Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.

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Quasinormal modes of black holes and black branes

Perturbations of black holes, initially considered in the context of possible observations of astrophysical effects, have been studied for the past 10 years in string theory, brane-world models, and quantum gravity. Through the famous gauge/gravity duality, proper oscillations of perturbed black holes, called quasinormal modes, allow for the description of the hydrodynamic regime in the dual finite temperature field theory at strong coupling, which can be used to predict the behavior of quark-gluon plasmas in the nonperturbative regime. On the other hand, the brane-world scenarios assume the existence of extra dimensions in nature, so that multidimensional black holes can be formed in a laboratory experiment. All this stimulated active research in the field of perturbations of higher-dimensional black holes and branes during recent years. In this review recent achievements on various aspects of black hole perturbations are discussed such as decoupling of variables in the perturbation equations, quasinormal modes (with special emphasis on various numerical and analytical methods of calculations), late-time tails, gravitational stability, anti--de Sitter/conformal field theory interpretation of quasinormal modes, and holographic superconductors. We also touch on state-of-the-art observational possibilities for detecting quasinormal modes of black holes.

Quasinormal modes of black holes: From astrophysics to string theory

Quasinormal frequencies of electromagnetic and gravitational perturbations in asymptotically anti-de Sitter spacetime can be identified with poles of the corresponding real-time Green's functions in a holographically dual finite temperature field theory. The quasinormal modes are defined for gauge-invariant quantities which obey an incoming-wave boundary condition at the horizon and a Dirichlet condition at the boundary. As an application, we explicitly find poles of retarded correlation functions of R-symmetry currents and the energy-momentum tensor in strongly coupled finite temperature N=4 supersymmetric SU(Nc) Yang-Mills theory in the limit of large Nc.

Quasinormal modes and holography

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body. Recapitulating investigations of this question, this article states that in 25 years of experience with radar, there have been no known incidents of pilots being affected by radar waves. In the future, however, involvement by some pilots with Vessel Traffic Service shore-based radar could affect pilots somewhat differently from limited exposure to pilot launch radar. Pilots who find themselves in new working conditions close to an emitting source should exercise care all times.

About heat waves

In this contribution we summarize some of our recent results about
 the matching of collapsing inhomogeneous dust-filled spacetimes to
 vacuum static exteriors with a negative cosmological constant in
 planar or hyperbolic symmetry. This models gravitational collapse to
 toroidal or higher genus asymptotically AdS black holes.

Gravitational Collapse to Toroidal and Higher Genus Black Holes

We present an idealised model of gravitational collapse, describing a collapsing rotating cylindrical shell of null dust in flat space, with the metric of a spinning cosmic string as the exterior. We find that the shell bounces before closed timelike curves can be formed. Our results also suggest slightly different definitions for the mass and angular momentum of the string.

Avoiding closed timelike curves with a collapsing rotating null dust shell

In this chapter we study the Schwarzschild metric, corresponding to the gravitational field of a spherically symmetric body of mass M. Unlike Minkowski space-time, it contains a preferred class of observers, called stationary observers, who are not moving with respect to the central mass. We compute their proper time in terms of the time coordinate t and use it to obtain the exact redshift formula (“gravity delays time”). We also explain how by measuring distances between them stationary observers are led to the conclusion that space is curved (“gravity curves space”). Next we write the differential equations for the geodesics and see how they differ subtly from the Newtonian differential equations for free-falling motion. This leads to orbits that are approximately ellipses but whose axes slowly rotate. This effect, which had been observed for the planet Mercury, was the first triumph of the general theory of relativity. The differential equations for the null geodesics (light rays) also predict two new effects: a bending of the light rays passing near the spherically symmetric body (whose experimental confirmation in 1919 led to a wide acceptance of the theory), and a delay in the time of arrival of the light, known as the Shapiro delay (experimentally confirmed in 1966). Finally, we analyze the surface $r=2M,$ where the Schwarzschild metric is not defined. We show that this is a problem with the choice of coordinates, and not the space-time itself, and introduce the so-called Painleve coordinates, which remove this problem. The surface $r=2M,$ however, remains special: it marks the boundary of a region from which nothing, not even light, can escape–a black hole.

The Schwarzschild Solution

We introduce a rigorous and general framework to study systematically self-gravitating elastic materials within general relativity, and apply it to investigate the existence and viability, including radial stability, of spherically symmetric elastic stars. We present the mass-radius ($M-R$) diagram for various families of models, showing that elasticity contributes to increase the maximum mass and the compactness up to a ${\cal O}(10\%)$ factor, thus supporting compact stars with mass well above two solar masses. Some of these elastic stars can reach compactness as high as $GM/(c^2R)\approx 0.35$ while remaining stable under radial perturbations and satisfying all energy conditions and subluminal wave propagation, thus being physically viable models of stars with a light ring. We provide numerical evidence that radial instability occurs for central densities larger than that corresponding to the maximum mass, as in the perfect-fluid case. Elasticity may be a key ingredient to build consistent models of exotic ultracompact objects and black-hole mimickers, and can also be relevant for a more accurate modelling of the interior of neutron stars.

Compact elastic objects in general relativity

We present a time-dependent solution of the Maxwell equations in the Einstein universe, whose electric and magnetic fields, as seen by the stationary observers, are aligned with the Clifford parallels of the $3$-sphere $S^3$. The conformal equivalence between Minkowski's spacetime and (a region of) the Einstein cylinder is then exploited in order to obtain a knotted, finite energy, radiating solution of the Maxwell equations in flat spacetime. We also discuss similar electromagnetic fields in expanding closed Friedmann models, and compute the matter content of such configurations.

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José Natário

Papers

Gravitational Collapse to Toroidal and Higher Genus Black Holes

Avoiding closed timelike curves with a collapsing rotating null dust shell

The Schwarzschild Solution

Compact elastic objects in general relativity

On a remarkable electromagnetic field in the Einstein Universe