Abstract:
This paper reviews the basic features of the theory of curvature perturbations in Kerr spacetime, which is customarily written in terms of gauge invariant components of the Weyl tensor which satisfy a perturbation equation known as the Teukolsky equation. I will describe how to evolve generic perturbations about the Kerr metric and the separable form of the wave solutions that one obtains, and the relation of the Teukolsky function to the energy of gravitational waves emitted by the black hole. A discussion of a numerical scheme to evolve perturbations as a function of time and some preliminary results of our research project implementing it for matter sources falling into the black hole is included.

Abstract:
A pair of wave equations for the electromagnetic and gravitational perturbations of the charged Kerr black hole are derived. The perturbed Einstein-Maxwell equations in a new gauge are employed in the derivation. The wave equations refer to the perturbed Maxwell spinor $\Phi_0$ and to the shear $\sigma$ of a principal null direction of the Weyl curvature. The whole construction rests on the tripod of three distinct derivatives of the first curvature $\kappa$ of a principal null direction.

Abstract:
The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition. The Robin BCs on the Teukolsky variables will allow for a quantitative study of instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.

Abstract:
The most general stationary black-hole solution of Einstein-Maxwell theory in vacuum is the Kerr-Newman metric, specified by three parameters: mass M, spin J and charge Q. Within classical general relativity, the most important and challenging open problem in black-hole perturbation theory is the study of gravitational and electromagnetic fields in the Kerr-Newman geometry, because of the indissoluble coupling of the perturbation functions. Here we circumvent this long-standing problem by working in the slow-rotation limit. We compute the quasinormal modes up to linear order in J for any value of Q and provide the first, fully-consistent stability analysis of the Kerr-Newman metric. For scalar perturbations the quasinormal modes can be computed exactly, and we demonstrate that the method is accurate within 3% for spins J/Jmax<~0.5, where Jmax is the maximum allowed spin for any value of Q. Quite remarkably, we find numerical evidence that the axial and polar sectors of the gravito-electromagnetic perturbations are isospectral to linear order in the spin. The extension of our results to nonasymptotically flat space-times could be useful in the context of gauge/gravity dualities and string theory.

Abstract:
We compute numerically the quasinormal modes of Kerr-Newman black holes in the scalar case, for which the perturbation equations are separable. Then we study different approximations to decouple electromagnetic and gravitational perturbations of the Kerr-Newman metric, computing the corresponding quasinormal modes. Our results suggest that the Teukolsky-like equation derived by Dudley and Finley gives a good approximation to the dynamics of a rotating charged black hole for Q

Abstract:
In Einstein-Maxwell theory, according to classic uniqueness theorems, the most general stationary black-hole solution is the axisymmetric Kerr-Newman metric, which is defined by three parameters: mass, spin and electric charge. The radial and angular dependence of gravitational and electromagnetic perturbations in the Kerr-Newman geometry do not seem to be separable. In this paper we circumvent this problem by studying scalar, electromagnetic and gravitational perturbations of Kerr-Newman black holes in the slow-rotation limit. We extend (and provide details of) the analysis presented in a recent Letter [arXiv:1304.1160]. Working at linear order in the spin, we present the first detailed derivation of the axial and polar perturbation equations in the gravito-electromagnetic case, and we compute the corresponding quasinormal modes for any value of the electric charge. Our study is the first self-consistent stability analysis of the Kerr-Newman metric, and in principle it can be extended to any order in the small rotation parameter. We find numerical evidence that the axial and polar sectors are isospectral at first order in the spin, and speculate on the possible implications of this result.

Abstract:
The ``close limit,'' a method based on perturbations of Schwarzschild spacetime, has proved to be a very useful tool for finding approximate solutions to models of black hole collisions. Calculations carried out with second order perturbation theory have been shown to give the limits of applicability of the method without the need for comparison with numerical relativity results. Those second order calculations have been carried out in a fixed coordinate gauge, a method that entails conceptual and computational difficulties. Here we demonstrate a gauge invariant approach to such calculations. For a specific set of models (requiring head on collisions and quadrupole dominance of both the first and second order perturbations), we give a self contained gauge invariant formalism. Specifically, we give (i) wave equations and sources for first and second order gauge invariant wave functions; (ii) the prescription for finding Cauchy data for those equations from initial values of the first and second fundamental forms on an initial hypersurface; (iii) the formula for computing the gravitational wave power from the evolved first and second order wave functions.

Abstract:
A gauge-independent, invariant theory of linear scalar perturbations of inflation and gravitational fields has been created. This invariant theory allows one to compare gauges used in the work of other researchers and to find the unambiguous criteria to separate the physical and coordinate effects. It is shown, in particular, that the so-called longitudial gauge, commonly used when considering inflation instability, leads to a fundamental overestimation of the effect because of non-physical perturbations of the proper time in the frame of reference specified by this gauge. Back reaction theories employing this sort of gauge [1] also involve coordinate effects. The invariant theory created here shows that the classical Lifshitz (1946) [2] gauge does not lead to non-physical perturbations of the proper time and can be used to analyze the inflation regime and the back reaction of perturbations on this regime properly. The first theory of back reaction on background of all types of perturbations (scalar, vector and tensor) based on this gauge was published in 1975 [3] and has been applied recently to the inflation [4]. The investigation of long-length perturbations, which characterize the stability of the inflationary process, and quantum fluctuations, which form the Harrison-Zel'dovich spectrum at the end of inflation, is performed in the invariant form. The invariant theory proposed allows one to examine the effect of quantum fluctuations on the inflationary stage when the periodic regime changes to an aperiodic one. That only the invariant theory must be used to analyze space experiments is one of the conclusions of the present work.

Abstract:
Using recently developed efficient symbolic manipulations tools, we present a general gauge-invariant formalism to study arbitrary radiative $(l\geq 2)$ second-order perturbations of a Schwarzschild black hole. In particular, we construct the second order Zerilli and Regge-Wheeler equations under the presence of any two first-order modes, reconstruct the perturbed metric in terms of the master scalars, and compute the radiated energy at null infinity. The results of this paper enable systematic studies of generic second order perturbations of the Schwarzschild spacetime. In particular, studies of mode-mode coupling and non-linear effects in gravitational radiation, the second-order stability of the Schwarzschild spacetime, or the geometry of the black hole horizon.

Abstract:
This paper presents a complete analysis of the effects of second order gravitational perturbations on Cosmic Microwave Background anisotropies, taking explicitly into account scalar, vector and tensor modes. We also consider the second order perturbations of the metric itself obtaining them, for a universe dominated by a collision-less fluid, in the Poisson gauge, by transforming the known results in the synchronous gauge. We discuss the resulting second order anisotropies in the Poisson gauge, and analyse the possible relevance of the different terms. We expect that, in the simplest scenarios for structure formation, the main effect comes from the gravitational lensing by scalar perturbations, that is known to give a few percent contribution to the anisotropies at small angular scales.