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

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662, 185 (2003)]. The basic idea consists in considering the direct product between an Abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S×g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D’Auria-Fre-like superalgebra, are considered. Finally, we find explicit, nontrace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of supergravity theories in arbitrary space-time dimensions.

Expanding Lie (super)algebras through abelian semigroups

https://www.arxiv.org/pdf/hep-th/0701002.pdf

Hawking radiation from rotating black holes in anti-de Sitter spaces via gauge and gravitational anomalies

We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an inter ...

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

Non commutative superspaces can be introduced as the Moyal-Weyl quan- tization of a Poisson bracket for classical superflelds. Difierent deformations are studied corresponding to constant background flelds in string theory. Supersymmetric and non supersymmetric deformations can be deflned, depending on the difierential operators used to deflne the Poisson bracket. Some examples of deformed, 4 dimensional lagrangians are given. For extended superspace (N > 1), some new deformations can be deflned, with no analogue in the N = 1 case.

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Supersymmetry in noncommutative superspaces

It is well known that the perturbation equations of massless fields for the Kerr-de Sitter geometry can be written in the form of separable equations. The equations have five definite singularities so that the analysis has been expected to be difficult. We show that these equations can be transformed to Heun's equations, for which we are able to use known technique for the analysis of the solutions. We reproduce results known previously for the Kerr geometry and de Sitter geometry in the confluent limits of the Heun's functions. Our analysis applies can be extended to Kerr-Newman-de Sitter geometry for massless fields with spin 0 and 1/2.

Perturbations of kerr-de sitter black holes and heun's equations

Perturbations of Kerr-de Sitter Black Hole and Heun's Equations

Noncommutative superspace, supermatrix and lowest Landau level

An analytic solution of the Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries is presented, and the properties of this solution are examined. In particular, we show that our solution satisfies the Teukolsky-Starobinsky identities explicitly and fix the relative normalization between solutions with spin weights s ands. alytic solution of the perturbation equation of massless fields in Kerr geometries which is commonly called the Teukolsky equation. 3) This solution enabled us to in- vestigate various properties of black holes analytically, 4) the scattering problem of a particle emitted in a black hole, an analytical expression for the absorption rate by a Kerr black hole, which was one of the main interests of Chandrasekhar, 5) and the Teukolsky-Starobinsky identities. Our solution is expressed in the form of a series of hypergeometric functions and Coulombwave functions. The coefficients of this series are determined b y solving three-term recurrence relations. Two series have different convergence regions, and they are matched in the region where both series converge. This method turns out to be quite powerful in practical calculations and is successfully applied to examine gravitational waves from a binary in a post-Newtonian expansion; to construct the template of the gravitational wave emitted from a particle moving around a Kerr black hole 6) and the rate of absorption of a gravitational wave by the black hole. 7) In a previous paper, 8) Suzuki, Takasugi and Umetsu extended this method to solve the perturbation equations of massless fields (the Teukolsky equation) in Kerr- de Sitter and Kerr-Newman-de Sitter geometries. We found transformations such that both the angular and the radial equations are reduced to Heun's equation. 9) The solution of Heun's equation is expressed in the form of a series of hypergeometric functions, and its coefficients are determined by three-term recurrence relations, similarly to the Kerr geometry case. It should be noted that electromagnetic fields and gravitational fields couple to each other in Kerr-Newman-de Sitter geometries and do not obey the equation we considered there, although the fields obey the

Analytic Solutions of the Teukolsky Equation in Kerr-de Sitter and Kerr-Newman-de Sitter Geometries

The analytic solution of Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries is presented and the properties of the solution are examined. In particular, we show that our solution satisfies the Teukolsky-Starobinsky identities explicitly and fix the relative normalization between solutions with the spin weight $s$ and $-s$.

Hiroshi Umetsu

Papers

Perturbations of kerr-de sitter black holes and heun's equations

Perturbations of Kerr-de Sitter Black Hole and Heun's Equations

Noncommutative superspace, supermatrix and lowest Landau level

Analytic Solutions of the Teukolsky Equation in Kerr-de Sitter and Kerr-Newman-de Sitter Geometries

Analytic Solutions of Teukolsky Equation in Kerr-de Sitter and Kerr-Newman-de Sitter Geometries