Trajectories of Light Beams in a Kerr Metric: the Influence of the Rotation of an Observer on the Shadow of a Black Hole

Christodoulou et al have shown that the interior volume of a Schwarzschild black hole grows linearly with time. Subsequently, their conclusion has been extended to the Reissner–Nordström (RN) and Kerr black holes. Meanwhile, the entropy of the scalar field inside a Schwarzschild black hole has also been calculated. In this paper, a general method calculating the number of quantum states of the scalar field inside the black hole is given, which can be used in an arbitrary black hole. After introducing the two important assumptions as the black-body radiation assumption and the quasi-static process assumption, the entropy of the scalar field inside a Kerr black hole is calculated using the differential form, and we find that the variation of the entropy is proportional to the variation of the Bekenstein–Hawking entropy except the ending of the black hole evaporation. Similarly, we recalculate the entropy of the scalar field inside a Schwarzschild black hole and demonstrate that the entropy inside a Kerr black hole can exactly degenerate to the Schwarzschild black hole. As well as, we find that the proportionality coefficient between the entropy of the scalar field and the Bekenstein–Hawking entropy in Schwarzschild case, which is obtained using the differential form, is half of that given in the previous literature. Furthermore, we investigate the total entropy of Kerr and Schwarzschild black holes and find that they all increase with time. It means that the black holes, evolution with Hawking radiation satisfies the second law of thermodynamics. Finally, the black hole information paradox is brought up again and discussed.

Purpose of Study: The study's objective is to investigate the function of entropy in black holes, with a particular emphasis on the ways in which entropy aids in the comprehension of the properties of various varieties of black holes, such as Schwarzschild, Kerr, and charged black holes (Reissner-Nordström and Kerr-Newman). The objective of the investigation is to examine the unique entropy characteristics that are associated with each form of black hole within the context of black hole thermodynamics. Methodology: The entropy of black holes is examined through a theoretical approach that utilizes the principles of thermodynamics and information theory. The analysis entails a comparison of the entropy properties of Schwarzschild, Kerr, and charged black holes, taking into account their distinctive characteristics and the implications for black hole thermodynamics. Results: The analysis demonstrates that the inherent properties of each form of black hole are directly correlated with their distinctive entropy characteristics. Schwarzschild black holes, Kerr black holes, and charged black holes exhibit unique entropy patterns, which contribute to a more exhaustive comprehension of black hole thermodynamics and provide more profound insights into their thermodynamic behavior. Applications: The results have substantial implications for the advancement of theoretical physics, particularly in the field of black hole thermodynamics. The development of more precise models and predictions regarding black hole behavior can be facilitated by an understanding of the entropy characteristics of various varieties of black holes. This knowledge has the potential to inform future research in quantum gravity and cosmology.

In this paper, we investigate thick accretion structures around a Kerr black hole in a swirling background, that is, a rotating black hole immersed in a rotating background. This is a novel solution characterized by the black hole mass in addition to two distinct rotational parameters, the Kerr parameter a, which identifies the rotation of the black hole, and the swirling parameter j, which describes the background rotation. The swirling background is characterized by an odd Z2 symmetry, where the northern and southern hemispheres rotate in opposite directions. The rotation of the black hole embedded into this swirling background leads to non-trivial spin–spin interactions with the background rotation. The spacetime properties in the vicinity of the black hole are significantly influenced by this spin–spin interaction. In order to study the influence on the basic properties of this spacetime, we analyze circular orbits and geometrically thick disks for different spacetime solutions, which are classified by the black hole and swirling spins. We identify stabilizing effects on prograde circular orbits and destabilizing effects on retrograde circular orbits, which originate from the spin–spin interaction and depend mainly on the Kerr rotation. Furthermore, we discover the emergence of static orbits, which appears due to the background rotation. The symmetry breaking of the spacetime rotation with regard to the equatorial plane highly influences the spatial distribution of circular orbits. This asymmetry causes a concave (convex) distribution of the prograde (retrograde) circular orbits and accordingly, bowl-like deformations of the accretion disk solutions. Moreover, due to the destabilizing effect of the swirling rotation, an outer marginally stable orbit appears, which heavily limits the range of the parameter space in which disk solutions can exist. Due to the possibility of an outer and inner disk cusp, different types of disk solutions are possible. We classify the different types of disk solutions, which differ from each other by the properties of their cusps. Four different scenarios can be identified in which different accretion dynamics could arise.

This is the second in a series of papers whose aim is to generate adiabatic gravitational waveforms from the inspiral of stellar-mass compact objects into massive black holes. In earlier work, we presented an accurate (2+1)D finite-difference time-domain code to solve the Teukolsky equation, which evolves curvature perturbations near rotating (Kerr) black holes. The key new ingredient there was a simple but accurate model of the singular source term based on a discrete representation of the Dirac-delta function and its derivatives. Our earlier work was intended as a proof of concept, using simple circular, equatorial geodesic orbits as a test bed. Such a source is effectively static, in that the smaller body remains at the same coordinate radius and orbital inclination over an orbit. (It of course moves through axial angle, but we separate that degree of freedom from the problem. Our numerical grid has only radial, polar, and time coordinates.) We now extend the time-domain code so that it can accommodate dynamic sources that move on a variety of physically interesting world lines. We validate the code with extensive comparison to frequency-domain waveforms for cases in which the source moves along generic (inclined and eccentric) bound geodesic orbits. We also demonstrate the ability of the time-domain code to accommodate sources moving on interesting nongeodesic worldlines. We do this by computing the waveform produced by a test mass following a kludged inspiral trajectory, made of bound geodesic segments driven toward merger by an approximate radiation loss formula.

The nonminimal coupling of the nonzero vacuum expectation value of the self-interacting antisymmetric Kalb-Ramond field with gravity leads to a power-law hairy black hole having a parameter $s$, which encompasses the Reissner$-$Nordstrom black hole ($s=1$). We obtain the axially symmetric counterpart of this hairy solution, namely, the rotating Kalb-Ramond black hole, which encompasses, as special cases, Kerr ($s=0$) and Kerr-Newman ($s=1$) black holes. Interestingly, for a set of parameters ($M, a$, and $\Gamma$), there exists an extremal value of the Kalb-Ramond parameter ($s=s_{e}$), which corresponds to an extremal black hole with degenerate horizons, while for $s<s_{e}$, it describes a nonextremal black hole with Cauchy and event horizons, and no black hole for $s>s_{e}$. We find that the extremal value $s_e$ is also influenced by these parameters. The black hole shadow size decreases monotonically and the shape gets more distorted with an increasing $s$; in turn, shadows of rotating Kalb-Ramond black holes are smaller and more distorted than the corresponding Kerr black hole shadows. We investigate the effect of the Kalb-Ramond field on the rotating black hole spacetime geometry and analytically deduced corrections to the light deflection angle from the Kerr and Schwarzschild black hole values. The deflection angle for Sgr A* and the shadow caused by the supermassive black hole M87* are included and compared with analogous results of Kerr black holes. The inferred circularity deviation $\Delta C\leq 0.10$ for the M87* black hole merely constrains the Kalb-Ramond field parameter, whereas shadow angular diameter $\theta_d=42\pm 3\, \mu$as, within the $1\sigma$ region, places bounds $\Gamma\leq 0.09205$ for $s=1$ and $\Gamma\leq 0.02178$ for $s=3$.

Q-balls are regular extended ‘objects’ that exist for some non-gravitating, self-interacting, scalar field theories with a global, continuous, internal symmetry, on Minkowski spacetime. Here, analogous objects are also shown to exist around rotating (Kerr) black holes, as non-linear bound states of a test scalar field. We dub such configurations Q-clouds. We focus on a complex massive scalar field with quartic plus hexic self-interactions. Without the self-interactions, linear clouds have been shown to exist, in synchronous rotation with the black hole horizon, along 1-dimensional subspaces – existence lines – of the Kerr 2-dimensional parameter space. They are zero modes of the superradiant instability. Non-linear Q-clouds, on the other hand, are also in synchronous rotation with the black hole horizon; but they exist on a 2-dimensional subspace, delimited by a minimal horizon angular velocity and by an appropriate existence line, wherein the non-linear terms become irrelevant and the Q-cloud reduces to a linear cloud. Thus, Q-clouds provide an example of scalar bound states around Kerr black holes which, generically, are not zero modes of the superradiant instability. We describe some physical properties of Q-clouds, whose backreaction leads to a new family of hairy black holes, continuously connected to the Kerr family.

We apply the counterterm subtraction technique to calculate the action and other quantities of the Kerr-AdS black hole in five dimensions using two boundary metrics: the Einstein universe and the rotating Einstein universe with an arbitrary angular velocity. In both cases, the resulting thermodynamic quantities satisfy the first law of thermodynamics. We point out that the reason for the violation of the first law in previous calculations was that the rotating Einstein universe, used as a boundary metric, was rotating with an angular velocity that depended on the black hole rotation parameter. Using a new coordinate system with a boundary metric that has an arbitrary angular velocity, one can show that the resulting physical quantities satisfy the first law.

The electromagnetic field of a general stationary source, occurring in the vicinity of a rotating (Kerr) black hole, is obtained by solving the Maxwell and Teukolsky equations. The field is expressed both outside and inside the radius at which the source is located. As examples the fields of point charges, charged rings, current loops, and magnetic dipoles not necessarily located in axisymmetric positions are calculated. The electromagnetic field occurring when a Kerr black hole is placed in an originally uniform magnetic field is derived without assuming the alignment of the direction of the magnetic field and the axis of symmetry of the black hole.

A toy model for the Blandford-Znajek mechanism is investigated: a Kerr black hole with a toroidal electric current residing in a thin disk around the black hole. The toroidal electric current generates a poloidal magnetic field threading the black hole and disk. Due to the interaction of the magnetic field with remote charged particles, the rotation of the black hole and disk induces an electromotive force, which can power an astrophysical load at remote distance. The power of the black hole and disk is calculated. It is found that, for a wide range of parameters specifying the rotation of the black hole and the distribution of the electric current in the disk, the power of the disk exceeds the power of the black hole. The torque provided by the black hole and disk is also calculated. The torque of the disk is comparable to the torque of the black hole. As the disk loses its angular momentum, the mass of the disk gradually drifts towards the black hole and gets accreted. Ultimately the power comes from the gravitational binding energy between the disk and the black hole, as in the standard theory of accretion disk, instead of the rotational energy of the black hole. This suggests that the Blandford-Znajek mechanism may be less efficient in extracting energy from a rotating black hole with a thin disk. The limitations of our simple model and possible improvements deserved for future work are also discussed.

The choked accretion model consists of a purely hydrodynamical mechanism in which, by setting an equatorial to polar density contrast, a spherically symmetric accretion flow transitions to an inflow-outflow configuration. This scenario has been studied in the case of a (non-rotating) Schwarzschild black hole as central accretor, as well as in the non-relativistic limit. In this article, we generalize these previous works by studying the accretion of a perfect fluid onto a (rotating) Kerr black hole. We first describe the mechanism by using a steady-state, irrotational analytic solution of an ultrarelativistic perfect fluid, obeying a stiff equation of state. We then use hydrodynamical numerical simulations in order to explore a more general equation of state. Analyzing the effects of the black hole's rotation on the flow, we find in particular that the choked accretion inflow-outflow morphology prevails for all possible values of the black hole's spin parameter, showing the robustness of the model.

Shadows of black hole surrounded by anisotropic fluid in Rastall theory

Very-long baseline interferometric observations have resolved structure on scales of only a few Schwarzschild radii around the supermassive black holes at the centers of our Galaxy and M87. In the near future, such observations are expected to image the shadows of these black holes together with a bright and narrow ring surrounding their shadows. For a Kerr black hole, the shape of this photon ring is nearly circular unless the black hole spins very rapidly. Whether or not, however, astrophysical black holes are truly described by the Kerr metric as encapsulated in the no-hair theorem still remains an untested assumption. For black holes that differ from Kerr black holes, photon rings have been shown numerically to be asymmetric for small to intermediate spins. In this paper, I calculate semi-analytic expressions of the shapes of photon rings around black holes described by a new Kerr-like metric which is valid for all spins. I show that photon rings in this spacetime are affected by two types of deviations from the Kerr metric which can cause the ring shape to be highly asymmetric. I argue that the ring asymmetry is a direct measure of a potential violation of the no-hair theorem and that both types of deviations can be detected independently if the mass and distance of the black hole are known. In addition, I obtain approximate expressions of the diameters, displacements, and asymmetries of photon rings around Kerr and Kerr-like black holes.

Despite its success in the weak gravity regime, General Relativity (GR) has\nyet to be verified in the regime of strong gravity. In this paper, we present\nthe results of detailed ray tracing simulations aiming at clarifying if the\ncombined information from X-ray spectroscopy, timing, and polarization\nobservations of stellar mass and supermassive black holes can be used to test\nGR's no-hair theorem. The latter states that stationary astrophysical black\nholes are described by the Kerr-family of metrics with the black hole mass and\nspin being the only free parameters. We use four "non-Kerr metrics", some\nphenomenological in nature and others motivated by alternative theories of\ngravity, and study the observational signatures of deviations from the Kerr\nmetric. Particular attention is given to the case when all the metrics are set\nto give the same Innermost Stable Circular Orbit (ISCO) in quasi-Boyer\nLindquist coordinates. We give a detailed discussion of similarities and\ndifferences of the observational signatures predicted for BHs in the Kerr\nmetric and the non-Kerr metrics. We emphasize that even though some regions of\nthe parameter space are nearly degenerate even when combining the information\nfrom all observational channels, X-ray observations of very rapidly spinning\nblack holes can be used to exclude large regions of the parameter space of the\nalternative metrics. Although it proves difficult to distinguish between the\nKerr and non-Kerr metrics for some portions of the parameter space, the\nobservations of very rapidly spinning black holes like Cyg X-1 can be used to\nrule out large regions for several black hole metrics.\n

The exact mechanism by which astrophysical jets are formed is still unknown. It is believed that the necessary elements consist of a rotating (Kerr) black hole and a magnetized accreting plasma. We model the accreting plasma as a collection of magnetic flux tubes/strings. If such a tube falls into a Kerr black hole, then the leading portion loses angular momentum and energy as the string brakes. To compensate for this loss, momentum and energy is redistributed to the trailing portion of the tube. We found that buoyancy creates a pronounced helical magnetic field structure aligned with the spin axis. Along the field lines, the plasma is centrifugally accelerated close to the speed of light. This process leads to unlimited stretching of the flux tube since one part of the tube continues to fall into the black hole and, simultaneously, the other part of the string is pushed outward. Eventually, reconnection cuts the tube. The inner part is filled with new material and the outer part forms a collimated bubble-structured relativistic jet. Each plasmoid can be considered as an outgoing particle in the Penrose mechanism: it carries extracted rotational energy away from the black hole while the falling part, with corresponding negative energy, is left inside the ergosphere.

We present the first results from fully general relativistic numerical studies of thick-disk accretion onto a rapidly-rotating (Kerr) black hole with a spin axis that is tilted (not aligned) with the angular momentum vector of the disk. We initialize the problem with the solution for an aligned, constant angular momentum, accreting thick disk around a black hole with spin a/M = J/M{sup 2} = +0.9 (prograde disk). The black hole is then instantaneously tilted, through a change in the metric, by an angle {beta}{sub 0}. In this Letter we report results with {beta}{sub 0} = 0, 15, and 30{sup o}. The disk is allowed to respond to the Lense-Thirring precession of the tilted black hole. We find that the disk settles into a quasi-static, twisted, warped configuration with Lense-Thirring precession dominating out to a radius analogous to the Bardeen-Petterson transition in tilted Keplerian disks.