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Maxwell-like gravitational equations

 

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In 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 equations 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 standard theory of gravitation could be written in the form of Maxwell equations with two gravitational constants. [4] [5]

In 1984 Maxwell-like equations were considered in 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

Unlike electromagnetic wave representation used in Lorentz-invariant theory of gravitation (LITG), Maxwell-like equations are written in gravitational wave representation. Accordingly, instead of the speed of light, the speed of gravity ~c_{{g}}  appears in all expressions:

~ \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 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 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 dominant role in all radiation processes. One example being, a comparison of radiation impedance of gravitational wave antennas to the value of characteristic impedance of free space in order to estimate the coupling efficiency of antennas to free space. The numerical value of characteristic impedance of free space is extremely small and therefore it is very difficult to make gravitational radiation receivers with appropriate impedance matching.

Applications

Wave equations in vacuum

The gravitational vacuum wave equation is a second-order partial differential equation that describes propagation of gravitational waves through vacuum in absence of matter. The homogeneous form of the equation, written in terms of either gravitational field strength  ~ \mathbf{\Gamma }  or 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.

From these equations it follows that in gravitational wave representation a gravitational wave propagates with the speed ~c_{{g}} , and in this representation all space-time measurements must be made using gravitational waves.

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 ({\mathbf {r}},t)=(\omega t-{\mathbf {k}}\cdot {\mathbf {r}})  where

~ \omega   is angular frequency (in radians per second),

 

~ \mathbf{k} = ( k_x, k_y, k_z)   is wave vector (in radians per meter), ~\omega =kc_{g}  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, without exception the total dipole gravitational radiation of each system of bodies, when viewed from infinity tends to zero, due to mutual compensation of radiations of individual bodies. As a result, the main components of of gravitational radiation are quadrupole and higher harmonics. With this in mind, the wave equation in Lorentz-invariant theory of gravitation, calculated in quadrupole approximation, are sufficiently accurate approximation to results of general relativity and covariant theory of gravitation.

 

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. This fluid plays the same role as electrons in a conductor or charged particles moving under influence of an electric field. 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 is equivalent to a spiral inductance in an electric circuit. In another part of the circuit is a section that accumulates the fluid. 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 gravitational energy of the fluid. The pistons with springs act like a capacitor in the circuit, and gravitational voltage  ~ V_g   is then equal to difference of gravitational potentials, and the gravitational mass current  ~ I_g   is equivalent to 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 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 relationships between gravitational voltage and mass of the fluid:

~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 gravitational characteristic impedance of LC circuit, which is equal to the ratio of 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 matter in rotation: [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 normal vector to the element area ~dS.

Let’s take the 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)

 

This integration formula used the Stokes theorem, replacing integration of the rotor vector over area on integration of this vector over a closed circuit. On the right side of (3) there 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) there is the time derivative of flux ~\Phi . According to (3), gravitational induction occurs when the flux of field through a certain area changes and is expressed in the occurrence of rotational forces acting on particles of matter.

See also

References

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