Gravitoelectromagnetic Field Strength Tensor in Flat Spacetime Frame: Dynamics of a Gyroscope in the Earth’s Gravitational Field

In this article, we develop gravitoelectromagnetism (GEM) as a formal relativistic theory of gravitation characterized by a field potential fourvector 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 η = 4 as an internal dynamical property of the gyroscope.

Introduction

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 [1]–[4]. The subsequent interpretation that the weak metric tensor constitutes scalar and vector components of a four-vector potential led to a two-component theory of gravitation composed of gravitoelectric and gravitomagnetic fields, collectively referred to as gravitoelectromagnetism (GEM) [4]–[8].

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 [9] (ii) frame-dragging (Lense-Thirring) and geodetic (spin-orbit interaction) effects, associated with gravitomagnetism, which have been confirmed in LARES, LAGEOS and Gravity Probe-B experimens [10]–[12].

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 [4], [13]–[19]. Elements of the metric tensor derived from an appropriately defined spacetime metric, notably the Lense-Thirring spacetime metric for GEM [20], have been used to determine the Christoffel symbols for explicit solutions of the gyroscope spin equations.

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 Gem Field Potential Four-Vector and Field Strength Tensor

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 $d{s}^2$, which has been used to extract elements of weak gravity metric tensor for determining the Christoffel symbols in studies of dynamical properties such as frame-dragging and geodetic effects in linearized relativity theory. We adopt the Lense-Thirring spacetime metric in the form [4], [6]–[8], [15]–[20]: which we reorganize to bring the terms involving $d\tau$ together as: where $\tau$ is the proper time and $\overrightarrow{v}$ is the velocity of a moving mass, such as a gyroscope in the GEM field. Introducing the speed of light $c$ as appropriate in the temporal component of (2) gives the form: where we now recognize the GEM field scalar and vector components $1/c(1+2\varnothing )c,2\overrightarrow{h}$ together with the corresponding velocity four-vector components $c,\overrightarrow{v}$. This allows us to express the coefficient of the temporal component in (3) in standard spacetime covariant form: where we identify $A_g^{\mu }$ as the GEM field potential four-vector and $V^{\mu }$ as the velocity four-vector of the moving mass obtained in respective forms:

(1) $$\begin{array}{r}d{s}^2=(1+2\varnothing )d{\tau }^2+2\overrightarrow{h}.d\overrightarrow{r}d\tau -(1-2\varnothing )d{\overrightarrow{r}}^2\end{array}$$

(2) $$\begin{array}{r}d{s}^2=\{(1+2\varnothing )+2\overrightarrow{h}.\overrightarrow{v}\}d{\tau }^2-(1-2\varnothing )d{\overrightarrow{r}}^2;\overrightarrow{v}=\frac{d\overrightarrow{r}}{d\tau }\end{array}$$

(3) $$\begin{array}{r}d{s}^2=\{\frac{1}{c}(1+2\varnothing )c+2\overrightarrow{h}.\overrightarrow{v}\}d{\tau }^2-(1-2\varnothing )d{\overrightarrow{r}}^2\end{array}$$

(4) $$\begin{array}{r}d{s}^2=A_g^{\mu }{V}_{\mu }d{\tau }^2-(1-2\varnothing )d{\overrightarrow{r}}^2;\mu =0,1,2,3\end{array}$$

$$\begin{array}{r}{A}_g^{\mu }=({\varnothing }_g,{\overrightarrow{A}}_g);{\varnothing }_g=\frac{1}{c}(1+2\varnothing );\end{array}$$

(5) $$\begin{array}{r}{\overrightarrow{A}}_g=2\overrightarrow{h};V^{\mu }=(c,-\overrightarrow{v})\end{array}$$

We have thus used the Lense-Thirring spacetime metric to determine the appropriate form of the GEM field potential four-vector $A_g^{\mu }$ and the corresponding form of the velocity four-vector $V^{\mu }$ of a mass moving in the GEM field. The form of the velocity four-vector reveals an important property that in the GEM field, masses move with velocity $-\overrightarrow{v}$ directed in the opposite sense relative to the direction of the GEM vector potential or gravitomagnetic field derivable from the vector potential. Note that in the model developed in [4], [6]–[8], [17], the GEM vector potential is defined with a negative sign and the velocity of the moving mass would then be in the opposite direction, specified as $\overrightarrow{v}$, thus characterizing the same dynamical property that the velocity of the moving mass and the GEM vector potential or gravitomagnetic field are oppositely directed.

The Gem Field Strength Tensor and Field Equations

Having determined the appropriate form of the field potential four-vector, we can now introduce the desired GEM field potential strength tensor $F_g^{\mu v}$, defined in flat spacetime frame as the general curl of $A_g^{\mu }$ in the form: composed of gravitoelectric and gravitomagnetic field intensities ${\overrightarrow{E}}_g,{\overrightarrow{B}}_g$ respectively, obtained as: where $A_g^{\mu },{\varnothing }_g,{\overrightarrow{A}}_g$ are defined in (5).

(6) $$\begin{array}{r}{F}_g^{\mu v}={\mathrm{\partial }}^{\mu }{A}_g^v-{\mathrm{\partial }}^v{A}_g^{\mu }\end{array}$$

(7) $$\begin{array}{r}{\overrightarrow{E}}_g=-\overrightarrow{\mathrm{\nabla }}{\varnothing }_g-\frac{1}{c}\frac{\mathrm{\partial }{\overrightarrow{A}}_g}{\mathrm{\partial }t};{\overrightarrow{B}}_g=\overrightarrow{\mathrm{\nabla }}\times {\overrightarrow{A}}_g\end{array}$$

The GEM field equations follow from $A_g^{\mu },{\varnothing }_g,{\overrightarrow{A}}_g,F_g^{\mu v}$ in (6) and its dual ${\tilde{F}}_g^{\mu v}$ in the form: where ${\zeta }^{\mu v\alpha \lambda }$ is the totally anti-symmetric Levi-Civita rank-4 tensor defined in standard form [21], while $J^v$ is the source current density, i.e., the linear momentum density four-vector of the source of the gravitational field (e.g., the earth) with mass density $\rho$ and mass current density, $\overrightarrow{j}=\overrightarrow{p}=\rho \overrightarrow{u}$ where $\overrightarrow{u}$ is the velocity and $\overrightarrow{p}$ is the linear momentum density of the source. The GEM field (8) provide the field potential wave equations obtained as:

$$\begin{array}{r}{\mathrm{\partial }}_{\mu }{F}_g^{\mu v}=\frac{1}{c}{J}^v;{\mathrm{\partial }}_{\mu }{\tilde{F}}_g^{\mu v}=0;{\tilde{F}}_g^{\mu v}=\frac{1}{2}{\zeta }^{\mu v\alpha \lambda }{F}_{g\alpha \lambda };J^v=(\rho c,\overrightarrow{j});\overrightarrow{j}=\rho \overrightarrow{u}\end{array}$$

(8) $$\begin{array}{r}\mu ,v,\alpha ,\lambda =0,1,2,3\end{array}$$

(9) $$\begin{array}{r}{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }{A}_g^v=\frac{1}{c}{j}^v;{\mathrm{\partial }}_{\mu }{A}_g^v=0\end{array}$$

And Maxwell-type equations obtained as: where the second of (9) is the Lorentz gauge condition.

$$\begin{array}{r}\overrightarrow{\mathrm{\nabla }}.{\overrightarrow{E}}_g=\rho ;\overrightarrow{\mathrm{\nabla }}\times {\overrightarrow{E}}_g=-\frac{1}{c}\frac{\mathrm{\partial }{\overrightarrow{B}}_g}{\mathrm{\partial }t};\end{array}$$

(10) $$\begin{array}{r}\overrightarrow{\mathrm{\nabla }}.{\overrightarrow{B}}_g=0;\overrightarrow{\mathrm{\nabla }}\times {\overrightarrow{B}}_g=\frac{1}{c}\frac{\mathrm{\partial }{\overrightarrow{B}}_g}{\mathrm{\partial }t}+\frac{1}{c}\overrightarrow{j}\end{array}$$

Dynamics in the Gem Field

The equation of motion of a body of mass $m$ velocity-four vector $V^{\mu }$ and linear momentum four-vector $P^{\mu }$ the GEM field is generated by the field strength tensor $F_g^{\mu v}$ in the form: where $\tau$ is the proper time defined as usual, while $m_g$ is the gravitational charge, which is related to the inertial mass of the moving body according to Einstein’s equivalence principle as will be specified when defining the gravitational magnetic moment in section 4.

(11) $$\begin{array}{r}\frac{d{p}^{\mu }}{d\tau }=\frac{m_g}{c}{F}_g^{\mu v}{V}_v;p^{\mu }=m{V}^{\mu };m_g=\text{gravitational}\text{charge}\end{array}$$

The equation of motion in the rest frame of the body is generated by the field strength tensor $F_g^{\mu v^{{}^{\prime }}}$ in the rest frame obtained through a Lorentz transformation of $F_g^{\mu v}$ through the Lorentz operator $L$, with transpose $L^T$, according to:

(12) $$\begin{array}{r}{F}_g^{\mu v^{{}^{\prime }}}=L{F}_g^{\mu v}{L}^T\end{array}$$

For dynamics in the electromagnetic (EM) field, J D Jackson obtained the Lorentz transformation operator in a general three-dimensional form in [21]. Noting that in the corresponding GEM field, the velocity of the moving body is directed in the opposite sense according to the definition of $V^{\mu }$ in (5), we set, $\overrightarrow{v}\to -\overrightarrow{v}$, such that, $\overrightarrow{\beta }\to -\overrightarrow{\beta },{\beta }_j\to -{\beta }_j,j=1,2,3$ in the Jackson Lorentz transformation operator in [21] to obtain the Lorentz transformation operator in the GEM field dynamics in the form where the Lorentz transformation factor $\gamma ={\displaystyle \frac{1}{\sqrt{1-{\beta }^2}}}$.

$$\begin{array}{r}L=\left(\begin{array}{c}\begin{array}{c}\gamma \gamma {\beta }_1\gamma {\beta }_2\gamma {\beta }_3\\ \gamma {\beta }_{1}1+{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_1^2{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_1{\beta }_2{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_1{\beta }_3\end{array}\\ \gamma {\beta }_2{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_2{\beta }_{1}1+{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_2^2{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_2{\beta }_3\\ \gamma {\beta }_3{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_3{\beta }_1{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_3{\beta }_{2}1+{\displaystyle \frac{\gamma -1}{{\beta }^2}}{\beta }_3^2\end{array}\right);\end{array}$$

(13) $$\begin{array}{r}\overrightarrow{\beta }=\frac{\overrightarrow{v}}{c};{\beta }_j=\frac{v_j}{c};j=1,2,3;\beta =\left|\overrightarrow{\beta }\right|\end{array}$$

In performing the Lorentz transformation according to (12), we use the definitions of the field intensities ${\overrightarrow{E}}_g=(E_g^1,E_g^2,E_g^3),{\overrightarrow{B}}_g=(B_g^1,B_g^2,B_g^3)$ in (7) to express the field strength tensor $F_g^{\mu v}$ in (6) and the transformed field strength tensor $F_g^{\mu v^{{}^{\prime }}}$ in the rest frame in the matrix form:

$$\begin{array}{r}{F}_g^{\mu v}=\left(\begin{array}{c}0-E_g^1-E_g^2-E_g^3\\ E_g^{1}0-B_g^3{B}_g^2\\ E_g^2{B}_g^{3}0-B_g^1\\ E_g^3-B_g^2{B}_g^{1}0\end{array}\right);\end{array}$$

(14) $$\begin{array}{r}{F}_g^{\mu v^{{}^{\prime }}}=\left(\begin{array}{c}0-E_g^{{}^{\prime }1}-E_g^{{}^{\prime }2}-E_g^{{}^{\prime }3}\\ E_g^{{}^{\prime }1}0-B_g^{{}^{\prime }3}{B}_g^{{}^{\prime }2}\\ E_g^{{}^{\prime }2}{B}_g^{{}^{\prime }3}0-B_g^{{}^{\prime }1}\\ E_g^{{}^{\prime }3}-B_g^{{}^{\prime }2}{B}_g^{{}^{\prime }1}0\end{array}\right)\end{array}$$

Substituting $L,L^T=L,F_g^{\mu v}$ from (13), (14) into (12) gives $F_g^{{}^{\prime }\mu v}$ as expressed in (14). The transformed components $E_g^{{}^{\prime }j},B_g^{{}^{\prime }j},j=1,2,3$ constitute the gravitoelectric and gravitomagnetic field intensities $E_g^{{}^{\prime }j},B_g^{{}^{\prime }j},$in the rest frame in the form:

$$\begin{array}{r}\overrightarrow{E_g^{{}^{\prime }}}=\gamma \{{\overrightarrow{E}}_g-\overrightarrow{\beta }\times {\overrightarrow{B}}_g-\frac{\gamma }{\gamma +1}\overrightarrow{\beta }(\overrightarrow{\beta }.{\overrightarrow{E}}_g)\}\end{array}$$

(15) $$\begin{array}{r}\overrightarrow{B_g^{{}^{\prime }}}=\gamma \{{\overrightarrow{B}}_g+\overrightarrow{\beta }\times {\overrightarrow{E}}_g-\frac{\gamma }{\gamma +1}\overrightarrow{\beta }(\overrightarrow{\beta }.{\overrightarrow{B}}_g)\}\end{array}$$

It is important to note the sign differences, especially in the first components$,{\overrightarrow{E}}_g-\overrightarrow{\beta }\times {\overrightarrow{B}}_g$, ${\overrightarrow{B}}_g+\overrightarrow{\beta }\times {\overrightarrow{E}}_g$ of the GEM field intensities in the rest frame as compared to the corresponding components of the electromagnetic field obtained in [21]. Here, the sign differences are due to the property that in the GEM field dynamics, the velocity of a moving mass is directed in the opposite sense as determined from Lense-Thirring spacetime metric in (1)–(5). We observe that the form ${\overrightarrow{E}}_g-\overrightarrow{\beta }\times {\overrightarrow{B}}_g$ agrees with the gravito-Lorentz force obtained in [4], [17].

We observe that, in the non-relativistic case, $\gamma =1$ the gravitomagnetic field intensity ${\overrightarrow{B}}_g^{{}^{\prime }}$ in the rest frame in (15) takes the same form as the corresponding frame-dragging and geodetic field intensity obtained in the gyroscope spin equation of motion in [18], [19], noting that the geodetic component $\overrightarrow{\beta }\times {\overrightarrow{E}}_g$ is weaker by a factor $3/2$ compared to that in [18], [19] due to the modification of the spacetime metric by Eddington parameters, which we have not included in the choice of the original form of the Lense-Thirring metric in (1). We shall see how the factor $3/2$ arises through the introduction of a gyromagnetic ratio in the definition of the gravitational magnetic moment in the full relativistic treatment of the gyroscope spin equation of motion in the next section.

The Spin Equation of Motion in Gem Field

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 [21]–[23]. However, we provide a full relativistic form of the spin equation of motion in the GEM field, following closely the standard approach for the relativistic spin equation of motion in electrodynamics (BMT equation) well developed by Jackson in [21].

We begin by noting that, apart from the torque generated by the GEM field strength tensor $F_g^{\mu v}$, the property that the gyroscope rest frame is a rotating spacetime frame means that the time evolution of the spin angular momentum depends also on forces proportional to the velocity four-vector $V^{\mu }$, where we consider that the form of the proportionality parameter may be determined by both the field strength tensor and the acceleration four-vector ${\displaystyle \frac{d{V}^{\mu }}{d\tau }}$ where $\tau$ is the proper time defined in the gyroscope’s rest frame. The dynamics of the gyroscope’s spin angular momentum four-vector $S^{\mu }$ is therefore driven by a combination of the torque generated by the field strength tensor and turning forces proportional to the velocity four-vector in the form: where $A_1$, $A_2$, $A_3$ are constant parameters. The first term is the torque generated directly by the field strength tensor, while the last two terms are the forms of torque proportional to the velocity four-vector with field strength tensor and acceleration four-vector dependent proportionality parameters.

(16) $$\begin{array}{r}\frac{d{S}^{\mu }}{d\tau }=A_1{F}_g^{\mu v}{S}_v+\frac{A_2}{c^2}\left(S_{\alpha }{F}_g^{\alpha \lambda }{V}_{\lambda }\right)V^{\mu }+\frac{A_g}{c^2}\left(S_v\frac{d{V}^v}{d\tau }\right)V^{\mu }\end{array}$$

For later convenience, we introduce a dimensionless four-vector ${\beta }^{\mu }$ consistent with the definitions $V^{\mu },\overrightarrow{\beta }={\displaystyle \frac{\overrightarrow{v}}{c}}$ in (5), (13) in the form: which we substitute into (16) to express the spin equation in the form:

(17) $$\begin{array}{r}{\beta }^{\mu }=\frac{v^{\mu }}{c}=>{\beta }^{\mu }=(1,-\overrightarrow{\beta })\end{array}$$

(18) $$\begin{array}{r}\frac{d{s}^{\mu }}{d\tau }=A_1{F}_g^{\mu v}{S}_v+A_2\left(S_{\alpha }{F}_g^{\alpha \lambda }{\beta }_{\lambda }\right){\beta }^{\mu }+A_3\left(S_v\frac{d{\beta }^v}{d\tau }\right){\beta }^{\mu }\end{array}$$

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

$(S_{\mu }{V}^{\mu }=0)$ gives the relations: which can be used with (18) to obtain physical constrains yielding the parameter relations $A_1=A_2,A_3=-1$. Substituting into (18) and reorganizing provides the spin equation in general covariant form:

(19) $$\begin{array}{r}{S}_{\mu }{\beta }^{\mu }=0=>\frac{d{S}_{\mu }}{d\tau }{\beta }^{\mu }=-S_{\mu }\frac{d{\beta }^{\mu }}{d\tau }\end{array}$$

(20) $$\begin{array}{r}\frac{d{S}^{\mu }}{d\tau }=A_1{\{F_g^{\mu v}{S}_v+\left(S_{\alpha }{F}_g^{\alpha \lambda }{\beta }_{\lambda }\right){\beta }^{\mu }\}}^{\mu }-\left(S_v\frac{d{\beta }^v}{d\tau }\right){\beta }^{\mu }\end{array}$$

To determine the physical value of the parameter $A_1$, we assume that the spinning gyroscope has a gyromagnetic ratio $\eta$ and introduce a gravitational magnetic moment $M_g$ of the gyroscope in the form: where $m_g$ is the gyroscope gravitational charge introduced earlier in (11) and $\overrightarrow{s}$ is the three-component spin angular momentum vector in the rest frame. The gyroscope spin equation of motion in the rest frame takes the form: where ${\overrightarrow{B}}_g^{{}^{\prime }}$is the gravitomagnetic field in the rest frame, obtained in (15). Reducing the general covariant spin equation of motion (20) to the rest frame, noting the Lorentz transformations of the field strength in Equations (12), (15) and the spin angular momentum four-vector in the form [21]: and comparing with (22) gives the parameter $A_1$ and the general covariant spin equation in the final form:

(21) $$\begin{array}{r}{M}_g=\frac{\eta m_g}{2mc}\overrightarrow{s}\end{array}$$

(22) $$\begin{array}{r}\frac{d\overrightarrow{s}}{d\tau }=M_g\times {\overrightarrow{B}}_g^{{}^{\prime }}=>\frac{d\overrightarrow{s}}{d\tau }=\frac{\eta m_g}{2mc}\overrightarrow{s}\times {\overrightarrow{B}}_g^{{}^{\prime }}\end{array}$$

$$\begin{array}{r}{S}^{\mu }=(S_0,\overrightarrow{S});S_0=\gamma \overrightarrow{\beta }.\overrightarrow{s};\end{array}$$

(23) $$\begin{array}{r}\overrightarrow{S}=\overrightarrow{s}+\frac{{\gamma }^2}{\gamma +1}(\overrightarrow{\beta }.\overrightarrow{s}\overrightarrow{\beta })\end{array}$$

(24) $$\begin{array}{r}{A}_1=\frac{\eta m_g}{2mc};\frac{d{s}^{\mu }}{d\tau }=\frac{\eta m_g}{2mc}\{F_g^{\mu v}{S}_v+\left(S_{\alpha }{F}_g^{\alpha \lambda }{\beta }_{\lambda }\right){\beta }^{\mu }{\}}^{\mu }-\left(S_v\frac{d{\beta }^v}{d\tau }\right){\beta }^{\mu }\end{array}$$

Introducing the dimensionless four-vector ${\beta }^{\mu }$ as defined in (17) into the equation of motion in (11) and substituting the resulting form into the last term in (24) turns the equation into a form corresponding to the BMT equation describing the dynamics of the electron spin angular momentum four-vector in [21].

Following the procedure well elaborated by Jackson in [21], the covariant (24) provides the equation of motion for the gyroscope spin angular momentum $\overrightarrow{s}$ in the rest frame in the general form:

(25) $$\begin{array}{r}\frac{d\overrightarrow{s}}{d\tau }=\frac{m_g}{mc}\overrightarrow{s}\times \{(\frac{\eta }{2}-1+\frac{1}{\gamma }){\overrightarrow{B}}_g-(\frac{\eta }{2}-1)\frac{\gamma }{\gamma +1}(\overrightarrow{\beta }.{\overrightarrow{B}}_g)\overrightarrow{\beta }+(\frac{\eta }{2}-\frac{\gamma }{\gamma +1})\overrightarrow{\beta }\times {\overrightarrow{B}}_g\}\end{array}$$

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 [21], differing in form only in the sign of the last terms according to the relation $\overrightarrow{\beta }\to -\overrightarrow{\beta }$. The specification of the spin equation of motion in covariant form in (24) and in the gyroscope rest frame in (25) is completed by defining the gyroscope gravitational charge $m_g$ and gyromagnetic ratio $\eta$ to compare with the corresponding equations obtained through the geodesic equation of motion for the gyroscope spin angular momentum four-vector in [15], [16], [18], [19].

The gyroscope gravitational charge $m_g$ determines its response to the gravitational field and is generally interpreted as the gravitational mass. According to Einstein’s equivalence principle, the gravitational mass equals the inertial mass of a body. In [6]–[8], [24], the property that gravitational force (gravitoelectric field) is always negative has been applied to interpret gravitational charge as negative and, noting further that the gravitational field quantum, the graviton, is a spin-2 particle, the authors have introduced two forms of gravitational charge corresponding to the electric and magnetic components of the GEM field, defining gravitoelectric charge as $m_g=-m$ and gravitomagnetic charge as $m_g=-2m$. In contrast to the interpretation in [6]–[8], [24], we assign the same gravitational charge $m_g=\pm m$ to both gravitoelectric and gravitomagnetic components, where we may adopt the negative sign assignment for further physical interpretation.

An important fundamental physical property that we have introduced in the definition of the gravitational magnetic moment in (21) is the gyroscope gyromagnetic ratio η which has not been considered in earlier studies of the gyroscope spin motion [4], [5], [13]–[19]. It is clear that in defining the gravitational magnetic moment in [5], [8], [24], the authors have assumed that the gyroscope gyromagnetic ratio is 1, even though they do not specifically refer to gyromagnetic ratio as a dynamical property of the gyroscope.

It emerges in the present work that if we take the gyroscope gravitational charge $m_g=-m$ $m_g=-m$ and set the gyromagnetic ratio $\eta =4$ then the slow-motion non-relativistic case, $({\beta }^2\approx 0,\gamma =1)$ the relativistic (25) reduces to the form of the corresponding gyroscope spin equation obtained in [15], [16], [18], [19] according to: where in the bracket, the first term $2{\overrightarrow{B}}_g$ is the gravitomagnetic field which generates the frame-dragging effect, while the second term $3/2(\overrightarrow{\beta }\times {\overrightarrow{B}}_g)$ is the induced (spin-orbit interaction) gravitomagnetic field which generates the geodetic effect.

$$\begin{array}{r}{\beta }^2=0;\gamma =1;m_g=m;\eta =4;\end{array}$$

(26) $$\begin{array}{r}\frac{d\overrightarrow{s}}{d\tau }=\frac{1}{c}(2{\overrightarrow{B}}_g+\frac{3}{2}\overrightarrow{\beta }\times {\overrightarrow{B}}_g)\times \overrightarrow{s}\end{array}$$

We clarify the comparison of the non-relativistic spin equation of motion (26) in the rest frame of the gyroscope with the corresponding equations on frame-dragging and geodetic effects generally obtained in the stationary field case where the definitions of the GEM field intensities ${\overrightarrow{E}}_g,{\overrightarrow{B}}_g$ in (7) take the form:

$$\begin{array}{r}{\overrightarrow{E}}_g=-{\varnothing }_g;{\overrightarrow{B}}_g=\overrightarrow{\mathrm{\nabla }}\times {\overrightarrow{A}}_g;\end{array}$$

where we have introduced a factor $3/2$ in the frame dragging (LT) component for ease of comparison with the results in [18], [19] after setting the Eddington parameters $\alpha =\gamma =1$ in the respective gyroscope spin equations. Noting that multiplying the solution of (9) for the stationary vector potential ${\overrightarrow{A}}_g$ by the factor 4 is equivalent to the solution for the vector potential $\overrightarrow{h}$ obtained with a factor 4 in [18], [19], it is easily established that the predicted values of the frame-dragging and geodesic effects provided by (27) are exactly equal to the values obtained in [5], [15], [16], [18], [19]. Relativistic corrections to the frame-dragging and geodetic effects can be determined by using the relativistic form characterized by the Lorentz transformation factor $\gamma$ in (24). In that case, increasing the speed of the satellite carrying the gyroscope, giving a corresponding value $\gamma >1$, yields appropriate relativistic corrections to the frame-dragging and geodetic effects.

(27) $$\begin{array}{r}\frac{d\overrightarrow{s}}{d\tau }=\frac{1}{c}(\frac{1}{2}\overrightarrow{\mathrm{\nabla }}\times 4{\overrightarrow{A}}_g+\frac{3}{2}\overrightarrow{\mathrm{\nabla }}{\varnothing }_g\times \overrightarrow{\beta })\times \overrightarrow{s}\end{array}$$

Conclusion

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 $\eta =4$ reduces the spin equation of motion to the precise form of the standard gyroscope in spin equation of motion derived through the geodesic equation of motion for the spin four-vector. The precise agreement in the form of the spin equations of motion in both approaches means that the frame-dragging and geodetic effects determined in the GEM field strength tensor driven dynamics have exactly the same estimated values which have been measured to good accuracy in the LARES, LAGEOS and Gravity Probe-B experiments. As explained above, the general form of the spin equation of motion in GEM allows determination of relativistic corrections to frame-dragging and geodesic effects by appropriately increasing the speed of the satellite carrying the gyroscope, thereby increasing the Lorentz transformation factor to a relativistic value larger than unity.

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.