**The gravitational torsion
field** is the force field acting
on the masses and bodies in translational or rotational motion, which is the
second component of the 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 the 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
the frequency, that is s^{-1}.

Torsion field plays an important
role in the gravitational model of strong
interaction.

## Contents·
1 The torsion
field in the Lorentz-invariant theory of gravitation o 1.1 Heaviside's equations o
1.2 The rotating
particle in torsion field o 1.3 Gravitational vector potential ·
2 The torsion
field (gravitomagnetic field) in the general theory of relativity ·
3 The effects
associated with the torsion field · 4 The analogies with electrodynamics · 5 See also · 6 References ·
7 External links |

In the Lorentz-invariant theory of gravitation (LITG)
the force of gravitation is considered as the two-component force, which
depends on the gravitational field strength
(gravitational acceleration) and the gravitational torsion field
:

where and
are the mass and the velocity of
the body moving in the gravitational field.

The
torsion in LITG up to a constant factor corresponds to the strength of the
so-called gravitomagnetic field in the general relativity (GTR). The
cause of emergence of the torsion in LITG is the necessity to comply with the
principle of Lorentz covariance for the gravitational field potentials in
inertial reference frames. ^{[1]}

As
the gravitational field strength, the torsion field contributes to the gravitational tensor, the gravitational
stress-energy tensor, as well as the energy density of gravitational field:

where
is the speed of propagation
of the gravitational influence or speed of gravity, is
the gravitational constant,

the
vector of energy flux density of gravitational field or Heaviside vector:

and Lagrangian for a particle in
gravitational field.

**Heaviside's
equations**

Torsion field is included in three of the four
differential Heaviside's equations:

where:

·
is the mass current density,

·
is the moving mass density,

·
is the velocity of the mass flux,
which creates the gravitational field and torsion.

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 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 LITG, the gravitational field strength and
torsion define
the components of the real physical gravitational force, which can be
substantiated at the quantum level like electromagnetic force. The torsion occurs
whenever there is any movement of the mass. Since any motion can be divided
into two parts – the rotational and translational, respectively, then we can
talk about two kinds of torsion. Torsion outside the rotating sphere with
angular momentum has a dipole form: ^{[2]}

The
presence of 1/2 in the formula for reflects the fact that the
gravitational moment of the axisymmetric body is equal to the half of its
angular momentum. In case of the rectilinear motion of the body, the torsion of
gravitational field equals:

where
is the velocity of the body, is the gravitational field strength
of the body at the point where the torsion
is determined, and the
strength is taken in view of the propagation
delay of the gravitational perturbation.

In
the general case, the torsion from the arbitrarily moving point mass can be
expressed through the gravitational field strength produced by it:

where is the unit vector, directed from
the point mass to the point where the torsion is determined, taken at earlier
time taking into account the delay.

**The
rotating particle in torsion field**

The formula for the moment of force acting on the rotating
particle with the spin in torsion field , is written as follows:

Since
the particle is a top with the spin , then in the presence
of the moment of forces the particle would precess along the direction of the field . This follows from the
equation of rotational motion:

Since
the moment of forces is perpendicular to the spin and the torsion , then the same is true
for the increment of the spin during the time . Perpendicularity of and leads to the precession of the spin
of the particle at the angular velocity around the
direction of . The last equality
follows from the fact that ,
and the quantity , where is the angle between and , and the increment of the angle is measured from the
projection of the vector on the plane perpendicular to the
vector up to the projection of the vector on this plane.

In
the presence of the non-uniform torsion field the particle with the spin will be dragged to the region of
the stronger field. From the equations of LITG the expression follows for such
force:

The
mechanical energy of the particle with the spin in the torsion field will be
equal to:

Torsion
field and gravitational field strength are closely related to the potentials of
the field and expressed by the formulas:

where
is the scalar potential, – the vector potential
of the 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 will set the additional energy of
such particle due to action of the field, and is a part of the generalized
momentum of the particle.^{[3]} The vector potential of the 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 , and the gravitational tensor, which consists of and is obtained as the four-curl of .

In contrast to Newtonian mechanics, in the general theory of relativity (GTR), the motion of the test particle
(and the rate of clock) in the 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 the 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 the weak gravitational field and at low velocities of
the particles we can consider separately the gravitomagnetic and the
gravitational forces acting on the test body, and the strength of the
gravitomagnetic field and the gravitomagnetic force are described by the
equations similar in the form to the corresponding equations of
electromagnetism.

We
shall consider the motion of the test particle in the vicinity of the rotating
spherically symmetric body with the mass
and the angular momentum . If the particle with
the mass is moving at the velocity ( 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 . In the CGS system of physical units we shall
have:

And
if the rotating mass is located at the origin of coordinates and is the radius vector to the observation
point, the strength of the gravitomagnetic field at this point is: ^{[4]}

where
is the gravitational constant.

The
last formula coincides (except for the coefficient) with the similar formula
for the field of the magnetic dipole with the magnetic
dipole moment equal to . 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. ^{[5][6]} The
same geometric meaning is obtained by the gravitomagnetic field .

In contrast
to this, in LITG it is assumed that the force of torsion arises already in the
Minkowski space, as a magnetic force. In GTR the equivalent gravitomagnetic
force is considered in the 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 the additional numerical factors
appear in comparison with the expressions for the force and the torsion field
in LITG. For rectilinear motion of bodies the formulas for the torsion in LITG
and in GTR coincide. ^{[7]}

In the case of strong fields and relativistic
velocities the torsion field can not be considered
separately from the gravitational field, since the dependence of the metric
tensor on the value of the fields is beginning to affect, and the field
equations become interrelated and nonlinear. In this case LITG turns into the covariant theory of gravitation (CTG). ^{[8]} 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 the
test particle near the rotating massive body. In the SI system of physical
units the angular velocity of precession is equal to and is directed against the
direction of torsion field .

·
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. ^{[9]} However, the
measurement errors were too large, in the range of 256-128%, impeding the
measurements. ^{[10] }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 is directed similarly to the spin
of the Earth , and from CTG it
follows:

where is the distance from the Earth’s
center to the gyroscope in the orbit near the pole.

Measuring
allows direct determination of the
speed of gravitation propagation in CTG. The same formula for
angular velocity of precession is valid in general relativity, but averaged
over the whole orbit. ^{[11]} With
for the we must have the value 0.0409 arc
seconds per year (here km for the satellite
Gravity Probe B, is the Earth's radius, the altitude
of the satellite is 640 km).

·
The geodetic precession (de Sitter effect) occurs in parallel transfer
of the angular momentum vector in the 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: ^{[12]}

where is the velocity of motion of the
gyroscope in the orbit of the Earth, is the mass of the
Earth, is the orbital angular velocity of the
gyroscope’s rotation.

The
angular velocity of the geodetic precession is perpendicular to the velocity of
the gyroscope and to the 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 the change of the rate of clock and is far beyond the
current accuracy of the experiment.

Since the gravitational torsion field is similar to
the magnetic field in electrodynamics, and the angular momentum of the particle
is similar to the dipole magnetic moment, then according to LITG it allows us
to interpret the Lense-Thirring effect at the example
of rotation of the electrically charged test particle around the 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 the direction of its
orbital angular momentum will precess relative to the
axis of the dipole magnetic moment of the body, as in the orbital Lense-Thirring effect.

If
the particle has its proper magnetic moment and spin, then there will be
interaction of the magnetic moments of the particle and the body. The energy of
this interaction depends on the mutual orientation of the magnetic moments. The
magnetic field of the body will tend to establish the magnetic moment of the
particle in the 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 the dipole magnetic moment of the body,
similar to the Schiff effect.

Another
effect is associated with the orbital motion of the particle with the magnetic
moment and spin relative to the charged body. In the rest frame of the center
of mass of the particle, the particle itself is rotating, and the body is
rotating around this particle. The orbital rotation of the charged body creates
the magnetic field acting on the magnetic moment of the particle. This leads to
the precession of the magnetic moment and the spin of the particle relative to
the 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 the geodetic
precession (the other two thirds arise from the curvature of the spacetime
around the body due to changing of the metric).

·
Gravitational field strength

·
Maxwell-like gravitational equations

·
Lorentz-invariant theory of gravitation

·
Covariant theory of gravitation

·
Gravitational stress-energy tensor

·
Magnetic field

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413.

2.
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3.
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5.
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- In
Search of gravitomagnetism, NASA, 20 April 2004.
- Gravitomagnetic London
Moment-New test of General Relativity?
- Measurement of Gravitomagnetic
and Acceleration Fields Around Rotating Superconductors M. Tajmar, et. al., 17 October 2006.
- Gravitational
torsion field in Russian