1. All the most known theories of gravity are built on the principle of long-range action.
When approximately "point" body or mass density (also electromagnetic energy) distributed
phenomena on the principle of proximity (locality). The cause of interactions is some spatial state of fundamental fields,
the consequence is change of these fields over time (first derivative in time at point of continuum).

Presumably, following fundamental gravitational fields exist:
(SI units in parentheses are m-metre, s-second, k-kilogram, A-Ampere)
scalar potential g (m2/s2)
vector potential G (m/s)
scalar strain f (m2/s3)
vector strain F (m/s2)

The gravitational constant g0 = 6.6742-11 (m3/s2/k) is also used,
local energy density u (k/m/s2), for example electromagnetic = ε0/2 • E2 + μ0/2 • H2
and Poynting vector S (k/s3) = [E × H]

Time derivatives are expressed as follows:
g' = - f - c2 • div G
G' = - F - grad g
f' = - c2 • div grad g + fu • u
F' = c2 • rot rot G - fs • S

The constants fu (m3/s2/k) and fs (m/k) are positive, signs are selected so that scalar potential g
becomes negative in presence of positive density u in vicinity of point.

The equations are similar to electromagnetic equations expressed in potentials:
a' = - c2 • div A
A' = - E - grad a
E' = c2 • rot rot A

In stationary state, for example, during formation of gravitational fields by stable elementary particle
or single celestial body:
S = 0, G = 0, f = 0
div grad g = fu • u / c2 = 4 • π • g0 • ρ, according to Newton's potential

Hence we get at ρ = u / c2: fu = 4 • π • g0

The effect of gravitational fields on other fundamental ones can manifest itself as a curvature of space,
and direct effect on velocity vector V, mentioned in this topic:
Hypothesis about the formation of particles from fields (thescienceforum.com)

With zero u and S, following types of "pure" gravitational waves can exist:
1. Longitudinal potential-potential: g' = - c2 • div G, G' = - grad g
2. Longitudinal with phase shift of 90 degrees: g' = - f, f' = - c2 • div grad g
3. Transverse: g' = - c2 • div G, G' = - F - grad g, F' = c2 • rot rot G
Transverse ones are probably easier to detect in experiments.

Also you can read the topic about particles internal structure, with the similar approach:
Hypothesis about the formation of particles from fields (thescienceforum.com)

2.

3. I believe that one can construct a theory of gravitation that is intermediate between Newton and Einstein by incorporating a gravitomagnetic field into Newtonian gravitation, analogous to the magnetic field of classical electrodynamics. This gravitomagnetic field is associated with mass current (aka momentum), similar to the association of the magnetic field with charge current. I also believe that one could construct a full set of equations that are analogous to Maxwell's equations of classical electrodynamics (such equations include Gauss's law for gravity). And these from equations, one could derive wave equations describing gravitational radiation. The correspondence between gravitation and electrodynamics is as follows:

gravitation <–> electrodynamics

mass <–> charge
momentum <–> current
gravitational field <–> electric field
gravitomagnetic field <–> magnetic field
gravitational potential <–> electric potential
gravitomagnetic vector potential <–> magnetic vector potential
G <–> –1/4πε0

4. Originally Posted by KJW
I believe that one can construct a theory of gravitation that is intermediate between Newton and Einstein by incorporating a gravitomagnetic field into Newtonian gravitation, analogous to the magnetic field of classical electrodynamics. This gravitomagnetic field is associated with mass current (aka momentum), similar to the association of the magnetic field with charge current. I also believe that one could construct a full set of equations that are analogous to Maxwell's equations of classical electrodynamics (such equations include Gauss's law for gravity). And these from equations, one could derive wave equations describing gravitational radiation. The correspondence between gravitation and electrodynamics is as follows:

gravitation <–> electrodynamics

mass <–> charge
momentum <–> current
gravitational field <–> electric field
gravitomagnetic field <–> magnetic field
gravitational potential <–> electric potential
gravitomagnetic vector potential <–> magnetic vector potential
G <–> –1/4πε0
This is about what I have been doing in praxis. Mostly I have renewed gravitation since the shell theory turned out to be a "magic" dream. This has led to rewriting most of the practical formulas during past few years. There I have a urgency to get direct contact with NASA, because Moon's an Mars's profile of grav. fields at all altitudes differ much from mainstream. That are paid for on Mars with many wrecks, and any plans to lift "cargo" from Moon surface will be dangerously challenging.
The tidying up of electric fields "basics" had only a few small lapsus. Convincing any intent scientists of these in principle small changes probably have a halftime greater than C14.
For gravity I have years ago used indifferent Transformation factor Ti. Beginning from 7,43*10^-11 to 7,462*10^-11 till I surprisingly found a good value for mass of nucleon mn=1,67136105*10^-27, from my electric calcs an go Ti=mn*c^2/2=7,510722*10^-11 Nm2/kg2, I was banned from the Finnish Astronomical Societys discussion site for heresy saying G=7,43 *10^-11.
The only "error" I found in the electric constants was permittivity (epsylon)0=1/(4*pi*ke). I would have guessed 4pi and there is no surprise it took time to find out 12 ke is the correct. From that we get exact 1/4piE0=>ks=4/(3*mn*c^2)=8,8762526*10^9Nm2/Ci2. From here towards quantum physics is wise to look out for when "qualities" refere to mols and when to kmols.
If math is written like compoAi writes, it can mean anything or nothing, no one will bother. Sorry AI

5. Originally Posted by Timo Moilanen
Mostly I have renewed gravitation since the shell theory turned out to be a "magic" dream. This has led to rewriting most of the practical formulas during past few years.
Shell theory? I'm not familiar with such a thing. Reference, please.

If you mean shell theorem, it is, in fact, a theorem, as in something that has been proven mathematically, and therefore not a '"magic" dream'.

6. Originally Posted by KJW
I believe that one can construct a theory of gravitation that is intermediate between Newton and Einstein by incorporating a gravitomagnetic field
All these theories are based on long-action. As if mass of point-body or even density extends its effect to the entire universe in some miraculous way. I think it doesn't correspond to the realities of nature.

7. Originally Posted by compuAI
All these theories are based on long-action. As if mass of point-body or even density extends its effect to the entire universe in some miraculous way. I think it doesn't correspond to the realities of nature.
What you are actually saying is that it doesn't conform to your wishes about how nature should behave.

"Argument from incredulity" (or "argument from personal taste") is a classic logical fallacy. It says much more about the biases and limited imagination of the incredulous disputant than about anything scientific. Do you have a scientific argument to offer?

8. Originally Posted by compuAI
Originally Posted by KJW
I believe that one can construct a theory of gravitation that is intermediate between Newton and Einstein by incorporating a gravitomagnetic field
All these theories are based on long-action. As if mass of point-body or even density extends its effect to the entire universe in some miraculous way. I think it doesn't correspond to the realities of nature.
I wrote what I wrote above in this thread because it looked to me like you were attempting to write something along the lines of a gravitational version of Maxwell's equations. But that would require a gravitational version of the magnetic field. It exists because it is the gravitational field viewed from a relatively moving frame of reference just as the magnetic field is the electric field viewed from a relatively moving frame of reference. However, the gravitomagnetic field is very much weaker than the corresponding gravitational field, too weak to appear in Newton's theory of gravitation. In general relativity, it is responsible for frame-dragging around a rotating mass, a tiny effect that was confirmed by the Gravity Probe B experiment. But what I wrote above was not general relativity, it was strictly Newtonian gravitation, but applying the principle of special relativity to deduce the gravitomagnetic field.

9. Originally Posted by tk421
Originally Posted by Timo Moilanen
Mostly I have renewed gravitation since the shell theory turned out to be a "magic" dream. This has led to rewriting most of the practical formulas during past few years.
Shell theory? I'm not familiar with such a thing. Reference, please.

If you mean shell theorem, it is, in fact, a theorem, as in something that has been proven mathematically, and therefore not a '"magic" dream'.
Yes I mean shell theorem. In 2016 I managed to integrate -RtoR for dV cos(a)/s^2 dx,dy,dz, which I read should be next to impossible, and after a few weeks got V/r^2 not depending on R. I must admit it felt like magic especially since it was just the answer I tried not to get. I also did same -RtoR for dV/s^2 dx,dy,dz that of course give "more"V/r^2 and for r/R=p I got V*((3/2*p^2)-(3/4)*(p^3-p)*ln((p+1)/(p-1))). For both to be true at same time I "had to" rewrite F=G*M/r^2 to F=M*Ti/(r^2 *((3/2*p^2)-(3/4)*(p^3-p)*ln((p+1)/(p-1))) Where the indifferent transformation coefficient (not same as G) Ti=7,5*10^-11 Nm2/kg2 to satisfy Cavendish type of experiments. And for shells, a shell R at dist R give F=M*Ti/(2R^2) MARK! this only for R.
The same for force between charged spheres where assumed C*C/r^2 turned out to be 4/3*C*C/r^2 helping me to pinpoint Couloumbs constant. Citating a textbook and understanding it is two different things.

10. Originally Posted by KJW
you were attempting to write something along the lines of a gravitational version of Maxwell's equations. But that would require a gravitational version of the magnetic field
Yes, there are some similarities between gravity and electromagnetism. But there are significant differences also. For mathematically valid minimal description of gravity new value had to be introduced, "scalar strain" in my text. Also in electromagnetic equations of pure field (without point-charges or currents) charge density is "internal" and always proportional to div(E), and changes with E'. In gravitational equations mass or energy density is "external", created by other fundamental fields.

The analogue of magnetic field in gravitation equations is the curl of vector potential, like electromagnetic equations. The difference is that analogues of electric and magnetic fields do not carry conservative energy. In the presence of "external" mass density in equations, it is impossible to draw up an adequate energy conservation law so that time derivative of energy density would be equal to minus divergence of some flux. In this sense, gravitational waves are energy-free.

11. Originally Posted by compuAI
Also in electromagnetic equations of pure field (without point-charges or currents) charge density is "internal" and always proportional to div(E), and changes with E'. In gravitational equations mass or energy density is "external", created by other fundamental fields.
In the gravitational analogue of Maxwell's equations, one has Gauss's law for gravity:

div(g) = –4πGρ

Originally Posted by compuAI
The difference is that analogues of electric and magnetic fields do not carry conservative energy. In the presence of "external" mass density in equations, it is impossible to draw up an adequate energy conservation law so that time derivative of energy density would be equal to minus divergence of some flux. In this sense, gravitational waves are energy-free.
I believe that all the equations of electrodynamics have a corresponding equation for the gravitational analogue. Given the Newtonian force law for gravitation, and analogous to the case of electrodynamics, one can obtain the energy density of the gravitational field. But because like masses attract, the energy density of the gravitational field is negative.

12. Originally Posted by tk421
Originally Posted by Timo Moilanen
Mostly I have renewed gravitation since the shell theory turned out to be a "magic" dream. This has led to rewriting most of the practical formulas during past few years.
Shell theory? I'm not familiar with such a thing. Reference, please.

If you mean shell theorem, it is, in fact, a theorem, as in something that has been proven mathematically, and therefore not a '"magic" dream'.
From Wiki" https://en.wikipedia.org/wiki/Shell_theorem " after Outside a shell one screen beneath the mowing image, there is a integral modified to a "primitive function" . But integrating (1+(r^2-R^2)/s^2)ds give F= s-(r^2+R^2)/s putting in s=r-R resp s=r+R => 2R and Fr= GMm/(2r^2). for R=r the outermost layer of a sphere contribute with only half its mass to the gravity as I have established. I never used this path integrating the grav. of sphere but "primitive function" seem to give "magic" results.
Holding on too hard to mainstream will allow minor "shortcuts" and may put science of track for decades.

13. Originally Posted by KJW
But because like masses attract, the energy density of the gravitational field is negative.
Gravitational potential generally is considered negative. I never saw expressions, describing gravitation field energy.

14. Originally Posted by compuAI
Originally Posted by KJW
But because like masses attract, the energy density of the gravitational field is negative.
Gravitational potential generally is considered negative. I never saw expressions, describing gravitation field energy.
Although it is probably true that the negative energy density of the gravitational field and the negative gravitational potential are connected, that connection is not the basis of my statement that the energy density of the gravitational field is negative. The energy density of the gravitational field gives rise to the gravitational force law just as the energy density of the electromagnetic field gives rise to the electromagnetic force law. Consider the total electromagnetic field around a pair of identical charges. This is the vector sum of the electromagnetic field of each individual charge. The energy density of the total electromagnetic field is the "square" of the total electromagnetic field, and the total energy is the integral of the energy density over all space. The repulsive force between the two charges is the increase in total energy that results from decreasing the separation between them. In the case of the gravitational field, the two masses are identical and thus the total gravitational field is the same as in the case of the total electromagnetic field around two identical charges. Therefore, the magnitude of the total energy must also increase as the separation between the masses is decreased. But because the force between the two masses is attractive, the total energy must decrease and therefore must be negative.

15. Originally Posted by KJW
Although it is probably true that the negative energy density of the gravitational field and the negative gravitational potential are connected, that connection is not the basis of my statement that the energy density of the gravitational field is negative. The energy density of the gravitational field gives rise to the gravitational force law just as the energy density of the electromagnetic field gives rise to the electromagnetic force law. Consider the total electromagnetic field around a pair of identical charges. This is the vector sum of the electromagnetic field of each individual charge. The energy density of the total electromagnetic field is the "square" of the total electromagnetic field, and the total energy is the integral of the energy density over all space. The repulsive force between the two charges is the increase in total energy that results from decreasing the separation between them. In the case of the gravitational field, the two masses are identical and thus the total gravitational field is the same as in the case of the total electromagnetic field around two identical charges. Therefore, the magnitude of the total energy must also increase as the separation between the masses is decreased. But because the force between the two masses is attractive, the total energy must decrease and therefore must be negative.
In the world of "big" bodies and even point-like particles the laws are quite different than at "pure field" level. Interactions represent some integral values, very complicated even in case of simplest particles. We cannot overlap two identical particles so that surrounding electric field becomes double, since the energy must increase by a factor of 4, where it will come from? Except that we accelerate each to the speed of light, so that magnetic field appears around in addition to electric one with the same energy, and direct particles at each other.

16. Originally Posted by compuAI
We cannot overlap two identical particles so that surrounding electric field becomes double, since the energy must increase by a factor of 4, where it will come from?
I didn't suggest overlapping two identically charged particles. But bringing two identically charged particles closer together even by a small amount increases the energy of the surrounding electromagnetic field. And this increase in the energy comes from the work done pushing these particles closer to each other.

17. At pure field level no external work or other type of direct forceful manipulation exist. All possible energy is represented by electric and magnetic field (vector all-coordinates squares sum). It is a bit tricky to imaginate, getting used to think about "big bodies" world, but I don't see an alternative to this view.

18. Originally Posted by compuAI
At pure field level no external work or other type of direct forceful manipulation exist. All possible energy is represented by electric and magnetic field (vector all-coordinates squares sum). It is a bit tricky to imaginate, getting used to think about "big bodies" world, but I don't see an alternative to this view.
It's not clear to me what you are disagreeing with. Suppose f and g are the electromagnetic fields associated with two sources, here represented as vectors in Hilbert space. Then the energy of the combined fields:

(f + g) • (f + g) = ff + 2 fg + gg

ff and gg are the energies that each source brings to the party. 2 fg is the energy of the interaction between the two sources (either positive or negative). fg > 0 implies a repulsive force between the two sources, fg < 0 implies an attractive force between the two sources. In the case that fg = 0, the vectors f and g are orthogonal in Hilbert space, there is no overall force between the sources. This is usually the case with two sources of light (different frequencies and/or different locations).

19. Originally Posted by KJW
Suppose f and g are the electromagnetic fields associated with two sources, here represented as vectors in Hilbert space.
This is concerning "material points", like "infinitely small" particles, as they are treated by quantum mechanics. Calculations are based on long-action principle and really describe very complicated physical processes. My equations are written in wave-like manner, without any sources. Even more, I think particles themselves are built of some fields. There is no essential conflict between two approaches, but different areas of usage and approximation levels.

20. Originally Posted by compuAI
Originally Posted by KJW
Suppose f and g are the electromagnetic fields associated with two sources, here represented as vectors in Hilbert space.
This is concerning "material points", like "infinitely small" particles, as they are treated by quantum mechanics.
What I'm saying is purely classical electrodynamics. Perhaps, you took my mention of Hilbert space to be referring to quantum mechanics. But in fact, I mentioned Hilbert space to enable a compact description of the electromagnetic field as a vector.

Also, there was no mention of point particles.

Originally Posted by compuAI
My equations are written in wave-like manner, without any sources.
Charges exist. If you are only dealing with sourceless fields, then you are missing a large part of electrodynamics. However, the idea of describing the force between two charges in terms of the change in energy of their combined electromagnetic field is indeed a field-only description. But even a field-only description has sources... its divergence.

Implicit in my previous post is the notion that there are separate individual sources with their own electromagnetic fields. The electromagnetic field of an individual source is the electromagnetic field of that source in the absence of all other sources. By having separate individual electromagnetic fields, one can formulate a combination of those electromagnetic fields, and define the energy associated with the interaction of those electromagnetic fields, separate from the energy of the electromagnetic fields of the sources themselves.

21. Originally Posted by KJW
What I'm saying is purely classical electrodynamics. Perhaps, you took my mention of Hilbert space to be referring to quantum mechanics.
The best practical results, for example, in computer simulation, gives some hybrid combination of quantum and classical approach, with many specific programming tricks.
Originally Posted by KJW
But even a field-only description has sources... its divergence.
In the short-action local sense, as continuous scalar field on the physical continuum.

22. You am find more information from Journal of High Energy Physics, Gravitation and Cosmology(ISSN Online: 2380-4335), is a cutting edge research periodical aimed to be forward looking and innovative and, at the same time, remaining in the mainstream.

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