In this article, we develop gravitoelectromagnetism (GEM) as a formal relativistic theory of gravitation characterized by a field potential four-vector in a flat (Minkowski) spacetime frame. By reorganizing the Lense-Thirring spacetime metric of linearized general relativity, we have determined the appropriate field potential four-vector and the field strength tensor which generates dynamics in the GEM field. The general covariant equation of motion of the spin four-vector of a gyroscope driven by the GEM field strength tensor provides the expected form of the frame-dragging and geodetic effects after introducing a gyromagnetic ratio

Linearization of Einstein’s general theory of relativity yielded a linear wave equation for the weak metric tensor (i.e., weak gravitational field potential), describing propagating gravitational waves [

Characterized by the weak metric tensor defining a four-vector gravitational field potential, GEM may be understood as the basic relativistic theory of gravitation within a four-dimensional flat (Minkowski) spacetime frame. This basic relativistic theory of gravitation may now be identified as gravitoelectrodynamics. Two fundamental dynamical properties of gravitoelectrodynamics are (i) gravitational waves which have been detected and confirmed in LGO experiment [

There are two important features of GEM as generally formulated. The first feature is that gravitational waves in Minkowski spacetime frame are a clear manifestation of an underlying gravitational field potential four-vector, which as stated above, has been defined in terms of the weak metric tensor. The second feature is that the dynamical properties of moving masses in GEM are generally determined through the geodesic equation generated by Christoffel symbols within the standard framework of general relativity theory.

Calculations of frame-dragging and geodetic effects have generally been based on the geodesic equation for the spin tensor or spin four-vector of a gyroscope in covariant form [

It follows from the standard approaches referred to above that the Christoffel symbols derivable from the weak metric tensor provide both gravitoelectric and gravitomagnetic field intensity components, such that the Christoffel symbols may be correctly interpreted as the general form of gravitational field strength, analogous to the electromagnetic field strength tensor. Hence in linearized general relativity theory, the geodetic equation of motion generated by the Christoffel symbols may be considered equivalent to a corresponding equation of motion generated by a GEM field strength tensor defined within the flat spacetime frame.

It is particularly important to note that in the four-dimensional flat spcetime frame where the linearized general relativity theory applies, covariant equations of motion are generated by a field strength tensor derived from an appropriate field potential four-vector. Determining an appropriate GEM field potential four-vector and the corresponding field strength tensor may then prove useful in finding alternative forms of the covariant equation of motion for the spin four-vector of a gyroscope in a weak gravitational field.

The main purpose of the present article is to determine an appropriate GEM potential four-vector and the corresponding field strength tensor to develop a formal theory of gravitoelectomagnetism comparable to the analogous standard theory of electromagnetism. Dynamical properties of masses e.g., gyroscopes, in a gravitational field may then be determined directly through the GEM equations of motion.

This article is organized as follows. The GEM field potential four-vector and field strength tensor are determined in section 2, where the appropriate form of the velocity four-vector is obtained. Dynamics in the GEM field and the Lorentz transformation of the field strength tensor to the rest frame of moving masses are presented in section 3. The equations of motion and spin dynamics of a gyroscope in the gravitational field of the earth are derived in section 4 where a gyromagnetic ratio is introduced in the definition of the gravitational magnetic moment of the gyroscope. The spin equation of motion is interpreted in relation to frame-dragging and geodetic effects. The conclusion is presented in section 5.

The field potential four-vector is the basic element of the GEM field. To determine the appropriate form of the field potential four-vector, the natural starting point is the Lense-Thirring spacetime metric

We have thus used the Lense-Thirring spacetime metric to determine the appropriate form of the GEM field potential four-vector

Having determined the appropriate form of the field potential four-vector, we can now introduce the desired GEM field potential strength tensor

The GEM field equations follow from

And Maxwell-type equations obtained as:

The equation of motion of a body of mass

The equation of motion in the rest frame of the body is generated by the field strength tensor

For dynamics in the electromagnetic (EM) field, J D Jackson obtained the Lorentz transformation operator in a general three-dimensional form in [

In performing the Lorentz transformation according to

Substituting

It is important to note the sign differences, especially in the first components

We observe that, in the non-relativistic case,

As a first application of the model of dynamics generated by the GEM field strength tensor, we consider the time evolution of the spin angular momentum of a gyroscope in the gravitational field of the earth. Noting that the Newtonian gravity force, i.e., the GEM gravitoelectric force of the earth, is a centripetal force that keeps the gyroscope rotating around the center of the earth, we arrive at the interpretation that the gyroscope’s rest frame is a rotating frame. Hence, the equation of motion of the gyroscope spin angular momentum may be determined in terms of its rate of change with time in a rotating frame [

We begin by noting that, apart from the torque generated by the GEM field strength tensor

For later convenience, we introduce a dimensionless four-vector

The property that the spin angular momentum and velocity four-vector are orthogonal:

To determine the physical value of the parameter

Introducing the dimensionless four-vector

Following the procedure well elaborated by Jackson in [

This is a fully relativistic gyroscope spin equation of motion in the GEM field, which corresponds directly to the Thomas precession equation for the electron spin in the electromagnetic field in [

The gyroscope gravitational charge

An important fundamental physical property that we have introduced in the definition of the gravitational magnetic moment in

It emerges in the present work that if we take the gyroscope gravitational charge

We clarify the comparison of the non-relativistic spin equation of motion

The formal theory of gravitoelectrodynamics which we have developed here as the basic relativistic theory of (weak) gravitation in a flat (Minkowski) spacetime frame is consistent and agrees fully with the well-established approaches based on analogy with electromagnetism and on approaches based on the geodesic equation of motion driven by the Christoffel field strengths (symbols) within the framework of linearized general relativity theory.

The property that the GEM gravitational field potential four-vector and field strength tensor have been determined from the standard Lense-Thirring spacetime means that gravitoelectrodynamics as formulated in the present work has precisely the same physical features of the familiar theories of linearized general relativity based on electromagnetic analogy or on the Christoffel field strength driven geodesic equations of motion. The advantage of gravitoelectrodynamics is that it provides a simpler systematic Lorentz invariant picture of motion of masses in (weak) gravitational fields in flat spacetime frames.

The covariant forms of the equations of motion generated by the GEM field strength tensor allow Lorentz transformations from the gravitational field to the rest frame of the moving mass, thus determining the effective gravitoelectric Lorentz force driving the general motion of the mass and the effective spin-orbit interaction gravitomagnetic force driving the dynamics of the spin angular momentum of the moving mass.

The covariant equation of motion of a gyroscope spin four-vector in the GEM field is derivable in a form similar to the BMT or Thomas precession equation of motion of spin angular momentum in standard classical electrodynamics. Expressed in component forms, the resulting equation of motion of the three-component spin angular momentum takes the general form generated by a combination of frame-dragging and geodesic components of the effective gravitomagnetic field in the rest frame of the gyroscope. The general form of the equation of motion is specified by the Lorentz transformation factor which determines relativistic effects depending on the speed and a gyromagnetic ratio of the gyroscope. In the non-relativistic approximation where the Lorentz transformation factor is unity, setting the gyromagnetic ratio to a value

We note that, besides the frame-dragging and geodesic effects arising in the dynamics of spin angular momentum of a gyroscope (moving mass) which we have treated here as an example, the formal theory of gravitoelectrodynamics as formulated in the present work now provides a systematic mathematical framework for studying a number of important phenomena in the gravitational field, energy loss by colliding masses such as black holes, neutron stars, etc., investigations of such phenomena may yield exciting results in the quest towards a deeper understanding of dynamics in the gravitational field and related events in cosmology.

We would like to acknowledge Maseno University for providing us with an enabling environment to do research. Great thanks go all to members of the academic staff who took time to read our work and gave valuable contributions.