Frame-dragging Research Articles

Deflection of light and particles by moving gravitational lenses

Various authors have investigated the problem of light deflection by radially moving gravitational lenses, but the results presented so far do not appear to agree on the expected deflection angles. Some publications claim a scaling of deflection angles with $1\ensuremath{-}v$ to first order in the radial lens velocity $v,$ while others obtained a scaling with $1\ensuremath{-}2v.$ In this paper we generalize the calculations for arbitrary lens velocities and show that the first result is the correct one. We discuss the seeming inconsistency of relativistic light deflection with the classical picture of moving test particles by generalizing the lens effect to test particles of arbitrary velocity, including light as a limiting case. We show that the effect of radial motion of the lens is very different for slowly moving test particles and light and that a critical test particle velocity exists for which the motion of the lens has no effect on the deflection angle to first order. An interesting and not immediately intuitive result is obtained in the limit of a highly relativistic motion of the lens towards the observer, where the deflection angle of light reduces to zero. This phenomenon is elucidated in terms of moving refractive media. Furthermore, we discuss the dragging of inertial frames in the field of a moving lens and the corresponding Lense-Thirring precession, in order to shed more light on the geometrical effects in the surroundings of a moving mass. In a second part we discuss the effect of transversal motion on the observed redshift of lensed sources. We demonstrate how a simple kinematic calculation explains the effects for arbitrary velocities of the lens and test particles. Additionally we include the transversal motion of the source and observer to show that all three velocities can be combined into an effective relative transversal velocity similar to the approach used in microlensing studies.

Relativistic Stars with Poloidal and Toroidal Magnetic Fields and Meridional Flow

We investigate stationary axisymmetric configurations of magnetized stars in the framework of general relativistic ideal magnetohydrodynamics. Our relativistic stellar model incorporates a toroidal magnetic field and meridional flow in addition to a poloidal magnetic field for the first time. The magnetic field and meridional flow are treated as perturbations, but no other approximation is made. We find that the stellar shape can be prolate rather than oblate when a toroidal field exists. We also find that, for fixed baryonic mass and total magnetic helicity, the more spherical the star is, the lower the energy it has. Further, we find two new types of the frame-dragging effect that differ from the standard one in a rotating star or Kerr geometry. They may violate the reflection symmetry about the equatorial plane.

Gravitational lensing by stars with angular momentum

Gravitational lensing by spinning stars, approximated as homogeneous spheres, is discussed in the weak field limit. Dragging of inertial frames, induced by angular momentum of the deflector, breaks spherical symmetry. We examine how the gravito-magnetic field affects image positions, caustics and critical curves. Distortion in microlensing-induced light curves is also considered.

The electric fields of radio pulsars with asymmetric nondipolar magnetic fields

The effect of the curvature of open magnetic field lines on the generation of electric fields in radio pulsars is considered in the framework of a Goldreich-Julian model, for both a regime with a free outflow of electrons from the neutron-star surface and the case of a small thermoemission current. An expression for the electron thermoemission current in a strong magnetic field is derived. The electric field associated with the curvature of the magnetic flux tubes is comparable to the field generated by the relativistic dragging of the inertial frames.

Extended Acceleration in Slot Gaps and Pulsar High‐Energy Emission

We revise the physics of primary electron acceleration in the slot (SG) above the pulsar polar caps (PCs), a regime originally proposed by Arons and Scharlemann (1979) in their electrodynamic model of pulsar PCs. We employ the standard definition of the SG as a pair-free space between the last open field lines and the boundary of the pair plasma column which is expected to develop above the bulk of the PC. The rationale for our revision is that the proper treatment of primary acceleration within the pulsar SGs should take into account the effect of the narrow geometry of the gap on the electrodynamics within the gap and also to include the effect of inertial frame dragging on the particle acceleration. We show that the accelerating electric field within the gap, being significantly boosted by the effect of frame dragging, becomes reduced because of the gap geometry by a factor proportional to the square of the SG width. The combination of the effects of frame dragging and geometrical screening in the gap region naturally gives rise to a regime of extended acceleration, that is not limited to favorably curved field lines as in earlier models, and the possibility of multiple-pair production by curvature photons at very high altitudes, up to several stellar radii. We present our estimates of the characteristic SG thickness across the PC, energetics of primaries accelerated within the gap, high-energy bolometric luminosities emitted from the high altitudes in the gaps, and maximum heating luminosities produced by positrons returning from the elevated pair fronts. The estimated theoretical high-energy luminosities are in good agreement with the corresponding empirical relationships for gamma-ray pulsars. We illustrate the results of our modeling of the pair cascades and gamma-ray emission from the high altitudes in the SG for the Crab pulsar. The combination of the frame-dragging field and high-altitude SG emission enables both acceleration at the smaller inclination angles and a larger emission beam, both necessary to produce widely-spaced double-peaked profiles.

Covariant calculation of general relativistic effects in an orbiting gyroscope experiment

We carry out a covariant calculation of the measurable relativistic effects in an orbiting gyroscope experiment. The experiment, currently known as Gravity Probe B, compares the spin directions of an array of spinning gyroscopes with the optical axis of a telescope, all housed in a spacecraft that rolls about the optical axis. The spacecraft is steered so that the telescope always points toward a known guide star. We calculate the variation in the spin directions relative to readout loops rigidly fixed in the spacecraft, and express the variations in terms of quantities that can be measured, to sufficient accuracy, using an Earth-centered coordinate system. The measurable effects include the aberration of starlight, the geodetic precession caused by space curvature, the frame-dragging effect caused by the rotation of the Earth and the deflection of light by the Sun.

On the aberration of light from a moving retroreflector

Laser ranging observations on a succession of spacecraft bearing corner retroreflector prisms have favored a model for velocity aberration that depends on the Fresnel–Fizeau frame-dragging effect in the material medium of the reflector. The present analysis, based manifestly on the principle of relativity, supports the classical medium-independent Bradley aberration model. Possibilities for resolving the conflict are discussed and a proposed improved experiment is outlined.

Spectral Lines from Rotating Neutron Stars

The line profiles from rotating neutron stars are affected by a number of relativistic processes such as Doppler boosts, strong self-lensing, frame dragging, and the differential gravitational redshift arising from the stellar oblateness. In this Letter, we calculate line profiles, taking into account the first two effects; this approach is accurate for rotation rates less than the breakup frequency. We show that the line profiles are not only broadened and weakened but are also significantly asymmetric, and allow for an independent measurement of both the mass and the radius of the neutron star. Furthermore, we investigate the case when a fraction of the neutron star surface contributes to the emission and find that the line profiles are typically doubly peaked. We discuss the implications of our results for searches for line features in the spectra of isolated neutron stars and X-ray bursters. We finally assess the systematic uncertainties introduced by the line asymmetry in inferring the compactness of neutron stars from the detection of redshifted lines.

Magnetohydrodynamic Shock Conditions for Accreting Plasma onto Kerr Black Holes. I.

We extend the work by Appl and Camenzind for special relativistic magnetohydrodynamic (MHD) jets to fully general relativistic studies of standing shock formation for accreting MHD plasma in a rotating, stationary, and axisymmetric black hole magnetosphere. All the postshock physical quantities are expressed in terms of the relativistic compression ratio, which can be obtained in terms of preshock quantities. Then, the downstream state of a shocked plasma is determined by the upstream state of the accreting plasma. In this paper sample solutions are presented for slow magnetosonic shocks for accreting flows in the equatorial plane. We find that some properties of the slow magnetosonic shock for the rotating magnetosphere can behave like a fast magnetosonic shock. In fact, it is confirmed that in the limit of weak gravity for the upstream nonrotating accretion plasma where the magnetic field lines are leading and rotating, our results are very similar to the fast magnetosonic shock solution by Appl and Camenzind. However, we find that the situation becomes far more complicated because of the effects of strong gravity and rotation, such as the frame-dragging effects. We show the tendency that the large spin of the black hole makes the slow magnetosonic shock strong for the accretion solutions with the same energy flux.

Three‐dimensional Hydrodynamic Simulations of Accretion Tori in Kerr Spacetimes

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

Regimes of Pulsar Pair Formation and Particle Energetics

We investigate the conditions required for the production of electron-positron pairs above a pulsar polar cap (PC) and the influence of pair production on the energetics of the primary particle acceleration. Assuming space-charge limited flow acceleration including the inertial frame-dragging effect, we allow both one-photon and two-photon pair production by either curvature radiation (CR) photons or photons resulting from inverse-Compton scattering of thermal photons from the PC by primary electrons. We find that, while only the younger pulsars can produce pairs through CR, nearly all known radio pulsars are capable of producing pairs through non-resonant inverse-Compton scatterings. The effect of the neutron star equations of state on the pair death lines is explored. We show that pair production is facilitated in more compact stars and more massive stars. Therefore accretion of mass by pulsars in binary systems may allow pair production in most of the millisecond pulsar population. We also find that two-photon pair production may be important in millisecond pulsars if their surface temperatures are above about three million degrees K. Pulsars that produce pairs through CR will have their primary acceleration limited by the effect of screening of the electric field. In this regime, the high-energy luminosity should follow a $L_{HE} \propto \dot E_{rot}^{1/2}$ dependence. The acceleration voltage drop in pulsars that produce pairs only through inverse-Compton emission will not be limited by electric field screening. In this regime, the high-energy luminosity should follow a $L_{HE} \propto \dot E_{rot}$ dependence. Thus, older pulsars will have significantly lower gamma-ray luminosity.

Rotating charged mass shell: Dragging, antidragging, and the gyromagnetic ratio

We calculate explicitly the system of a spherical shell of radius R, carrying (nearly) arbitrary mass M and charge q, and rotating slowly around an axis through its center. We discuss, mainly graphically in the plane of the model parameters $M/R$ and $q/R,$ the following properties of this system: The dragging of inertial frames which turns over to antidragging in part of the parameter space, the induced magnetic field, the angular momentum, the magnetic moment, and the gyromagnetic ratio. The latter is very near to the value 2 in an overwhelming part of the parameter space, and we argue that this signals a deep connection between general relativity and quantum theory which could be important in the search for quantum gravity.

General relativistic electromagnetic fields of a slowly rotating magnetized neutron star - II. Solution of the induction equations

We have solved numerically the general relativistic induction equations in the interior background space-time of a slowly rotating magnetized neutron star. The analytic form of these equations was discussed recently (Paper I), where corrections due to both the space-time curvature and the dragging of reference frames were shown to be present. Through a number of calculations we have investigated the evolution of the magnetic field with different rates of stellar rotation, different inclination angles between the magnetic moment and the rotation axis, as well as different values of the electrical conductivity. All of these calculations have been performed for a constant-temperature relativistic polytropic star and make use of a consistent solution of the initial-value problem which avoids the use of artificial analytic functions. Our results show that there exist general relativistic effects introduced by the rotation of the space-time which tend to decrease the decay rate of the magnetic field. The rotation-induced corrections are however generally hidden by the high electrical conductivity of the neutron star matter, and when realistic values for the electrical conductivity are considered, these corrections become negligible even for the fastest known pulsar.

Precession of Magnetically Driven Warped Disks and Low‐Frequency Quasi‐periodic Oscillations in Low‐Mass X‐Ray Binaries

An accretion disk around a rotating magnetized star is subjected to magnetic torques that induce disk warping and precession. These torques arise generically from interactions between the stellar field and induced surface currents on the disk. Applying these new effects to weakly magnetized (B ~ 107-109 G) neutron stars in low-mass X-ray binaries, we study the global hydrodynamical warping/precession modes of the disk under the combined influences of relativistic frame dragging, classical precession due to the oblateness of the neutron star, and the magnetic torques. Under quite general conditions, the magnetic warping torque can overcome the Bardeen-Petterson viscous damping and make the modes grow. The modes are confined to the inner region of the disk and have frequencies equal to 0.3-0.95 (depending on the mass accretion rate ) times the sum of the Lense-Thirring frequency, the classical precession frequency, and the magnetically driven precession frequency evaluated at the inner disk radius rin. As increases, the mode frequency is reduced relative to the total precession frequency at rin, since the mode becomes less concentrated around rin due to the increasing viscous stress associated with the large . Because of this, and because the magnetically driven precession is retrograde (opposite to the Lense-Thirring precession) and depends strongly on , the mode frequency can have a nonmonotonic dependence on the mass accretion rate. This may account for several observed features of low-frequency (10-60 Hz) quasi-periodic oscillations (LFQPOs) in low-mass X-ray binaries that are otherwise difficult to explain, such as the flattening/turnover in the LFQPO frequency- correlation or in the LFQPO frequency-kHz QPO frequency correlation (e.g., as seen clearly in GX 17+2).

Relativistic Self‐similar Disks

We formulate and solve by semi-analytic means the axisymmetric equilibria of relativistic self-similar disks of infinitesimal vertical thickness. These disks are supported in the horizontal directions against their self-gravity by a combination of isothermal (two-dimensional) pressure and a flat rotation curve. The dragging of inertial frames restricts possible solutions to rotation speeds that are always less than 0.438 times the speed of light, a result first obtained by Lynden-Bell and Pineault in 1978 for a cold disk. We show that prograde circular orbits of massive test particles exist and are stable for all of our model disks, but retrograde circular orbits cannot be maintained with particle velocities less than the speed of light once the disk develops an ergoregion. We also compute photon trajectories, planar and non-planar, in the resulting spacetime, for disks with and without ergoregions. We find that all photon orbits, except for a set of measure zero, tend to be focused by the gravity of the flattened mass-energy distribution toward the plane of the disk. This result suggests that strongly relativistic, rapidly rotating, compact objects may have difficulty ejecting collimated beams of matter or light along the rotation axes until the flows get well beyond the flattened parts of the relativistic mass distribution (which cannot happen in the self-similar models considered in this paper).

Gravitational waves, gyroscopes and frame dragging

We study the behaviour of a small gyroscope moving on acircular trajectory in the field of a weak gravitationalplane wave. In order to monitor the precessional velocitywe introduce an observer comoving with the gyroscope andan observer-adapted frame optically defined bymeans of light rays and not Fermi-Walker transported.From such a frame we infer some physical informationabout the gravitational-wave-induced frame dragging,which the observer will detect looking at the smallperturbations on the gyroscopic precession. We findevidence of gyroscope sensitivity to the polarization state ofthe gravitational wave. As expected, the generalrelativistic precession of the compass of inertia and the`memory effect' which are discussed in this paper arevery small and hardly detectable with modern technology;however, they are operationally well defined andtherefore not ignorable for future considerations.

General relativistic electromagnetic fields of a slowly rotating magnetized neutron star - I. Formulation of the equations

We present analytic solutions of Maxwell equations in the internal and external background spacetime of a slowly rotating magnetized neutron star. The star is considered isolated and in vacuum, with a dipolar magnetic field not aligned with the axis of rotation. With respect to a flat spacetime solution, general relativity introduces corrections related both to the monopolar and the dipolar parts of the gravitational field. In particular, we show that in the case of infinite electrical conductivity general relativistic corrections due to the dragging of reference frames are present, but only in the expression for the electric field. In the case of finite electrical conductivity, however, corrections due both to the spacetime curvature and to the dragging of reference frames are shown to be present in the induction equation. These corrections could be relevant for the evolution of the magnetic fields of pulsars and magnetars. The solutions found, while obtained through some simplifying assumption, reflect a rather general physical configuration and could therefore be used in a variety of astrophysical situations.

Frame dragging in the spacetime of a superconducting cosmic string

In this paper we calculate particle geodesics in the spacetime of a supermassive superconducting cosmic string $(G\ensuremath{\mu}>{10}^{\ensuremath{-}3}).$ Numerical techniques are used to compare the spacetime of a string supporting a fermionic supercurrent with a string supporting a stationary spin-0 current condensate. In both cases it is found that frame dragging leads to exotic behavior of geodesics, in which particles are trapped by the string spacetime, without the string violating the dominant energy condition. In the case of a string with a fermionic current it is also found that the string core is completely isolated from nonrelativistic particles in the ``outside'' universe.

Neutron Stars Imply Relativity's a Drag

A new finding, based on x-rays from distant neutron stars, could be the first clear evidence of a weird relativistic effect called frame dragging, in which a heavy chunk of spinning matter wrenches the space-time around it like an eggbeater. Using data from NASA's Rossi X-ray Timing Explorer, three astronomers in Amsterdam found circumstantial evidence for frame dragging in the flickering of three neutron stars in binary systems. They announced their results in the 1 September issue of The Astrophysical Journal.

Slowly rotating voids incosmology

We consider a spacetime consisting of an empty void separated from analmost Friedmann-Lemaître-Robertson-Walker (FLRW) dustuniverse by a spherically symmetric, slowly rotating shell which iscomoving with the cosmic dust. We treat in a unified manner all types ofFLRW universes. The metric is expressed in terms of a constantcharacterizing the angular momentum of the shell, and parametrized bythe comoving radius of the shell. Treating the rotation as a first-orderperturbation, we compute the dragging of inertial frames as well as theapparent motion of distant stars within the void. Finally, we discuss, interms of in principle measurable quantities, `Machian' features of themodel.