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 vacuum permittivity);
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