Physics/Essays/Fedosin/Maxwell-like gravitational equations

In the weak gravitational field approximation, Maxwell-like gravitational equations are a 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 are presented in the same form as gravitoelectromagnetism and Lorentz-invariant theory of gravitation. They are used here to show that gravitational waves determine the speed of gravity which is close to the speed of light just as speed of electromagnetic waves determine the speed of light.

History
Due to McDonald, the first who used Maxwell equations to describe gravitation was Oliver Heaviside. 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.

In the 80-ties Maxwell-like equations were considered in the Wald book of general relativity. Since the 90s this approach has been used by Sabbata, Lano, Sergey Fedosin.

The ways of experimental determination of gravitational waves properties are developed in papers of Raymond Y. Chiao.

Maxwell-like equations can be found in many other recent works:

Field equations
Field equations in Lorentz-invariant theory of gravitation and field equations in a weak gravitational field according to the Einstein field equations for general relativity have the form:
 * $$~ \nabla \cdot \mathbf{\Gamma } = -4 \pi G \rho, $$


 * $$~ \nabla \times \mathbf{\Gamma } = - \frac{\partial \mathbf{\Omega}} {\partial t}, $$


 * $$~ \nabla \cdot \mathbf{\Omega} = 0, $$


 * $$~ \nabla \times \mathbf{\Omega} = \frac{1}{c^2_{g}} \left( -4 \pi G \mathbf{J} + \frac{\partial \mathbf{\Gamma }} {\partial t} \right) = \frac{1}{c^2_{g}} \left( -4 \pi G \rho \mathbf{v}_{\rho} + \frac{\partial \mathbf{\Gamma }} {\partial t} \right), $$

where:
 * $$~ \mathbf{\Gamma } $$ is gravitational field strength or the gravitational acceleration,
 * $$~ G $$ is the gravitational constant,
 * $$~ \mathbf{\Omega}$$ is the gravitational torsion field or simply torsion,
 * $$~ \mathbf{J}$$ – mass current density,
 * $$~ \rho $$ – the moving mass density,
 * $$~ \mathbf{v}_{\rho} $$ – speed of mass current density,
 * $$~ c_{g}$$ – speed of gravity.

From these equations the wave equations are derived:


 * $$ ~\nabla^2 \mathbf{\Gamma }- \frac {1}{c^2_{g}}\frac{\partial^2 \mathbf{\Gamma }} {\partial t^2} =-4 \pi G \nabla \rho - \frac {4 \pi G }{ c^2_{g}} \frac{\partial \mathbf{J}} {\partial t}, $$


 * $$ ~\nabla^2 \mathbf{\Omega }- \frac {1}{c^2_{g}}\frac{\partial^2 \mathbf{\Omega }} {\partial t^2} = \frac {4 \pi G }{ c^2_{g}} \nabla \times \mathbf{J}. $$

These equations describing gravitoelectromagnetism are the gravitational analogs to Maxwell's equations for electromagnetism.

Gravitational constants
Proceeding from the analogy of both gravitational and Maxwell's equations, the following values can be entered: $$~\varepsilon_g = \frac{1}{4\pi G } = 1.192708\cdot 10^9 \, \mathrm {kg \cdot s^2 \cdot m^{-3}} $$ as the gravitoelectric permittivity (like electric constant);

$$~\mu_g = \frac{4\pi G }{ c^2_{g}}$$ as the gravitomagnetic permeability (like vacuum permeability). If the speed of gravitation is equal to the speed of light, $$~ c_{g}=c,$$ then $$~\mu_g = 9.328772\cdot 10^{-27} \mathrm {m / kg}, $$ and


 * $$~\frac{1}{\sqrt{\mu_g\varepsilon_g}} = c = 2.99792458\cdot 10^8 \, \mathrm {m/s}. $$

The gravitational characteristic impedance of free space for gravitational waves would then be defined as:


 * $$~\sqrt{\frac{\mu_g}{\varepsilon_g}} = \rho_{g} = \frac{4\pi G }{c_g}. $$

If $$~ c_{g}=c,$$  then the gravitational characteristic impedance of free space is equal to:


 * $$~ \rho_{g0} = \frac{4\pi G }{c} =2.796696\cdot 10^{-18} \, \mathrm {m^2/(s\cdot kg)} $$.

As in electromagnetism, the characteristic impedance of free space plays the dominant role in all radiation processes. One example being, a comparison of the radiation impedance of gravitational wave antennas to the value of said impedance in order to estimate the coupling efficiency of antennas to free space. The numerical value of this impedance is extremely small, therefore it became exceedingly difficult even unto the present to construct receivers with proper impedance matching.

Wave equations in vacuum
The gravitational vacuum wave equation is a second-order 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 $$~ \mathbf{\Gamma } $$ or the gravitational torsion field $$~ \mathbf{\Omega}$$, has the form:


 * $$ ~\nabla^2 \mathbf{\Gamma }- \frac {1}{c^2_{g}}\frac{\partial^2 \mathbf{\Gamma }} {\partial t^2} = 0, $$


 * $$ ~\nabla^2 \mathbf{\Omega }- \frac {1}{c^2_{g}}\frac{\partial^2 \mathbf{\Omega }} {\partial t^2} = 0. $$

For waves in one direction the general solution of the gravitational wave equation is a linear superposition of flat waves of the form
 * $$~ \mathbf{\Gamma }( \mathbf{r}, t ) =  f(\phi( \mathbf{r}, t ))  =  f( \omega t  -  \mathbf{k} \cdot \mathbf{r}   )  $$

and
 * $$~ \mathbf{\Omega }( \mathbf{r}, t ) =  q(\phi( \mathbf{r}, t ))  =  q( \omega t  -  \mathbf{k} \cdot \mathbf{r}) $$

for virtually any well-behaved functions $$~ f $$ and $$~ q $$ of dimensionless argument $$~\phi, $$ where
 * $$~ \omega $$ is the angular frequency (in radians per second),
 * $$~ \mathbf{k} = ( k_x, k_y, k_z) $$ is the wave vector (in radians per meter), and $$~ \Gamma =c_g \Omega. $$

Considering the following relationships between inductions and strengths of gravitational fields:


 * $$~\mathbf{\Omega } = \mu_g \mathbf{H_g}, \qquad \Omega=\frac {\Gamma }{c_g}, $$


 * $$~\mathbf{D_g} = \varepsilon_g\mathbf{ \Gamma }, $$

where $$~\mathbf{D_g}$$ is gravitational displacement field, $$~\mathbf{ H_g}$$ is torsion (gravitomagnetic) field strength, we could obtain the following interconnected values:
 * $$~\sqrt{\mu_g}H_g = \sqrt{\varepsilon_g}\Gamma. $$

This equation determines the wave impedance (gravitational characteristic impedance of free space) in a standard form similar to the case of electromagnetism:
 * $$~\rho_g = \sqrt{\frac{\mu_g}{\varepsilon_g}}=c_g \mu_g = \frac{\Gamma }{H_g}. $$

In practice, without exception the total dipole gravitational radiation of each system of bodies, when 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 Lorentz-invariant theory of gravitation, calculated in the quadrupole approximation, are sufficiently accurate approximations to the results of general relativity and covariant theory of gravitation.

If in the system of bodies, are bodies with an electric charge which radiate electromagnetically, the balance is disrupted along with some uncompensated dipole gravitational radiation.

Gravitational LC circuit
As a model of LC circuit, consider the case of motion of an ideal liquid fluid in a closed pipe under influence of gravitational and other forces. Suppose that this circuit has a tubular coil through which passes a fluid, due to its rotation creates a 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 section that accumulates the liquid. For the possibility of fluid motion in two opposite directions in this circuit, on both sides of the section are pistons with springs. This allows for periodical conversion fluid motion energy into energy of compression springs,which is approximately equated to changes in the gravitational energy of the fluid. The pistons with springs act like a capacitor in a circuit, and gravitational voltage $$~ V_g $$ is then equal to the difference of gravitational potentials, and the gravitational mass current $$~ I_g $$  is equivalent to the mass of liquid per unit time through a section of the pipe.

Gravitational voltage on gravitational inductance $$~ L_g $$ is:
 * $$~V_{gL} = -L_g \cdot \frac{d I_{gL}}{d t}. $$

Gravitational mass current through gravitational capacitance $$~ C_g $$ is:
 * $$~I_{gC} = C_g \cdot \frac{d V_{gC}}{d t}. $$

Differentiating these equations with respect to the time variable, we obtain:
 * $$~\frac{d V_{gL}}{d t} = -L_g \frac{d^2I_{gL}}{dt^2}, \qquad \frac{d I_{gC}}{d t} = C_g \frac{d^2V_{gC}}{dt^2}.$$

Considering the following relationships for gravitational voltages and currents:
 * $$~V_{gL} = V_{gC}=V_g; \qquad I_{gL} = I_{gC}=I_g, $$

we obtain the following differential equations for gravitational oscillations:
 * $$~\frac{d^2 I_g}{dt^2} + \frac{1}{L_g C_g}I_g = 0, \qquad \frac{d^2 V_g}{dt^2} + \frac{1}{L_g C_g}V_g = 0.  $$

Furthermore, considering the following relationships between gravitational voltage and mass of the liquid:
 * $$~m = C_g V_g $$

and mass current with flux of torsion field:
 * $$~\Phi = L_g I_g $$

the above oscillation equation for $$~ V_g $$ could be rewritten in the form:
 * $$~\frac{d^2 m}{dt^2} + \frac{1}{L_g C_g}m = 0. $$

This equation has the partial solution:
 * $$~m(t) = m_0 \sin (\omega_{g}t), $$

where
 * $$~\omega_{g} = \frac{1}{\sqrt{L_g C_g}} $$

is the resonance frequency in absence of energy loss, and
 * $$~\rho_{LC} = \sqrt{\frac{L_g}{C_g}}= \frac {V_{g0}}{I_{g0}} $$

then describes the gravitational characteristic impedance of LC circuit, which is equal to the ratio of the gravitational voltage amplitude to the mass current amplitude.

Gravitational induction
According to the second equation for gravitational fields, after a change in time of $$~\mathbf{\Omega }$$ there appears a circular field (rotor) of $$~\mathbf{\Gamma }$$, having the opportunity to lead in the rotation of matter:
 * $$\nabla \times \mathbf{\Gamma } = -\frac{\partial \mathbf{\Omega } }{\partial t }. \qquad\qquad (1) $$

If the vector field $$~\mathbf{\Omega }$$ crosses a certain area $$~ S $$, then we can calculate the flux of this field through this area:
 * $$~\Phi = \int   \mathbf{ \Omega }\cdot \mathbf{n }ds,  \qquad\qquad (2) $$

where $$~\mathbf{n} $$ – The normalized vector to the element area $$~dS$$.

To find the partial derivative in the equation $$(2)$$ with respect to time with a minus sign and integrate over the area, taking into account the equation $$(1)$$:


 * $$~ -\int \frac{\partial \mathbf{\Omega} }{\partial t} \cdot \mathbf{n }ds = -\frac{\partial \Phi }{\partial t}= \int  [\nabla \times \mathbf{\Gamma }] \cdot \mathbf{n }ds = \int \mathbf{\Gamma }\vec d\ell.  \qquad\qquad (3)  $$

Thi integration formula used the Stokes theorem, replacing the integration of the rotor vector over the area on the integration of this vector over a closed circuit. On the right side of $$(3)$$ is a term, equal to the work on transfer of a unit mass of matter on a closed loop $$~\ell$$, covering an area $$~S$$. As a parallel comparison to electromagnetism, this work could be called gravitomotive force. In the middle of $$(3)$$ is the time derivative of flux such that$$~\Phi $$. According to $$(3)$$, gravitational induction occurs when the flux of fields through a certain area changes and is expressed in the occurrence of rotational forces acting on particles of matter. The direction of motion of the matter will be such that field $$~\mathbf{\Omega}$$ of the matter will be sent in the same direction as the initial torsion field which created the circulation of the matter (this is in contrast to Lenz's_law in electromagnetic theory).