Frame-dragging Research Articles

Frame dragging in the field of Kerr family

The interesting phenomenon of frame dragging which is associated with the rotation of the source in the field of Kerr family is discussed and the angular velocity of an uncharged test particle is obtained with a straightforward mathematical method.

El aumento de la masa inercial: Einstein y las coordenadas

Durante los años de gestación de la Teoría General de la Relatividad (TGR) y de la subsiguiente cosmología relativista, Albert Einstein esperaba que su teoría de gravitación satisficiera las ideas de Mach sobre la inercia. Para esto calculó un par de efectos que debían garantizar la consonancia de TGR con el origen material de la inercia à la Mach. Estos efectos fueron: (A) El aumento de la masa inercial cuando se aglomera materia en su vecindad y (B) el arrastre de los marcos inerciales. En este articulo se hace un estudio histórico-conceptual del primero(A). Mientras que el arrastre es considerado como una predicción real de TGR, el aumento de la masa inercial ha llegado a desecharse como un artificio de coordenadas no impugnable a la variedad espacio-temporal estudiada. Se presenta una revisión técnica con clarificaciones conceptuales es presentada.

Frame dragging and bending of light in Kerr and Kerr–(anti) de Sitter spacetimes

The equations of general relativity in the form of timelike and null geodesics that describe motion of test particles and photons in Kerr spacetime are solved exactly including the contribution from the cosmological constant. We then perform a systematic application of the exact solutions obtained, to the following cases. The exact solutions derived for null, spherical, polar and non-polar orbits are applied for the calculation of frame dragging (Lense–Thirring effect) for the orbit of a photon around the galactic centre, assuming that the latter is a Kerr black hole for various values of the Kerr parameter including those supported by recent observations. Unbound null polar orbits are investigated, and an analytical expression for the deviation angle of a polar photon orbit from the gravitational Kerr field is derived. In addition, we present the exact solution for timelike and null equatorial orbits. In the former case, we derive an analytical expression for the precession of the point of closest approach (perihelion, periastron) for the orbit of a test particle around a rotating mass whose surrounding curved spacetime geometry is described by the Kerr field. In the latter case, we calculate an exact expression for the deflection angle for a light ray in the gravitational field of a rotating mass (the Kerr field). We apply this calculation for the bending of light from the gravitational field of the galactic centre, for various values of the Kerr parameter, and the impact factor.

A note on frame dragging

The measurement of spin effects in general relativity has recently taken centre stage with the successfully launched Gravity Probe B experiment coming towards an end, coupled with recently reported measurements using laser ranging. Many accounts of these experiments have been in terms of frame dragging. We point out that this terminology has given rise to much confusion and that a better description is in terms of spin–orbit and spin–spin effects. In particular, we point out that the de Sitter precession (which has been measured to a high accuracy) is also a frame-dragging effect and provides an accurate benchmark measurement of spin–orbit effects which GPB needs to emulate.

Drag-free control system for frame dragging measurements based on cold atom interferometry

This paper describes the design and analysis of the drag-free and attitude control system (DFACS) for frame dragging measurements using a cold atom interferometer. The general architecture of the DFACS is derived and important aspects of the control system are discussed. This includes requirements and constraints, design and equipment issues. The actual control algorithms derived in this paper are designed as decoupled SISO controllers using modern optimal control theory (LQG, H ∞ ). The performance of the control system is analyzed in a comprehensive simulation campaign. From the results it can be concluded that the proposed system can meet the requirements for frame dragging measurements, with very little additional technology adaptation for the drag-free sensor and micro-propulsion system.

Gravitomagnetic Field and Penrose Scattering Processes

In this paper we present theoretical model calculations involving Monte Carlo computer simulations of Compton scattering and electron-positron (e-e+) pair production processes in the ergosphere of a supermassive rotating black hole. Particles from an accretion disk surrounding the rotating black hole fall into the ergosphere and are scattered by particles that are confined in equatorial and nonequatorial orbits. The energy-momentum vectors are calculated for the scattered escaping particles. Particles escape with energies of about 3 GeV or greater. Importantly, these model calculations show that the Lense-Thirring effect, that is, the dragging of local inertial frames into rotation, inside the ergosphere, caused by the angular momentum of the rotating black hole, results in a gravitomagnetic force being exerted on the scattered escaping particles. Effects of this force on the Penrose scattered particles are analyzed and discussed.

Spectral Constraints for Millisecond Pulsars Due to General Relativistic Frame Dragging

Spectral Constraints for Millisecond Pulsars Due to General Relativistic Frame Dragging

On the measurement of the Lense–Thirring effect using the nodes of the LAGEOS satellites, in reply to “On the reliability of the so-far performed tests for measuring the Lense–Thirring effect with the LAGEOS satellites” by L. Iorio

In this paper, we provide a detailed description of our recent analysis and determination of the frame-dragging effect obtained using the nodes of the satellites LAGEOS and LAGEOS 2, in reply to the paper “On the reliability of the so-far performed tests for measuring the Lense–Thirring effect with the LAGEOS satellites” by L. Iorio (doi: 10.1016/j.newast.2005.01.001). First, we discuss the impact of the J ˙ 2 n uncertainties on our measurement and we show that the corresponding error is of the order of 1% of frame-dragging only. We report the result of the orbital simulations and analyses obtained with and without J ˙ 4 and with a J ˙ 4 equal to its EIGEN-GRACE02S value plus 12 times its published error, i.e., a J ˙ 4 equal to about 611% of the value adopted in EIGEN-GRACE02S, that is J ˙ 4 = 6 . 11 × ( - 1 . 41 × 10 - 11 ) ≃ - 8 . 61 × 10 - 11 . In all these three cases, by also fitting the final combined residuals with a quadratic, we obtain the same value of the measured Lense–Thirring effect. This value differs by only 1% with respect to our recent measurement of the Lense–Thirring effect. Therefore, the error due to the uncertainties in the J ˙ 2 n in our measurement of the gravitomagnetic effect can at most reach 1%, in complete agreement with our previously published error budget. Our total error budget in the measurement of frame-dragging is about 5% of the Lense–Thirring effect, alternatively even by simply considering the published errors in the J ˙ 2 n and their recent determinations we get a total error budget of the order of 10%, in complete agreement with our previously published error budget. Furthermore, we explicitly give the results and plot of a simulation clearly showing that the claim of Iorio’s paper that the J ˙ 2 n uncertainty may contribute to up a 45% error error in our measurement is clearly unsubstantiated. We then present a rigorous proof that any “imprint” or “memory” effect of the Lense–Thirring effect is completely negligible on the even zonal harmonics produced using the GRACE satellites only and used on the orbits of the LAGEOS satellites to measure the frame-dragging effect. In this paper we do not discuss the problem of the correlation of the Earth’s even zonal harmonics since it only refers to our previous, 1998, analysis with EGM96 and it will be the subject of a different paper; nevertheless, we stress that in the present analysis with EIGENGRACE02S the total error due to the static Earth gravity field has been calculated by pessimistically summing up the absolute values of the errors due to each Earth’s even zonal harmonic uncertainty, i.e., we have not used any covariance matrix to calculate the total error but we have just considered the worst possible contribution of each even zonal harmonic uncertainty to the total error budget. We also present and explain our past work on the technique of measuring the Lense–Thirring effect using the LAGEOS nodes and give its main references. Finally we discuss some other minor points and misunderstandings of the paper by Iorio, including some obvious mistakes contained both in this paper and in some other previous papers of Iorio. In conclusion, the criticisms in Iorio’s paper are completely unfounded and misdirected: the uncertainties arising from the possible variations of the J ˙ 2 n are fully accounted for in the error budget that we have published.

ELECTROMAGNETIC EFFECTS IN SUPERCONDUCTORS IN STATIONARY GRAVITATIONAL FIELD

The general relativistic modifications to the resistive state in superconductors of second type in the presence of a stationary gravitational field are studied. Some superconducting devices that can measure the gravitational field by its red-shift effect on the frequency of radiation are suggested. It has been shown that by varying the orientation of a superconductor with respect to the earth gravitational field, a corresponding varying contribution to AC Josephson frequency would be added by gravity. A magnetic flux (being proportional to angular velocity of rotation Ω) through a rotating hollow superconducting cylinder with the radial gradient of temperature ∇rT is theoretically predicted. The magnetic flux is assumed to be produced by the azimuthal current arising from Coriolis force effect on radial thermoelectric current. Finally the magnetic flux through the superconducting ring with radial heat flow located at the equatorial plane interior of the rotating neutron star is calculated. In particular it has been shown that nonvanishing magnetic flux will be generated due to the general relativistic effect of dragging of inertial frames on the thermoelectric current.

CHARGED PARTICLE DYNAMICS IN THE FIELD OF A SLOWLY ROTATING COMPACT STAR

We study the dynamics of a charged particle in the field of a slowly rotating compact star in the gravitoelectromagnetic approximation to the geodesic equation. The star is assumed to be surrounded by an ideal, highly conducting plasma (taken as a magneto-hydrodynamic fluid) with a stationary, axially symmetric electromagnetic field. The general relativistic Maxwell equations are solved to obtain the effects of the background spacetime on the electromagnetic field in the linearized Kerr spacetime. The equations of motion are then set up and solved numerically to incorporate the gravitational as well as the electromagnetic effects. The analysis shows that in the slow rotation approximation, the frame dragging effects on the electromagnetic field are absent. However the particle is directly effected by the rotating gravitational source such that close to the star the gravitational and electromagnetic field produce contrary effects on the particle trajectories.

Empirical Constraints on the General Relativistic Electric Field Associated with PSR J0437-4715

We simulate the magnetosphere of the nearby millisecond pulsar PSR J0437-4715, which is expected to have an unscreened electric potential due to the lack of magnetic pair production. We incorporate general relativistic (GR) effects and study curvature radiation (CR) by primary electrons but neglect inverse Compton scattering of thermal X-ray photons by these electrons. We find that the CR spectrum cuts off at energies below ~17 GeV, well below the threshold of the High Energy Stereoscopic System (H.E.S.S.) telescope (100 GeV), while other models predict a much higher cutoff of 100 GeV. GR theory also predicts a relatively narrow pulse (βo ~ 0.2 phase width) centered on the magnetic axis. EGRET observations above 100 MeV significantly constrain the application of the Muslimov & Harding model for γ-ray production as a result of GR frame dragging and ultimately its polar cap (PC) current and accelerating potential. Whereas the standard prediction of this pulsar's γ-ray luminosity due to GR frame dragging is ~10% of the spin-down power, a nondetection by forthcoming H.E.S.S. observations will constrain it to 0.3%, enforcing an even more severe revision of the accelerating electric field and PC current.

CHARGE CONJUGATION AND LENSE–THIRRING EFFECT: A NEW ASYMMETRY

This essay presents a new asymmetry that arises from the interplay of charge conjugation and Lense–Thirring effect. When applied to Majorana neutrinos, the effects predicts [Formula: see text] oscillations in gravitational environments with rotating sources. Parameters associated with astrophysical environments indicate that the presented effect is presently unobservable for solar neutrinos. But, it will play an important role in supernova explosions, and carries relevance for the observed matter–antimatter asymmetry in the universe.

General Relativistic Magnetohydrodynamic Simulations of Collapsars: Rotating Black Hole Cases

We have performed 2.5-dimensional general relativistic magnetohydrodynamic (MHD) simulations of collapsars including a rotating black hole. This paper is an extension of our previous paper. The current calculation focuses on the effect of black hole rotation using general relativistic MHD with simplified microphysics; i.e., we ignore neutrino cooling, physical equation of state, and photodisintegration. Initially, we assume that the core collapse has failed in this star. A rotating black hole of a few solar masses is inserted by hand into the calculation. We consider two cases, a corotating case and a counterrotating case with respect to the black hole rotation. Although the counterrotating case may be unrealistic for collapsars, we perform it as the maximally dragging case of a magnetic field. The simulation results show the formation of a disklike structure and the generation of a jetlike outflow near the central black hole. The jetlike outflow propagates outwardly with the twisted magnetic field and becomes collimated. We have found that the jets are generated and accelerated mainly by the magnetic field. The total jet velocity in the rotating black hole case is comparable to that of the nonrotating black hole case, ~0.3c. When the rotation of the black hole is faster, the magnetic field is twisted strongly owing to the frame-dragging effect. The magnetic energy stored by the twisting magnetic field is directly converted to kinetic energy of the jet rather than propagating as an Alfven wave. Thus, as the rotation of the black hole becomes faster, the poloidal velocity of the jet becomes faster. In the rapidly rotating black hole case the jetlike outflow can be produced by the frame-dragging effect only through twisting of the magnetic field, even if there is no stellar rotation.

Frame Dragging and Lense-Thirring Effect

We first describe some phenomena due to spin and mass-energy currents on clocks and photons, in particular we derive the delay in the travel time of photons due to the spin of a body both inside a rotating shell and outside a rotating body. We then review the 1995–2001 measurements of the gravitomagnetic field of Earth and Lense-Thirring effect obtained by analyzing the orbits of the two laser-ranged satellites LAGEOS (LAser GEOdynamics Satellite) and LAGEOS II. Finally, we report the latest measurement of the Lense-Thirring effect, obtained in 2003 with the two LAGEOS satellites over more than 10 years of data. This 2003 result fully confirms and improves our previous measurements of the Earth frame-dragging: the Lense-Thirring effect exists and its experimental value is within ~20% of what is predicted by Einstein's theory of general relativity.

Precise relativistic orbits in Kerr and Kerr–(anti) de Sitter spacetimes

The timelike geodesic equations resulting from the Kerr gravitational metric element are derived and solved exactly including the contribution from the cosmological constant. The geodesic equations are derived, by solving the Hamilton–Jacobi partial differential equation by separation of variables. The solutions can be applied in the investigation of the motion of a test particle in the Kerr and Kerr–(anti) de Sitter gravitational fields. In particular, we apply the exact solutions of the timelike geodesics: (i) to the precise calculation of dragging (Lense–Thirring effect) of a satellite's spherical polar orbit in the gravitational field of Earth assuming Kerr geometry; (ii) assuming the galactic centre is a rotating black hole we calculate the precise dragging of a stellar polar orbit around the galactic centre for various values of the Kerr parameter including those supported by recent observations. The exact solution of non-spherical geodesics in Kerr geometry is obtained by using the transformation theory of elliptic functions. The exact solution of spherical polar geodesics with a nonzero cosmological constant can be expressed in terms of Abelian modular theta functions that solve the corresponding Jacobi's inversion problem.

Gravitational Experiments in Space: Gravity Probe B and STEP

We describe two space based gravitational physics experiments, the Gravity Probe B Relativity Mission (GPB) and the Satellite Test of the Equivalence Principle (STEP). GP-B will perform precision tests of two independent predictions of general relativity, the geodetic effect and frame dragging. STEP will provide a precision test of a foundation of general relativity, the Equivalence Principle.

The traditional approximation in general relativity

We discuss the generalisation of the so-called traditional approximation, well known in geophysics, to general relativity. We show that the approximation is applicable for rotating relativistic stars provided that one focuses on relatively thin radial shells. This means that the framework can be used to study waves in neutron star oceans. We demonstrate that, once the effects of the relativistic frame-dragging are accounted for, the angular problem reduces to Laplace's tidal equation. We derive the dispersion relation for various classes of waves in a neutron star ocean and show that the combined effects of the frame-dragging and the gravitational redshift typically lower the frequency of a mode by about 20%.

Magnetically Driven Accretion Flows in the Kerr Metric. II. Structure of the Magnetic Field

We present a detailed analysis of the magnetic field structure found in a set of four general relativistic three-dimensional MHD simulations of accreting tori in the Kerr metric with different black hole spins. Among the properties analyzed are the field strength as a function of position and black hole spin, the shapes of field lines, the degree to which they connect different regions, and their degree of tangling. Strong magnetic field is found toward small radii, and field strength increases with black hole spin. In the main disk body, inner torus, and corona the field is primarily toroidal. Most field lines passing through a given radius in these regions wander through a narrow radial range, suggesting an overall tightly wound spiral structure. In the main disk body and inner torus sharp field-line bends on small spatial scales are superposed on the spirals, but the field lines are much smoother in the corona. The magnetic field in the plunging region is also comparatively smooth, being stretched out radially by the infalling gas. The magnetic field in the axial funnel resembles a split monopole, but with evidence of frame dragging of the field lines near the poles of the black hole. We investigate prior speculations about the structure of the magnetic fields and discuss how frequently certain configurations are seen in the simulations. For example, coronal loops are very rare, and field lines connecting high latitudes on the event horizon to the disk are not found at all. Almost the entire system is matter-dominated; the only force-free regions are in the axial funnel. We also analyze the distribution of current density, with a view toward identifying possible locations of magnetic energy dissipation. Regions of high current density are concentrated toward the inner torus and plunging region. Dissipation inside the marginally stable orbit may provide a new source of energy for radiation, supplementing the dissipation associated with torques in the stably orbiting disk body.

MULTI-BLACK HOLES SOLUTION WITH COSMIC STRINGS

Some black hole-cosmic string models such as Reissner–Nordström, RN–de Sitter, Kerr–Newman and multi-black holes with cosmic string are given. Energy and angular momentum of a timelike particle in circular orbits in multi-black hole space–time are calculated. The geodesic equations for the timelike particles for the far region of the multi-black hole sources are calculated and small oscillations around the circular orbit obtained. It is seen that the particle's orbit precesses like the Lens–Thirring effect.

Do rotations beyond the cosmological horizon affect the local inertial frame?

If perturbations beyond the horizon have the velocities prescribed everywhere then the dragging of inertial frames near the origin is suppressed by an exponential factor. However if perturbations are prescribed in terms of their angular momenta there is no such suppression. We resolve this paradox and in doing so give new explicit results on the dragging of inertial frames in closed, flat and open universe with and without a cosmological constant.