The expressions for the quasinormal modes (QNMs) of black holes with nonlinear electrodynamics, calculated in the eikonal approximation, are presented. In the eikonal limit QNMs of black holes are determined by the parameters of the circular null geodesics. The unstable circular null orbits are derived from the effective metric that is the one obeyed by light rays under the influence of a nonlinear electromagnetic field. As an illustration we calculate the QNMs of four nonlinear electromagnetic black holes, two singular and two regular, namely from Euler-Heisenberg and Born-Infeld theories, for singular, and the magnetic Bardeen black hole and the one derived by Bronnikov for regular ones. Comparison is shown with the QNMs of the linear electromagnetic counterpart, their Reissner-Nordström black hole.
We study the dynamics of test particle and stability of circular geodesics in the gravitational field of a non-commutative geometry inspired Schwarzschild black hole spacetime (NCSBH). The coordinate time Lyapunov exponent ($λ_{c}$) is crucial to investigate the stability of equatorial circular geodesics of massive and massless test particles. The stability or instability of circular orbits are discussed by analysing the variation of Lyapunov exponent with radius of these orbits for different values of non-commutative parameter ($α$). In the case of null circular orbits, the instability expone…
It has long been known that null unstable geodesics are related to the characteristic modes of black holes-- the so called quasinormal resonances. The basic idea is to interpret the free oscillations of a black hole in the eikonal limit in terms of null particles trapped at the unstable circular orbit and slowly leaking out. The real part of the complex quasinormal resonances is related to the angular velocity at the unstable null geodesic. The imaginary part of the resonances is related to the instability timescale (or the inverse Lyapunov exponent) of the orbit. While this geometric-optics d…
We investigate the stability of both timelike as well as null circular geodesics in the vicinity of a dual (3+1) dimensional stringy black hole (BH) spacetime by using an excellent tool so-called Lyapunov exponent. The proper time ($τ$) Lyapunov exponent ($λ_{p}$) and coordinate time ($t$) Lyapunov exponent~($λ_{c}$) are explicitly derived to analyze the stability of equatorial circular geodesics for the stringy BH spacetime with \emph{electric charge} parameter ($α$) and \emph{magnetic charge} parameter~($Q$). By computing these exponents for both the cases of BH spacetime, it is observed tha…
Stability criteria of nearly extreme black holes in the perturbations level is one of the interesting issues in gravitational systems. Considering the nearly extreme conformal-de Sitter black holes, in this paper, we obtain an exact relation for the quasinormal modes of scalar perturbations. As for stability criteria, we find a lower bound on the event horizon radius which is corresponding to an upper bound on the value of the quasinormal frequencies. In addition, we show that the asymptotic behavior of quasinormal modes gives highly damped modes which is important due to possible connection b…
In Phys. Rev. D 79, 064016 (2009) it was claimed that quasinormal modes which any stationary, spherically symmetric and asymptotically flat black hole emits in the eikonal regime are determined by the parameters of the circular null geodesic: the real and imaginary parts of the quasinormal mode are multiples of the frequency and instability timescale of the circular null geodesics respectively. We shall consider asymptotically flat black hole in the Einstein-Lovelock theory, find analytical expressions for gravitational quasinormal modes in the eikonal regime and analyze the null geodesics. Co…
Null geodesics, quasinormal modes of a massless scalar field perturbation and the correspondence with shadow radii are investigated in the background spacetime of high-dimensional Einstein-Yang-Mills black holes. Based on the properties of null geodesics, we obtain the connection between the radius of a photon sphere and the radius of a horizon in the five- and six-dimensional Einstein-Yang-Mills spacetimes. Especially in the five-dimensional case, there exist two branches for the radius of a photon sphere, but only the branch outside the event horizon satisfies the condition of circular null …
The well-known duality between quasinormal modes of any stationary, spherically symmetric and asymptotically flat or de Sitter black hole and parameters of the circular null geodesic was initially claimed for gravitational and test field perturbations. According to this duality the real and imaginary parts of the $\ell \gg n$ quasinormal mode (where $\ell$ and $n$ are multipole and overtone numbers respectively) are multiples of the frequency and instability timescale of the circular null geodesics respectively. Later it was shown that the duality is guaranteed only for test fields and may be …
We use Penrose limits to approximate quasinormal modes with large real frequencies. The Penrose limit associates a plane wave to a region of spacetime near a null geodesic. This plane wave can be argued to geometrically realize the geometrical optics approximation. Therefore, when applied to the bound null orbits around black holes, the Penrose limit can be used to study quasinormal modes. For instance, this Penrose limit point of view makes manifest the symmetry that emerges in the geometrical optics approximation of quasinormal modes in terms of isometries of the resulting plane wave spaceti…
In this contribution we study the birefringence and the quasinormal modes (QNM) in the eikonal approximation, of the Einstein-Euler-Heisenberg black hole (EEH-BH) . The EEH-BH is an exact solution of the Einstein equations coupled to the Euler-Heisenberg nonlinear electrodynamics. It is well known that in the presence of strong electromagnetic fields light rays modify their trajectories and do follow the null geodesics of an effective metric as a result of the nonlinear interaction between light and the intense magnetic or electric background. In the Euler-Heisenberg theory the phenomenon of b…
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 electro…
For a static and spherically symmetric black hole, a photon sphere is composed of circular null geodesics of fixed radius, and plays an important role in observing the black hole. Recently, in an Einstein-Maxwell-scalar model with a non-minimal coupling between the scalar and electromagnetic fields, a class of hairy black holes has been found to possess two unstable and one stable circular null geodesics on the equatorial plane, corresponding to three photon spheres outside the event horizon. In this paper, we study quasinormal modes of the scalar field, which are associated with these circula…
The numerical study indicates that there exists a relation between the quasinormal modes and the Davies point for a black hole. In this paper, we analytically study this relation for the charged Reissner-Nordström black holes in asymptotically flat and dS spacetimes. In the eikonal limit, the angular velocity $Ω$ and the Lyapunov exponent $λ$ of the photon sphere, respectively, corresponding to the real and imaginary parts of the quasinormal modes are obtained from the null geodesics. Both in asymptotically flat and dS spacetimes, we observe the spiral-like shapes in the complex quasinormal mo…
Geodesic motion determines important features of spacetimes. Null unstable geodesics are closely related to the appearance of compact objects to external observers and have been associated with the characteristic modes of black holes. By computing the Lyapunov exponent, which is the inverse of the instability timescale associated with this geodesic motion, we show that, in the eikonal limit, quasinormal modes of black holes in any dimensions are determined by the parameters of the circular null geodesics. This result is independent of the field equations and only assumes a stationary, spherica…
Realistic black holes are usually dynamical, noticeable or sluggish. The Vaidya metric is a significant and tractable model for simulating a dynamical black hole. In this study, we consider scalar perturbations in a dynamical Vaidya black hole, and explore the quasinormal modes by employing the matrix method. We find the proper boundary conditions of the quasinormal modes from physical analysis in the background of a dynamical black hole for the first time. The results show that the eigenfrequencies become different at the apparent horizon and null infinity, because the physical interactions p…
The impact of an antisymmetric tensor on charged black holes: evaporation process, geodesics, deflection angle, scattering effects and quasinormal modes
The role of the wandering null geodesic is studied in a black hole spacetime. Based on the continuity of the solution of the geodesic equation, the wandering null geodesics commonly exist and explain the typical phenomena of the optical observation of event horizons. Moreover, a new concept of `black room' is investigated to relate the wandering null geodesic to the black hole shadow more closely.
In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Tod and Luebbe.
We study the quasinormal mode eigenvalues and eigenfunctions for the Teukolsky equation in a horizon penetrating, hyperboloidally compactified (HPHC) coordinate system. Following earlier work by Zenginoğlu (arXiv:1102.2451), we show that the quasinormal eigenfunctions for the Teukolsky equation are regular from the black hole horizon to future null infinity in these coordinates. We then present several example quasinormal eigenfunction solutions, and study some of their properties in the near-extremal Kerr limit.
We consider here scalar and electromagnetic perturbations for the Vaidya metric in double-null coordinates. Such an approach allows one to go a step further in the analysis of quasinormal modes for time-dependent spacetimes. Some recent results are refined, and a new non-stationary behavior corresponding to some sort of inertia for quasinormal modes is identified. Our conclusions can enlighten some aspects of the wave scattering by black holes undergoing some mass accretion processes.
Based on the geometric-optics correspondence between the parameters of a quasinormal mode and the conserved quantities along geodesics, we propose an equation to calculate the typical shadow radius for asymptotically flat and rotating black holes when viewed from the equatorial plane given by \begin{equation}\notag \bar{R}_s=\frac{\sqrt{2}}{2}\left(\sqrt{\frac{ r_0^{+}}{f'(r)|_{r_0^{+}}}}+\sqrt{\frac{ r_0^{-}}{f'(r)|_{r_0^{-}}}}\right), \end{equation} with $r_0^{\pm}$ being the radius of circular null geodesics for the corresponding mode. Furthermore we have explicitly related the shadow radiu…
Extremely compact objects containing a region of trapped null geodesics could be of astrophysical relevance due to trapping of neutrinos with consequent impact on cooling processes or trapping of gravitational waves. These objects have previously been studied under the assumption of spherical symmetry. In the present paper, we consider a simple generalization by studying trapping of null geodesics in the framework of the Hartle-Thorne slow-rotation approximation taken to first order in the angular velocity, and considering a uniform-density object with uniform emissivity for the null geodesics…
The scalar wave equation is considered in the background of a charged Vaidya metric in double null coordinates $(u,v)$ describing a non-stationary charged black hole with varying mass $m(v)$ and charge $q(v)$. The resulting time-dependent quasinormal modes are presented and analyzed. We show, in particular, that it is possible to identify some signatures in the quasinormal frequencies from the creation of a naked singularity.
We show that positivity of energy for stationary, or strongly uniformly Schwarzschildian, asymptotically flat, non-singular domains of outer communications can be proved using Galloway's null rigidity theorem.
The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.
It is well established that the response of a black hole to a generic perturbation is characterized by a spectrum of damped resonances, called quasinormal modes; and that, in the limit of large angular momentum ($l \gg 1$), the quasinormal mode frequency spectrum is related to the properties of unstable null orbits. In this paper we develop an expansion method to explore the link. We obtain new closed-form approximations for the lightly-damped part of the spectrum in the large-$l$ regime. We confirm that, at leading order in $l$, the resonance frequency is linked to the orbital frequency, and …
Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation.
We perform a complete rho-integration of the GHP equations for all spacetimes that admit a geodesic shear-free expanding null congruence, whose Ricci spinor is aligned to the congruence and whose Ricci scalar is constant. We also deduce the system of GHP equations after the integration is completed, and discuss a few applications.
We discuss critical gravitational collapse on the threshold of apparent horizon formation as a model both for the discussion of global aspects of critical collapse and for numerical studies in a compactified context. For our matter model we choose a self-gravitating massless scalar field in spherical symmetry, which has been studied extensively in the critical collapse literature. Our evolution system is based on Bondi coordinates, the mass function is used as an evolution variable to ensure regularity at null infinity. We compute radiation quantities like the Bondi mass and news function and …
We consider a wide class of static spherically symmetric black holes of arbitrary dimension with a photon sphere (a hypersurface on which a massless particle can orbit the black hole on unstable circular null geodesics). This class includes various spacetimes of physical interest such as Schwarzschild, Schwarzschild-Tangherlini and Reissner-Nordström black holes, the canonical acoustic black hole or the Schwarzschild-de Sitter black hole. For this class of black holes, we provide general analytical expressions for the Regge poles of the $S$-matrix associated with a massless scalar field theory…
We analyze the null geodesics of regular black holes. A detailed analysis of geodesic structure both null geodesics and time-like geodesics have been investigated for the said black hole. As an application of null geodecics, we calculate the radius of photon sphere and gravitational bending of light. We also study the shadow of the black hole spacetime. Moreover, we determine the relation between radius of photon sphere~$(r_{ps})$ and the shadow observed by a distance observer. Furthermore, We discus the effect of various parameters on the radius of shadow $R_s$. Also we compute the angle of d…
We study the properties of the null geodesics of four-dimensional Einstein-Gauss-Bonnet black holes surrounded by quintessence matter. Due to the quintessence correction, we discuss the radial and non-radial geodesics. For the case of non-radial geodesics we obtain the effective potential, the photon sphere and the impact parameter associated to the null geodesics in details. We analyze the different possible orbits of massless particles (i.e. photons) such as unstable circular orbits and unbounded orbits and investigate the role of the Gauss-Bonnet coupling $α$ and the quintessence parameter …
The paper deals with an extensive study of null and timelike geodesics in the background of wormhole geometries. Starting with a spherically symmetric spacetime, null geodesics are analyzed for the Morris-Thorne wormhole(WH) and photon spheres are examined in WH geometries. Both bounded and unbounded orbits are discussed for timelike geodesics. A similar analysis has been done for trajectories in a dynamic spherically symmetric WH and for a rotating WH. Finally, the invariant angle method of Rindler and Ishak has been used to calculate the angle between radial and tangential vectors at any poi…
The issue of the local visibility of the shell-focussing singularity in marginally bound spherical dust collapse is considered from the point of view of the existence of future-directed null geodesics with angular momentum which emanate from the singularity. The initial data (i.e. the initial density profile) at the onset of collapse is taken to be of class $C^3$. Simple necessary and sufficient conditions for the existence of a naked singularity are derived in terms of the data. It is shown that there exist future-directed non-radial null geodesics emanating from the singularity if and only i…
Region of trapped null geodesics hidden inside of extremely compact objects is of astrophysical importance because of trapping of gravitational waves, or neutrinos. The trapping effect of null geodesics was extensively studied for spherically symmetric extremely compact objects. Recently, influence of rotation of the extremely compact objects on the trapping of null geodesics was treated in the simplest possible model of internal linearised \mbox{Hartle--Thorne} spacetime with uniform energy density distribution and uniform emissivity distribution of null geodesics. Here we extend the study of…
The purpose of this paper is to study quasinormal modes (QNM) of the Bardeen black hole due to scalar perturbations. We have done a thorough analysis of the QNM frequencies by varying the charge $q$, mass $M$ and the spherical harmonic index $l$. The unstable null geodesics are used to compute the QNM's in the eikonal limit. Furthermore, massive scalar field modes are also studied by varying the mass of the field. Comparisons are done with the QNM frequencies of the Reissner-Nordstrom black hole.
The quasinormal modes (QNMs) of a regular black hole with charge are calculated in the eikonal approximation. In the eikonal limit the QNMs of black hole are determined by the parameters of the unstable circular null geodesics. The behaviors of QNMs are compared with QNMs of Reisner-Nordström black hole, it is done by fixing some of the parameters that characterize the black holes and varying another. We observed that the parameter that is related one effective cosmological constant at small distances , determines the behaviors of the QNMs of regular black hole with charge.
We derive an expression for the expansion of outgoing null geodesics in spherical dust collapse and compute the limiting value of the expansion in the approach to singularity formation. An analogous expression is derived for the spherical collapse of a general form of matter. We argue on the basis of these results that the covered as well as the naked singularity solutions arising in spherical dust collapse are stable under small changes in the equation of state.
We present dynamical properties of linear waves and null geodesics valid for Kerr and Kerr-de Sitter black holes and their stationary perturbations. The two are intimately linked by the geometric optics approximation. For the nullgeodesic flow the key property is the r-normal hyperbolicity of the trapped set and for linear waves it is the distribution of quasi-normal modes: the exact quantization conditions do not hold for perturbations but the bounds on decay rates and the statistics of frequencies are still valid.
We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing a canonical choice of null tetrad and coordinates on a NEH. By looking for particular classes of null geodesic congruences which live exterior to NEHs but have the special property that their shear vanishes at the intersection with the horizon, a good cut formalism for NEHs is developed which closely mirrors asymptotic theory. In particular, we show that suc…
A direct and more general calculation of the limb effect connected with red-shift observations is obtained. Result agrees completely, in its general form, with that obtained from observations of the solar spectra but with different value for the maximum effect.
We show that, though they are rare, there are asymptotically flat space-times that possess null geodesic congruences that are both asymptotically shear- free and twist-free (surface forming). In particular, we display the class of space-times that possess this property and demonstrate how these congruences can be found. A special case within this class are the Robinson- Trautman space-times. In addition, we show that in each case the congruence is isolated in the sense that there are no other neighboring congruences with this dual property.
We find exact solutions to Maxwell equations written in terms of four-vector potentials in non--rotating, as well as in Gödel and Kerr spacetimes. We show that Maxwell equations can be reduced to two uncoupled second-order differential equations for combinations of the components of the four-vector potential. Exact electromagnetic waves solutions are written on given gravitational field backgrounds where they evolve. We find that in non--rotating spherical symmetric spacetimes, electromagnetic waves travel along null geodesics. However, electromagnetic waves on Gödel and Kerr spacetimes do not…
We study photon orbits within (1+2)-dimensional Einstein-power-Maxwell non-linear electrodynamics assuming a static and circularly symmetric background. An exact analytical solution to the null geodesic equations for light rays is obtained in terms of the Weierstraß function. We investigate in detail the impact of the photon energy, the electric charge of the black hole, and the integration constant (initial condition) on the shape of the orbits.
We investigate the geodesic motion of massless test particles in the background of a noncommutative geometry inspired Schwarzschild black hole. The behaviour of effective potential is analysed in the equatorial plane and the possible motions of massless particles (i.e. photons) for different values of impact parameter are discussed accordingly. We have also calculated the frequency shift of photons in this spacetime. Further, the mass parameter of a non-commutative inspired Schwarzschild black hole is computed in terms of the measurable redshift of photons emitted by massive particles moving a…
