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

We formulate a prescription for computing Minkowski-space correlators from AdS/CFT correspondence. This prescription is shown to give the correct retarded propagators at zero temperature in four dimensions, as well as at finite temperature in the two-dimensional conformal field theory dual to the BTZ black hole. Using the prescription, we calculate the Chern-Simons diffusion constant of the finite-temperature = 4 supersymmetric Yang-Mills theory in the strong coupling limit. We explain why the quasinormal frequencies of the asymptotically AdS background correspond to the poles of the retarded Green's function of the boundary conformal field theory.

Minkowski-space correlators in AdS/CFT correspondence: recipe and applications

We compute leading-order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form −k ln(Area). For BTZ black holes, k = 3/2, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner–Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black-hole entropy and its corrections.

https://iopscience.iop.org/article/10.1088/0264-9381/19/9/302/pdf

General logarithmic corrections to black-hole entropy

Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical approximation is analysed. We compute all quantum corrections in the single particle action revealing that these are proportional to the usual semiclassical contribution. We show that a simple choice of the proportionality constants reproduces the one loop back reaction effect in the spacetime, found by conformal field theory methods, which modifies the Hawking temperature of the black hole. Using the law of black hole mechanics we give the corrections to the Bekenstein-Hawking area law following from the modified Hawking temperature. Some examples are explicitly worked out.

Quantum tunneling beyond semiclassical approximation

Quantum Tunneling Beyond Semiclassical Approximation

We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is -3/2log area), in agreement with that for the Schwarzschild black hole obtained in the four-dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the correct expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toroidal boundary of the black hole.

https://iopscience.iop.org/article/10.1088/0264-9381/18/15/303/pdf

Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole

We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is $-3/2 log(Area)$, in agreement with that for the Schwarzschild black hole obtained in the canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the right expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toral boundary of the black hole.

A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-de Sitter black holes. This is inspired by the case of the three-dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five-dimensional Schwarzschild anti-de Sitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well known instability of small black holes in anti-de Sitter space.

Quasi-normal modes of AdS black holes: a superpotential approach

A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-de Sitter black holes. This is inspired by the case of the three-dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five dimensional Schwarzschild anti-deSitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well-known instability of small black holes in anti-deSitter space.

Quasi-normal modes of AdS black holes : A superpotential approach

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.

V Suneeta

Papers

Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole

Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole

Quasi-normal modes of AdS black holes: a superpotential approach

Quasi-normal modes of AdS black holes : A superpotential approach

Note on the action with the Schwarzian at the stretched horizon