Rotational Dragging Research Articles

Rotation of inertial frames by angular momentum of matter and waves

We elucidate the dynamics of a thin spherical material shell with a tangential pressure, using a new approach. This is both simpler than the traditional method of extrinsic curvature junction conditions (which we also employ), and suggests an expression for a ‘gravitational potential energy’ of the shell. Such a shell, if slowly spinning, can rotationally drag the inertial frames within it through a finite angle as it collapses and rebounds from a minimum radius. Rebounding ‘spherical’ and cylindrical pulses of rotating gravitational waves were studied previously. Here we calculate their angular momentum and show that their rotational frame dragging is in agreement with that of the rotating spherical shell and a rotating cylindrical dust shell. This shows that Machian effects occur equally for material and analogous ‘immaterial’ sources.

Geometry of deformed black holes. I. Majumdar-Papapetrou binary

Although black holes are eminent manifestations of very strong gravity, the geometry of space-time around and even inside them can be significantly affected by additional bodies present in their surroundings. We study such an influence within static and axially symmetric (electro-)vacuum space-times described by exact solutions of Einstein's equations, considering astrophysically motivated configurations (such as black holes surrounded by rings) as well as those of pure academic interest (such as specifically "tuned" systems of multiple black holes). The geometry is represented by the simplest invariants determined by the metric (the lapse function) and its gradient (gravitational acceleration), with special emphasis given to curvature (the Kretschmann and Ricci-square scalars). These quantities are analyzed and their level surfaces plotted both above and below the black-hole horizons, in particular near the central singularities. Estimating that the black hole could be most strongly affected by the other black hole, we focus, in this first paper, on the Majumdar--Papapetrou solution for a binary black hole and compare the deformation caused by "the other" hole (and the electrostatic field) with that induced by rotational dragging in the well-known Kerr and Kerr--Newman solutions.

Gyratonicppwaves and their impulsive limit

We investigate a class of gravitational pp-waves which represent the exterior vacuum field of spinning particles moving with the speed of light. Such exact spacetimes are described by the original Brinkmann form of the pp-wave metric including the often neglected off-diagonal terms. We put emphasis on a clear physical and geometrical interpretation of these off-diagonal metric components. We explicitly analyze several new properties of these spacetimes associated with the spinning character of the source, such as rotational dragging of frames, geodesic deviation, impulsive limits and the corresponding behavior of geodesics.

Dynamical and thermodynamical aspects of interacting Kerr black holes

We consider the asymptotically flat double-Kerr solution for two equal mass black holes with either the same or opposite angular momentum and with a massless strut between them. For fixed angular momentum and mass, the angular velocity of two corotating Kerr black holes decreases as they approach one another, from the Kerr value at infinite separation to the value of a single Kerr black hole with twice the mass and the angular momentum at the horizons merging limit. We show that the ratio $J/{M}^{2}$ for extremal corotating Kerr black holes varies from unity at infinite separation to two at the merging limit. These results are interpreted in terms of rotational dragging and compared with the case of counterrotating Kerr black holes. We then analyze the merging of ergoregions. In the corotating case the merger point occurs at an angle of $\ensuremath{\pi}/2$, in agreement with recent general arguments. In the counterrotating case the ergoregions never merge. We study the horizon geometry for both cases as a function of the distance and provide embedding diagrams. Finally, we study the thermodynamical evolution of the corotating double-Kerr system, showing that, in the canonical ensemble, it is thermodynamically stable for fast spinning black holes. As for single Kerr black holes the stable and unstable phases are separated by a second order phase transition. We show that for large fixed angular momentum two Kerr black holes reach a minimum distance, before horizon merging has occurred, where the thermodynamical approximation breaks down. We also consider the microcanonical ensemble to study the maximal energy that can be extracted from the double-Kerr system as a function of the separation between the black holes.

Gravitational waves and dragging effects

Linear and rotational dragging effects of gravitational waves on local inertial frames are studied in purely vacuum spacetimes. First, the linear dragging caused by a simple cylindrical pulse is investigated. Surprisingly strong transverse effects of the pulse are exhibited. The angular momentum in cylindrically symmetric spacetimes is then defined and confronted with some results in the literature. In the main part, a general procedure is developed for studying weak gravitational waves with translational but not axial symmetry which can carry angular momentum. After a suitable averaging the rotation of local inertial frames due to such rotating waves can be calculated explicitly and illustrated graphically. This is done in detail in the accompanying paper. Finally, the rotational dragging is given for strong cylindrical waves interacting with a rotating cosmic string with a small angular momentum.

A model for linear dragging

We try to carry over, as closely as possible, the well-known results for rotational dragging (Thirring, Brill and Cohen) to dragging due to linearly accelerated masses. To this end, a spherical, charged mass shell is linearly accelerated by a (weak) external, axisymmetric and dipolar charge distribution. It is shown that the interior of this (Reissner–Nordström-like) shell stays flat. The dragging of neutral test particles inside the shell, defined by their acceleration, scaled by the overall acceleration of the (rigid) shell, is calculated for the weak field case for a highly massive but weakly charged shell and for the general strong field case. The results compare favourably with the corresponding results for rotational dragging.

Bounds on the dragging rate of slowly and differentially rotating relativistic stars

For relativistic stars rotating slowly and differentially with a positive angular velocity, some properties in relation to the positiveness of the rate of rotational dragging and of the angular momentum density are derived. Moreover, the proof for the bounds on the rotational mass-energy, which we have generalized (outside the slow rotation limit) in a previous paper, is briefly exposed.

Extraction of black hole rotational energy by a magnetic field and the formation of relativistic jets.

Using numerical simulations, we modeled the general relativistic magnetohydrodynamic behavior of a plasma flowing into a rapidly rotating black hole in a large-scale magnetic field. The results show that a torsional Alfvén wave is generated by the rotational dragging of space near the black hole. The wave transports energy along the magnetic field lines outward, causing the total energy of the plasma near the hole to decrease to negative values. When this negative energy plasma enters the horizon, the rotational energy of the black hole decreases. Through this process, the energy of the spinning black hole is extracted magnetically

Late-time evolution of realistic rotating collapse and the no-hair theorem

We study analytically the asymptotic late-time evolution of realistic rotating collapse. This is done by considering the asymptotic late-time solutions of Teukolsky's master equation, which governs the evolution of gravitational, electromagnetic, neutrino and scalar perturbations fields on Kerr spacetimes. In accordance with the no-hair conjecture for rotating black-holes we show that the asymptotic solutions develop inverse power-law tails at the asymptotic regions of timelike infinity, null infinity and along the black-hole outer horizon (where the power-law behaviour is multiplied by an oscillatory term caused by the dragging of reference frames). The damping exponents characterizing the asymptotic solutions at timelike infinity and along the black-hole outer horizon are independent of the spin parameter of the fields. However, the damping exponents at future null infinity are spin dependent. The late-time tails at all the three asymptotic regions are spatially dependent on the spin parameter of the field. The rotational dragging of reference frames, caused by the rotation of the black-hole (or star) leads to an active coupling of different multipoles.

Some aspects of motion in the Kerr field

We find that the Kerr field, the simplest relativistic gravitational field of a rotating source, may help in (pre-)collimation of cosmic jets emanating from many active galactic nuclei. Solving the geodesic motion in the Kerr field, we use in part a tetrad approach — the description of quantities by their physically measured components in the locally Cartesian frame of a local observer (rather than by their coordinate components). The stationary frames, the most important local frames in the Kerr spacetime, are reviewed and their properties discussed. We use the Newtonian “force” language to interpret the occurrence of the “rotospheres”, the toroidal regions that circle round the Kerr sources; in these regions, the 4-acceleration of a stationary observer depends on his orbital angular velocity in a way going against common intuition. We give the rotospheres in various Kerr spacetimes and introduce a new class of privileged observers (“extremally accelerated observers”). We then generalize this approach to any motion in any spacetime. In the Kerr spaces, we construct the cones generated by directions along which photons can escape to infinity from a given point. The results are consistent with the expected influence of rotational dragging for black holes, whereas they are rather complicated for naked singularities.

Kaluza-klein description of the electrical field due to an infinitely long, straight line charge

It is shown that according to the Kaluza-Klein theory the electromagnetic scalar potential is equal to the angular velocity of the rotational inertial dragging effect in five-dimensional space-time. The gravitational vacuum field equations for a five-dimensional space-time outside a rotating cylindrical shell are solved. The angular velocity of the inertial dragging field outside the shell is determined in terms of the angular momentum of the shell by means of Israel's geometric description of surface layers in general relativity. Projecting the solution into four-dimensional space-time, the inertial dragging field due to the rotation of the shell is interpreted as an electrical field. We conjecture that according to the Kaluza-Klein theory all electromagnetic phenomena are four-dimensional manifestations of the five-dimensional inertial dragging effect.

Rotational perturbations and frame dragging in a Friedmann universe

Classical work on frame dragging effects near a slowly rotating mass shell and later improvements thereof are extended to a model with cosmological boundary conditions: a mass shell with flat interior in an asymptotic Friedmann universe. The metric describing a shell, which follows the motion of the surrounding cosmic dust, can be given in terms of analytic functions. Its energy-momentum tensor fulfils the energy conditions. Rotation of the shell and the cosmic matter is treated in the first order of the angular velocity and a perturbation of this exact solution. The dragging of the inertial frames inside the shell is defined in an invariant manner by, in principle, measurable quantities. It is shown that frame dragging occurs in this rotationally perturbed Friedmann universe and that the inertial frames are influenced by the shell as well as by the cosmic matter. The behaviour of the dragging coefficient is discussed and compared with results of previous work.

Electrodynamical Properties of Empty Space in General Relativity

The general-relativistic formulation of the unipolar induction problem (massive, magnetized, rotating conductor) shows that the external vacuum solution attributes a virtual charge density and a virtual toroidal current to the surrounding empty space. The explanation thereof illuminates a new aspect of the rotational dragging of the inertial frame. If direct consequences of these phenomena could be detected in pulsars or in other astrophysical structures where relativistic gravitation is important, new important verifications of general relativity would be achieved. The main thrust of these considerations lies, however, in their applicability to any rotating metric with a superposed magnetic field.