Nonlinear Gravitodynamics

The following sections are included:

• FOREWORD

• CONTENTS

The following sections are included:

• From electrodynamics to nonlinear gravitodynamics
  • From electricty and magnetism to electrodynamics
  • Classical and quantum electrodynamics
  • Newtonian gravitational theory
  • The general theory of relativity
  • The birth of relativistic astrophysics
  • The "Nonlinear gravitodynamics"

• The gravitational analogue of Coulomb-like interactions

• The gravitational analogue of Hertz-like wave solutions

• The gravitational analogue of the Oersted-Ampère-like magnetic interaction

• The goal of this volume

• The contributions to the WFM3/INW1

• The Classic Papers

• Contemporary papers

• Acknowledgments

• References

A major consequence of General Relativity and related theories of gravity is that all inertial frames are local. These local frames are accelerated, warped and stretched, and rotated with respect to each other due to the surrounding mass-energy distributions. While only their relative rotations are typically called frame dragging effects, this phrase describes a broader range of gravitational influences on inertia. Discussing the rotational effects on inertial frames within the wider framework, two conclusions are reached. Concerning gravito-magnetism, the relativistic gravitational force between two sources of moving matter, this interaction is already well verified through its contributions to a number of experimentally measured phenomena, and its presence is essential to the local Lorentz invariance of gravity which is also well confirmed by observation. So explicit observation of the gravito-magnetic interaction between the spinning Earth and an orbitting gyroscope will, at the expected one percent level of precision, add little to our existing empirical knowledge concerning relativistic gravity. However, measurement to a part in 10⁵ of the larger form of frame dragging on such a gyroscope — the deSitter geodetic precession which results from the motion of the gyroscope through the Earth's gravitational field — will result in a most significant quantitative improvement in our knowledge of relativistic gravity.

We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields, to first order in the external field, but to an arbitrary order in spin. The noncovariant spin formalism is crucial for the correct description of the influence of the spin on the particle trajectory. We show that the true coordinate of a relativistic spinning particle is its naive, common coordinate . Concrete calculations are performed up to second order in spin included. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a specific spin effect in general relativity. It is shown that for the Kerr black hole the gravimagnetic ratio, i.e. the coefficient at the GM, equals unity (just as for the charged Kerr hole the gyromagnetic ratio equals two). The equations of motion obtained for relativistic spinning particle in external gravitational field differ essentially from the Papapetrou equations.

The generalization of the gravitational field equations is explored that give rise to the orbits of structureless (geodesic) particle motion as well as the orbits of particles with spin. In the simplest case these equations are equivalent to the Einstein equations with cosmological member of the Cartan-Killing metric of the manifold of the Anti De Sitter group, projected on the Anti De Sitter universe as base space. These equations are generally Einstein's equations with a source nonlinear in the curvature, combined with the equations suggested by C.N. Yang for a gauge theory of gravitation with third derivatives of the metric. The general case is found to be geometrically more complex, with torsion in the higher dimensional space.

Mathisson-Papapetrou equations are solved numerically to obtain trajectories of spinning test particles in (the meridional section of) the Kerr spacetime. The conditions p_σS^μσ = 0 are used to close the system of equations. A spin-curvature interaction may in principle lead to considerable deviations from geodesic motion. However, in astrophysical situations probably no large spin effects can be expected for the values of spin consistent with a pole-dipole test-particle approximation.

The energy first integral for a spinning particle in the Papapetrou model, in the Schwarzschild background, is calculated and then transformed according to Barker & O'Connell center-of-mass shift law. Thus its expression reduces approximatively to that of non-spinning test particle.

We study the Papapetrou-Dixon-Souriau model for the motion of a charged massive particle. We discuss supplementary conditions and the role of different families of observers. In this framework we consider spin-gravity and spin-electromagnetism coupling terms for an orbit in the Reissner-Nordström background, generalizing the picture given in the literature.

The Frenet-Serret approach is applied in several ways to a familiar but still not geometrically well understood example: circular orbits in black hole spacetimes. An invariant spacetime Frenet-Serret frame approach is useful in understanding the properties of these orbits and of Fermi-Walker transport along them, and provides a visual interpretation of the geometry of this family of orbits. Closely related to the spacetime frame for these special curves are the relative Frenet-Serret frames that may be defined with respect to a family of test observers on the spacetime. The latter connect more directly to our 3-dimensional intuition about the tangent, normal, and binormal to a curve in ordinary space. These absolute and relative frames together help interpret the effects of space curvature, and the gravitoelectric and gravitomagnetic effects on circular orbiting test particles and on their gyroscopic frames of reference.

In nonrelativistic mechanics noninertial observers studying accelerated test particle motion experience a centripetal acceleration which, once interpreted as a centrifugal force acting on the particle, allows writing the particle's equation of motion in a Newtonian form, simply by adding the inertial force contribution to that of the external forces in the acceleration-equals-force equation. In general relativity centripetal and centrifugal acceleration generalizing the classical concepts must be properly (geometrically) defined. This requires a relative Frenet-Serret frame approach based on a family of test observers.

Due to the resemblance between Maxwell and the gravitomagnetic equations obtained in the weak field and slow motion limit of General Relativity, one can ask if it is possible to amplify a seed intrinsic rotation or spin motion by a gravitomagetic dynamo, in analogy with the well-known dynamo effect. Using the Galilean limits of the gravitomagnetic equations, the answer to this question is negative, due to the fact that a "magnetic" Galilean limit for the gravitomagnetic equations is physically inconsistent.

For observers in a rotating reference system the polarization plane of linearly polarized light undergoes an apparent rotation. This effect, which is of relevance in the gravitational field of rotating bodies, is sometimes called the "gravitational Faraday effect". Here I investigate whether the gravitational Faraday effect would cause a rotation of the polarization plane relative to the image of the light source.

We give a general derivation of the metric of a spinning body of any shape and composition using linearized general relativity theory (LGRT), and also obtain the same metric using a simple transformation argument. The latter derivation makes it clear that the linearized metric contains only the Eddington γ and α (≡ 1) parameters, so no new parameter is involved in any frame–dragging or Lense–Thirring (LT) effects. We then calculate the precession of an orbiting gyroscope in a general gravitational field, described by a Newtonian potential (gravito-electric field) and a vector potential (gravito-magnetic field). Finally we do a multipole analysis and give the general spherical harmonics expansion of the precession in terms of multipoles of the scalar and vector potentials, i.e., moments of the density distribution. In particular, in regard to the Gravity Probe B (GP-B) experiment, we find that the effect of the Earth's quadrupole moment J₂ on the geodetic precession is large enough to be measured by GP-B (a previously known result), but the effect on the LT precession is somewhat beyond the expected GP-B accuracy.

Classical torques are the first to come into one's mind when thinking about sources of systematic errors in Gravity Probe B (GP-B) Relativity Science Mission, which is why they have been most thoroughly investigated. In this paper we give a brief survey of these studies and their results. First, the classification of all known non-relativistic torques that is used in the GP-B Error Tree is given, and the Error Tree spreadsheet is described. Second, the theory of electrostatic support dependent torques is outlined, including the universal expression for the housing-fixed components of the torques, the definition of the fifteen torque coefficients and their relation to the rotor shape. Examples of the final formulas for the drift rates of several particular support dependent and support independent torques are presented. Finally, contributions of different groups of torques to the total expected experimental error are compared, and the top contributing torques are discussed.

Data analysis is one of the most essential components of the Gravity Probe B (GP-B) experiment. We discuss three main problems that need to be resolve in the data reduction: 1)direct observation of the GP-B gyroscope's precession angle; 2)precise estimation of the gyroscope's geodetic and frame-draggig average drift rates; 3)gyroscope-telescope matching: elimination of the reference direction pointing error. We formulate these problems from the point of view of the modern control theory and describe our approach to the GP-B data analysis as the "bank" of filters that estimate the model-dependent system state vectors and calculate corresponding covariance matrices. For the problems 1), 2), and 3) estimation recursive algorithms are obtained based on the two-step nonlinear filtering approach and these filters will be used in the GP-B data reduction.

This paper is structured in four sections. In the first one we briefly describe the gravitomagnetic field and the Lense-Thirring effect in general relativity. In the second and third sections we review the method proposed to measure the Lense-Thirring effect by analyzing the orbits of the two laser-ranged satellites LAGEOS and LAGEOS II. We then report on our measurement of the Lense-Thirring effect by analyzing the data of LAGEOS and LAGEOS II with the orbital programs GEODYN-SOLVE, using the Earth's Gravitational Models JGM-3 and EGM-96, and the method explained in section 2. The first detection was obtained in 1995, the most accurate measurements were obtained in 1998 using EGM-96. Finally, in the fourth section, we briefly describe the proposed LARES experiment to measure the Lense-Thirring effect with accuracy of about 2%–3%.

Detection of the Lense-Thiiring precession of the orbit of an Earth satellite has been a challenge for decades, but as yet, no dedicated mission has been flown to accomplish this goal. The difficulty is not the ability to observe the precession of an orbit with sufficient precision, but rather, it is the uncertainty in the knowledge of the even zonal harmonics of the Earth's gravity field which limits the experiment using existing satellites. As the gravity models for the Earth have improved, there has been a re-examination of whether these errors still limit this fundamental test of general relativity. While this goal is tantalizingly close, it is argued here that the uncertainties are still too large for a confident and robust test of general relativity. However, the prospects for a successful test will improve significantly with the launch of the GRACE gravity mission.

A possible new galvanogyroscopic effect which is the rotational (gravitomagnetic) analogue of Hall effect in electromagnetism has been proposed. As a consequence of a generalized Ohm's law, the effect of the Coriolis force on the conduction current is predicted to give rise to an azimuthal potential difference V_gg in a spinning rotor carrying radial electric current i_r. The potential difference developed by the galvanogyroscopic effect is proportional to angular velocity Ω and to the electric current both. An experiment is proposed for measuring the earth's gravitomagnetic field by using galvanogyroscopic effect. Due to the Lense-Thirring effect potential difference about 10^-19V will be developed in a current carrying conductor on a platform which is nonrotating relative to the distant stars.

The analysis of a large data set (from 1990 to 1995) of VLBI group delays has provided a new γ estimate (γ = 1.0008 ± 0.0012), highlighting some critical issues in treating data which have undergone processing in many different steps and places.

The possibility of quantum tests of the Lense–Thirring effect is analyzed. These quantum tests are (i) atomic interferometry with spin flip, (ii) atomic interferometric measurement of the acceleration, and (iii) Hughes–Drever type experiments. With the present day techniques it is not possible to observe the Lense–Thirring effect quantum mechanically.

The relativistic precession model for quasi periodic oscillations, QPOs, in low mass X-ray binaries is reviewed. The behaviour of three simultaneous types of QPOs is well matched in terms of the fundamental frequencies for geodesic motion in the gravitational field of the accreting compact object, for reasonable star masses and spin frequencies. The model ascribes the higher frequency kHz QPOs, the lower frequency kHz QPOs and the horizontal branch oscillations to the Keplerian, periastron precession and nodal precession frequencies of matter inhomogeneities orbiting close to the inner edge of the accretion disk. The remarkable correlation between the centroid frequency of QPOs in both neutron star and black hole candidate low mass X-ray binaries is very well fit by the model. Some testable predictions are described. QPOs from low mass X-ray binaries might provide an unprecedented laboratory to test general relativity in the strong field regime.

The X-ray observation of black hole candidates provides a valuable tool to probe regions very close to the central black hole, where strong-field relativistic effects become important. Recent studies have shown that these effects seem to manifest themselves in the observed X-ray spectrum, in terms of the shape of X-ray continuum and the profile of emission lines, and in the X-ray light curves, in terms of certain quasi-periodic oscillations (QPOs). The latter will be the focus of this paper. We will review the proposed observational evidence for gravitomagnetic precession in black hole candidates, in light of new observational data and alternative models. Quantitative comparison will be made between the data and the models. We will comment on recent theoretical and observational efforts to address whether the gravitomagnetic precession of accreted, orbiting matter around a rotating black hole is a viable process for producing the observed QPOs. The results are inconclusive. We will mention a few areas where further progress can be made to possibly shed more light on the issue.

Since the initial discoveries with the Rossi X-ray Timing Explorer (RXTE) in 1996 of kilohertz quasi-periodic oscillations (kHz QPOs) and burst oscillations in a number of low-mass X-ray binaries (LMXBs) containing low-magnetic-field neutron stars, a very active field has developed. I briefly summarize some of the developments since those early days, which include the discovery of the first accreting millisecond pulsar, further claims of the detection of strong-field general-relativistic effects, intriguing correlations of the kHz QPO properties with QPOs at lower frequencies, strong challenges for the beat-frequency interpretation of the twin kHz QPO peaks and a number of new theoretical ideas, some of which involving frame dragging. Twenty LMXBs have now been seen to exhibit periodic or quasi-periodic phenomena with frequencies exceeding 10^2.5Hz. Although the commensurabilities between the observed frequencies that suggest a beat-frequency interpretation have conclusively been shown to be not precise, the preponderance of the evidence still is in favour of the idea that there exists some kind of beat-frequency relation between the twin kHz peaks. However, that evidence is only coming from 4 of the 17 sources showing twin kHz QPOs.

We discuss a possibility that observed radiation from active galactic nuclei (AGN) can be visibly modulated by collisions between stars (or secondary black holes) and an accretion disc around a central black hole. It is proposed that material swept out of the disc partly obscures the innermost region of the disc, while contributing by its own radiation. Under suitable circumstances, an individual orbiter can produce periodic modulation. If the central black hole is rotating and the companion star is at a low orbit, the Lense-Thirring orbital precession will show up in the signal and parameters of the central black hole can be thus determined.

Influential and recent selected works concerning mutual interaction between electromagnetic fields and a rotating (Kerr) black hole are summarized. Vacuum solutions (exact and perturbative), electrodynamic approximation of test particles in electrovacuum fields, and force-free magnetohydrodynamic (MHD) flows are mentioned. Equations and corresponding boundary conditions which govern waves in the background magnetosphere are formulated, keeping properly all the dragging terms which become important near the black-hole horizon.

In this work we analyze the most significant manifestations of the Lense-Thirring effect in the superfluid component of a neutron star, which consist in the line vortex diffraction, the lowering of the vortex density and the covariant Magnus force. These provide a very interesting framework for studying the Lense-Thirring effect in quantum coherent systems.

Neutron stars offer a unique opportunity to explore the interplay of strong gravitational fields and matter at super-nuclear density via relativistic effects such as frame-dragging. I will explain how a detailed description of the Lense-Thirring precession of orbits can be obtained using the equations of general relativity. The calculations presented go beyond the usual weak field approximations for the frame-dragging and quadrupole precessions and provide an approximate expression for the precession of orbits in the region close to a neutron star. The results in this report show that the exact strong field effects are very important for neutron stars which are described by a stiff equation of state: the precession frequencies for these stars differ greatly from the usual weak field approximations.

We will briefly review how we investigate the modes of oscillation trapped within the inner region of accretion disks by the strong-field gravitational properties of a black hole (or a compact, weakly-magnetized neutron star). The focus here will be on the 'corrugation'(c)–modes, nearly incompressible perturbations of the inner disk. The lowest c–mode has an eigenfrequency which corresponds to the Lense–Thirring frequency in the weak-field or slow-rotation limit. Its radial extent is a decreasing function of the angular momentum, so a major part of the disk is excited for slowly rotating black holes. We compare the c and g('gravity ')–mode frequencies with stable frequency features detected in the X-ray power spectra of two galactic 'microquasars'.

I study the generation and evolution of magnetic fields in the plasma surrounding a rotating black hole. Attention is focused on effects of the gravitomagnetic potential. The gravitomagnetic force appears as battery term in the generalized Ohm's law. The generated magnetic field should be stronger than fields generated by the classical Biermann battery. The coupling of the gravitomagnetic potential with electric fields appears as gravitomagnetic current in Maxwell's equations. In the magnetohydrodynamic induction equation, this current re-appears as source term for the poloidal magnetic field, which can produce closed magnetic structures around an accreting black hole. In principle, even self-excited axisymmetric dynamo action is possible, which means that Cowling's anti dynamo theorem does not hold in the Kerr metric. Finally, the structure of a black hole driven current is studied.

It has recently been suggested that gravitomagnetic precession of the inner part of the accretion disk, possibly driven by radiation torques, may be responsible for some of the 20–300 Hz quasi-periodic X-ray brightness oscillations (QPOs) observed in some low-mass binary systems containing accreting neutron stars and black hole candidates. We have explored warping modes of geometrically thin disks in the presence of gravitomagnetic and radiation torques. We have found a family of overdamped, low-frequency gravitomagnetic (LFGM) modes all of which have precession frequencies lower than a certain critical frequency ω_crit, which is ~1 Hz for a compact object of solar mass. A radiation warping torque can cause a few of the lowest-frequency LFGM modes to grow with time, but even a strong radiation warping torque has essentially no effect on the LFGM modes with frequencies ≳10^-4 Hz. We have also discovered a second family of high-frequency gravitomagnetic (HFGM) modes with precession frequencies that range from ω_crit up to slightly less than the gravitomagnetic precession frequency of a particle at the inner edge of the disk, which is 30 Hz if the disk extends inward to the innermost stable circular orbit around a 2M_⊙ compact object with dimensionless angular momentum cJ/GM² = 0.2. The highest-frequency HFGM modes are very localized spiral corrugations of the inner disk and are weakly damped, with Q values as large as 50. We discuss the implications of our results for the observability of Lense-Thirring precession in X-ray binaries.

The purpose of this work is to provide a critical analysis of the classical papers of H. Thirring [Phys. Z., 19, 33 (1918); Phys. Z., 22, 29 (1921)] and J. Lense and H. Thirring [Phys. Z., 19, 156 (1918)] on rotating masses in the relativistic theory of gravitation and to render them accessible to a wider circle of scholars. An English translation of these papers is presented which follows the original German text as closely as possible. This is followed by a concise account of the significance of the results of these papers as well as the possibility of measuring the gravitational effects of rotating masses.

The following sections are included:

• THE MAIN PROPERTIES OF THE NEW SOLUTION

• DEFINITION OF THE LINEAR ELEMENT AND PROOF THAT IT SATISFIES THE FIELD EQUATIONS

• PROOFS FOR THE PROPERTIES ENUMERATED

• SOME ADDITIONAL THEOREMS AND CONSIDERATIONS ABOUT THE SOLUTION

A method for the derivation of the equations of motion of test particles in a given gravitational field is developed. The equations of motion of spinning test particles are derived. The transformation properties are discussed and the equations of motion are written in a covariant form.

The equations of motion for spinning test particles are discussed for particles characterized by the condition Sⁱ⁴ = 0.

A technique is proposed to use the gyroscopic stability of a spinning artificial satellite to measure the apparent Coriolis force (Lense–Thirring effect) predicted by general relativity. It is established that the primary satellite ought to be shielded from atmospheric buffeting by a secondary or tender satellite that would accompany the primary test vehicle.

Other perturbations considered are interaction with the magnetic field of the earth, and interaction of any gravitational quadrupole moment of the satellite with the gradient of the earth's gravitational field. These interactions are sufficiently small that design of a vehicle to avoid excessive interaction should be possible.

The spinning shielded satellite technique proposed here could make possible a rather accurate measurement of the curvature of space near the earth. This curvature manifests itself in a precession of the spin (the de Sitter–Fokker effect), which is about a factor of 12 larger than the Lense–Thirring effect and thus could provide a logical first step for such a program.

Development of the exterior, or tender, satellite alone would offer a number of subsidiary advantages. It could be used with a dummy interior satellite ot obtain drag-free trajectories and thus provide a very accurate tool to study air drag, as well as gravitation and geodetics. By using an interior satellite with very different nuclear composition the tender could be used to establish the equivalence of gravitational and inertial mass to greater accuracy.

The Experimental Basis of Einstein's Theory.—Einstein's theory of gravitation, the general theory of relativity, has been accepted as the most satisfactory description of gravitational phenomena for more than forty years. It is a theory of great conceptual and structural elegance, and it is designed so that it automatically agrees in the appropriate limits with Galileo's observation of the equality of gravitational and inertial mass, with Newton's mechanics of gravitating bodies, and with Einstein's special theory of relativity. Leaving aside the very important matter of elegance, we wish in this section to examine the experimental basis of the theory. This basis consists of the three points of limiting agreement with earlier results just mentioned, together with certain astronomical evidence…

The following sections are included:

• Introduction

• Description of the experiment

• Analysis of gyro errors

• Summary and conclusion

• REFERENCES

The bound geodesies (orbits) of a particle in the Kerr metric are examined. (By "bound" we signify that the particle ranges over a finite interval of radius, neither being captured by the black hole nor escaping to infinity.) All orbits either remain in the equatorial plane or cross it repeatedly. A point where a nonequatorial orbit intersects the equatorial plane is called a node. The nodes of a spherical (i.e., constant radius) orbit are dragged in the sense of the spin of the black hole. A spherical orbit near the one-way membrane traces out a helix-like path lying on a sphere enclosing the black hole.

Wilkins has pointed out that, in addition to the periodicities associated with circular orbits in the equatorial plane of a Kerr geometry, there exist periodicities connected with the longitudinal motion of particles. We extend this result to spherical orbits of charged particles in a Kerr-Newman geometry. We give explicit examples of nonspherical orbits, illustrating dragging of inertial frames and these longitudinal periodicities. The relevance of these results for the physics of collapsed objects is discussed.

Cette Note montre comment la récente découverte d'un pulsar membre d'un système binaire pourrait fournir des informations très importantes pour la vérification de la Relativité générale ainsi que pour la connaissance de la structure et des processus d'émission des pulsars. On présente des estimations des principaux effets relativistes que l'on peut espéres observer dans un tel système.

In 1918, Lense and Thirring calculated that a moon orbiting a rotating planet would experience a nodal dragging effect due to general relativity. We describe an experiment to measure this effect to 1% with two counter-orbiting drag-free satellites in polar earth orbit. In addition to tracking data from existing ground stations, satellite-to-satellite Doppler ranging data are taken near the poles. New geophysical information is inherent in the polar data.

We describe a new method of measuring the Lense-Thirring relativistic nodal drag using LAGEOS together with another similar high-altitude, laser-ranged satellite with appropriately chosen orbital parameters. We propose, for this purpose, that a future satellite such as LAGEOS II have an inclination supplementary to that of LAGEOS. The experiment proposed here would provide a method for experimental verification of the general relativistic formulation of Mach's principle and measurement of the gravitomagnetic field.

According to general relativity, the calculated rate of motion of lunar perigee should include a contribution of 19.2 msec/yr from geodetic precession. We show that existing analyses of lunar–laser-ranging data confirm the general-relativistic rate for geodetic precession with respect to the planetary dynamical frame. In addition, the comparison of Earth-rotation results from lunar laser ranging and from very long-baseline interferometry (VLBI) shows that the relative drift of the planetary dynamical frame and the extragalactic VLBI reference frame is small. The estimated accuracy is about 10%.

We analyzed lunar laser-ranging data, accumulated between 1970 and 1986, to estimate the deviation of the precession of the Moon's orbit from the predictions of general relativity. We found no deviation from this predicted de Sitter precession rate of nearly 2 angular sec per century (sec/cy), to within our estimated standard error of 0.04 sec/cy. This standard error, 2% of the predicted effect, incorporates our assessment of the likely contributions of systematic errors, and is about threefold larger than the statistical standard error.

New observations of the binary pulsar B1913 + 16 are presented. Since 1978 the leading component of the pulse profile has weakened dramatically by about 40%. For the first time, a decrease in component separation is observed, consistent with expectations of geodetic precession. Assuming the correctness of general relativity and a circular hollow-cone–like beam, a fully consistent model for the system geometry is developed. The misalignment angle between pulsar spin and orbital momentum is determined, giving direct evidence for an asymmetric kick during the second supernova explosion. It is argued that the orbital inclination angle is 132°.8 (rather than 47°.2). A prediction of this model is that PSR B1913 + 16 will no longer be observable after the year 2025.