Gravitational waves emitted by perturbed black holes or relativistic stars are dominated by `quasinormal ringing', damped oscillations at single frequencies which are characteristic of the underlying system. These quasinormal modes have been studied for a long time, often with the intent of describing the time evolution of a perturbation in terms of these modes in a way very similar to a normal-mode analysis. In this review, we summarize how quasinormal modes are defined and computed. We will see why they have been regarded as closely analogous to normal modes, and discover why they are actually quite different. We also discuss how quasinormal modes can be used in the analysis of a gravitational wave signal, such as will hopefully be detected in the near future.

https://iopscience.iop.org/article/10.1088/0264-9381/16/12/201/pdf

Quasinormal modes: the characteristic `sound' of black holes and neutron stars

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

In this paper we consider generalizations in 4 dimensions of the Einstein-Maxwell equations which typically arise from Kaluza-Klein theories. We specify conditions such that stationary solutions lead to non-linearσ-models for symmetric spaces. Using both this group theoretic structure and some properties of harmonic maps we are able to generalize many of the known existence and uniqueness theorems for black holes in Einstein-Maxwell theory to this more general setting.

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$4$-dimensional black holes from Kaluza-Klein theories

The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.

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Stationary Black Holes: Uniqueness and Beyond

Rotating relativistic stars have been studied extensively in recent years, both theoretically and observationally, because of the information they might yield about the equation of state of matter at extremely high densities and because they are considered to be promising sources of gravitational waves. The latest theoretical understanding of rotating stars in relativity is reviewed in this updated article. The sections on equilibrium properties and on nonaxisymmetric oscillations and instabilities in f-modes and r-modes have been updated. Several new sections have been added on equilibria in modified theories of gravity, approximate universal relationships, the one-arm spiral instability, on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity including both hydrodynamic and magnetohydrodynamic studies of these objects.

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Rotating Stars in Relativity

The superposition of two Kerr solutions

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 (part 3 A).

Eine Methode zur Konstruktion stationärer EINSTEIN‐MAXWELL‐Felder

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.

https://iopscience.iop.org/article/10.1088/0305-4470/16/9/017/pdf

Einstein-Maxwell solitons

This paper presents the gravitational field of a uniformly rotating stationary and axisymmetric disk consisting of dust particles as a rigorous global solution to the Einstein equations. The problem is formulated as a boundary value problem of the Ernst equation and solved by means of inverse methods. The solution is given in terms of linear integral equations and depends on two parameters: the angular velocity Ω and the relative redshift z from the center of the disk. The Newtonian limit z<<1 represents the MacLaurin solution of a rotating fluid in the disk limit. For z→∞ the exterior solution is given by the extreme Kerr solution. This proves a conjecture of Bardeen & Wagoner (1969, 1971)

The Einsteinian Gravitational Field of the Rigidly Rotating Disk of Dust

The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This article is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust (“galaxies”). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

G. Neugebauer

Papers

The superposition of two Kerr solutions

Eine Methode zur Konstruktion stationärer EINSTEIN‐MAXWELL‐Felder

Einstein-Maxwell solitons

The Einsteinian Gravitational Field of the Rigidly Rotating Disk of Dust

Progress in relativistic gravitational theory using the inverse scattering method