Gravitational induction is a property of gravitational field to
drive matter as a result of changing of torsion field flux.
Contents

One of the four equations of Lorentzinvariant theory of gravitation
(see also gravitomagnetism and Maxwelllike gravitational equations) has
the following form: ^{[1]}
where:
According to (1),
after a change in time of there appear circular field (rotor) of , having the opportunity to lead matter in rotation.
If the vector of torsion crosses a certain surface , then we can calculate the flux of this
field through this surface:
where –
the vector normal to the element of surface .
Let’s take the partial derivative
in equation (2) with
respect to time with the minus sign and integrate
over the surface, taking into account the equation (1):
In the integration was used
Stokes' theorem, replacing the integration over a surface of the curl of vector
on the integration of this vector over the boundary of the surface. In the right
side of (3) is a term, equal to the work on transfer
of a unit mass of matter on closed path , bounding the surface . By analogy with electromagnetism, this
work could be called gravitomotive force. In the middle of (3) is time derivative of the flux . According to (3),
gravitational induction occurs when the flux of field through a certain surface
changes and is expressed in occurrence of rotational forces acting on particles
of matter.
Gravitational induction can be
regarded as gravitational analogue of the law of electromagnetic induction. ^{[2]} ^{[3] [4]}
Just as in electromagnetism,
there are two different cases of gravitational induction. In the first case the
flux is
changed due to variable with a constant flow surface, bounded by a loop.
In the second case, the
torsion remains constant, but the flux changes due to changes in the area occupied
by the matter of the loop. For example, consider rubber hose filled with liquid
and arranged in a closed rectangular loop in the torsion field . Let the three sides of the loop are
fixed, while the fourth side extends with speed , increasing the area of the loop. Since
the flux through the loop changes, the liquid in the
hose begins to circulate. The direction of motion of the fluid will be such
that torsion field of the fluid will be sent in the
same direction as initial torsion field created the circulation of fluid (this
is contrary to the Lenz's_law in electromagnetism).
The second case, with expanding
of the loop, can also be considered using the expression for full gravitational
force:
where:
From (4)
for the unit mass, located only in torsion field , should:
The integral of field strength (5) around the contour of the loop gives
gravitomotive force, as the work of gravitational force on the displacement of
unit mass. This integral will be:
where is
the vector describing change in the area of the loop during time , arises due to the movement of one side
of the loop in the direction of velocity .
Expression (6) is the rate of
change of the flux of torsion field when the contour of the area changes.
Comparing (3), (5) and (6) we find for the induced field strength: . Thus, during of changing the flux liquid inside the hose comes in motion and
begins to circulate in the direction specified by the vector of induced field strength . Gravitational induction regards to the matter of the hose too, so that if
the hose is not attached, it will rotate synchronously together with its
contents.
The theory of phenomena of
gravitational induction can be explained also by means of differential
quantities. ^{[5]} If we assume that the flux of
torsion field instead of (2) is determined by the expression , where is the vector of a certain small
area, and torsion is homogeneous in this area, then the rate of change of the flux of torsion
field can be written:
Substituting (1) and (6):
From this, taking into account
(3) in the general case follows:
so in the case of changing of
field torsion , or in the case of changing of vector of
area when contour is intersecting the torsion
field, the flux of torsion field is changing and gravitomotive force is
creating. When the vector of area is changing gravitomotive force arises in the
sides of the loop, which move at the speed crossing lines of torsion field. The direction
of the force acting on matter of the loop is determined by vector product .
In covariant theory of gravitation (CTG) gravitational stressenergy tensor has the form: ^{[5]}
,
where –
speed of gravity, – gravitational
constant, – metric tensor, and gravitational tensor is calculated through
gravitational
fourpotential as
follows:
.
In weak field approximation, when
the curvature of spacetime can be set almost equal to zero, the equations of
CTG become close to equations of Lorentzinvariant
theory of gravitation. This causes the wave equations ^{[6]}
for potentials of gravitational field ( –
scalar potential, –
vector potential), and for field strength
and torsion (gravitomagnetic) field . In stationary case, the wave equations
of gravitational field become Poisson's equations of classical physics. In this
approximation components of gravitational stressenergy tensor can be written explicitly:
– energy
density of gravitational field,
, where
index and is the
vector of energy flux density of gravitational field or Heaviside vector.
Negative energy density and
energy flux lead to unique property inherent to gravitational field. This
property lies in the fact that the gravitational effect of induction between
two masses under certain conditions is not damped, and may increase in
amplitude, as in systems with positive feedback. For example, if two bodies are
attracted by gravitation and rotate in the same direction, then the change of
potential energy of gravitational field will transform into rotational energy of the bodies through gravitational induction.
Thus, the bodies will rotate each other, increasing torsion field around
them.
Described mechanism is proposed
to explain the nuclear forces between nucleons in atomic nuclei. ^{[5]} With proper arrangement of nucleons in nucleus
due to the gravitational induction nucleons spin up to a maximum angular
velocity. The result is a repulsive force of nucleons spins (in gravitoelectromagnetism these forces are
called gravitomagnetic forces) of such magnitude that are enough to compensate
the force of attraction of the nucleons from the field of strong gravitation. In such evaluating of
the forces acting in atomic nuclei, is used strong
gravitational constant.^{[7]}