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.

## HistoryEdit

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]}
Since the 90s this approach has been used 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 equationsEdit

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:

where:

- is gravitational field strength or the gravitational acceleration,
- is the gravitational constant,
- is the gravitational torsion field or simply torsion,
- – mass current density,
- – the moving mass density,
- – speed of mass current density,
- – speed of gravity.

From these equations the wave equations are derived: ^{[11]}

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

## Gravitational constantsEdit

Proceeding from the analogy of both gravitational and Maxwell's equations, the following values can be entered:
as the gravitoelectric permittivity (like *electric constant*);

as the gravitomagnetic permeability (like *vacuum permeability*). If the speed of gravitation is equal to the speed of light, then ^{[49]} and

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

If then the gravitational characteristic *impedance of free space* is equal to: ^{[39]}

- .

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.

## ApplicationsEdit

### Wave equations in vacuumEdit

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 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* well-behaved 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:
^{[50]}

where is gravitational displacement field, is torsion (gravitomagnetic) field strength, we could obtain the following interconnected values:

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

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 circuitEdit

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 is then equal to the difference of gravitational potentials, and the gravitational mass current is equivalent to the mass of liquid per unit time through 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

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 inductionEdit

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

If the vector field crosses a certain area , then we can calculate the flux of this field through this area:

where – The normalized vector to the element area .

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

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 is a term, equal to the work on transfer of a unit mass of matter on a closed loop , covering an area . As a parallel comparison to electromagnetism, this work could be called gravitomotive force. In the middle of is the time derivative of flux such that . According to , 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 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).

## See alsoEdit

## ReferencesEdit

- ↑ K.T. McDonald, Am. J. Phys. 65, 7 (1997) 591-2.
- ↑ O. Heaviside, Electromagnetic Theory (”The Electrician” Printing and Publishing Co., London, 1894) pp. 455-465.
- ↑ OLIVER HEAVISIDE. A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY, Part I, The Electrician,
**31**, 281-282 (1893). - ↑ W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1955), p. 168, 166.
- ↑ R. L. Forward, Proc. IRE 49, 892 (1961).
- ↑ R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
- ↑ V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, Singapore,1985).
- ↑ V. de Sabbata and C.Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore,1994)
- ↑ R.P. Lano (1996-03-12). "Gravitational Meissner Effect".
*arXiv: hep-th 9603077*. http://arxiv.org/PS_cache/hep-th/pdf/9603/9603077v1.pdf. - ↑
^{10.0}^{10.1}Fedosin S.G. Fizika i filosofiia podobiia: ot preonov do metagalaktik, Perm, (1999-06-09) 544 pp. ISBN 5-8131-0012-1. - ↑
^{11.0}^{11.1}Fedosin S.G. Electromagnetic and Gravitational Pictures of the World. Apeiron, Vol. 14, No. 4, pp. 385- 413 (2007). http://dx.doi.org/10.5281/zenodo.891124. - ↑ Fedosin S.G. Mass, Momentum and Energy of Gravitational Field. // Journal of Vectorial Relativity, Vol. 3, No. 3, pp.30-35 (2008). http://dx.doi.org/10.5281/zenodo.890899
- ↑ Fedosin S.G. Model of Gravitational Interaction in the Concept of Gravitons. // Journal of Vectorial Relativity, Vol. 4, No. 1, pp.1-24 (2009). http://dx.doi.org/10.5281/zenodo.890886.
- ↑ Fedosin S.G. (2009), written at Perm, 844 pages, Tabl. 21, Pic. 41, Ref. 289., Fizicheskie teorii i beskonechnaia vlozhennost’ materii, ISBN 978-5-9901951-1-0. (in Russian).
- ↑ Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, Vol. 35, No. 1, pp. 35-70 (2012). http://dx.doi.org/10.5281/zenodo.889804.
- ↑ Fedosin S.G. The Hamiltonian in Covariant Theory of Gravitation. Advances in Natural Science, Vol. 5, No. 4, pp. 55-75 (2012). http://dx.doi.org/10.3968%2Fj.ans.1715787020120504.2023.
- ↑ Fedosin S.G. 4/3 Problem for the Gravitational Field. Advances in Physics Theories and Applications, Vol. 23, pp. 19-25 (2013). http://dx.doi.org/10.5281/zenodo.889383.
- ↑ Fedosin S.G. The General Theory of Relativity, Metric Theory of Relativity and Covariant Theory of Gravitation. Axiomatization and Critical Analysis. International Journal of Theoretical and Applied Physics (IJTAP), Vol. 4, No. I , pp. 9-26 (2014). http://dx.doi.org/10.5281/zenodo.890781.
- ↑ Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.
- ↑ Fedosin S.G. The Procedure of Finding the Stress-Energy Tensor and Equations of Vector Field of Any Form. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.
- ↑ Fedosin S.G. Relativistic Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.
- ↑ Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.
- ↑ Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304.
- ↑ Fedosin S.G. Estimation of the physical parameters of planets and stars in the gravitational equilibrium model. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.
- ↑ Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.
- ↑ Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics, Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8.
- ↑ Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947.
- ↑ Fedosin S.G. The Gravitational Field in the Relativistic Uniform Model within the Framework of the Covariant Theory of Gravitation. International Letters of Chemistry, Physics and Astronomy, Vol. 78, pp. 39-50 (2018). http://dx.doi.org/10.18052/www.scipress.com/ILCPA.78.39.
- ↑ Fedosin S.G. The covariant additive integrals of motion in the theory of relativistic vector fields. Bulletin of Pure and Applied Sciences, Vol. 37 D (Physics), No. 2, pp. 64-87 (2018). http://dx.doi.org/10.5958/2320-3218.2018.00013.1.
- ↑ Fedosin S.G. The integral theorem of generalized virial in the relativistic uniform model. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). http://dx.doi.org/10.1007/s00161-018-0715-x.
- ↑ Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19.
- ↑ Fedosin S.G. The Integral Theorem of the Field Energy. GAZI UNIVERSITY JOURNAL OF SCIENCE, Vol. 32, Issue 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783.
- ↑ Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06.
- ↑ Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.
- ↑ Fedosin S.G. The Mass Hierarchy in the Relativistic Uniform System. Bulletin of Pure and Applied Sciences, Vol. 38 D (Physics), No. 2, pp. 73-80 (2019). http://dx.doi.org/10.5958/2320-3218.2019.00012.5.
- ↑ Raymond Y. Chiao. "Conceptual tensions between quantum mechanics and general relativity: Are there experimental consequences, e.g., superconducting transducers between electromagnetic and gravitational radiation?" arXiv:gr-qc/0208024v3 (2002). [PDF
- ↑ R.Y. Chiao and W.J. Fitelson. Time and matter in the interaction between gravity and quantum fluids: are there macroscopic quantum transducers between gravitational and electromagnetic waves? In Proceedings of the “Time & Matter Conference” (2002 August 11-17; Venice, Italy), ed. I. Bigi and M. Faessler (Singapore: World Scientific, 2006), p. 85. arXiv: gr-qc/0303089. PDF
- ↑ R.Y. Chiao. Conceptual tensions between quantum mechanics and general relativity: are there experimental consequences? In Science and Ultimate Reality, ed. J.D. Barrow, P.C.W. Davies, and C.L. Harper, Jr. (Cambridge: Cambridge University Press, 2004), p. 254. arXiv:gr-qc/0303100.
- ↑
^{39.0}^{39.1}Raymond Y. Chiao. "New directions for gravitational wave physics via “Millikan oil drops” arXiv:gr-qc/0610146v16 (2009). PDF - ↑ Stephen Minter, Kirk Wegter-McNelly, and Raymond Chiao. Do Mirrors for Gravitational Waves Exist? arXiv:gr-qc/0903.0661v10 (2009). PDF
- ↑ Flanders W.D., Japaridze G.S. Photon deflection and precession of the periastron in terms of spatial gravitational fields. Class. Quant. Gravit. Vol. 21, pp. 1825-1831 (2004). https://doi.org/10.1088/0264-9381/21/7/007.
- ↑ Borodikhin V.N. Vector theory of gravity. Gravit. Cosmol. Vol. 17, pp. 161-165 (2011). https://doi.org/10.1134/S0202289311020071.
- ↑ Nyambuya G.G. Fundamental Physical Basis for Maxwell-Heaviside Gravitomagnetism. Journal of Modern Physics, Vol. 6, pp. 1207-1219 (2015). http://dx.doi.org/10.4236/jmp.2015.69125.
- ↑ Behera H., Barik N. Attractive Heaviside-Maxwellian (Vector) Gravity from Special Relativity and Quantum Field Theory. arXiv: 1709.06876v2. (2017).
- ↑ Behera H. Comments on gravitoelectromagnetism of Ummarino and Gallerati in “Superconductor in a weak static gravitational field” vs other versions. Eur. Phys. J. C. Vol. 77, Article number 822 (2017). https://doi.org/10.1140/epjc/s10052-017-5386-4.
- ↑ Ummarino G.A., Gallerati A. Superconductor in a weak static gravitational field. Eur. Phys. J. C. Vol. 77, Article number 549 (2017). https://doi.org/10.1140/epjc/s10052-017-5116-y.
- ↑ Antoine Acke. Gravitation explained by the Theory of Informatons. academia.edu/43626998. (2020).
- ↑ Timkov V.F., Timkov S.V., Zhukov V.A., Afanasiev K.E. The quantum of the gravitational field. The gravitational-electromagnetic resonance. Physical nature of the quantum of the gravitational field. Conference: Collection of scientific articles of the X-th International Scientific and Practical Conference "Technical Regulation, Metrology, Information and Transport Technologies", Odessa State Academy of Regulation and Quality, Odessa, 2020.At: Ukraine, Odessa, [1].
- ↑ Kiefer, C.; Weber, C. On the interaction of mesoscopic quantum systems with gravity. Annalen der Physik, 2005, Vol. 14, Issue 4, Pages 253 – 278.
- ↑ Sergey Fedosin. The physical theories and infinite hierarchical nesting of matter, Volume 2. – LAP LAMBERT Academic Publishing, pages: 420, ISBN-13: 978-3-659-71511-2. (2015).