are simply obtained as gravitational solitons from... characterization of the gravitational solitons as the soliton waves of nongravitational physics It is true that in some restricted cases soliton solutions can be topologically characterized in a mathematical sense, but this characterization is then blurred in the physics of the gravitational spacetime the solutions describe Things like the velocity of propagation, energy of the solitons, shape persistence... they present the most clear nonlinear features of soliton physics such as the time delay when solitons interact Solutions representing finite perturbations of isotropic cosmologies are also derived and studied In chapter 6 we describe gravitational solitons with cylindrical symmetry Mathematically most of the gravitational solutions in this context are easily derived from the cosmological solution of... be of interest to represent gravitational radiation around a straight cosmic string Also some soliton solutions were found illustrating the gravitational analogue of the electromagnetic Faraday rotation, which is a typical nonlinear effect of gravity Some of this work was reviewed in ref [288] In this book we give a comprehensive review of the ISM in gravitation and of the gravitational soliton solutions... this context are easily derived from the cosmological solution of the two previous chapters but, of course, they describe different physics In chapter 7 we describe the connection of gravitational solitons with exact gravitational plane waves and the head-on collision of plane waves We illustrate the physically more interesting properties of the spacetimes describing plane waves and the head-on collision... time shift after collision have been only partially characterized, and this has represented a clear obstruction in any attempt to the quantization of gravitational solitons We feel that more work along these lines should lead to a better understanding of gravitational physics at the classical and, even possibly, the quantum levels As regards to the level of presentation of this book we believe that its... Pullin, H Sato, A Shabat and G Neugebauer for stimulating discussions or suggestions Rome Barcelona September 2000 Vladimir A Belinski Enric Verdaguer 1 Inverse scattering technique in gravity The purpose of this chapter is to describe the Inverse Scattering Method (ISM) for the gravitational field We begin in section 1.1 with a brief overview of the ISM in nonlinear physics In a nutshell the procedure... sine-Gordon equation, which will be of use later We then go on to generalize and adapt the ISM in the gravitational context to solve the Einstein equations in vacuum when the spacetimes admit an orthogonally transitive two-parameter group of isometries We describe in detail the procedure for obtaining gravitational soliton solutions The ISM is generalized to solve vacuum Einstein equations in an arbitrary . parameter λ. This parameter is the energy of the scattered particle and positive values are the continuous spectrum for the problem (1.2). Moreover, a discrete set of negative eigenvalues of. spectrum (in the one-dimensional problem these are the transmission and reflection coefficients, T (λ) and R(λ ), respectively ), and the negative eigenvalues λ n of the discrete spectrum together. Vladimir A. Belinski Barcelona Enric Verdaguer September 2000 1 Inverse scattering technique in gravity The purpose of this chapter is to describe the Inverse Scattering Method (ISM) for the gravitational