Angular Momentum Of Black Hole Research Articles

An Extended Scheme for Fitting X‐Ray Data with Accretion Disk Spectra in the Strong Gravity Regime

Accreting black holes are believed to emit X-rays which then mediate information about strong gravity in the vicinity of the emission region. We report on a set of new routines for the Xspec package for analysing X-ray spectra of black-hole accretion disks. The new computational tool significantly extends the capabilities of the currently available fitting procedures that include the effects of strong gravity, and allows one to systematically explore the constraints on more model parameters than previously possible (for example black-hole angular momentum). Moreover, axial symmetry of the disk intrinsic emissivity is not assumed, although it can be imposed to speed up the computations. The new routines can be used also as a stand-alone and flexible code with the capability of handling time-resolved spectra in the regime of strong gravity. We have used the new code to analyse the mean X-ray spectrum from the long XMM--Newton 2001 campaign of the Seyfert 1 galaxy MCG--6-30-15. Consistent with previous findings, we obtained a good fit to the broad Fe K line profile for a radial line intrinsic emissivity law in the disk which is not a simple power law, and for near maximal value of black hole angular momentum. However, equally good fits can be obtained also for small values of the black hole angular momentum. The code has been developed with the aim of allowing precise modelling of relativistic effects. Although we find that current data cannot constrain the parameters of black-hole/accretion disk system well, the approach allows, for a given source or situation, detailed investigations of what features of the data future studies should be focused on in order to achieve the goal of uniquely isolating the parameters of such systems.

Entropy of a Kerr–de Sitter black hole due to arbitrary spin fields

The Newman-Penrose formalism is used to derive the Teukolsky master equations controlling massless scalar, neutrino, electromagnetic, gravitino, and gravitational field perturbations of the Kerr--de Sitter spacetime. Then the quantum entropy of a nonextreme Kerr--de Sitter black hole due to arbitrary spin fields is calculated by the improved thin-layer brick wall model. It is shown that the subleading order contribution to the entropy is dependent on the square of the spins of particles and that of the specific angular momentum of black holes as well as the cosmological constant. The logarithmic correction of the spins of particles to the entropy relies on the rotation of the black hole and the effect of the cosmological constant.

Advection-Dominated Accretion Flow in the Kerr Metric: Propagation of a Wave Pattern

Abstract The time evolution of an advection-dominated accretion flow is explored in terms of relativistic fluid equations. An axially symmetric, vertically averaged mean flow is constructed and perturbed. An axisymmetric pattern is followed as it propagates in the form of a wave towards the horizon of a rotating black hole. Several assumptions are relaxed in comparison with previous works (Manmoto et al. 1996, AAA 65.067.206). The wave reflection and steepening are examined and the influence of black hole angular momentum is discussed.

Three‐dimensional Hydrodynamic Simulations of Accretion Tori in Kerr Spacetimes

This paper presents results of three-dimensional simulations of global hydrodynamic instabilities in black hole tori, extending earlier work by Hawley to Kerr spacetimes. This study probes a three-dimensional parameter space of torus angular momentum, torus size, and black hole angular momentum. We have observed the growth of the Papaloizou-Pringle instability for a range of torus configurations and the resultant formation of m=1 planets. We have also observed the quenching of this instability in the presence of early accretion flows; however, in one simulation both early accretion and planet formation occurred. Though most of the conclusions reached in Hawley's earlier work on Schwarzschild black holes carry over to Kerr spacetime, the presence of frame dragging in the Kerr geometry adds an element of complexity to the simulations; we have seen especially clear examples of this phenomenon in the accretion flows that arise from retrograde tori.

Off‐Axis Neutrino Scattering in Gamma‐Ray Burst Central Engines

The search for an understanding of an energy source great enough to explain the gamma-ray burst (GRB) phenomena has attracted much attention from the astrophysical community since its discovery. In this paper we extend the work of K. Asano and T. Fukuyama, and J. D. Salmonson and J. R. Wilson, and analyze the off-axis contributions to the energy-momentum deposition rate (MDR) from the neutrino anti-neutrino collisions above a rotating black hole/thin accretion disk system. Our calculations are performed by imaging the accretion disk at a specified observer using the full geodesic equations, and calculating the cumulative MDR from the scattering of all pairs of neutrinos and anti-neutrinos arriving at the observer. Our results shed light on the beaming efficiency of GRB models of this kind. Although we confirm Asano and Fukuyama's conjecture as to the constancy of the beaming for small angles away from the axis; nevertheless, we find the dominant contribution to the MDR comes from near the surface of the disk with a tilt of approximately \pi/4 in the direction of the disk's rotation. We find that the MDR at large radii is directed outward in a conic section centered around the symmetry axis and is larger, by a factor of 10 to 20, than the on-axis values. By including this off-axis disk source, we find a linear dependence of the MDR on the black hole angular momentum (a). In addition, we find that scattering is directed back onto the black hole in regions just above the horizon of the black hole. This gravitational ``in scatter'' may provide an observable high energy signature of the central engine, or at least another channel for accretion.

Critical gravitational collapse of a perfect fluid: Nonspherical perturbations

Continuously self-similar (CSS) solutions for the gravitational collapse of a spherically symmetric perfect fluid, with the equation of state p=kappa rho, with 0<kappa<1 a constant, are constructed numerically and their linear perturbations, both spherical and nonspherical, are investigated. The l=1 axial perturbations admit an analytical treatment. All others are studied numerically. For intermediate equations of state, with 1/9<kappa<0.49, the CSS solution has one spherical growing mode, but no nonspherical growing modes. That suggests that it is a critical solution even in (slightly) nonspherical collapse. For this range of kappa we predict the critical exponent for the black hole angular momentum to be 5(1+3kappa)/3(1+kappa) times the critical exponent for the black hole mass. For kappa=1/3 this gives an angular momentum critical exponent of mu=0.898, correcting a previous result. For stiff equations of state, 0.49<kappa<1, the CSS solution has one spherical and several nonspherical growing modes. For soft equations of state, 0<kappa<1/9, the CSS solution has 1+3 growing modes: a spherical one, and an l=1 axial mode (with m=-1,0,1).

Relativistic Diskoseismology. II. Analytical Results for c‐modes

We first briefly review how we investigate the modes of oscillation trapped within the inner region of accretion disks by the strong-field gravitational properties of a black hole (or a compact, weakly magnetized neutron star). Then we focus on the "corrugation" (c-) modes, nearly incompressible perturbations of the inner disk. The fundamental c-modes have eigenfrequencies (ordered by radial mode number) which correspond to the Lense-Thirring frequency, evaluated at the outer trapping radius of the mode, in the slow rotation limit. This trapping radius is a decreasing function of the black hole angular momentum, so a significant portion of the disk is modulated only for slowly rotating black holes. The eigenfrequencies are thus strongly increasing functions of black hole angular momentum. The dependence of the eigenfrequencies on the speed of sound (or the luminosity) within the disk is very weak, except for slowly rotating black holes.

Accretion disk spectra of super-luminal jet sources and ultra-luminous compact X-ray sources in nearby spiral galaxies

The Ultra-luminous Compact X-ray Sources (ULXs)in nearby spiral galaxies and the Galactic super-luminaljet sources sharethe common spectral characteristic that they haveextremely high disk temperatures which cannot be explainedin the framework of the standard accretion disk modelin the Schwarzschild metric. We have calculated an extreme Kerr disk model to examine if the Kerr disk model can instead explain the observed `too hot' accretion disk spectra.We found that the Kerr disk spectrum becomes significantly hardercompared to the Schwarzschild disk only when the disk is highlyinclined.For super-luminal jet sources, which are known to beinclined systems, the Kerr disk model may thuswork if we choose proper values for the black hole angular momentum. For the ULXs, however, the Kerr disk interpretation will be problematic,as is is highly unlikely that their accretion disks are preferentiallyinclined.

On the radial dependence of the disk FeK α line emissivity in active galactic nuclei

The iron K α fluorescent line detected in many Seyfert 1 galaxies by ASCA and BeppoSAX is usually considered to arise in the innermost region of an accretion disk around a supermassive black hole. The line profile is usually modelled by assuming that the disk emissivity follows a power law, ϵ( R)∝ R − q . We present a new technique to invert the line profile and infer: (a) the free-format ϵ( R), (b) the disk inclination to the line of sight and (c) the black hole angular momentum. The application of this technique to published K α data shows that the line shapes are consistent with the accretion disk model and that the major contribution to the emissivity comes from the innermost disk regions. We also show that a sharp distinction between different black hole angular momenta can usually not be made on the basis of present data.

Effects of Kerr space-time on spectral features from X-ray illuminated accretion discs

We performed detailed calculations of the relativistic effects acting on both the reflection continuum and the iron line from accretion discs around rotating black holes. Fully relativistic transfer of both illuminating and reprocessed photons has been considered in Kerr space-time. We calculated overall spectra, line profiles and integral quantities, and present their dependences on the black hole angular momentum. We show that the observed EW of the lines is substantially enlarged when the black hole rotates rapidly and/or the source of illumination is near above the hole. Therefore, such calculations provide a way to distinguish between different models of the central source.

RXTESpectral Observations of the 1996–1997 Outburst of the Microquasar GRO J1655−40

Excellent coverage of the entire 16 month 1996-1997 outburst cycle of GRO J1655-40 was provided by the Rossi X-Ray Timing Explorer (RXTE). We present a full spectral analysis of these data, which includes 52 Proportional Counter Array spectra from 2.5 to 20 keV and High Energy X-Ray Timing Experiment spectra above 20 keV. We also include a nearly continuous All-Sky Monitor light curve with several intensity measurements per day. The data are interpreted in the context of the multicolor blackbody disk/power-law model. The source exhibits two principal outburst states which we associate with the very high and the high/soft states. During the very high state, the spectrum is often dominated by a power-law component with photon index (Γ) ~ 2.3-2.7. The source exhibits intense hard flares on timescales of hours to days which are correlated with changes in both the fitted temperature and radius of the inner accretion disk. During the high/soft state, the spectrum is dominated by the soft thermal emission from the accretion disk with spectral parameters that suggest approximately constant inner disk radius and temperature. The power-law component is relatively weak with Γ ~ 2-3. During the last few observations, the source undergoes a transition to the low/hard state. We find that a tight relationship exists between the observed inner radius of the disk and the flux in the power-law component. During intense hard flares, the inner disk radius is observed to decrease by as much as a factor of 3 on a timescale of days. The apparent decrease of the inner disk radius observed during the flares may be due to the failure of the multicolor disk model caused by a steepening of the radial temperature profile in the disk coupled with increased spectral hardening and not physical changes of the inner disk radius. The distortion of the inner disk spectrum by the power-law flares indicates that the physical mechanism responsible for producing the power-law emission is linked to the inner disk region. Assuming that our spectral model is valid during periods of weak power-law emission, our most likely value for the inner disk radius implies a* < 0.7. Such a low value for the black hole angular momentum is inconsistent with the relativistic frame dragging and the "diskoseismic" models as interpretations for the 300 Hz X-ray QPO seen during some of these RXTE observations.

A Magnetically Switched, Rotating Black Hole Model for the Production of Extragalactic Radio Jets and the Fanaroff and Riley Class Division

A model is presented in which both Fanaroff and Riley class I and II extragalactic jets are produced by magnetized accretion disk coronae in the ergospheres of rotating black holes. While the jets are produced in the accretion disk itself, the output power still is an increasing function of the black hole angular momentum. For high enough spin, the black hole triggers the magnetic switch, producing highly-relativistic, kinetic-energy-dominated jets instead of Poynting-flux-dominated ones for lower spin. The coronal mass densities needed to trigger the switch at the observed FR break power are quite small ($\sim 10^{-15} g cm^{-3}$), implying that the source of the jet material may be either a pair plasma or very tenuous electron-proton corona, not the main accretion disk itself. The model explains the differences in morphology and Mach number between FR I and II sources and the observed trend for massive galaxies to undergo the FR I/II transition at higher radio power. It also is consistent with the energy content of extended radio lobes and explains why, because of black hole spindown, the space density of FR II sources should evolve more rapidly than that of FR I sources. If the present model is correct, then the ensemble average speed of parsec-scale jets in sources distinguished by their FR I morphology (not luminosity) should be distinctly slower than that for sources with FR II morphology. The model also suggests the existence of a population of high-redshift, sub-mJy FR I and II radio sources associated with spiral or pre-spiral galaxies that flared once when their black holes were formed but were never again re-kindled by mergers.

Angular momentum near the black hole threshold in scalar field collapse

For the formation of a black hole in the gravitational collapse of a massless scalar field, we calculate a critical exponent that governs the black hole angular momentum for slightly non-spherical initial data near the black hole threshold. We calculate the scaling law by second-order perturbation theory. We then use the numerical results of a previous first-order perturbative analysis to obtain the numerical value mu ~ 0.76 for the angular momentum critical exponent. A quasi-periodic fine structure is superimposed on the overall power law.

Quantum properties of the electron field in Kerr–Newman black hole manifolds

We study some spectral features of the one-particle electron Hamiltonian obtained by separating the Dirac equation in a Kerr–Newman black hole background. We find that the essential spectrum coincides with the whole real line. As a consequence, there is no gap in the spectrum and discrete eigenvalues are not allowed for any value of the black hole charge Q and angular momentum J. Our spectral analysis will be also related to the dissipation of the black hole angular momentum and charge.

Gravitational wave background from a cosmological population of core-collapse supernovae

We analyse the stochastic background of gravitational radiation emitted by a cosmological population of core-collapse supernovae. The supernova rate as a function of redshift is deduced from an observation-based determination of the star formation rate density evolution. We then restrict our analysis to the range of progenitor masses leading to black hole collapse. In this case, the main features of the gravitational-wave emission spectra have been shown to be, to some extent, independent of the initial conditions and of the equation of state of the collapsing star, and to depend only on the black hole mass and angular momentum. We calculate the overall signal produced by the ensemble of black-hole collapses throughout the Universe, assuming a flat cosmology with vanishing cosmological constant. Within a wide range of parameter values, we find that the spectral strain amplitude has a maximum at a few hundred Hz with an amplitude between $10^{-28}$ and $10^{-27} Hz^{-1/2}$; the corresponding closure density, $\Omega_{GW}$, has a maximum amplitude ranging between $10^{-11}$ and $10^{-10}$ in the frequency interval $\sim 1.5-2.5$ kHz. Contrary to previous claims, our observation-based determination leads to a duty cycle of order 0.01, making our stochastic backgound a non-continuous one. Although the amplitude of our background is comparable to the sensitivity that can be reached by a pair of advanced LIGO detectors, the characteristic shot-noise structure of the predicted signal might be in principle exploited to design specific detection strategies.

Statistical Entropy of Four-Dimensional Rotating Black Holes from Near-Horizon Geometry

We show that a class of four-dimensional rotating black holes allow five-dimensional embeddings as black rotating strings. Their near-horizon geometry factorizes locally as a product of the three-dimensional anti-deSitter space-time and a two-dimensional sphere (AdS_3 x S^2), with angular momentum encoded in the global space-time structure. Following the observation that the isometries on the AdS_3 space induce a two-dimensional (super)conformal field theory on the boundary, we reproduce the microscopic entropy with the correct dependence on the black hole angular momentum.

Diskoseismology: Probing black holes and their accretion disks

We review the relativistic results for diskoseismic modes of oscillation which are trapped within thin accretion disks by non-Newtonian gravitational properties of a black hole. Predicted frequencies are calculated for the potentially most observable modes, “internal gravity” modes and “corrugation” modes. The most definitive property of these two classes of modes is that the resulting eigenfrequencies depend almost entirely upon only the mass and angular momentum of the black hole. Such features may have been detected by RXTE in the power spectra of the luminosity modulations of the two galactic microquasars, GRS 1915+105 and GRO J1655-40. In the former system, we consider the possibility that this 67 Hz feature can be attributed to a g-mode in an accretion disk about a 10.6 M ⊙ (nonrotating) to 36.3 M ⊙ (maximally rotating) black hole. In the latter system, identification of the fundamental g-mode with the 300 Hz feature implies a black hole angular momentum approximately 93% of maximum.

Angular momentum at the black hole threshold

Near the black hole threshold in phase space, the black hole mass as a function of the initial data shows the “critical scaling” M?C(p-p*)?, where p labels a family of initial data, p* is the value of p at the threshold, and the critical exponent ? is universal for a given matter model. The black hole charge Q obeys a similar law. To complete the picture, we include the angular momentum as a perturbation. For the black hole angular momentum L?, we find the oscillating behavior L??Re[(A?+iB?)(p-p*)?+i?]. The assumptions of the calculation hold for p=?/3 perfect fluid matter, and we calculate ??0.799 and ??0.231.

Evidence for Frame-Dragging around Spinning Black Holes in X-Ray Binaries

In the context of black hole spin in X-ray binaries, we propose that certain type of quasi-period oscillations (QPOs) observed in the light curves of black hole binaries (BHBs) are produced by X-ray modulation at the precession frequency of accretion disks, due to relativistic dragging of inertial frames around spinning black holes. These QPOs tend to be relatively stable in their centroid frequencies. They have been observed in the frequency range of a few to a few hundred Hz for several black holes with dynamically determined masses. By comparing the computed disk precession frequency with that of the observed QPO, we can derive the black hole angular momentum, given its mass. When applying this model to GRO J1655-40, GRS 1915+105, Cyg X-1, and GS 1124-68, we found that the black holes in GRO J1655-40 and GRS 1915+105, the only known BHBs that occasionally produce superluminal radio jets, spin at a rate close to the maximum limit, while Cyg X-1 and GS 1124-68, typical (persistent and transient) BHBs, contain only moderately rotating ones. Extending the model to the general population of black hole candidates, the fact that only low-frequency QPOs have been detected is consistent with the presence of only slowly spinning black holes in these systems. Our results are in good agreement with those derived from spectral data, thus strongly support the classification scheme that we proposed previously for BHBs.

Measuring black-hole parameters and testing general relativity using gravitational-wave data from space-based interferometers.

Among the expected sources of gravitational waves for the Laser Interferometer Space Antenna (LISA) is the capture of solar-mass compact stars by massive black holes residing in galactic centers. We construct a simple model for such a capture, in which the compact star moves freely on a circular orbit in the equatorial plane of the massive black hole. We consider the gravitational waves emitted during the late stages of orbital evolution, shortly before the orbiting mass reaches the innermost stable circular orbit. We construct a simple model for the gravitational-wave signal, in which the phasing of the waves plays the dominant role. The signal's behavior depends on a number of parameters, including $\ensuremath{\mu}$, the mass of the orbiting star, $M$, the mass of the central black hole, and $J$, the black hole's angular momentum. We calculate, using our simplified model, and in the limit of large signal-to-noise ratio, the accuracy with which these quantities can be estimated during a gravitational-wave measurement. For concreteness we consider a typical system consisting of a $10{M}_{\ensuremath{\bigodot}}$ black hole orbiting a nonrotating black hole of mass ${10}^{6}{M}_{\ensuremath{\bigodot}}$, whose gravitational waves are monitored during an entire year before the orbiting mass reaches the innermost stable circular orbit. Defining $\ensuremath{\chi}\ensuremath{\equiv}\frac{\mathrm{cJ}}{G{M}^{2}}$ and $\ensuremath{\eta}\ensuremath{\equiv}\frac{\ensuremath{\mu}}{M}$, we find $\ensuremath{\Delta}\ensuremath{\chi}\ensuremath{\simeq}5\ifmmode\times\else\texttimes\fi{}\frac{{10}^{\ensuremath{-}2}}{\ensuremath{\rho}}$, $\frac{\ensuremath{\Delta}\ensuremath{\eta}}{\ensuremath{\eta}}\ensuremath{\simeq}6\ifmmode\times\else\texttimes\fi{}\frac{{10}^{\ensuremath{-}2}}{\ensuremath{\rho}}$, and $\frac{\ensuremath{\Delta}M}{M}\ensuremath{\simeq}2\ifmmode\times\else\texttimes\fi{}\frac{{10}^{\ensuremath{-}3}}{\ensuremath{\rho}}$. Here, $\ensuremath{\rho}$ denotes the signal-to-noise ratio associated with the signal and its measurement. That these uncertainties are all much smaller than $\frac{1}{\ensuremath{\rho}}$, the signal-to-noise ratio level, is due to the large number of wave cycles received by the detector in the course of one year. These are the main results of this paper. Our simplified model also suggests a method for experimentally testing the strong-field predictions of general relativity.