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Maxwell-like gravitational equations

 

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In the weak gravitational field approximation, Maxwell-like 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 Lorentz-invariant 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

  • 1 History
  • 2 Field equations
  • 3 Gravitational constants
  • 4 Applications
    • 4.1 Wave equations in vacuum
    • 4.2 Gravitational LC circuit
    • 4.3 Gravitational induction
  • 5 See also
  • 6 References
  • 7 External links

History

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 80-ties Maxwell-like equations were considered in the Wald book of general relativity.[6] In the 90-ties this approach was developed by Sabbata,[7] [8] Lano, [9] Sergey Fedosin. [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

The ways of experimental determination of gravitational waves properties are developed in papers of Raymond Y. Chiao. [36] [37] [38] [39] [40]

Maxwell-like equations can be found in many other recent works: [41] [42] [43] [44] [45] [46] [47] [48]

Field equations

Main sources: Lorentz-invariant theory of gravitation and Gravitoelectromagnetism

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:

From these equations the wave equations are followed: [11]

~\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 as in gravitoelectromagnetism are the gravitational analogs to Maxwell's equations for electromagnetism.

Gravitational constants

Main source: Selfconsistent gravitational constants and Vacuum constants

 

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

~\mu_g = \frac{4\pi G }{ c^2_{g}} as gravitomagnetic permeability (like vacuum permeability). If the speed of gravitation is equal to the speed of light, ~ c_{g}=c,  then [49]   ~\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 the gravitational waves could 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: [39]

~ \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, 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.

Applications

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: [50]

~\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 interconnections:

~\sqrt{\mu_g}H_g = \sqrt{\varepsilon_g}\Gamma. 

This equation determines the wave impedance (gravitational characteristic impedance of free space) in standard form like the case of electromagnetism:

~\rho_g = \sqrt{\frac{\mu_g}{\varepsilon_g}}=c_g \mu_g = \frac{\Gamma }{H_g}.

 

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 Lorentz-invariant 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.

 

Gravitational LC circuit

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  ~ V_g   is equal to difference of gravitational potentials, and gravitational mass current  ~ I_g   is mass of liquid per unit time throw 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}}

is 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

Main source: Gravitational induction

According to the second equation for gravitational fields, after a change in time of ~\mathbf{\Omega } there appear circular field (rotor) of ~\mathbf{\Gamma }, having the opportunity to lead in rotation matter: [10]

~ \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 vector normal to the element area ~dS.

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):

~ -\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)

 

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 ~\ell, covering an area ~S. By analogy with electromagnetism, this work could be called gravitomotive force. In the middle of (3) is time derivative of the flux ~\Phi . 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 ~\mathbf{\Omega}  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).

See also

References

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  2. O. Heaviside, Electromagnetic Theory (”The Electrician” Printing and Publishing Co., London, 1894) pp. 455-465.
  3. OLIVER HEAVISIDE. A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY, Part I, The Electrician, 31, 281-282 (1893).
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