The WKB approach for finding quasinormal modes of black holes, suggested in [1] by Schutz and Will at the first order and later developed to higher orders [2-4], became popular during the past decades, because, unlike more sophisticated numerical approaches, it is automatic for different effective potentials and mostly provides sufficient accuracy. At the same time, the seeming simplicity of the WKB approach resulted in appearance of a big number of partially misleading papers, where the WKB formula was used beyond its scope of applicability. Here we review various situations in which the WKB formula can or cannot bring us to reliable conclusions. As the WKB series converges only asymptotically, there is no mathematically strict criterium for evaluation of an error. Therefore, here we are trying to introduce a number of practical recipes instead and summarize cases in which higher WKB orders improve accuracy. We show that averaging of the Pade approximations, suggested first by J. Matyjasek and M. Opala [4], leads to much higher accuracy of the WKB approach, estimate the error and present the automatic code [5] which computes quasinormal modes and grey-body factors.

https://iopscience.iop.org/article/10.1088/1361-6382/ab2e25/pdf

Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations

In this paper we have studied the extended phase space thermodynamics in the canonical ensemble of black holes in massive gravity in anti--de Sitter space. The black holes considered here belong to a theory of massive gravity where the graviton gain a mass due to Lorentz symmetry breaking. We have computed various thermodynamical quantities such as temperature, pressure, Gibbs free energy, and specific heat capacity. The local and the global thermodynamical stability of the black holes are studied in detail. For a specific value of the parameter in the theory, the black holes undergo a first order phase transition similar to the van der Waals phase transitions between gas and liquid under constant temperature. This transition is between the small and the large black holes. The critical exponents are computed at the critical values and shown to be the same as for the van der Waals critical exponents.

P -V criticality in AdS black holes of massive gravity

Weak gravitational lensing by black holes and wormholes in the context of massive gravity (Bebronne and Tinyakov, JHEP 0904:100, 2009) theory is studied. The particular solution examined is characterized by two integration constants, the mass M and an extra parameter S namely ‘scalar charge’. These black hole reduce to the standard Schwarzschild black hole solutions when the scalar charge is zero and the mass is positive. In addition, a parameter $$\lambda $$
 in the metric characterizes so-called ‘hair’. The geodesic equations are used to examine the behavior of the deflection angle in four relevant cases of the parameter $$\lambda $$
 . Then, by introducing a simple coordinate transformation $$r^\lambda =S+v^2$$
 into the black hole metric, we were able to find a massless wormhole solution of Einstein–Rosen (ER) (Einstein and Rosen, Phys Rev 43:73, 1935) type with scalar charge S. The programme is then repeated in terms of the Gauss–Bonnet theorem in the weak field limit after a method is established to deal with the angle of deflection using different domains of integration depending on the parameter $$\lambda $$
 . In particular, we have found new analytical results corresponding to four special cases which generalize the well known deflection angles reported in the literature. Finally, we have established the time delay problem in the spacetime of black holes and wormholes, respectively.

/pdf/deflection-of-light-by-black-holes-and-massless-wormholes-in-3ud4lhx8rb.pdf

Deflection of light by black holes and massless wormholes in massive gravity

In this paper, we use a suitable conformal rescaling to construct static and rotating regular black holes in conformal massive gravity. The new metric is characterized by the mass $M$, the ``scalar charge'' $Q$, the angular momentum parameter $a$, the ``hair parameter'' $\ensuremath{\lambda}$, and the conformal scale factor encoded in the parameter $L$. We explore the shadow images and the deflection angles of relativistic massive particles in the spacetime geometry of a rotating regular black hole. For $\ensuremath{\lambda}\ensuremath{\ne}0$ and $Qg0$, the shadow is larger than the shadow of a Kerr black hole. In particular, if $\ensuremath{\lambda}l0$, the shadow radius increases considerably. For $\ensuremath{\lambda}\ensuremath{\ne}0$ and $Ql0$, the shadow is smaller than the shadow of a Kerr black hole. Additionally, we put observational constraints on the parameter $Q$ using the latest Event Horizon Telescope observation of the supermassive black hole M87*. Lastly, using the Gauss-Bonnet theorem, we show that the deflection angle of massive particles is strongly affected by the parameter $L$. The deflection angle might be used to distinguish rotating regular black holes from rotating singular black holes.

Rotating regular black holes in conformal massive gravity

The first observational run of the Advanced LIGO detectors, from September 12, 2015 to January 19, 2016, saw the first detections of gravitational waves from binary black hole mergers. In this paper we present full results from a search for binary black hole merger signals with total masses up to 100M⊙ and detailed implications from our observations of these systems. Our search, based on general-relativistic models of gravitational wave signals from binary black hole systems, unambiguously identified two signals, GW150914 and GW151226, with a significance of greater than 5σ over the observing period. It also identified a third possible signal, LVT151012, with substantially lower significance, which has a 87% probability of being of astrophysical origin. We provide detailed estimates of the parameters of the observed systems. Both GW150914 and GW151226 provide an unprecedented opportunity to study the two-body motion of a compact-object binary in the large velocity, highly nonlinear regime. We do not observe any deviations from general relativity, and place improved empirical bounds on several high-order post-Newtonian coefficients. From our observations we infer stellar-mass binary black hole merger rates lying in the range 9−240Gpc−3yr−1. These observations are beginning to inform astrophysical predictions of binary black hole formation rates, and indicate that future observing runs of the Advanced detector network will yield many more gravitational wave detections.

Binary Black Hole Mergers in the First Advanced LIGO Observing Run

We have studied quasinormal modes of scalar perturbations of a black hole in massive gravity. The parameters of the theory, such as the mass of the black hole, the scalar charge of the black hole and the spherical harmonic index is varied to see how the corresponding quasinormal frequencies change. We have also studied the massive scalar field perturbations. Most of the work is done using WKB approach while sections are devoted to compute quasinormal modes via the unstable null geodesics approach and the Poschl–Teller approximation. Comparisons are done with the Schwarzschild black hole.

Black holes in massive gravity: quasi-normal modes of scalar perturbations

In this paper, we have studied QNM modes and absorption cross-sections of Born–Infeld–de Sitter black holes. WKB approximation is employed to compute the QNM modes of massless scalar fields. We have also used null geodesics to compute quasinormal modes in the eikonal approximation. In the eikonal limit, QNMs of black holes are determined by the parameters of the circular null geodesics. Unstable circular null orbits are derived from the effective metric which is obeyed by light rays under the influence of a nonlinear electromagnetic field. Comparison is shown with the QNM of the linear electromagnetic counterpart, the Reissner–Nordstrom black hole. Furthermore, the null geodesics are employed to compute the absorption cross-sections in the high frequency limit via the sinc approximation.

Quasinormal modes and absorption cross-sections of Born-Infeld-de Sitter black holes

In this paper, we have studied QNM modes and absorption cross sections of Born-Infeld-de Sitter black holes. WKB approximation is employed to compute the QNM modes of massless scalar fields. We have also used null geodesics to compute quasi-normal modes in the eikonal approximation. In the eikonal limit QNMs of black holes are determined by the parameters of the circular null geodesics. Unstable circular null orbits are derived from the effective metric which is obeyed by light rays under the influence of a nonlinear electromagnetic field. Comparison is shown with the QNM of the linear electromagnetic counterpart, the Reissner-Nordstrom black hole. Furthermore, the null geodesics are employed to compute the absorption cross sections in the high frequency limit via the sinc approximation.

Quasinormal modes and absorption cross sections of Born-Infeld-de Sitter black holes

We have studied quasinormal modes of scalar perturbations of a black hole in massive gravity. The parameters of the theory, such as the mass of the black hole, the scalar charge of the black hole and the spherical harmonic index is varied to see how the corresponding quasinormal frequencies change. We have also studied the massive scalar field perturbations. Most of the work is done using WKB approach while sections are devoted to compute quasinormal modes via the unstable null geodesics approach and the Poschl-Teller approximation. Comparisons are done with the Schwarzschild black hole.

Tyler Clark

Papers

Black holes in massive gravity: quasi-normal modes of scalar perturbations

Quasinormal modes and absorption cross-sections of Born-Infeld-de Sitter black holes

Quasinormal modes and absorption cross sections of Born-Infeld-de Sitter black holes

Black holes in massive gravity: Quasi-normal modes of scalar perturbations