The Frenet Serret Description of Gyroscopic Precession

The phenomenon of gyroscopic precession is studied within the framework of Frenet-Serret formalism adapted to quasi-Killing trajectories. Its relation to the congruence vorticity is highlighted with particular reference to the irrotational congruence admitted by the stationary, axisymmetric spacetime. General precession formulae are obtained for circular orbits with arbitrary constant angular speeds. By successive reduction, different types of precessions are derived for the Kerr - Schwarzschild - Minkowski spacetime family. The phenomenon is studied in the case of other interesting spacetimes, such as the De Sitter and G\"{o}del universes as well as the general stationary, cylindrical, vacuum spacetimes.Comment: 37 pages, Paper in Late

Kerr black hole quasinormal frequencies

Black-hole quasinormal modes (QNM) have been the subject of much recent attention, with the hope that these oscillation frequencies may shed some light on the elusive theory of quantum gravity. We compare numerical results for the QNM spectrum of the (rotating) Kerr black hole with an {\it exact} formula Re$\omega \to T_{BH}\ln 3+\Omega m$, which is based on Bohr's correspondence principle. We find a close agreement between the two. Possible implications of this result to the area spectrum of quantum black holes are discussed.Comment: 3 pages, 2 figure

Quasinormal modes for tensor and vector type perturbation of Gauss Bonnet black holes using third order WKB approach

We obtain the quasinormal modes for tensor perturbations of Gauss-Bonnet (GB) black holes in $d=5, 7, 8$ dimensions and vector perturbations in $d = 5, 6, 7$ and 8 dimensions using third order WKB formalism. The tensor perturbation for black holes in $d=6$ is not considered because of the fact that it is unstable to tensor mode perturbations. In the case of uncharged GB black hole, for both tensor and vector perturbations, the real part of the QN frequency increases as the Gauss-Bonnet coupling ($\alpha'$) increases. The imaginary part first decreases upto a certain value of $\alpha'$ and then increases with $\alpha'$ for both tensor and vector perturbations. For larger values of $\alpha'$, the QN frequencies for vector perturbation differs slightly from the QN frequencies for tensorial one. It has also been shown that as $\alpha' \to 0$, the quasinormal mode frequency for tensor and vector perturbation of the Schwarzschild black hole can be obtained. We have also calculated the quasinormal spectrum of the charged GB black hole for tensor perturbations. Here we have found that the real oscillation frequency increases, while the imaginary part of the frequency falls with the increase of the charge. We also show that the quasinormal frequencies for scalar field perturbations and the tensor gravitational perturbations do not match as was claimed in the literature. The difference in the result increases if we increase the GB coupling.Comment: 17 pages, 11 figures, change in title and abstract, new equations and results added for QN frequencies for vector perturbations, new referencees adde

Quasinormal modes from potentials surrounding the charged dilaton black hole

We clarify the purely imaginary quasinormal frequencies of a massless scalar perturbation on the 3D charged-dilaton black holes. This case is quite interesting because the potential-step appears outside the event horizon similar to the case of the electromagnetic perturbations on the large Schwarzschild-AdS black holes. It turns out that the potential-step type provides the purely imaginary quasinormal frequencies, while the potential-barrier type gives the complex quasinormal modes.Comment: 19 pages, 8 figure

Numerical analysis of quasinormal modes in nearly extremal Schwarzschild-de Sitter spacetimes

We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasinormal frequencies with large imaginary parts, on the other hand, this formula becomes inaccurate even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes, which are similar to those observed in the case of a Reissner-Nordstr{\" o}m black hole. The amplitude of oscillating ${\rm Re(\omega)}$ as a function of ${\rm Im}(\omega)$ approaches a non-zero constant value for gravitational perturbations and zero for electromagnetic perturbations in the limit of highly damped modes, where $\omega$ denotes the quasinormal frequency. This means that for gravitational perturbations, the real part of quasinormal modes of the nearly extremal Schwarzschild-de Sitter spacetime appears not to approach any constant value in the limit of highly damped modes. On the other hand, for electromagnetic perturbations, the real part of frequency seems to go to zero in the limit.Comment: 9 pages, 7 figures, to appear in Physical Review

3D simulations of linearized scalar fields in Kerr spacetime

We investigate the behavior of a dynamical scalar field on a fixed Kerr background in Kerr-Schild coordinates using a 3+1 dimensional spectral evolution code, and we measure the power-law tail decay that occurs at late times. We compare evolutions of initial data proportional to f(r) Y_lm(theta,phi) where Y_lm is a spherical harmonic and (r,theta,phi) are Kerr-Schild coordinates, to that of initial data proportional to f(r_BL) Y_lm(theta_BL,phi), where (r_BL,theta_BL) are Boyer-Lindquist coordinates. We find that although these two cases are initially almost identical, the evolution can be quite different at intermediate times; however, at late times the power-law decay rates are equal.Comment: 12 pages, 9 figures, revtex4. Major revision: added figures, added subsection on convergence, clarified discussion. To appear in Phys Rev

Stability analysis of f(R)-AdS black holes

We study the stability of f(R)-AdS (Schwarzschild-AdS) black hole obtained from f(R) gravity. In order to resolve the difficulty of solving fourth order linearized equations, we transform f(R) gravity into the scalar-tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a dynamical scalaron, showing that all linearized equations are second order. Using the positivity of gravitational potentials and S-deformed technique allows us to guarantee the stability of f(R)-AdS black hole if the scalaron mass squared satisfies the Breitenlohner-Freedman bound. This is confirmed by computing quasinormal frequencies of the scalaron for large f(R)-AdS black hole.Comment: 17 pages, 1 figure, version to appear in EPJ

Thermodynamic and gravitational instability on hyperbolic spaces

We study the properties of anti--de Sitter black holes with a Gauss-Bonnet term for various horizon topologies (k=0, \pm 1) and for various dimensions, with emphasis on the less well understood k=-1 solution. We find that the zero temperature (and zero energy density) extremal states are the local minima of the energy for AdS black holes with hyperbolic event horizons. The hyperbolic AdS black hole may be stable thermodynamically if the background is defined by an extremal solution and the extremal entropy is non-negative. We also investigate the gravitational stability of AdS spacetimes of dimensions D>4 against linear perturbations and find that the extremal states are still the local minima of the energy. For a spherically symmetric AdS black hole solution, the gravitational potential is positive and bounded, with or without the Gauss-Bonnet type corrections, while, when k=-1, a small Gauss-Bonnet coupling, namely, \alpha << {l}^2 (where l is the curvature radius of AdS space), is found useful to keep the potential bounded from below, as required for stability of the extremal background.Comment: Shortened to match published (PRD) version, 18 pages, several eps figure