# Physics/Essays/Fedosin/Gravitational torsion field

Gravitational torsion field is a force field acting on masses and bodies in translational or rotational motion, which is the second component of gravitational field in the Lorentz-invariant theory of gravitation and in the covariant theory of gravitation. By its action the torsion field is similar to magnetic field in electromagnetism (see Maxwell-like gravitational equations). The term torsion field in this meaning was introduced by Sergey Fedosin in 1999. The torsion field dimension in the system of physical units SI is the same as for frequency, that is s$^{-1}$.
Torsion field plays an important role in gravitational model of strong interaction.

## Torsion field in Lorentz-invariant theory of gravitation

In Lorentz-invariant theory of gravitation (LITG) the force of gravitation is considered as two-component force, which depends on gravitational field strength (gravitational acceleration) $~\mathbf {\Gamma }$ and gravitational torsion field $~\mathbf {\Omega }$:
    $~\mathbf {F} =M\mathbf {\Gamma } +M\left[\mathbf {v} \times \mathbf {\Omega } \right],$
where $~M$ and $~\mathbf {v}$ are the mass and velocity of a body moving in gravitational field.
The torsion field $~\mathbf {\Omega }$ in LITG up to a constant factor corresponds to the strength of so-called gravitomagnetic field $~\mathbf {H_{g}}$ in general relativity (GTR). The cause of emergence of the torsion field in LITG is the necessity to comply with the principle of Lorentz covariance for gravitational field potentials in inertial reference frames.
As the gravitational field strength, the torsion field contributes to gravitational tensor, gravitational stress-energy tensor, the energy density of gravitational field
    $~u=-{\frac {1}{8\pi G}}\left(\Gamma ^{2}+c^{2}\Omega ^{2}\right),$
where $~c$ is the speed of light, $~G$ is the gravitational constant,
as well as to the vector of energy flux density of gravitational field or Heaviside vector:
    $~\mathbf {H} =-{\frac {c^{2}}{4\pi G}}[\mathbf {\Gamma } \times \mathbf {\Omega } ],$
and Lagrangian for a particle in gravitational field.

### Heaviside's equations

Torsion field is included in three of the four differential Heaviside's equations:
    $~\nabla \cdot \mathbf {\Omega } =0,$
    $~\nabla \times \mathbf {\Omega } ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right)={\frac {1}{c^{2}}}\left(-4\pi G\rho \mathbf {v_{\rho }} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),$
    $~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},$
where:
* $~\mathbf {J}$ is the mass current density,
* $~\rho$ is the moving mass density,
* $~\mathbf {v_{\rho }}$ is the velocity of the mass flux, which creates the gravitational field and torsion field.
From the first equation it follows, that the torsion field has no sources, and hence, the torsion field lines are always closed as in case of magnetic field. According to the second equation, the torsion field is produced by the motion of matter and the change in time of gravitational field strength. The third equation implies effect of gravitational induction.
According to LITG, the gravitational field strength $~\mathbf {\Gamma }$ and torsion field $~\mathbf {\Omega }$ define the components of real physical gravitational force, which can be substantiated at the quantum level like electromagnetic force. The torsion field occurs whenever there is any movement of mass. Since any motion can be divided into two parts – the rotational and translational, respectively, then we can talk about two kinds of torsion field. Torsion field outside a rotating sphere with angular momentum $~\mathbf {L}$ has a dipole form:
    $~\mathbf {\Omega } ={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {r} /r)\mathbf {r} /r}{r^{3}}}.$
The presence of 1/2 in the formula for $~\mathbf {\Omega }$ reflects the fact that the gravitational moment of axisymmetric body is equal to the half of its angular momentum. In case of rectilinear motion of a body, the torsion field equals:
    $~\mathbf {\Omega } ={\frac {\mathbf {V} }{c^{2}}}\times \mathbf {\Gamma } ,$
where $~\mathbf {V}$ is velocity of the body, $~\mathbf {\Gamma }$ is gravitational field strength of the body at the point where the torsion field $~\mathbf {\Omega }$ is determined, and the strength $~\mathbf {\Gamma }$ is taken in view of propagation delay of gravitational perturbation.
In general case, the torsion field of arbitrarily moving point mass can be expressed through gravitational field strength $~\mathbf {\Gamma }$ produced by it:
    $~\mathbf {\Omega } ={\frac {1}{c}}\mathbf {e} _{r}\times \mathbf {\Gamma } ,$
where $\mathbf {e} _{r}$ is the unit vector, directed from the point mass to the point where the torsion field is determined, taken at earlier time (see retarded time).

### Rotating particle in torsion field

The formula for the moment of force acting on a rotating particle with the spin $~\mathbf {L}$ in torsion field $~\mathbf {\Omega }$, is written as follows:
    $~\mathbf {K} ={\frac {1}{2}}\mathbf {L} \times \mathbf {\Omega } .$
Since the particle is a top with the spin $~\mathbf {L}$, then in the presence of the moment of forces $~\mathbf {K}$ the particle would precess along the direction of the field $~\mathbf {\Omega }$. This follows from the equation of rotational motion:
    $~\mathbf {K} ={\frac {d\mathbf {L} }{dt}}.$
Since the moment of forces $~\mathbf {K}$ is perpendicular to the spin $~\mathbf {L}$ and the torsion $~\mathbf {\Omega }$, then the same is true for the increment of the spin $~d\mathbf {L}$ during the time $~dt$. Perpendicularity of $~\mathbf {L}$ and $~d\mathbf {L}$ leads to precession of the spin of the particle at angular velocity $~\mathbf {w} =-{\frac {\mathbf {\Omega } }{2}}$ around direction of $~\mathbf {\Omega }$. The last equality follows from the fact that $~\mathbf {K} ={\frac {d\mathbf {L} }{dt}}={\frac {1}{2}}\mathbf {L} \times \mathbf {\Omega }$, and the quantity $~w={\frac {dL}{L\sin Qdt}}={\frac {d\varphi }{dt}}$, where $~Q$ is the angle between $~\mathbf {\Omega }$ and $~\mathbf {L}$, and the increment of the angle $d\varphi$ is measured from the projection of the vector $\mathbf {L}$ on the plane perpendicular to the vector $~\mathbf {\Omega }$ up to the projection of the vector $\mathbf {L} +d\mathbf {L}$ on this plane.
In the presence of the non-uniform torsion field the particle with the spin $~\mathbf {L}$ will be dragged to the region of the stronger field. From the equations of LITG the expression follows for such force:
    $~\mathbf {F} ={\frac {1}{2}}\nabla \left(\mathbf {L} \cdot \mathbf {\Omega } \right).$
The mechanical energy of the particle with the spin in torsion field will be equal to:
    $~U=-{\frac {1}{2}}\mathbf {L} \cdot \mathbf {\Omega } .$

### Gravitational vector potential

Torsion field and gravitational field strength are closely related to the potentials of the field and expressed by the formulas:
    $~\mathbf {\Omega } =\nabla \times \mathbf {D} ,$
    $~\mathbf {\Gamma } =-\nabla \psi -{\frac {\partial \mathbf {D} }{\partial t}},$
where $~\psi$ is the scalar potential, $~\mathbf {D}$ – the vector potential of gravitational field.
The significance of potentials is in the fact that if a test particle of unit mass is placed in an external gravitational field, then $~\psi$ will set the additional energy of such particle due to action of the field, and $~\mathbf {D}$ is a part of the generalized momentum of the particle. The vector potential of gravitational field contributes to the torsion field and gravitational field strength, but the torsion field is not directly dependent on the scalar potential. When equations are wrote in four-dimensional form, field potentials form the gravitational four-potential $~D_{\mu }$, and gravitational tensor, which consists of $~\mathbf {\Omega }$ and $~\mathbf {\Gamma } ,$ is obtained as the four-curl of $~D_{\mu }$.

## Torsion field (gravitomagnetic field) in general theory of relativity

In contrast to Newtonian mechanics, in the general theory of relativity (GTR), the motion of test particle (and the rate of clock) in gravitational field depends on whether the body, the source of the field, rotates or not. The influence of rotation affects even in the case when distribution of masses in the source in its reference frame does not change with time (for example, there is cylindrical symmetry with respect to the axis of rotation). Gravitomagnetic effects in weak fields are extremely small. In a weak gravitational field and at low velocities of particles we can consider separately gravitomagnetic and gravitational forces acting on the test particles, and the strength of gravitomagnetic field and the gravitomagnetic force are described by the equations similar in the form to corresponding equations of electromagnetism.
Let us consider the motion of a test particle in the vicinity of a rotating spherically symmetric body with the mass $~M$ and the angular momentum $~\mathbf {L}$. If the particle with the mass $~m$ is moving at the velocity $~v\ll c$ ($~c$ is the speed of light), then the particle would be influenced, in addition to the gravitational force, by the gravitomagnetic force, directed (like the Lorentz force) perpendicular to both the velocity and the strength of the gravitomagnetic field $~\mathbf {H_{g}}$. In the CGS system of physical units we shall have:
    $~\mathbf {F} ={\frac {m}{c}}\left[\mathbf {v} \times 2\mathbf {H_{g}} \right].$
And if the rotating mass is located at the origin of coordinates and $~\mathbf {r}$ is the radius vector to the observation point, the strength of the gravitomagnetic field at this point is:
    $~\mathbf {H_{g}} ={\frac {G}{c}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {r} /r)\mathbf {r} /r}{r^{3}}},$
where $~G$ is the gravitational constant.
The last formula coincides (except for the coefficient) with the similar formula for the field of magnetic dipole with the magnetic dipole moment equal to $~\mathbf {L}$. In GTR gravitation is not an independent physical force. Gravitation in GTR is rather reduced to the curvature of spacetime and is treated as a geometric effect, and is equated to the metric field. The same geometric meaning is obtained by the gravitomagnetic field $~\mathbf {H_{g}}$.
In contrast to this, in LITG it is assumed that the force of torsion field arises already in Minkowski space, as a magnetic force. In GTR the equivalent gravitomagnetic force is considered in Riemannian space, where gravitation has tensor, not vector character. Therefore, the spin of gravitons in GTR is assumed twice greater than in the vector theory of LITG. Hence, in a number of works on gravitoelectromagnetism in GTR, in the expressions for the force and the gravitomagnetic field additional numerical factors appear in comparison with the expressions for the force and torsion field in LITG. For rectilinear motion of bodies the formulas for torsion field in LITG and in GTR coincide.

## Effects associated with torsion field

In the case of strong fields and relativistic velocities, the torsion field cannot be considered separately from the gravitational field, since dependence of metric tensor on the magnitude of fields begins to have an effect, and the field equations become interconnected and nonlinear. In this case LITG turns into the covariant theory of gravitation (CTG). In weak fields as separate effects of the torsion field the following effects are considered:
* Dragging of inertial reference frames. This is precession of the spin and orbital moments of test particle near the rotating massive body. In the SI system of physical units the angular velocity of precession is equal to $~\mathbf {w} =-{\frac {\mathbf {\Omega } }{2}}$ and is directed against the direction of torsion field $~\mathbf {\Omega }$.
** The orbital Lense-Thirring effect, which leads to precession, that is to the rotation of the normal of the elliptical orbit of the particle relative to the vector of gravitational torsion field of the rotating body. This effect is vectorially added to the standard general relativistic precession of the pericenter (43" per century for Mercury), which does not depend on the rotation of the central body. The orbital Lense-Thirring precession was first measured for the satellites LAGEOS and LAGEOS II.
** The spin Lense-Thirring effect (or the Schiff precession) is expressed in the precession of gyroscope, located near the rotating body. If we consider the gyroscope as a spinning top, then the axis of this top will periodically change its direction in space with the precession frequency. Checking this precession was one of the goals of the experiment of Gravity Probe B, conducted by NASA in 2005-2007 on the satellite with the orbit passing through the pole of the Earth. However, the measurement errors were too large, in the range of 256-128%, impeding the measurements. The experimental measurement of the Schiff precession is the test for the theories of GTR and CTG with respect to the formulas for the precession. At the Earth’s pole the angular velocity of precession $~w$ is directed similarly to the spin of the Earth $~L$, and from CTG it follows:
    $~w=-{\frac {\Omega }{2}}={\frac {GL}{2c^{2}r^{3}}},$
where $~r$ is distance from the Earth’s center to the gyroscope in the orbit near the pole.
The same formula for angular velocity of precession is valid in general relativity, but averaged over the whole orbit. For the $~w$ there is the value 0.0409 arc seconds per year (here $~r=R_{e}+640=6378+640=7018$ km for the satellite Gravity Probe B, $~R_{e}$ is the Earth's radius, the altitude of the satellite is 640 km).
* The geodetic precession (de Sitter effect) occurs in parallel transport of the angular momentum vector in curved spacetime. For the Earth-Moon system, moving in the field of the Sun, the geodetic precession is 1.9" per century; the precise astrometric measurements revealed this effect, which coincided with the predicted within the error range of 1 %. The geodetic precession of gyroscopes on the satellite Gravity Probe B coincided with the predicted value (the rotation of the axis by 6.606 arc seconds per year in the plane of the satellite’s orbit) with the accuracy more than 1 %. The formula for the de Sitter precession in GTR has the form:
    $~\mathbf {w} ={\frac {3}{2}}{\frac {GM}{c^{2}r^{3}}}\mathbf {r} \times \mathbf {V} ={\frac {3}{2}}{\frac {GM}{c^{2}r^{2}}}\mathbf {\omega } _{o},$
where $~\mathbf {V}$ is velocity of motion of the gyroscope in the orbit of the Earth, $~M$ is the mass of the Earth, $~\mathbf {\omega } _{o}$ is the orbital angular velocity of the gyroscope’s rotation.
The angular velocity of geodetic precession is perpendicular to velocity of gyroscope and to acceleration of Earth's gravitation, coinciding with the direction of the orbital angular velocity of rotation. Therefore, at the pole of the Earth the geodetic precession is perpendicular to the spin precession in the Schiff effect, the angular velocity of which is directed along the axis of the Earth’s rotation.
* The gravitomagnetic time shift. In the weak fields (for example, near the Earth) this effect is masked by the standard special and general relativistic effects of change of the rate of clock and is far beyond the current accuracy of experiments.

## Analogies with electrodynamics

Since the gravitational torsion field is similar to magnetic field in electrodynamics, and the angular momentum of a particle is similar to dipole magnetic moment, then according to LITG it allows us to interpret the Lense-Thirring effect as an example of rotation of electrically charged test particle around a body attracting it, if this body has the dipole magnetic moment. Due to the law of conservation of the orbital angular momentum, the orbital plane of the particle tends not to change its position in space. However, during the rotation of the particle around the body, the charge of the particle will be influenced by the additional Lorentz force from the magnetic field of the body, which is perpendicular to the particle velocity. In the stationary state, the velocity and the orbital angular momentum of the particle will not change in the magnitude, but the orbit of the rotating particle and direction of its orbital angular momentum will precess relative to the axis of dipole magnetic moment of the body, as in orbital Lense-Thirring effect.
If the particle has its proper magnetic moment and spin, then there will be interaction of magnetic moments of the particle and the body. The energy of this interaction depends on the mutual orientation of magnetic moments. The magnetic field of the body will tend to establish the magnetic moment of particle in direction of the field, but the particle has the spin, tending to preserve its direction in space. Therefore there will be precession of the particle’s spin relative to the axis of dipole magnetic moment of the body, similar to the Schiff effect.
Another effect is associated with the orbital motion of a particle with magnetic moment and spin relative to a charged body. In the rest frame of the center of momentum of the particle, the particle itself is rotating, and the body is rotating around this particle. The orbital rotation of charged body creates the magnetic field acting on magnetic moment of the particle. This leads to the precession of the magnetic moment and the spin of the particle relative to magnetic field from the orbital rotation of the body. Now we shall replace the magnetic field by the torsion of the gravitational field, and the charges by masses. Then it turns out that the angular velocity of the spin precession of the particle would be one third of the angular velocity of geodetic precession (the other two thirds arise from the curvature of spacetime around the body due to changing of space time metric).