Article

Maximal Analytic Extension of the Kerr Metric

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Abstract

Kruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a2 < m2, this extension consists of an infinite sequence Einstein‐Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.

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... [1]). For null ray propagation in the equatorial plane, the solutions to the equations of motion can be written explicitly in the form of the elliptic integral [39]. When the impact parameter for the null ray is much larger than the gravitational radius GM of the Kerr black hole, one can decompose the expression for the ray's scattering angle in powers of the small parameter GM/ . ...
... The first terms of this expression coincide with the result presented in [39] 4 See also [40,41]. 4 Let us note that the sign of the rotation parameter in the paper [39] is chosen to be opposite to the sign adopted in [1,4]. ...
... The first terms of this expression coincide with the result presented in [39] 4 See also [40,41]. 4 Let us note that the sign of the rotation parameter in the paper [39] is chosen to be opposite to the sign adopted in [1,4]. ...
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... In this study, we will relate the change in the action variables to the energy transfer between the EMRI and perturber during a resonance crossing. This transfer of energy can be calculated by looking at the superposition of the gravitational perturbations generated by the EMRI and perturber, which is contained in the gauge-invariant Weyl curvature scalar, ψ 4 [24,37]. This Weyl scalar satisfies a separable partial differential equation on the Kerr background spacetime, and allows us to compute some types of multi-body interactions without computing metric perturbations [38]. ...
... We work in the Boyer-Lindquist [37] coordinate system (t, r, θ, φ), with the −+++ signature. Four-vector quantities will be denoted with a boldface print J with components indexed by Greek letters. ...
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... We describe the gravitational field of the black hole in the framework of Einsteinian gravity by using the classical Kerr metric [12,[24][25][26][27][28][29][30][31][32] describing the rotating black hole with a mass M and with a black-hole-specific angular momentum (spin) a = J/M in standard Boyer-Lindquist coordinates (t, r, θ, φ) [25]: ...
... We describe the gravitational field of the black hole in the framework of Einsteinian gravity by using the classical Kerr metric [12,[24][25][26][27][28][29][30][31][32] describing the rotating black hole with a mass M and with a black-hole-specific angular momentum (spin) a = J/M in standard Boyer-Lindquist coordinates (t, r, θ, φ) [25]: ...
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... We begin with the standard form of the Kerr metric [20] in the Boyer-Lindquist coordinates [21]: ...
... or, by use of eqs. (6), (7), (17), (18), (21), (24), (26), (27), ...
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... Then, we discuss the perturbations of a static rotating axisymmetric BH as characterized generally by the Kerr solution; Kerr (1963) in terms of the Boyer-Lindquist coordinate; Boyer and Lindquist (1967): ...
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Quasi-normal modes (QNMs) of a black hole (BH) are the eigen modes describing the dissipative oscillation of various fields in that spacetime, which can be intrinsically produced by the linear perturbation theory. With the discovery of the first gravitational waves (GWs) event, GW150914, a new window into the universe has been opened, allowing for the detection of QNMs associated to the ringdown process, which will enable more accurate measurements of the BHs parameters as well as further testing of general relativity. This article discusses the linear perturbation theory of BHs and provides review of several QNMs calculation methods including the newly developed methods. We will also focus on the connection between QNMs and the detection of GWs as well as some recent advancements in this area.
... which is half the Schwarzchild radius r s . The radius of the BH's outer horizon is Boyer & Lindquist [1967] ...
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Ultralight bosons are a class of hypothetical particles that could potentially solve critical problems in fields ranging from cosmology to astrophysics and fundamental physics. If ultralight bosons exist, they form clouds around spinning black holes with sizes comparable to their particle Compton wavelength through superradiance, a well-understood classical wave amplification process that has been studied for decades. After these clouds form, they dissipate and emit continuous gravitational waves through the annihilation of ultralight bosons into gravitons. These gravitons could be detected with ground-based gravitational-wave detectors using continuous-wave searches. However, it is conceivable for other continuous-wave sources to mimic the emission from the clouds, which could lead to false detections. Here we investigate how one can use continuous waves from clouds formed around known merger remnants to alleviate this problem. In particular, we simulate a catalogue of merger remnants that form clouds around them and demonstrate with select "golden" merger remnants how one can perform a Bayesian cross-verification of the ultralight boson hypothesis that has the potential to rule out alternative explanations. Our proof-of-concept study suggest that, in the future, there is a possibility that a merger remnant exists close enough for us to perform the analysis and test the boson hypothesis if the bosons exist in the relevant mass range. Future research will focus on building more sophisticated continuous-wave tools to perform this analysis in practice.
... where ρ 2 = r 2 + a 2 cos 2 θ + b 2 sin 2 θ, and a and b are arbitrary constants. It could further be transformed into the Boyer-Lindquist form [18], ...
Preprint
We wish to carry forward to higher dimensions the insightful and novel method of obtaining the Kerr metric proposed by one of us in \cite{Dadhich} for deriving Myers-Perry rotating black hole metric. We begin with a flat spacetime metric written in oblate spheroidal coordinates (ellipsoidal geometry) appropriate for inclusion of rotation, and then introduce arbitrary functions to bring in gravitational potential due to mass which are then determined by requiring that massless particle experiences no acceleration while massive particle feels the Newtonian acceleration at large $r$. We have further generalized the method to include the cosmological constant $\Lambda$ to obtain the Myers-Perry-dS/AdS black hole metric.
... Since we have considered MBH = 1, the present analysis is applicable for black holes of all mass scales. In this work, we investigate the accretion flow around a Kerr black hole and hence, we consider Kerr metric in Boyer-Lindquist coordinates (Boyer & Lindquist 1967) as, ...
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... The metric on its exterior E A in traditional Boyer-Lindquist coordinates [107] (t, r, θ, φ) is ...
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This thesis is concerned with the black hole stability problem in general relativity. In particular, it presents stability and instability results associated to the linearised vacuum Einstein equation on black hole backgrounds. The first chapter of this thesis gives a direct rigorous mathematical proof of the Gregory--Laflamme instability for the 5-dimensional Schwarzschild black string. Under a choice of ansatz for the perturbation and a gauge choice, the linearised vacuum Einstein equation reduces to an ODE problem for a single function. In this work, the ODE is cast into a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution which has admissible behaviour at the future event horizon and spacelike infinity. After the addition of a pure gauge solution, this gives rise to a regular exponentially growing mode solution of the linearised vacuum Einstein equation in harmonic/transverse-traceless gauge. The remainder of this thesis is concerned with conservation laws associated to the linearised vacuum Einstein equation. For later application, chapter 2 of this thesis contains a review of the double null gauge for the vacuum Einstein equations. In chapter 3, the `canonical energy' conservation law of Hollands and Wald is studied. This canonical energy conservation law gives an appealing criterion for stability of black holes based upon a conserved current. The method is appealing in its simplicity as it requires one to 'simply’ check the sign of the canonical energy Ε with Ε>0 implying weak stability and Ε<0 implying instability. However, in practice establishing the sign of E proves difficult. Indeed, even for the 4-dimensional Schwarzschild black hole exterior the positivity was not previously established. In this thesis, a resolution to this issue for the Schwarzschild black hole is presented by connecting to another weak stability result of Holzegel which exploits the double null gauge. Further weak stability statements for the Schwarzschild black hole (including a proof of mode stability) arising from the canonical energy are also established. In chapter 4, some preliminary results associated to a novel conserved current associated to the linearised vacuum Einstein equation are presented. This can be viewed as a modification/simplification of the conserved current associated to the canonical energy. In particular, applications of this current to other black hole spacetimes are discussed.
... The QNMs of the Teukolsky equation have found use in astrophysics, theoretical physics, and mathematical relativity [3][4][5][6]. Teukolsky QNM mode calculations are typically computed using coordinates where the constant time hypersurfaces intersect the bifurcation sphere and spatial infinity (for example Boyer-Lindquist coordinates have this property [7]) [2,[8][9][10][11]. As was pointed out though by Zenginoglu, horizon-penetrating, hyperboloidally compactified (HPHC) coordinates-that is, coordinates where constant time hypersurfaces intersect both the black hole horizon and future null infinity (see figure 1)-can be considered a more 'natural' set of coordinates to study black hole perturbations [1]. ...
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We study the quasinormal mode eigenvalues and eigenfunctions for the Teukolsky equation in a horizon penetrating, hyperboloidally compactified coordinate system. Following earlier work by Zenginoğlu (2011 Phys. Rev. D 83 127502), we show that the quasinormal eigenfunctions (QNEs) for the Teukolsky equation are regular from the black hole horizon to future null infinity in these coordinates. We then present several example QNE solutions, and study some of their properties in the near-extremal Kerr limit.
... It would be quite interesting to apply the above technique for other spacetimes such as Kerr spacetime [43][44][45][46][47][48], Kerr-de Sitter spacetime [49][50][51], Kerr-Newman-de Sitter spacetimes [52][53][54], charged BTZ black hole spacetime [55][56][57][58] and black holes in Horava gravity [59,60]. ...
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For a single horizon spacetime, the Hawking temperature is proportional to the surface gravity at the horizon. However, for a multi-horizon spacetime, it is not fully understood whether the Hawking temperature for this spacetime does coincide with the conventional Hawking temperature related to the outer horizon or not. Here we show that the Hawking temperature for this spacetime does not coincide with the conventional Hawking temperature associated with the outer horizon. The contribution between the horizons determines the temperature. We consider the tunneling process to compute the Hawking temperature in the Schwarzschild-de Sitter spacetime, Reissner-Nordstrom-de Sitter spacetime, and rotating BTZ black hole spacetime here. There are two contributions to the tunneling process of radiation. These two contributions separately give the radiation with the two times the original Hawking temperature. But the combination of these two gives the radiation with the exact Hawking temperature.
... From the condition ds 2 ≥ 0 one obtains that if D is larger than D s = ct, then no physical object can be at rest in coordinates t, D, θ, φ. The value D s corresponds to χ = 1 and it plays the role of the static limit for the rotating black hole in Boyer-Lindquist [15] coordinates. The energy of the particle with mass m in coordinates t, D, θ, φ is ...
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Particles with negative energies are considered for three different cases: inside of horizon of nonrotating Schwarzschild black hole, Milne's coordinates in flat Minkowski space-time (Milne's universe using nonsynchronous coordinates) and in cosmological Godel model of the rotating universe. It is shown that differently from the Godel model with nondiagonal term where it occurs that negative energies are impossible they are present in all other cases considered in the paper. Particles with zero energy are also possible in first two cases.
... The subextremal Kerr black hole exterior is a manifold covered globally (modulo the usual degeneration of polar coordinates) by so-called Boyer-Lindquist coordinates (t, r, θ, φ) ∈ R × (r + , ∞) × S 2 [BL67], and endowed with the Lorentzian metric g = − ρ 2 (dt − a sin 2 θ dφ) 2 + ρ 2 dr 2 + ρ 2 dθ 2 + sin 2 θ ρ 2 adt − (r 2 + a 2 )dφ 2 , Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
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We uncover hidden spectral symmetries of the Teukolsky equation in Kerr(-de Sitter) black holes, recently conjectured by Aminov, Grassi and Hatsuda (Ann. Henri Poincaré 23, 1951-1977, 2022, and Gen. Relativ. Grav. 53(10):93, 2021) in the zero cosmological constant case. Using these symmetries, we provide a new, simpler proof of mode stability for subextremal Kerr black holes. We also present a partial mode stability result for Kerr–de Sitter black holes.
... After summarizing what is known about the bending angle in the Kerr metric, we identify and seek to resolve a subtle but important discrepancy that has appeared in the literature. The spacetime around a rotating compact object is described by the Kerr metric [8], which is typically written in Boyer-Lindquist [9] coordinates (t, r, θ, φ), where θ is measured with respect to the axis of rotation. For simplicity, we work in the equatorial plane (θ = π/2), where the line element takes the form ds 2 = g tt c 2 dt 2 + g rr dr 2 + g φφ dφ 2 + 2g tφ c dt dφ. ...
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The observation of the bending of light by mass, now known as gravitational lensing, was key in establishing general relativity as one of the pillars of modern physics. In the past couple of decades, there has been increasing interest in using gravitational lensing to test general relativity beyond the weak deflection limit. Black holes and neutron stars produce the strong gravitational fields needed for such tests. For a rotating compact object, the distinction between prograde and retrograde photon trajectories becomes important. In this paper, we explore subtleties that arise in interpreting the bending angle in this context and address the origin of seemingly contradictory results in the literature. We argue that analogies that cannot be precisely quantified present a source of confusion.
... impossible or at least intermittent (King and Pringle, 2021), a net charge has negligible effect on the general relativistic spacetime geometry (Zajaček et al., 2018;Zajacek and Tursunov, 2019). Therefore, the spacetime geometry of astrophysical BHs is usually described by that one of spinning, non-charged BHs, given by the Kerr metric (Kerr, 1963;Boyer and Lindquist, 1967). The radius of the event horizon from the singularity is then given by ...
Thesis
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The extragalactic gamma-ray sky is dominated by blazars, active galactic nuclei (AGN) with a relativistic jet that is closely aligned with the line of sight. Galaxies develop an active nucleus if the central supermassive black hole (BH) accretes large amounts of ambient matter and magnetic flux. The inflowing mass accumulates around the plane perpendicular to the accretion flow's angular momentum. The flow is heated through viscous friction and part of the released energy is radiated as blackbody or non-thermal radiation, with luminosities that can dominate the accumulated stellar luminosity of the host galaxy. A fraction of the accretion flow luminosity is reprocessed in a surrounding field of ionised gas clouds. These clouds, revolving around the central BH, emit Doppler-broadened atomic emission lines. The region where these broad-line-emitting clouds are located is called broad-line region (BLR). About one in ten AGN forms an outflow of radiation and relativistic particles, called a relativistic jet. According to the Blandford-Znajek mechanism, this is facilitated through electromagnetic processes in the magnetosphere of a spinning BH. The latter induces a magnetospheric poloidal current circuit, generating a decelerating torque on the BH and inducing a toroidal magnetic field. Consequently, rotational energy of the BH is converted to Poynting flux streaming away mainly along the rotational axis and starting the jet. One possibility for particle acceleration near the jet base is realised by magnetospheric vacuum gaps, regions temporarily devoid of plasma, such that an intermittent electric field arises parallel to the magnetic field lines, enabling particle acceleration and contributing to the mass loading of the jets. Magnetised structures, containing bunches of relativistic electrons, propagate away from the galactic nucleus along the jets. Assuming that these electrons emit synchrotron radiation and that they inverse-Compton (IC) up-scatter abundant target photons, which can either be the synchrotron photons themselves or photons from external emitters, the emitted spectrum can be theoretically determined. Additionally taking into account that these emission regions move relativistically themselves and that the emission is Doppler-boosted and beamed in forward direction, the typical two-hump spectral energy distribution (SED) of blazars is recovered. There are however findings that challenge this well-established model. Short-time variability, reaching down to minute scales at very high energy gamma rays, is today known to be a widespread phenomenon of blazars, calling for very compact emission regions. In most models of such optically thick emission regions, the gamma-ray flux is usually pair-absorbed exponentially, without considering the cascade evolving from the pair-produced electrons. From the observed flux, it is often concluded that emission emanates from larger distances where the region is optically thin, especially from outside of the BLR. Only in few blazars gamma-ray attenuation associated with pair absorption in the BLR was clearly reported. With the advent of sophisticated high-energy or very high energy gamma-ray detectors, like the Fermi Large Area Telescope or the Major Atmospheric Gamma-ray Imaging Cherenkov telescopes, besides the extraordinarily fast variability spectral features have been found that cannot be explained by conventional models reproducing the two-hump SED. Two such narrow spectral features are discussed in this work. For the nearby blazar Markarian 501, hints to a sharp peak around 3 TeV have been reported from a multi-wavelength campaign carried out in July 2014, while for 3C 279 a spectral dip was found in 2018 data, that can hardly be described with conventional fitting functions. In this work it is examined whether these spectral peculiarities of blazar jet emission can be explained, if the full radiation reprocessing through an IC pair cascade is accounted for. Such a cascade is the multiple concatenation of IC scattering events and pair production events. In the cascades generally considered in this work, relativistic electrons and high-energy photons are injected into a fixed soft target photon field. A mathematical description for linear IC pair cascades with escape terms is delivered on the basis of preliminary works. The steady-state kinetic equations for the electrons and for the photons are determined, whereby it is paid attention to an explicit formulation and to motivating the correct integration borders of all integrals from kinematic constraints. In determining the potentially observable gamma-ray flux, both the attenuated injected flux and the flux evolving as an effect of IC up-scattering, pair absorption and escape are incorporated, giving the emerging spectra very distinct imprints. Much effort is dedicated to the numerical solution of the electrons' kinetic equation via iterative schemes. It is explained why pointwise iteration from higher to lower Lorentz factors is more efficient than iterating the whole set of sampling points. The algorithm is parallelised at two positions. First, several workers can perform pointwise iterations simultaneously. Second, the most demanding integral is cut into a number of part integrals which can be determined by multiple workers. Through these measures, the Python code can be readily applied to simulate steady-state IC pair cascades with escape. In the case of Markarian 501 the developed framework is as follows. The AGN hosts an advection-dominated accretion flow with a normalised accretion rate of several \(10^{-4}\) and an electron temperature near \(10^{10}\) K. On the one hand, the accretion flow illuminates the few ambient gas clouds with approximate radius \(10^{11}\) m, which reprocess a fraction 0.01 of the luminosity into hydrogen and helium emission lines. On the other hand, the gamma rays from the accretion flow create electrons and positrons in a sporadically active vacuum gap in the BH magnetosphere. In the active gap, a power of roughly 0.001 of the Blandford-Znajek power is extracted from the rotating BH through a gap potential drop of several \(10^{18}\) V, generating ultra-relativistic electrons, which subsequently are multiplied by a factor of about \(10^6\) through interaction with the accretion flow photons. This electron beam propagates away from the central engine and encounters the photon field of one passing ionised cloud. The resulting IC pair cascade is simulated and the evolving gamma-ray spectrum is determined. Just above the absorption troughs due to the hydrogen lines, the spectrum exhibits a narrow bump around 3 TeV. When the cascaded emission is added to the emission generated at larger distances, the observed multi-wavelength SED including the sharp peak at 3 TeV is reproduced, underlining that radiation processes beyond conventional models are motivated by distinct spectral features. The dip in the spectrum of 3C 279 is addressed by a similar cascade model. Three types of injection are considered, varying in the ratio of the photon density to the electron density and varying in the spectral shape. The IC pair cascade is assumed to happen either in the dense BLR photon field with a luminosity of several \(10^{37}\) W and a radial size of few \(10^{14}\) m or in the diluted photon field outside of the BLR. The latter scenario is however rejected as the spectral slope around several 100 MeV and the dip at few 10 GeV cannot be reconciled within this model. The radiation cascaded in the BLR can explain the observational data, irrespective of the assumed injected rate. It is therefore concluded that for this period of gamma-ray emission, the radiation production happens at the edge of the BLR of 3C 279. Both investigations show that IC pair cascades can account for fine structure seen in blazar SEDs. It is insufficient to restrict the radiation transport to pure exponential absorption of an injection term. Pair production and IC up-scattering by all generations of photons and electrons in the optically thick regime critically shape the emerging spectra. As the advent of future improved detectors will provide more high-precision spectra, further observations of narrow spectral features can be expected. It seems therefore recommendable to incorporate cascading into conventional radiation production models or to extend the model developed in this work by synchrotron radiation.
... The initial equilibrium torus state is prescribed in the Boyer-Lindquist coordinates (Boyer & Lindquist 1967) in the original solutions and they are transformed into Kerr-Schild (KS) coordinates in the code (see Weinberg 1972 andVisser 2007 , and the azimuthal angle f remains the same, f = x [3] . Here the parameter h can be adjusted to concentrate the numerical resolution near to the midplane and we use a value of 0.5 for it in our models. ...
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We investigate the dependence of the gamma-ray burst (GRB) jet structure and its evolution on the properties of the accreting torus in the central engine. Our models numerically evolve the accretion disk around a Kerr black hole using three-dimensional general relativistic magnetohydrodynamic simulations. We use two different analytical hydrodynamical models of the accretion disk, based on the Fishbone–Moncrief and Chakrabarti solutions, as our initial states for the structure of the collapsar disk and the remnant after a binary neutron star (BNS) merger, respectively. We impose poloidal magnetic fields of two different geometries upon the initial stable solutions. We study the formation and evolution of the magnetically arrested disk state and its effect on the properties of the emitted jet. The jets produced in our models are structured and have a relatively hollow core and reach higher Lorentz factors at an angle ≳9° from the axis. The jet in our short GRB model has an opening angle of up to ∼25° while our long GRB engine produces a narrower jet, of up to ∼11°. We also study the time variability of the jets and provide an estimate of the minimum variability timescale in our models. The application of our models to the GRB jets in the BNS postmerger system and to the ultrarelativistic jets launched from collapsing stars are briefly discussed.
... We add a radial coordinate r and a polar angular coordinate, θ chosen so that the event horizon is at r = rH. This leads to the definition of Boyer & Lindquist (1967) ...
Preprint
We interpret the 1.3mm VLBI observations made by the Event Horizon Telescope of the black hole in M87. It is proposed that, instead of being a torus of accreting gas, the ring is a rotating, magnetically-dominated ergomagnetosphere that can transmit electromagnetic angular momentum and energy outward to the disc through a combination of large scale magnetic torque and small scale instabilities. It is further proposed that energy can be extracted by magnetic flux threading the ergosphere through the efficient emission of long wavelength electromagnetic disturbances onto negative energy orbits, when the invariant $B^2-E^2$ becomes negative. In this way, the spinning black hole and its ergosphere not only power the jets but also the ejection disc so as to drive away most of the gas supplied near the Bondi radius. This outflow takes the form of a MHD wind, extending over many decades of radius, with a unidirectional magnetic field, that is collimated by the infalling gas across a magnetopause. This wind, in turn, collimates the relativistic jets and the emission observed from the jet sheath may be associated with a return current. A model for the global flow of mass, angular momentum, energy and current, on scales from the horizon to the Bondi radius, is presented and discussed.
... Around a rotating, non-charged BH, the background geometry is described by the Kerr metric (Bardeen 1970). In the Boyer-Lindquist coordinates (Boyer & Lindquist 1967), the line element can be expressed as ds 2 = g tt dt 2 +2g tϕ dtdϕ+g ϕϕ dϕ 2 +g rr dr 2 +g θθ dθ 2 , (1) where ...
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We examine the temporary evolution of axisymmetric magnetospheres around rapidly rotating black holes (BHs), by applying our two-dimensional particle-in-cell simulation code. Assuming a stellar-mass BH, we find that the created pairs fail to screen the electric field along the magnetic field, provided that the mass accretion rate is much small compared to the Eddington limit. Magnetic islands are created by reconnection near the equator and migrate toward the event horizon, expelling magnetic flux tubes from the BH vicinity during a large fraction of time. When the magnetic islands stick to the horizon due to redshift and virtually vanish, a strong magnetic field penetrates the horizon, enabling efficient extraction of energy from the BH. During this flaring phase, a BH gap appears around the inner light surface with a strong meridional return current toward the equator within the ergosphere. If the mass accretion rate is 0.025 percent of the Eddington limit, the BH's spin-down luminosity becomes 16-19 times greater than its analytical estimate during the flares, although its long-term average is only 6 percent of it. We demonstrate that the extracted energy flux concentrates along the magnetic field lines threading the horizon in the middle latitudes. It is implied that this meridional concentration of the Poynting flux may result in the formation of limb-brightened jets from low-accreting BH systems.
... The metric of a Schwarzschild black hole spacetime [69] takes the form of g ab = − 2l (a n b) + 2m (amb) , (1.1) where (l a , n a , m a ,m a ) is a Hawking-Hartle null tetrad [40] and reads in the Boyer-Lindquist coordinates (t, r , θ, φ) [18] l a = 1 Here, the function μ = μ(r , M) = r −2 with = (r , M) = r 2 − 2Mr and M is the mass of the black hole. The larger root r = 2M of the function is the location of the event horizon H, and we define the domain of outer communication (DOC), denoted as D, of a Schwarzschild black hole spacetime to be the closure of {(t, r , θ, φ) ∈ R × (2M, ∞) × S 2 } in the Kruskal maximal extension. ...
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In this work, we derive the globally precise late-time asymptotics for the spin-\({\mathfrak {s}}\) fields on a Schwarzschild background, including the scalar field \(({\mathfrak {s}}=0)\), the Maxwell field \(({\mathfrak {s}}=\pm 1)\) and the linearized gravity \(({\mathfrak {s}}=\pm 2)\). The conjectured Price’s law in the physics literature which predicts the sharp rates of decay of the spin \(s=\pm {\mathfrak {s}}\) components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin \(+1, +2\) components have an extra power of decay at the event horizon than the conjectured Price’s law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.
... In both aforementioned works the properties of the metric were formulated in terms of the proper distance. Meanwhile, much more usual and convenient coordinate systems for rotating axially symmetric space-times represent natural generalization of the Boyer-Lindquist coordinates which were introduced for the Kerr metric [5]. Actually, the relevant metric components are characterized by two integers p and q that describe the rate with which these components approach zero (see below). ...
Preprint
We consider the metric of an axially symmetric rotating black hole. We do not specify the concrete form of a metric and rely on its behavior near the horizon only. Typically, it is characterized (in the coordinates that generalize the Boyer-Lindquist ones) by two integers $p$ and $q$ that enter asymptotic expansions of the time and radial metric coefficients in the main approximation. For given $p,$ $q$ we find a general form for which the metric is regular, and how the expansions of the metric coefficients look like. We compare two types of requirement: (i) boundedness of curvature invariants, (ii) boundedness of separate components of the curvature tensor in a free falling frame. Analysis is done for nonextremal, extremal and ultraextremal horizons separately.
... The Kerr metric of a rotating black hole [16] in the Boyer-Lindquist coordinates [17] has the form ...
Preprint
It has been shown that temperatures near the horizon of rotating black holes can be about the phase transition temperature in the Standard Model with the Higgs boson. The distance from the horizon and gravitational and electromagnetic radiation emitted in collisions between particles have been numerically estimated.
... In four dimensions, a rotating uncharged axially-symmetric black hole with a quasispherical event horizon can be described by the so-called Kerr solution [35]. In Boyer-Lindsquit coordinates, the Kerr metric can be written as follows [36]: ...
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The lightlike limit of boosted black hole solutions with one angular momentum is considered for $D \geq4$ dimensions. The boost is performed parallel to the angular momentum and the lightlike limit is done by means of perturbative expansions. We shown that for $D=4$ and $D> 5$ the lightlike limit cannot be extended inside the ring singularity. Then, for $D = 5$ we discuss the arising of trapped surfaces in the head-on collision. We find that, inside the validity of the perturbative analysis we do, a trapped surface with topology $\mathbb R \times \mathbb S_1 \times\mathbb S_1$ seems to appear over the past light cone of the collision below a critical value $a_c$ of the Kerr parameter $a$.
... One can use the Kerr-Newman metric (Eqs. (1), (2), (3) and (4)) to write [23] ...
Article
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Classical models of the electron have been predicted to have negative rest energy density in certain regions. Using the model of the electron by Blinder we show that there are regions containing negative energy density, although the integral of the energy density over all space gives the electron rest mass. If the spin of the electron is ignored, then all regions of space have positive energy density with the Blinder model. The existence of Poincaré stress for the Blinder model is also demonstrated. The classical model for the electron discussed here admittedly does not involve quantum electrodynamics, where the infinite self energy is made finite with renormalization methods.
... The bookkeeping parameters α ′ and χ ′ denote the expansion orders. In Boyer-Lindquist coordinate [89], (t,r,θ,φ), g (2,1) µν , g (2,2) µν , ϑ (1,1) and ϑ (1,2) are undetermined functions ofr andθ. The zero-order approximation g (K) µν is the Kerr metric [90]. ...
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We calculate the gravitational waveform radiated from spinning black holes (BHs) binary in dynamical Chern-Simons (dCS) gravity. The equation of motion (EOM) of the spinining binary BHs is derived based on the modified Mathisson-Papapetrou-Dixon equation for the spin-aligned circular orbits. The leading-order effects induced by the dCS theory contains spin-spin interaction and monopole-quadrupole interaction, which influence both the EOM of the binary system and corresponding gravitational waveform at the second post-Newtonian (PN) order (i.e., 2PN order). After reporting the waveforms, we investigate the polarization modes of gravitational waves (GWs) in dCS theory. None of the extra modes appears in this theory up to the considered PN order. Moreover, since the time scale of the binary merger is much smaller than that of the cosmological expansion, the parity-violating effect of the dCS theory does not appear in the process of GW generation. However, during the process of GW propagation, amplitude birefringence, a typical parity-violating effect, makes plus and cross modes convert to each other, which modifies the gravitational waveform at 1.5PN order.
... Now, the leading-order formula for the scattering angle in the Kerr metric is [56] θ ...
Preprint
The massless (or ultrarelativistic) limit of a Schwarzschild black hole with fixed energy was determined long ago in the form of the Aichelburg-Sexl shockwave, but the status of the same limit for a Kerr black hole is less clear. In this paper, we explore the ultrarelativistic limit of Kerr in the class of Kerr-Schild impulsive pp-waves by exploiting a relation between the metric profile and the eikonal phase associated with scattering between a scalar and the source of the metric. This gives a map between candidate metrics and tree-level, 4-point scattering amplitudes. At large distances from the source, we find that all candidates for the massless limit of Kerr in this class do not have spin effects. This includes the metric corresponding to the massless limit of the amplitude for gravitational scattering between a scalar and a massive particle of infinite spin. One metric, discovered by Balasin and Nachbagauer, does have spin and finite size effects at short distances, leading to a remarkably compact scattering amplitude with many interesting properties. We also discuss the classical single copy of the ultrarelativistic limit of Kerr in electromagnetism.
Article
We study the relativistic, inviscid, advective accretion flow around the black holes and investigate a key feature of the accretion flow, namely the shock waves. We observe that the shock-induced accretion solutions are prevalent and such solutions are commonly obtained for a wide range of the flow parameters, such as energy (${\cal E}$) and angular momentum (λ), around the black holes of spin value 0 ≤ ak < 1. When the shock is dissipative in nature, a part of the accretion energy is released through the upper and lower surfaces of the disc at the location of the shock transition. We find that the maximum accretion energies that can be extracted at the dissipative shock ($\Delta {\cal E}^{\rm max}$) are $\sim 1\%$ and $\sim 4.4\%$ for Schwarzschild black holes (ak → 0) and Kerr black holes (ak → 1), respectively. Using $\Delta {\cal E}^{\rm max}$, we compute the loss of kinetic power (equivalently shock luminosity, Lshock) that is enabled to comply with the energy budget for generating jets/outflows from the jet base (i.e., post-shock flow). We compare Lshock with the observed core radio luminosity (LR) of black hole sources for a wide mass range spanning 10 orders of magnitude with sub-Eddington accretion rate and perceive that the present formalism seems to be potentially viable to account LR of 16 Galactic black hole X-ray binaries (BH-XRBs) and 2176 active galactic nuclei (AGNs). We further aim to address the core radio luminosity of intermediate-mass black hole (IMBH) sources and indicate that the present model formalism perhaps adequate to explain core radio emission of IMBH sources in the sub-Eddington accretion limit.
Article
As a consequence of Birkhoff's theorem, the exterior gravitational field of a spherically symmetric star or black hole is always given by the Schwarzschild metric. In contrast, the exterior gravitational field of a rotating (axisymmetric) star differs, in general, from the Kerr metric, which describes a stationary, rotating black hole. In this paper I discuss the possibility of a quasi–stationary transition from rotating equilibrium configurations of normal matter to rotating bla ck holes.
Article
The gravitational lensing of relativistic neutral massive particles caused by a Kerr-Newman black hole is investigated systematically in the weak-field limit. Based on the Kerr-Newman metric in Boyer-Lindquist coordinates, we first derive the analytical form of the equatorial gravitational deflection angle of a massive particle in the third post-Minkowskian approximation. The resulting bending angle, which is found to be consistent with the result in the previous work, is adopted to solve the popular Virbhadra-Ellis lens equation. The analytical expressions for the main observable properties of the primary and secondary images of the particle source are thus obtained beyond the weak-deflection limit, within the framework of standard perturbation theory. The observables include the positions, magnifications, and gravitational time delays of the individual images, the differential time delay, and the total magnification and centroid position. The explicit forms of the correctional effects induced by the deviation of the initial velocity of the massive particle from the speed of light on the observables of the lensed images are then achieved. Finally, serving as an application of the formalism, the supermassive black hole at the Galactic Center, Sagittarius A*, is modeled to be a Kerr-Newman lens. The magnitudes of the velocity-induced correctional effects on the practical lensing observables as well as the possibilities to detect them in this scenario are also analyzed.
Article
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly confluent, and triconfluent. All the series are illustrated with concrete examples of use, and Python implementations are available for download. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.
Chapter
Die Schwarzschild-Metrik beschreibt eine statische sphärisch-symmetrischen Massenverteilung ausgegangen. Wenn aber ein rotierender Stern zu einem Schwarzen Loch kollabiert, so bleibt der Drehimpuls erhalten. Schwarze Löcher, die allein durch ihre Masse und ihren Drehimpuls definiert sind, werden durch die Kerr-Metrik beschrieben, deren Eigenschaften deutlich komplizierter sind, als die der Schwarzschild-Metrik.
Article
Extreme mass ratio inspirals (EMRIs)—systems with a compact object orbiting a much more massive (e.g., Galactic Center) black hole—are of interest as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and results in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar ψ4 at the horizon and at ∞) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.
Article
We interpret the 1.3mm VLBI observations made by the Event Horizon Telescope of the black hole in M87. It is proposed that, instead of being a torus of accreting gas, the observed annular ring is a rotating, magnetically-dominated ergomagnetosphere that can transmit electromagnetic angular momentum and energy outward to the disc through a combination of large scale magnetic torque and small scale instabilities. It is further proposed that energy can be extracted by magnetic flux threading the ergosphere through the efficient emission of long wavelength electromagnetic disturbances on to negative energy orbits, when the invariant B2 − E2 becomes negative. In this way, the spinning black hole and its ergosphere not only power the jets but also the ejection disc so as to drive away most of the gas supplied near the Bondi radius. This outflow takes the form of a MHD wind, extending over many decades of radius, with a unidirectional magnetic field, that is collimated by the infalling gas across a magnetopause. This wind, in turn, collimates the relativistic jets and the emission observed from the jet sheath may be associated with a return current. A model for the global flow of mass, angular momentum, energy and current, on scales from the horizon to the Bondi radius, is presented and discussed.
Article
Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole’s spacetime. Accurate models of such systems require postgeodesic corrections to this motion. Postgeodesic effects that drive the small body away from the geodesic include the gravitational self-force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body’s spin to the black hole’s spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body’s spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics (in a sense that can be made precise) and using closed-form results due to van de Meent describing precession of the small body’s spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit’s motion is O(S) out of the equatorial plane), but for which the small body’s spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits, contrasting them with geodesics, and show how the small body’s spin shifts the frequencies Ωr and Ωϕ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations.
Article
We wish to carry forward to higher dimensions the insightful and novel method of obtaining the Kerr metric proposed by one of us [Gen. Relativ. Gravit. 45, 2383 (2013)] for deriving the Myers-Perry rotating black hole metric. We begin with a flat spacetime metric written in oblate spheroidal coordinates (ellipsoidal geometry) appropriate for the inclusion of rotation, and then introduce arbitrary functions to introduce a gravitational potential due to mass, which are then determined by requiring that a massless particle experiences no acceleration, while a massive particle feels Newtonian acceleration at large $r$. We further generalize the method to include the cosmological constant $\Lambda$ to obtain the Myers–Perry– de Sitter/anti–de Sitter black hole metric.
Article
In this paper we discuss the gravitational field of ultrarelativistic extended spinning objects. For this purpose, we use a solution of the linearized gravitational equations obtained in the frame where such an object is translationally at rest, and boost this solution close to the speed of light. In order to obtain a regular limiting metric for nonspinning matter, it is sufficient to keep the energy of the boosted body fixed. This process is known as the Penrose limit. We demonstrate that in the presence of rotation, an additional rescaling is required for the angular momentum density components in the directions orthogonal to the boost. As a result of the Lorentz contraction, the thickness of the body in the direction of the boost shrinks. The body takes the form of a pancake, and its gravitational field is localized in the null plane. We discuss light and particle scattering in this gravitational field, and calculate the scattering parameters associated with the gravitational memory effect. We also show that by taking the inverse of the Penrose transform, one can use the obtained scattering map to study the gravitational lensing effect in the rest frame of a massive spinning object.
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It has been shown that temperatures near the horizon of rotating black holes can be about the phase transition temperature in the Standard Model with the Higgs boson. The distance from the horizon and gravitational and electromagnetic radiation emitted in collisions between particles have been numerically estimated.
Book
Persamaan yang mengikuti skema medan gravitasi adalah relativitas umum. Gaya antara partikel elementer dan cabang-cabangnya yang terkait, terutama tingkat energi yang tinggi, memerlukan teori ini sebagai prototipe dalam menjelaskan konstruksi dan persamaan yang kompleks, sambil memberikan paradigma baru, memisahkan sejelas mungkin variabel-variabel yang bersama-sama membentuk alasan yang penting secara fundamental dari Relativitas umum, dan lubang hitam, serta aplikasinya. Dalam buku ini, pertama-tama, kami menjelaskan secara singkat relativitas khusus, dan tahapan perkembangannya yang menyebabkan munculnya teori ini, dan bagaimana teori itu muncul. Dalam buku ini juga dijelaskan kurva koordinat dalam ruang-waktu, yang disebabkan oleh gravitasi, medan metrik juga ditambahkan untuk menjelaskan materi ini, dan menambah hubungan afin, sehingga dari sini dapat dengan jelas melihat struktur ruang-waktu itu sendiri, inilah dasar pembentukan persamaan Einstein. Topik penerapan relativitas umum berkaitan dengan kalkulus tensor dan fisika modern, di mana dapat menganalisis wawasan bidang akademik dan penelitian, khususnya relativitas umum. Buku ini juga memiliki keunggulan yang biasanya sebagian besar buku-buku terkait tidak didedikasikan untuk pembenaran, atau presentasi, validasi formalisme, dan interpretasi teori. Buku ini sejalan dengan tujuan yang telah ditetapkan sesuai dengan judul buku ini, yaitu “Pengantar teori relativitas umum dan lubang hitam” Adapun dalam buku ini berisi atas 21 bab yang menjelaskan secara singkat dan jelas, dan lebih jarang dari penjelasan buku lain yang serupa, dalam menjelaskan teori relativitas, dan lubang hitam. Buku ini menjelaskan materi sesingkat dan sejelas-jelasnya dengan penurunan rumus yang seperlunya, materi terkhusus relativitas umum dan lubang hitam ini dapat mudah dijangkau oleh para pembaca. Dalam membaca buku ini, diharapkan juga para pembaca memiliki dasar matematis yang cukup kuat, karena sebagian besar dalam buku ini menjelaskan materi dan penurunan persamaan yang eksakta dalam menjelaskan aplikasi dan pemodelamn dari persamaan, termasuk itu lubang hitam, kami para penulis berharap, para pembaca yang non fisika tidak kecewa dengan membaca buku ini, karena sesuai dengan judulnya adalah “Pengantar teori relativitas dan lubang hitam”.
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We investigate the quasibound states of charged massive scalar fields in the Kerr–Newman black hole spacetime by using a new approach recently developed, which uses the polynomial conditions of the Heun functions. We calculate the resonant frequencies related to the spectrum of quasibound states, as well as its corresponding angular and radial wave eigenfunctions. We also analyze the instability of the system. These results are particularized to the cases of Schwarzschild and Kerr black holes. Additionally, we compare our analytical results with the numerical ones known in the literature. Finally, we apply the obtained results to compute the characteristic times of growth and decay of bosonic particles around a supermassive black hole situated at the center of the M87 galaxy.
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Starting from the dynamics of a scalar field on a Kerr metric, we propose an analytical extension of spacetime coordinates (with Boyer–Lindquist coordinates) for a closed system with no flow at the boundary, and we apply the polynomial conditions for confluent Heun functions with the aim to find the distribution of the energy levels for the scalar field that describes the spacetime inside the ergosphere of a Kerr black hole. We obtain discretizations of the mass and angular momentum that can be motivated from the expression of each energy level.
Article
We present the global solutions of low angular momentum, inviscid, advective accretion flow around Kerr-Taub-NUT (KTN) black hole in presence and absence of shock waves. These solutions are obtained by solving the governing equations that describe the relativistic accretion flow in KTN spacetime which is characterized by the Kerr parameter ( a k ) and NUT parameter ( n ). During accretion, rotating flow experiences centrifugal barrier that eventually triggers the discontinuous shock transition provided the relativistic shock conditions are satisfied. In reality, the viability of shocked accretion solution appears more generic over the shock free solution as the former possesses high entropy content at the inner edge of the disc. Due to shock compression, the post-shock flow (equivalently post-shock corona, hereafter PSC) becomes hot and dense, and therefore, can produce high energy radiations after reprocessing the soft photons from the pre-shock flow via inverse Comptonization. In general, PSC is characterized by the shock properties, namely shock location ( r s ), compression ratio ( R ) and shock strength ( S ), and we examine their dependencies on the energy ( ξ ) and angular momentum (λ) of the flow as well as black hole parameters. We identify the effective domain of the parameter space in λ- ξ plane for shock and observe that shock continues to form for wide range of flow parameters. We also find that a k and n act oppositely in determining the shock properties and shock parameter space. Finally, we calculate the disc luminosity ( L ) considering free-free emissions and observe that accretion flows containing shocks are more luminous compared to the shock free solutions.
Article
In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Nonspinning test bodies follow geodesics of the spacetime in which they move. A test body’s spin couples to the curvature of that spacetime, introducing a “spin-curvature force” which pushes the body’s worldline away from a geodesic trajectory. The spin-curvature force is an important example of a postgeodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper’s predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit’s motion is no more than O(S) out of the equatorial plane). In this paper, we apply this formulation to the fully generic case—orbits which are inclined and eccentric, with the small body’s spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body’s spin affects important quantities such as the observable orbital frequencies Ωr, Ωθ and Ωϕ.
Article
In this work, we construct and study the Carter-Penrose diagram for sonic black hole and white hole analogs as manifested in the analog spacetime embedded inside the flow of hydrodynamic inviscid matter onto astrophysical rotating black holes. For general relativistic black hole accretion in the Kerr metric, we show that linear perturbation of the axially symmetric matter flow having certain geometrical configurations leads to the emergence of black-hole-like acoustic spacetime. Such an analog spacetime is shown to be endowed with one white-hole-like sonic horizon flanked by two black-hole-like acoustic horizons. We construct the compactified causal structures, i.e., the Carter Penrose diagrams for such emergent spacetime to study the corresponding horizon effects. For the first time in literature, the Carter-Penrose formalism is carried out for analog spacetime embedded within a natural large scale fluid flow under the influence of strong gravity.
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We present a novel approach to study the global structure of steady, axisymmetric, advective, magnetohydrodynamic (MHD) accretion flow around black holes in full general relativity (GR). Considering ideal MHD conditions and relativistic equation of state (REoS), we solve the governing equations to obtain all possible smooth global accretion solutions. We examine the dynamical and thermodynamical properties of accreting matter in terms of the flow parameters, namely energy (${\cal E}$), angular momentum (${\cal L}$), and local magnetic fields. For a vertically integrated GRMHD flow, we observe that toroidal component (bφ) of the magnetic fields generally dominates over radial component (br) at the disk equatorial plane. This evidently suggests that toroidal magnetic field indeed plays important role in regulating the disk dynamics. We further notice that the disk remains mostly gas pressure (pgas) dominated (β = pgas/pmag > 1, pmag refers magnetic pressure) except at the near horizon region, where magnetic fields become indispensable (β ∼ 1). We observe that Maxwell stress is developed that eventually yields angular momentum transport inside the disk. Towards this, we calculate the viscosity parameter (α) that appears to be radially varying. In addition, we examine the underlying scaling relation between α and β, which clearly distinguishes two domains coexisted along the radial extent of the disk. Finally, we discuss the utility of the present formalism in the realm of GRMHD simulation studies.
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We consider the asymptotics of a Maxwell field in Schwarzschild and Kerr spacetimes. In a Kerr spacetime, we utilize the basic energy and Morawetz estimates proven in earlier work to derive pointwise estimates where the total power of decay rate for all components of the Maxwell field towards a stationary solution is −7/2. Moreover, on a Schwarzschild background, if the constructed Newman–Penrose constant vanishes (or does not vanish), we prove almost sharp Price's law decay v−2−sτ−3+s+ (or v−2−sτ−2+s+) for Maxwell field and v−2−sτ−2+s−ℓ0+ (or v−2−sτ−1+s−ℓ0+) for the ℓ≥ℓ0 modes of the field towards a static solution, with s being the spin weight of the Newman–Penrose components of Maxwell field. All estimates are uniform in the exterior of the black hole.
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The approximate solution of the Klein-Gordon equation for a real scalar field of mass μ in the geometry of a Kerr black hole obtained by Detweiler [Detweiler, Phys. Rev. D 22, 2323 (1980)] is widely used in the analysis of the stability of black holes as well as the search of axionlike particles. In this work, we confirm a a missing factor 1/2 in this solution, which was first identified in [Pani et al., Phys. Rev. D 86, 104017 (2012)]. The corrected result has strange features that put questions on the power-counting strategy. We solve this problem by adding the next-to-leading order (NLO) contribution. Compared to the numerical results, the NLO solution reduces the percentage error of the LO solution by a factor of 2 for all important values of rgμ. Especially the percentage error is ≲10% in the region of rgμ≲0.35. The NLO solution also has a compact form and could be used straightforwardly.
Article
Gravitational Faraday Rotation (GFR) is a frame-dragging effect induced by rotating massive objects, which is one of the important, yet studied characteristics of lensed gravitational waves (GWs). In this work, we calculate the GFR angle χ g of GWs in the weak deflection limit, assuming it is lensed by a Kerr black hole (BH). We find that the GFR effect changes the initial polarization state of the lensed GW. Compared with the Einstein deflection angle, the dominant term of the rotation angle χ g is a second-order correction to the polarization angle, which depends on the light-of-sight component of BH angular momentum. Such a rotation is tiny and degenerates with the initial polarization angle. In some critical cases, the GFR angle is close to the detection capability of the third-generation GW detector network, although the degeneracy has to be broken.
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Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);
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The ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time-like hypersurfaces r = const which are left invariant by the group of motions are topologically three-spheres S3, in contrast to the topology S2 × R (or S2 × S1) for the r = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface t = const and an r = const surface is a sphere S2. If σ is any spacelike hypersurface in the generalized metric, then its (two-dimensional) intersection with an r = const surface is not any closed two-dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as r ↠ ∞, in fact Rμναβ = O(1∕r3) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which gμν − &eegr;μν = O(1∕r). An apparent singularity in the metric at small values of r, which appears to be similar to the spurious Schwarzschild singularity at r = 2m, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter.
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An analysis is given of the stress-energy tensor and geometry produced by slowly rotating bodies. The geometrized mass GM/c2 of the body is allowed to be comparable to its radius. The geometry is treated as a perturbation of the Schwarzschild geometry, which leads to considerable simplification of Einstein's equations. The rotation of the intertial frame induced by a rotating massive shell is calculated and discussed with particular attention to two limiting cases: (1) For small masses it reduces to Thirring's well-known result; (2) for large masses, whose Schwarzschild radius approaches the shell radius, the induced rotation approaches the rotation of the shell. These and the corresponding results for an expanding and recollapsing dust cloud are examined for their consistency with particular interpretations of Mach's principle. The analytic extension of the rotating exterior metric is a completely source-free rotating solution. It describes a slowly rotating, expanding, and recontracting Einstein-Rosen bridge which can be taken as a geometrodynamic model for a slowly rotating body.
Article
DOI:https://doi.org/10.1103/PhysRevLett.15.689
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The 2-dimensional metric on the symmetry axis of the Kerr solution is examined and it is shown that in the form usually given it is incomplete when ${a}^{2}<~{m}^{2}$. The method developed by Kruskal for completing the Schwarzschild solution is adapted to the distinct cases ${a}^{2}<{m}^{2}$ and ${a}^{2}={m}^{2}$. In each case a singularity-free metric is obtained which is periodic with respect to a timelike coordinate, and which is shown to be a complete analytic extension. The generalization to the full 4-dimensional Kerr solution is discussed, and finally the questions of uniqueness and causality are considered.
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The writers investigate the possibility of an atomistic theory of matter and electricity which, while excluding singularities of the field, makes use of no other variables than the gmunu of the general relativity theory and the varphimu of the Maxwell theory. By the consideration of a simple example they are led to modify slightly the gravitational equations which then admit regular solutions for the static spherically symmetric case. These solutions involve the mathematical representation of physical space by a space of two identical sheets, a particle being represented by a "bridge" connecting these sheets. One is able to understand why no neutral particles of negative mass are to be found. The combined system of gravitational and electromagnetic equations are treated similarly and lead to a similar interpretation. The most natural elementary charged particle is found to be one of zero mass. The many-particle system is expected to be represented by a regular solution of the field equations corresponding to a space of two identical sheets joined by many bridges. In this case, because of the absence of singularities, the field equations determine both the field and the motion of the particles. The many-particle problem, which would decide the value of the theory, has not yet been treated.
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In this paper, gravitational radiation is defined invariantly within the framework of general relativity theory. The definition is arrived at by assuming (a) that gravitational radiation is characterized by the Riemann tensor, and (b) that it is propagated with the fundamental velocity. Therefore a gravitational wave front should appear as a discontinuity in the Riemann tensor across a null 3-surface; the possible form of this discontinuity is here calculated from Lichnerowicz's continuity conditions.The concept of an observer who follows the gravitational field is defined in terms of the eigenbivectors of the Riemann tensor. It is shown that the 4-velocity of this observer is timelike for one of Petrov's three canonical types of Riemann tensor, but null for the other two types. The first type is identified with the absence of radiation, the other two with its presence. This constitutes the definition. It is shown that the difference between the no-radiation type and one of the radiation types can be made to correspond to the discontinuity possible across a null 3-surface; this demonstrates the consistency of the wave front and following-the-field concepts.A covariant approximation to the canonical energy-momentum pseudo-tensor is defined, using normal coordinates, which are given a physical interpretation. It is shown that when gravitational radiation is present, the approximate gravitational energy-flux cannot be removed by a local Lorentz transformation, which supports the definition of radiation.It is proved that, as would be demanded of a sensible definition, there can be no gravitational radiation present in a region of empty space-time where the metric is static.
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There is presented a particularly simple transformation of the Schwarzschild metric into new coordinates, whereby the "spherical singularity" is removed and the maximal singularity-free extension is clearly exhibited.
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A covariant formulation of the outgoing radiation condition for gravitational fields is proposed. The condition is based on a detailed examination of the geometry of null lines and of the algebraic and differential properties of the Riemann tensor. It relates the absence of incoming radiation, in a gravitational field with bounded sources and Euclidean topology, to the asymptotic behaviour of the Riemann tensor. Fields that are algebraically special in the Petrov classification are highly special examples of fields obeying the suggested condition.
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The scattering of plane electromagnetic waves by the gravitational field of an isolated physical system is studied. On the level of the geometrical optics approximation the general theory of light rays is formulated. In particular, the generalized formula for the Einstein deflection of light rays is obtained. On the level of the vectorial optics the problem of polarization is examined in detail. The formula obtained, describing a rotation of the plane of polarization due to the presence of the gravitational field, admits a direct geometrical interpretation. The theory is applied to the rotating body and a system of point masses. The physical results established concerning the asymptotic behavior of the electromagnetic waves are independent of the coordinate system used in the computations.
Article
The general-relativistic equations governing the motion of a large mass under the influence of its own gravitational field and its own pressure have been approximated by finite-difference equations. A spherically symmetric, co-moving frame of reference was used. The pressure was assumed to be zero at the outer boundary. Rest mass was assumed to be conserved and heat transfer by neutrinos, radiation, etc., was not taken into account. Numerical solutions were obtained on a computer for several simplified equations of state, chosen to bracket the behavior of stellar material in late stages of collapse, and several masses. The maximum stable masses obtained were of the same order of magnitude, but somewhat larger than the maximum stable masses calculated statically. The behavior of light signals, of the metric coefficients, and of the hydrodynamic quantities as functions of time is described for collapse past the Schwarzchild radius. Such collapse leads to regions where the surface area of concentric spheres decreases as the rest mass contained by the spheres increases.
Article
In what follows we shall derive some properties of the gravitational field of an isolated, axially symmetric, uniformly rotating mass of perfect fluid in a steady state, according to the general theory of relativity. Several exact models describing rotating fluids are known in Newtonian mechanics, the Maclaurin and Jacobi ellipsoids ((6)) being perhaps the most interesting. In general relativity, no such exact solution is known in its entirety, although Kerr ((4)) has exhibited a certain vacuum solution possessing features that one might expect of a space-time exterior to some rotating body. Throughout this paper we shall have Kerr's solution in mind. The question that we shall keep before us is whether a perfect fluid interior can be matched to any given exterior field. Our main results exhibit the class of all possible fluid boundaries, given the exterior field, and some relations between the pressure, density, 4-velocity, and interior metric tensor.(Received August 12 1964)
Article
The preceding paper ((1)) dealt with some general properties of the gravitational field of a rotating fluid mass. An interesting example of a vacuum solution that might be the exterior field of some rotating body was recently found by Kerr ((4)). It was natural to apply the preceding theory to the Kerr solution. This paper deals with other aspects of that solution, particularly the behaviour of its bounded geodesics (planetary orbits). It would seem desirable to know what sort of rotating body could be a source of the Kerr field. It will appear that one of the parameters in Kerr's solution can plausibly be related to the angular momentum per unit mass of a uniformly rotating sphere, the other parameter being a measure of the mass of the sphere.(Received August 12 1964)
Article
A method originally developed for the Kerr solution is adapted in order to obtain the complete analytic continuation of the Reissner-Nordström metric in the special case e2 = m2. This case is the exterior field of a static spherical charged dust cloud.
The collapse of a spherically symmetric mass has been studied in detail by They find that unless the collapse is halted before any fraction of the total mass has fallen within its own Schwarzschild radius, a singularity invariably ensues
  • M M May
  • R H White