Kerr Source Research Articles

Kerr–Schild form of the exact metric for a constantly moving Kerr black hole and null gravitational deflection

The exact metric of a moving Kerr black hole with an arbitrary constant velocity is derived in Kerr–Schild coordinates. We then calculate the null equatorial gravitational deflection caused by a radially moving Kerr source up to the second post-Minkowskian order, acting as an application of the weak-field limit of the metric. The bending angle of light is found to be consistent with the result given in the previous works.

Critical Tidal Currents in General Relativity

Relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source and their solutions are studied. The existence of certain relativistic critical tidal currents are thereby elucidated. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The general tidal equations for the free motion of test particles are worked out with respect to the Kerr background. The analytic solutions of these equations are investigated and the existence of a tidal acceleration mechanism is emphasized.

Metric for two arbitrary Kerr sources

The full metric describing a stationary axisymmetric system of two arbitrary Kerr sources, black holes or hyperextreme objects, located on the symmetry axis and kept apart in equilibrium by a massless strut is presented in a concise explicit form involving five physical parameters. The binary system composed of a Schwarzschild black hole and a Kerr source is a special limiting case of the general formulas, and we elaborate the metric for this physically interesting configuration too.

Black hole-naked singularity dualism and the repulsion of two Kerr black holes due to spin-spin interaction

We report about the possibility for interacting Kerr sources to exist in two different states – black holes or naked singularities – both states characterized by the same masses and angular momenta. Another surprising discovery reported by us is that in spite of the absence of balance between two Kerr black holes, the latter nevertheless can repel each other, which provides a good opportunity for experimental detection of the spin-spin repulsive force through the observation of astrophysical black-hole binaries.

Unequal binary configurations of interacting Kerr black holes

Stationary axisymmetric binary configurations of unequal Kerr sources with a massless strut among them are developed in a physical representation. In order to describe interacting black holes, the axis conditions in the most general case are solved analytically deriving the corresponding 5-parametric asymptotically flat exact solution. In addition, we obtain concise formulas for the black hole horizons, the interaction force, as well as the thermodynamical characteristics of each source in terms of physical Komar parameters: mass Mi, angular momentum Ji, and coordinate distance R, where such parameters are part of a cubic equation which can be interpreted as a dynamical law for interacting black holes with struts. Some limits are obtained and discussed.

Corotating two-body system of identical Kerr sources

A binary system of equal corotating Kerr sources is studied after deriving the corresponding 3-parametric asymptotically flat exact solution. Both sources are apart from each other by means of a massless strut (conical singularity). In the context of black holes, the analytical functional form of each horizon σ is expressed in terms of arbitrary Komar physical parameters: mass M, angular momentum J (with parallel spin), and the coordinate distance R. Later on, all the thermodynamical properties related to the horizon are depicted by concise formulae. Finally, the extreme limit case is obtained as a 2-parametric subclass of Kinnersley–Chitre metric.

Relativistic tidal acceleration of astrophysical jets

Within the framework of general relativity, we investigate the tidal acceleration of astrophysical jets relative to the central collapsed configuration ("Kerr source"). We neglect electromagnetic forces throughout. The rest frame of the Kerr source is locally defined via the set of hypothetical static observers in the spacetime exterior to the source. Relative to such a fiducial observer fixed on the rotation axis of the Kerr source, jet particles are tidally accelerated to almost the speed of light if their outflow speed is above a certain threshold, given roughly by one half of the Newtonian escape velocity at the location of the reference observer; otherwise, the particles reach a certain height, reverse direction and fall back toward the gravitational source.

Relativistic gravity gradiometry

In general relativity, relativistic gravity gradiometry involves the measurement of the relativistic tidal matrix, which is theoretically obtained from the projection of the Riemann curvature tensor onto the orthonormal tetrad frame of an observer. The observer's 4-velocity vector defines its local temporal axis and its local spatial frame is defined by a set of three orthonormal nonrotating gyro directions. The general tidal matrix for the timelike geodesics of Kerr spacetime has been calculated by Marck [Proc. R. Soc. A 385, 431 (1983)]. We are interested in the measured components of the curvature tensor along the inclined ``circular'' geodesic orbit of a test mass about a slowly rotating astronomical object of mass $M$ and angular momentum $J$. Therefore, we specialize Marck's results to such a ``circular'' orbit that is tilted with respect to the equatorial plane of the Kerr source. To linear order in $J$, we recover the gravitomagnetic beating phenomenon [B. Mashhoon and D. S. Theiss, Phys. Rev. Lett. 49, 1542 (1982)], where the beat frequency is the frequency of geodetic precession. The beat effect shows up as a special long-period gravitomagnetic part of the relativistic tidal matrix; moreover, the effect's short-term manifestations are contained in certain post-Newtonian secular terms. The physical interpretation of this effect is briefly discussed.

Circular motion in NUT space-time

We consider circular motion in the NUT (Newman–Unti–Tamburino) space-time. Among other things, we determine the location of circular time-like geodesic orbits, in particular of the innermost stable circular orbit (ISCO) and of the marginally bound circular orbit. Moreover, we discuss the von Zeipel cylinders with respect to the stationary observers and with respect to the zero angular momentum observers (ZAMOs). We also investigate the relation of von Zeipel cylinders to inertial forces, in particular in the ultra-relativistic limit. Finally, we generalise the construction of thick accretion tori (‘Polish doughnuts’) which are well known on the Schwarzschild or Kerr background to the case of the NUT metric. We argue that, in principle, an NUT source could be distinguished from a Schwarzschild or Kerr source by observing the features of circular matter flows in its neighbourhood.

Metric of two balancing Kerr particles in physical parametrization

The present paper aims at elaborating a completely physical representation for the general 4-parameter family of the extended double-Kerr spacetimes describing two spinning sources in gravitational equilibrium. This involved problem is solved in a concise analytical form by using the individual Komar masses and angular momenta as arbitrary parameters, and the simplest equatorially symmetric specialization of the general expressions obtained by us yields the physical representation for the well-known Dietz-Hoenselaers superextreme case of two balancing identical Kerr constituents. The existence of the physically meaningful "black hole-superextreme object" equilibrium configurations permitted by the general solution may be considered as a clear indication that the spin-spin repulsion force might actually be by far stronger than expected earlier, when only the balance between two superextreme Kerr sources was thought possible. We also present the explicit analytical formulas relating the equilibrium states in the double-Kerr and double-Reissner-Nordstr\"om configurations.

Stationary configurations of two extreme black holes obtainable from the Kinnersley-Chitre solution

Stationary axisymmetric systems of two extreme Kerr sources separated by a massless strut, which arise as subfamilies of the well-known Kinnersley-Chitre solution, are studied. We present explicit analytical formulas for the individual masses and angular momenta of the constituents and establish the range of the parameters for which such systems can be regarded as describing black holes. The mass-angular momentum relations and the interaction force in the black-hole configurations are also analyzed. Furthermore, we construct a charging generalization of the Kinnersley-Chitre metric and, as applications of the general formulas obtained, discuss two special cases describing a pair of identical co- and counter-rotating extreme Kerr-Newman black holes kept apart by a conical singularity. From our analysis it follows, in particular, that the equality ${m}^{2}\ensuremath{-}{a}^{2}\ensuremath{-}{e}^{2}=0$ relating the mass, angular momentum per unit mass and electric charge of a single Kerr-Newman extreme black hole is no longer verified by the analogous extreme black-hole constituents in binary configurations.

Erratum: Remarks on the mass-angular momentum relations for two extreme Kerr sources in equilibrium [Phys. Rev. D82, 124042 (2010)

The general analysis of the relations between masses and angular momenta in the configurations composed of two balancing extremal Kerr particles is made on the basis of two exact solutions arising as extreme limits of the well-known double-Kerr spacetime. We show that the inequality M^2 >= |J| characteristic of an isolated Kerr black hole is verified by all the extremal components of the Tomimatsu and Dietz-Hoenselaers solutions. At the same time, the inequality can be violated by the total masses and total angular momenta of these binary systems, and we identify all the cases when such violation occurs.

On a Simple Representation of the Kinnersley-Chitre Metric

A concise form of the Kinnersley-Chitre five-parameter metric for a spinning mass is obtained by exploiting a remarkable similarity between the metric's factor structure and the analogous structure of the Tomimatsu-Sato solutions with even distortion parameter delta. The corresponding general subfamily of asymptotically flat spacetimes containing four arbitrary real parameters is considered, and all configurations describing two extreme Kerr sources separated by a strut are identified and briefly discussed.

Remarks on the mass-angular momentum relations for two extreme Kerr sources in equilibrium

The general analysis of the relations between masses and angular momenta in the configurations composed of two balancing extremal Kerr particles is made on the basis of two exact solutions arising as extreme limits of the well-known double-Kerr spacetime. We show that the inequality ${M}^{2}\ensuremath{\ge}|J|$ characteristic of an isolated Kerr black hole is verified by all the extremal components of the Tomimatsu and Dietz-Hoenselaers solutions. At the same time, the inequality can be violated by the total masses and total angular momenta of these binary systems, and we identify all the cases when such violation occurs.

Gravitational duality and rotating solutions

We study how gravitational duality acts on rotating solutions, using the Kerr-NUT black hole as an example. After properly reconsidering how to take into account both electric (i.e. mass-like) and magnetic (i.e. NUT-like) sources in the equations of general relativity, we propose a set of definitions for the dual Lorentz charges. We then show that the Kerr-NUT solution has non-trivial such charges. Further, we clarify in which respect Kerr's source can be seen as a mass M with a dipole of NUT charges.

Nonequatorial charged particle confinement around Kerr black holes

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

Complex Kerr geometry, twistors and the Dirac electron

The Kerr spinning particle displays some remarkable relations to the Dirac electron, and has a reach spinor structure which is based on a twistorial description of the Kerr congruence determined by the Kerr theorem. We consider the relation between this spinor-twistorial structure and spinors of the Dirac equation, and show that the Dirac equation may naturally be incorporated into Kerr–Schild formalism as a master equation controlling the twistorial structure of Kerr geometry. As a result, the Dirac electron acquires an extended spacetime structure having a clear coordinate description with natural incorporation of a gravitational field. The relation between the Dirac wave function and Kerr geometry is realized via a chain of links: Dirac wave function ⇒ Complex Kerr Source ⇒ Kerr Theorem ⇒ Real Kerr geometry. As a result, the wave function acquires the role of an ‘order parameter’ which controls spin, dynamics and twistorial polarization of Kerr–Newman spacetime.

Tidal dynamics in Kerr spacetime

The motion of free nearby test particles relative to a stable equatorial circular geodesic orbit about a Kerr source is investigated. It is shown that the nonlinear generalized Jacobi equation can be transformed in this case to an autonomous form. Tidal dynamics beyond the critical speed is studied. We show, in particular, that a free test particle vertically launched from the circular orbit parallel or antiparallel to the Kerr rotation axis is tidally accelerated if its initial relative speed exceeds . Possible applications of our results to high-energy astrophysics are briefly mentioned.

Scattering of high-speed particles in the Kerr gravitational field

We calculate the angles of deflection of high-speed particles projected in an arbitrary direction into the Kerr gravitational field. This is done by first calculating the lightlike boost of the Kerr gravitational field in an arbitrary direction and then using this boosted gravitational field as an approximation to the gravitational field experienced by a high-speed particle. In the rest frame of the Kerr source the angles of deflection experienced by the high-speed test particle can then easily be evaluated.

Structure of spinning particle suggested by gravity, supergravity and low-energy string theory

The structure of spinning particle suggested by the rotating Kerr-Newman (black hole) solution, super-Kerr-Newman solution and the Kerr-Sen solution to low energy string theory is considered. Main peculiarities of the Kerr spinning particle are discussed: a vortex of twisting principal null congruence, singular ring and the Kerr source representing a rotating relativistic disk of the Compton size. A few stringy structures can be found in the real and complex Kerr geometry. Low-energy string theory predicts the existence of a heterotic string placed on the sharp boundary of this disk. The obtained recently supergeneralization of the Kerr-Newman solution suggests the existence of extra axial singular line and fermionic traveling waves concentrating near these singularities.We discuss briefly a possibility of experimental test of these predictions.