Kerr Parameter Research Articles

Precise analytic treatment of Kerr and Kerr-(anti) de Sitter black holes as gravitational lenses

The null geodesic equations that describe the motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant Λ. The exact solution for the deflection angle for generic light orbits (i.e. non-polar, non-equatorial) is calculated in terms of the generalized hypergeometric functions of Appell and Lauricella. We then consider the more involved issue in which the black hole acts as a ‘gravitational lens’. The constructed Kerr black hole gravitational lens geometry consists of an observer and a source located far away and placed at arbitrary inclination with respect to the black hole's equatorial plane. The resulting lens equations are solved elegantly in terms of Appell–Lauricella hypergeometric functions and the Weierstraß elliptic function. We then, systematically, apply our closed form solutions for calculating the image and source positions of generic photon orbits that solve the lens equations and reach an observer located at various values of the polar angle for various values of the Kerr parameter and the first integrals of motion. In this framework, the magnification factors for generic orbits are calculated in closed analytic form for the first time. The exercise is repeated with the appropriate modifications for the case of a non-zero cosmological constant.

Gamma-Ray Burst Physics

I have developed two-dimensional general relativistic magnetohydrodynamic (GRMHD) code. I have performed numerical simulations of collapsars using these codes and realistic progenitor models. In the GRMHD simulation, it is shown that a jet is launched from the center of the progenitor. We also performed simulations of collapsars with different Kerr parameters a = 0, 0.5, 0.9, 0.95. It is shown that a more rapidly rotating black hole is driving a more energetic jet. No jet is seen for the case of Schwartzschild black hole case, while the total energy of the jet is as large as 1050 erg for a rapidly rotating Kerr black hole case (a = 0.95). In order to explain the high luminosity of a GRB, it is concluded that a rapidly rotating black hole is favored ('faster is better'). We also performed two-dimensional hydrodynamic simulations in the context of collapsar model to investigate the explosive nucleosynthesis happened there. It is found that the amount of 56Ni is very sensitive to the energy deposition rate. This result means that the amount of synthesized 56Ni can be little even if the total explosion energy is as large as 1052 erg. Thus, some GRBs can associate with faint supernovae. Thus we consider it is quite natural to detect no underlying supernova in some X-ray afterglows.

The need for hypercritical accretion in massive black hole binaries with large Kerr parameters

Recent measurements of the Kerr parameters of the black holes in M33 X-7 and LMC X-1 yield a*=0.84\pm0.05 and a*=0.90^{+.04}_{-.09} respectively. We study massive binary evolution scenarios that can reproduce such high values for the Kerr parameters. We first discuss a model with Case C mass transfer leading to a common envelope and tidal synchronization of the primary before it collapses into a black hole. We also study a Case M evolution model (which involves tidally-locked, rotationally-mixed, chemically-homogeneous stars in a close binary). Our analysis suggests that, regardless of the specific scenario, the observed Kerr parameters for the black holes in M33 X-7 and LMC X-1 had to be obtained through hypercritical mass accretion.

Collision of an innermost stable circular orbit particle around a Kerr black hole

We derive a general formula for the center-of-mass (CM) energy for the near-horizon collision of two particles of the same rest mass on the equatorial plane around a Kerr black hole. We then apply this formula to a particle which plunges from the innermost stable circular orbit (ISCO) and collides with another particle near the horizon. It is found that the maximum value of the CM energy ${E}_{\mathrm{cm}}$ is given by ${E}_{\mathrm{cm}}/(2{m}_{0})\ensuremath{\simeq}1.40/\sqrt[4]{1\ensuremath{-}{a}_{*}^{2}}$ for a nearly maximally rotating black hole, where ${m}_{0}$ is the rest mass of each particle and ${a}_{*}$ is the nondimensional Kerr parameter. This coincides with the known upper bound for a particle which begins at rest at infinity within a factor of 2. Moreover, we also consider the collision of a particle orbiting the ISCO with another particle on the ISCO and find that the maximum CM energy is then given by ${E}_{\mathrm{cm}}/(2{m}_{0})\ensuremath{\simeq}1.77/\sqrt[6]{1\ensuremath{-}{a}_{*}^{2}}$. In view of the astrophysical significance of the ISCO, this result implies that particles can collide around a rotating black hole with an arbitrarily high CM energy without any artificial fine-tuning in an astrophysical context if we can take the maximal limit of the black hole spin or ${a}_{*}\ensuremath{\rightarrow}1$. On the other hand, even if we take Thorne's bound on the spin parameter into account, highly or moderately relativistic collisions are expected to occur quite naturally, for ${E}_{\mathrm{cm}}/(2{m}_{0})$ takes 6.95 (maximum) and 3.86 (generic) near the horizon and 4.11 (maximum) and 2.43 (generic) on the ISCO for ${a}_{*}=0.998$. This implies that high-velocity collisions of compact objects are naturally expected around a rapidly rotating supermassive black hole. Implications to accretion flows onto a rapidly rotating black hole are also discussed.

KERR PARAMETERS FOR STELLAR MASS BLACK HOLES AND THEIR CONSEQUENCES FOR GAMMA-RAY BURSTS AND HYPERNOVAE

Recent measurements of the Kerr parameters a⋆ for two black hole binaries in our Galaxy, GRO J1655−40 and 4U 1543−47, of a⋆ = 0.65–0.75 and a⋆ = 0.75–0.85, respectively, fit the predictions of Lee et al. of a⋆ ≅ 0.8. They predicted a⋆>0.5 for 80% of the soft X-ray transient (SXT) sources. The maximum available energy in the Blandford–Znajek formalism for a⋆>0.5 gives E>3 × 1053 erg, orders of magnitude larger than the energy needed for the gamma-ray burst (GRB) and hypernova explosion. We interpret the SXTs to be relics of GRBs and hypernovae. We find that most galactic SXTs were subluminous given that they could use only a small part of the available rotational energy.

A GENERAL RELATIVISTIC RAY-TRACING METHOD FOR ESTIMATING THE ENERGY AND MOMENTUM DEPOSITION BY NEUTRINO PAIR ANNIHILATION IN COLLAPSARS

Bearing in mind the application to the collapsar models of gamma-ray bursts (GRBs), we develop a numerical scheme and code for estimating the deposition of energy and momentum due to the neutrino pair annihilation ($\nu + {\bar \nu} \rightarrow e^{-} + e^{+}$) in the vicinity of accretion tori around a Kerr black hole. Our code is designed to solve the general relativistic neutrino transfer by a ray-tracing method. To solve the collisional Boltzmann equation in curved spacetime, we numerically integrate the so-called rendering equation along the null geodesics. For the neutrino opacity, the charged-current $\beta$-processes are taken into account, which are dominant in the vicinity of the accretion tori. The numerical accuracy of the developed code is certificated by several tests, in which we show comparisons with the corresponding analytic solutions. Based on the hydrodynamical data in our collapsar simulation, we estimate the annihilation rates in a post-processing manner. Increasing the Kerr parameter from 0 to 1, it is found that the general relativistic effect can increase the local energy deposition rate by about one order of magnitude, and the net energy deposition rate by several tens of percents. After the accretion disk settles into a stationary state (typically later than $\sim 9$ s from the onset of gravitational collapse), we point out that the neutrino-heating timescale in the vicinity of the polar funnel region can be shorter than the dynamical timescale. Our results suggest the neutrino pair annihilation has a potential importance equal to the conventional magnetohydrodynamic mechanism for igniting the GRB fireballs.

Nonequatorial charged particle confinement around Kerr black holes

We analyze the nonequatorial charged particle dynamics around a rotating black hole in the presence of an external magnetic field, the latter being given by Wald's exact analytical solution to the Maxwell's equations in the Kerr background. At variance with the corresponding Schwarzschild case, the behavior of the particle becomes here markedly charge-sign dependent, and the more so the more the Kerr parameter increases. The interplay between the rotating black hole and the magnetic field is shown to provide a mechanism both for selective charge ejection in axially collimated jetlike trajectories, and for selective charge confinement into nonequatorial bound orbits around the hole; the possibility of such a confinement allows the fate of an accreting particle to not necessarily be doomed: infall into the hole can be prevented, and the neutrality of the Kerr source could therefore be preserved, while the charge is safely parked into bound cross-equatorial orbits all around it.

MEASURING THE SPIN OF THE PRIMARY BLACK HOLE IN OJ287

The compact binary system in OJ287 is modelled to contain a spinning primary black hole with an accretion disk and a non-spinning secondary black hole. Using Post Newtonian (PN) accurate equations that include 2.5PN accurate non-spinning contributions, the leading order general relativistic and classical spin-orbit terms, the orbit of the binary black hole in OJ287 is calculated and as expected it depends on the spin of the primary black hole. Using the orbital solution, the specific times when the orbit of the secondary crosses the accretion disk of the primary are evaluated such that the record of observed outbursts from 1913 up to 2007 is reproduced. The timings of the outbursts are quite sensitive to the spin value. In order to reproduce all the known outbursts, including a newly discovered one in 1957, the Kerr parameter of the primary has to be $0.28 \pm 0.08$. The quadrupole-moment contributions to the equations of motion allow us to constrain the `no-hair' parameter to be $1.0\:\pm\:0.3$ where 0.3 is the one sigma error. This supports the `black hole no-hair theorem' within the achievable precision. It should be possible to test the present estimate in 2015 when the next outburst is due. The timing of the 2015 outburst is a strong function of the spin: if the spin is 0.36 of the maximal value allowed in general relativity, the outburst begins in early November 2015, while the same event starts in the end of January 2016 if the spin is 0.2

10.1007/s11443-008-3002-5

The MASTER robotic telescope has obtained the first optical images of GRB 060926. We have discovered an optical flare from GRB 060926, a recurrent brightening that closely follows its behavior in the X-ray range. We have determined the spectral slope for GRB 060926 from the X-ray to optical range in the first minutes. Based on the spinar model, we show that the parameters of the optical and X-ray flares suggest that the gamma-ray burst resulted from the core collapse of a 7M ⊙ star with an initial effective Kerr parameter of 7.6 and an initial magnetic-to-gravitational energy ratio of 10−4.

Accretion process onto super-spinning objects

The accretion process onto spinning objects in Kerr spacetimes is studied with numerical simulations. Our results show that accretion onto compact objects with Kerr parameter (characterizing the spin) |a| M is very different. In the superspinning case, for |a| moderately larger than M, the accretion onto the central object is extremely suppressed due to a repulsive force at short distance. The accreting matter cannot reach the central object, but instead is accumulated around it, forming a high density cloud that continues to grow. The radiation emitted in the accretion process will be harder and more intense than the one coming from standard black holes; e.g. {gamma}-rays could be produced as seen in some observations. Gravitational collapse of this cloud might even give rise to violent bursts. As |a| increases, a larger amount of accreting matter reaches the central object and the growth of the cloud becomes less efficient. Our simulations find that a quasisteady state of the accretion process exists for |a|/M > or approx. 1.4, independently of the mass accretion rate at large radii. For such high values of the Kerr parameter, the accreting matter forms a thin disk at very small radii. We provide some analytical argumentsmore » to strengthen the numerical results; in particular, we estimate the radius where the gravitational force changes from attractive to repulsive and the critical value |a|/M{approx_equal}1.4 separating the two qualitatively different regimes of accretion. We briefly discuss the observational signatures which could be used to look for such exotic objects in the Galaxy and/or in the Universe.« less

Transmission through a Kerr barrier in photonic crystal waveguides: dispersioneffects

The transmission of guided modes through a barrier of Kerr nonlinear optical mediacontained within a photonic crystal waveguide of linear dielectric media is studied in orderto determine the effects of the dispersion of the incident waveguide modes ontheir barrier transmission coefficients. In McGurn (2008 Phys. Rev. B 77 115105)the conditions under which resonances exist in the guided mode transmissionthrough the barrier were investigated for an incident waveguide mode having asingle fixed frequency and a wavevector near the edge of the Brillouin zone. Thetransmission coefficient maxima were determined as functions of two parameterscharacterizing the Kerr nonlinearity of the barrier media and shown to exhibit acomplex pattern in the two parameter space of the Kerr parameters, associated withvarious kinds of modes excited within the barrier. In the present paper the focusis on how the pattern of transmission resonance maxima in the two parameterKerr parameter space is affected by varying the wavevector and frequency ofthe guided modes incident on the barrier. In addition, the effects of the barriersize on the pattern are determined. The focus of the paper is on affirming theclassification scheme proposed in our previous papers upon the introduction ofdispersive effects. The dynamical equations of our model are quite general, so it isexpected that this scheme will be useful in studying the nonlinear dynamics of othernonlinear physical models which may or may not be based on photonic crystalwaveguides.

Two-temperature accretion around rotating black holes: a description of the general advective flow paradigm in the presence of various cooling processes to explain low to high luminous sources

We investigate the viscous two temperature accretion discs around rotating black holes. We describe the global solution of accretion flows with a sub-Keplerian angular momentum profile, by solving the underlying conservation equations including explicit cooling processes selfconsistently. Bremsstrahlung, synchrotron and inverse Comptonization of soft photons are considered as possible cooling mechanisms, for sub-Eddington, Eddington and super-Eddington mass accretion rates around Schwarzschild and Kerr black holes with a Kerr parameter 0.998. It is found that the flow, during its infall from the Keplerian to sub-Keplerian transition region to the black hole event horizon, passes through various phases of advection -- general advective paradigm to radiatively inefficient phase and vice versa. Hence the flow governs much lower electron temperature ~10^8-10^{9.5} K, in the range of accretion rate in Eddington units 0.01 <~ \mdot <~ 100, compared to the hot protons of temperature ~ 10^{10.2} - 10^{11.8}K. Therefore, the solution may potentially explain the hard X-rays and \gamma-rays emitted from AGNs and X-ray binaries. We then show that a weakly viscous flow is expected to be cooling dominated, particularly at the inner region of the disc, compared to its highly viscous counterpart which is radiatively inefficient. With all the solutions in hand, we finally reproduce the observed luminosities of the under-fed AGNs and quasars (e.g. Sgr A^*) to ultra-luminous X-ray sources (e.g. SS433), at different combinations of input parameters such as mass accretion rate, ratio of specific heats. The set of solutions also predicts appropriately the luminosity observed in the highly luminous AGNs and ultra-luminous quasars (e.g. PKS 0743-67).

Cross Section, Final Spin, and Zoom-Whirl Behavior in High-Energy Black-Hole Collisions

We study the collision of two highly boosted equal-mass, nonrotating black holes with generic impact parameter. We find such systems to exhibit zoom-whirl behavior when fine-tuning the impact parameter. Near the threshold of immediate merger the remnant black-hole Kerr parameter can be near maximal (a/M greater, similar 0.95) and the radiated energy can be as large as 35 +/- 5% of the center-of-mass energy.

Two temperature viscous accretion flows around rotating black holes: Description of under-fed systems to ultra-luminous X-ray sources

We discuss two temperature accretion disk flows around rotating black holes. As we know that to explain observed hard X-rays the choice of Keplerian angular momentum profile is not unique, we consider the sub-Keplerian regime of the disk. Without any strict knowledge of the magnetic field structure, we assume the cooling mechanism is dominated by bremsstrahlung process. We show that in a range of Shakura–Sunyaev viscosity parameter 0.2 ≳ α ≳ 0.0005 , flow behavior varies widely, particularly by means of the size of disk, efficiency of cooling and corresponding temperatures of ions and electrons. We also show that the disk around a rotating black hole is hotter compared to that around a Schwarzschild black hole, rendering a larger difference between ion and electron temperatures in the former case. With all the theoretical solutions in hand, finally we reproduce the observed luminosities ( L) of two extreme cases—the under-fed AGNs and quasars (e.g. Sgr A ∗ ) with L ≳ 10 33 erg / s to ultra-luminous X-ray sources with L ∼ 10 41 erg/s, at different combinations of mass accretion rate, ratio of specific heats, Shakura–Sunyaev viscosity parameter and Kerr parameter, and conclude that Sgr A ∗ may be an intermediate spinning black hole.

THE STANDING ACCRETION SHOCK INSTABILITY IN THE DISK AROUND THE KERR BLACK HOLE

This paper is a sequel to our previous work for accretion onto a Schwarzschild black hole and the so-called standing accretion shock instability (SASI), in this paper we investigate non-axisymmetric perturbations for a Kerr black hole. The linear and non-linear phases for the shock evolution are analyzed in detail by both 2D general relativistic hydrodynamical simulations and linear analysis. Since the structure of steady axisymmetric accretion flows with a standing shock wave is very sensitive to the inner transonic flow, their properties such as Mach numbers, which are important for the stability, depend on the Kerr parameter very much. Although the essential features of the instability do not differ from the previous results for the Schwarzschild black hole, the frame dragging effects specific to the Kerr black hole is also evident. Interestingly, the oscillation periods of the fundamental unstable modes are dependent only on the shock radius irrespective of the injection parameters.

Modeling of non-stationary accretion disks in X-ray novae A 0620–00 and GRS 1124–68 during outburst

Aims. We address the task of modeling soft X-ray and optical light curves of X-ray novae in the high/soft state. Methods. The analytic model of viscous evolution of an externally truncated accretion α-disk is used. Relativistic effects near a Kerr black hole and self-irradiation of an accretion disk are taken into account. Results. The model is applied to the outbursts of X-ray nova Monocerotis 1975 (A 0620−00) and X-ray nova Muscae 1991 (GRS 1124−68). Comparison of observational data with the model yields constraints on the angular momentum (the Kerr parameter) of the black holes in A 0620−00 and GRS 1124−68: 0.3−0. 6a nd≤0.4, and on the viscosity parameter α of the disks: 0.7−0.95 and 0.55−0.75. We also conclude that the accretion disks should have an effective geometrical thickness 1.5−2 times greater than the theoretical value of the distance between the photometric layers.

Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity

We study the transition from inspiral to plunge in general relativity by computing gravitational waveforms of nonspinning, equal-mass black-hole binaries. We consider three sequences of simulations, starting with a quasicircular inspiral completing 1.5, 2.3 and 9.6 orbits, respectively, prior to coalescence of the holes. For each sequence, the binding energy of the system is kept constant and the orbital angular momentum is progressively reduced, producing orbits of increasing eccentricity and eventually a head-on collision. We analyze in detail the radiation of energy and angular momentum in gravitational waves, the contribution of different multipolar components and the final spin of the remnant, comparing numerical predictions with the post-Newtonian approximation and with extrapolations of point-particle results. We find that the motion transitions from inspiral to plunge when the orbital angular momentum $L={L}_{\mathrm{crit}}\ensuremath{\simeq}0.8{M}^{2}$. For $Ll{L}_{\mathrm{crit}}$ the radiated energy drops very rapidly. Orbits with $L\ensuremath{\simeq}{L}_{\mathrm{crit}}$ produce our largest dimensionless Kerr parameter for the remnant, $j=J/{M}^{2}\ensuremath{\simeq}0.724\ifmmode\pm\else\textpm\fi{}0.13$ (to be compared with the Kerr parameter $j\ensuremath{\simeq}0.69$ resulting from quasicircular inspirals). This value is in good agreement with the value of 0.72 reported in [I. Hinder, B. Vaishnav, F. Herrmann, D. Shoemaker, and P. Laguna, Phys. Rev. D 77, 081502 (2008).]. These conclusions are quite insensitive to the initial separation of the holes, and they can be understood by extrapolating point-particle results. Generalizing a model recently proposed by Buonanno, Kidder and Lehner [A. Buonanno, L. E. Kidder, and L. Lehner, Phys. Rev. D 77, 026004 (2008).] to eccentric binaries, we conjecture that (1) $j\ensuremath{\simeq}0.724$ is close to the maximal Kerr parameter that can be obtained by any merger of nonspinning holes, and (2) no binary merger (even if the binary members are extremal Kerr black holes with spins aligned to the orbital angular momentum, and the inspiral is highly eccentric) can violate the cosmic censorship conjecture.

Extended monopole solution of the Blandford-Znajek mechanism: Higher order terms for a Kerr parameter

The Blandford-Znajek mechanism, by which the rotational energy of a black hole is extracted through electromagnetic fields, is one of the promising candidates as an essential process of the central engine of active compact objects such as gamma-ray bursts. The only known analytical solution of this mechanism is the perturbative monopole solution for Kerr parameter a up to the second order terms. In order to apply the Blandford-Znajek mechanism to rapidly rotating black holes, we try to obtain the perturbation solution up to the fourth order. As a result, we find that the fourth order terms of the vector potential diverge at infinity, and this implies that the perturbation approach breaks down at a large distance from the black hole. Although there are some uncertainties about the solution due to the unknown boundary condition at infinity for the fourth order terms, we can derive the evaluation of the total energy flux extracted from the black hole up to fourth order of a without any ambiguity. Furthermore, from the comparison between the numerical solution, which is valid for 0<a<1, and the fourth order solution, we find that the fourth order solution reproduces the numerical result better than the second order solution.more » At the same time, since the fourth order solution does not match well with the numerical result at large a, we conclude that more higher order terms are required to reproduce the numerical result.« less

Light Echoes in Kerr Geometry: A Source of High‐Frequency QPOs from Random X‐Ray Bursts

We propose that high-frequency quasi-periodic oscillations (HFQPOs) can be produced from randomly formed X-ray bursts (flashes) by plasma interior to the ergosphere of a rapidly rotating black hole. We show by direct computation of their orbits that the photons comprising the observed X-ray light curves, if due to a multitude of such flashes, are significantly affected by the black hole's dragging of inertial frames; the photons of each such burst arrive to an observer at infinity in multiple (double or triple), distinct "bunches" separated by a roughly constant time lag of Δ tlag/M ≃ 14, regardless of the bursts' azimuthal position. We argue that every other such "bunch" represents photons that follow trajectories with an additional orbit around the black hole at the photon circular orbit radius (a photon "echo"). The presence of this constant lag in the response function of the system leads to a QPO feature in its power density spectra, even though the corresponding light curve consists of a totally stochastic signal. This effect is by and large due to the black hole spin and is shown to gradually diminish as the spin parameter a decreases or the radial position of the burst moves outside the static limit surface (ergosphere). Our calculations indicate that for a black hole with Kerr parameter of a/M = 0.99 and mass of M = 10 M☉, the QPO is expected at a frequency of νQPO ∼ 1.3–1.4 kHz. We discuss the plausibility and observational implications of our model results, as well as its limitations.

Multipolar analysis of spinning binaries

We present a preliminary study of the multipolar structure of gravitational radiation from spinning black hole binary mergers. We consider three different spinning binary configurations: (1) one ‘hang-up’ run, where the black holes have equal masses and large spins initially aligned with the orbital angular momentum; (2) seven ‘spin-flip’ runs, where the holes have a mass ratio q ≡ M1/M2 = 4, the spins are anti-aligned with the orbital angular momentum, and the initial Kerr parameters of the holes j1 = j2 = ji (where j ≡ J/M2) are fine-tuned to produce a Schwarzschild remnant after merger; (3) three ‘super-kick’ runs where the mass ratio q = 1, 2, 4 and the spins of the two holes are initially located on the orbital plane, pointing in opposite directions. For all of these simulations we compute the multipolar energy distribution and the Kerr parameter of the final hole. For the hang-up run, we show that including leading-order spin–orbit and spin–spin terms in a multipolar decomposition of the post-Newtonian waveforms improves agreement with the numerical simulation.