In the weak gravitational field approximation, Maxwelllike gravitational equations are
the set of four partial differential equations that describe the properties of
two components of gravitational field and relate them to their sources, mass
density and mass current density. These equations have the same form as in gravitoelectromagnetism and in Lorentzinvariant theory of gravitation,
and are used here to show that gravitational wave has the speed of gravitation
which is close to the speed of light as speed of electromagnetic wave.
Contents

Due to McDonald, ^{[1]} the first
who used Maxwell equations to describe gravitation was Oliver Heaviside.^{[2]} ^{[3]} The point
is that in weak gravitational field the standard theory of gravitation could be
written in the form of Maxwell equations with two gravitational constants. ^{[4]}
^{[}^{5}^{]}
In the 80ties Maxwelllike equations were considered in
the Wald book of general relativity.^{[}^{6}^{]} In the 90ties this
approach was developed by Sabbata,^{[}^{7}^{]} ^{[}^{8}^{]} Lano, ^{[}^{9}^{]} Sergey Fedosin. ^{[}^{10}^{]} ^{[1}^{1}^{]} ^{[1}^{2}^{]} ^{[1}^{3}^{]} The ways of experimental determination of gravitational waves properties are developed in papers of Raymond Y. Chiao. ^{[1}^{4}^{]} ^{[1}^{5}^{]} ^{[1}^{6}^{]} ^{[1}^{7}^{]} ^{[1}^{8}^{]}
Main sources: Lorentzinvariant theory of gravitation
and Gravitoelectromagnetism
Field equations in Lorentzinvariant theory of
gravitation and field
equations in a weak
gravitational field according to the Einstein field equations for general
relativity have the form:
where:
From these equations the wave equations are followed: ^{[1}^{1}^{]}
These equations as in gravitoelectromagnetism are the
gravitational analogs to Maxwell's equations for electromagnetism.
Main source: Selfconsistent gravitational constants
and Vacuum constants
Proceeding from the analogy of gravitational and
Maxwell's equations, enter the following values: as gravitoelectric
permittivity (like electric
constant);
as
gravitomagnetic permeability (like vacuum
permeability). If the speed of gravitation is equal to the speed of
light, then ^{[18]} and
The gravitational characteristic impedance of free space
for the gravitational waves could be defined as:
If then
the gravitational characteristic impedance of free space is equal to: ^{[1}^{7}^{]}
.
As in electromagnetism, the characteristic impedance of
free space plays the dominant role in all radiation processes, such as in a
comparison of radiation impedance of gravitational wave antennas to the value
of this impedance in order to estimate the coupling efficiency of these
antennas to free space. The numerical value of this impedance is extremely small,
and therefore it is very hard up to now to make receivers with proper impedance
matching.
The gravitational
vacuum wave equation is a secondorder partial differential equation
that describes the propagation of gravitational waves through vacuum in absence
of matter. The homogeneous form of the equation, written in terms of either the
gravitational field strength or
the gravitational torsion field , has the form:
For waves in one direction the general solution of the
gravitational wave equation is a linear superposition of flat waves of the form
and
for virtually any
wellbehaved functions and of dimensionless argument where
is the angular
frequency (in radians per second),
is the wave vector (in radians per meter), and
Considering the following relationships between inductions and strengths of
gravitational fields: ^{[}^{20}^{]}
where is
gravitational displacement field, is
torsion (gravitomagnetic) field strength, we could obtain the following
interconnections:
This equation determines the wave impedance (gravitational characteristic
impedance of free space) in
standard form like the
case of electromagnetism:
In practice, always so that the
total dipole gravitational radiation of each system of bodies, viewed from
infinity tends to zero due to mutual compensation of emissions of individual
bodies. As a result, the main components of the emission of gravitational
radiation are quadrupole and higher harmonics. With this in mind, the wave
equation in Lorentzinvariant theory of
gravitation, recorded in the quadrupole approximation, are sufficiently
accurate approximation to the results of general relativity and covariant theory of gravitation.
If in the system of bodies are
bodies with electric charge which radiating electromagnetically, the balance is
disrupted and there is some uncompensated dipole gravitational radiation.
As a model of LC circuit,
consider the case of motion of ideal liquid fluid in a closed pipe under
influence of gravitational and some additional forces. Suppose that this
circuit has a tubular coil, passing through which the fluid due to its rotation
creates torsion field in the space and passes portion of its energy to the
field. The tubular coil plays the role of spiral inductance in
electromagnetism. In another part of the circuit is a source that accumulates
the liquid. For the possibility of fluid motion in two opposite directions in
this circuit on both sides of the source are pistons with springs. This allows
for periodical converting of energy of fluid motion into energy of compression
springs, approximately equated to change in gravitational energy of fluid. The
pistons with springs act like a capacitor in the
circuit, and gravitational voltage is
equal to difference of gravitational potentials, and gravitational mass current
is
mass of liquid per unit time throw a section of the pipe..
Gravitational voltage on gravitational inductance is:
Gravitational mass current through gravitational
capacitance is:
Differentiating these equations with respect to the time
variable, we obtain:
Considering the following relationships for gravitational
voltages and currents:
we obtain the following differential equations for
gravitational oscillations:
Furthermore, considering the following relationships
between gravitational voltage and mass of the liquid:
and mass current with flux of torsion field:
the above oscillation equation for could be rewritten in the form:
This equation has the partial solution:
where
is the resonance frequency in absence of energy loss, and
is the gravitational characteristic impedance of LC
circuit, which is equal to the ratio of the gravitational voltage amplitude to
the mass current amplitude.
Main source: Gravitational induction
According to the second equation for gravitational
fields, after a change in time of there appear circular field (rotor) of , having the opportunity to lead in
rotation matter: ^{[}^{10}^{]}
If the vector field crosses a certain area , then we can calculate the flux of this
field through this area:
where –
the vector normal to the element area .
Let’s find partial derivative in equation (2) with
respect to time
with a minus sign and integrate over the area, taking into account the equation
(1):
In the integration was used
Stokes theorem, replacing the integration of the rotor vector over the area on
the integration of this vector over a closed circuit. In the right side of (3)
is a term, equal to the work on transfer of a unit
mass of matter on closed loop , covering an area . 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 area
changing and is expressed in occurrence of rotational forces acting on particles
of matter. The direction of motion of the matter will be such that field of
the matter will be sent in the same direction as initial torsion field which created the circulation of the matter (this is
contrary to the Lenz's_law in electromagnetism).
Source:
http://sergf.ru/muen.htm