Uncommon GR derivation

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Uncommon GR derivation

Postby pogono » Wed Oct 03, 2012 3:00 am

I would like to discuss with you my extraordinary General Relativity derivation :-)

Starting from weird assumption that we may consider locally accelerated photon (what is mathematically possible with Rindler's transformation) I derived... Schwarschild metric.

Brief description of my article is here: http://www.dilationasfield.net/eng
How do you like it?


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Joined: Wed Oct 03, 2012 2:49 am

Gravity and Electromagnetism as spacetime structure disturba

Postby pogono » Sun Apr 21, 2013 2:26 pm

Hi everybody :-)

I have just put on arXiv 4th version of my article (after 2 reviews)

You my find there, that gravity and electromagnetism may be explained as two different consequences of one filed equation describing disturbances in spacetime structure. The field also produces valid quanta values (rest mass, photon energy, elementary charge).

I am open for your questions :-)
If you would like to discuss the idea in real live - come to my seminars.
I was invited to give open speech at Moscov Lemonosov University (may 14th).
I also wait for confirmation of my speech at big international GR20 conference in Warsaw (July).

For those who are interested in my idea - I put news page at my webpage:


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Re: Uncommon GR derivation

Postby pogono » Sun Apr 28, 2013 4:13 pm

I have prepared a presentation for one of my seminar that explains the article:

Take a look if you are curious how is it possible to:
- consider photon acceleration
- explain light as disturbances of spacetime structure

Posts: 1
Joined: Tue Jul 02, 2013 9:04 am

Re: Uncommon GR derivation

Postby do17 » Tue Jul 02, 2013 9:56 am

Hello! I have a different theory about the four main forces.
Gravity was the first main force documented and explained. Then electromagnetism, strong and weak nuclear forces.
What if there is only one force, and here I mean only gravity. Because:

1) Gravity- that makes everything come together, and we can not ignore that
2) Electromagnetism- holds the electrons on atom orbits.
3) Strong force - keeps atomic core particles together
4) Weak force – the force that makes one subatomic particle become another

Otherwise said, first comes gravity and it says: everything is attracted to everything. This is valid for matter. Matter is made from atoms, so atoms are attracted to one other, because gravity sais that everything attracts. This means that even subatomic parts should attract.
Then comes electromagnetism, that until then has said: particles of similar charge are rejecting one other, and those of inverse signs attract each other. And that was a given fact too. This should mean that electrons and protons attract each other. But still electrons remain far away form atomic core. ‘’Yes, but that is so, because the electromagnetic force keeps them there’’, even if it just said that particles of inverse sign attract each other.
And what about the protons and neutrons in the core? Shouldn’t they just reject one other? Why do they stay there, if electromagnetic force sais that they should go apart? ‘’Well they stay there because the strong force commands it so’’, and that is a given fact too.
And yes, from time to time a proton becomes a neutron, and a neutron becomes a proton, but that is so because of the weak force

Everything is attracted to one other. According to gravity. After the 3 seconds that followed the Big Bang, our astronomers describe the evolution of the known Universe. We have no why, just how. Before those 3 seconds, according to them, everything was incredible dens and hot, everything was just one point of unimaginable energy status. Then boom! It exploded, and since the moment 0+3 seconds begins the history of our known Universe. Since then it has enormously expanded itself, it cooled down and everything attract itself. Experiments and scientific observations confirm these things, and make less or more valid explanations for the ones that surround us. It is certain, that after explosion and dilation, it attracted itself and still does. Clouds of matter gathered together, in which particles were attracted to each other, suns came to life, planets, galaxies. This is gravity. It is out there, everywhere, we can not deny it, we can calculate it since Newton, but still has no found particle. Since 1930 physics attributes it a theoretic particle, the graviton, who still waits to be found. Nobel prize for the one who gets it. Further on, his antiparticle, the antigraviton is also still theoretic. Because every particle must have a counter particle. But now we must put gravity aside, because it is boring, because we just can not find that dam’ particle, and we will play on with electromagnetism. Here we have all the particles man can wish for. And interactions between then as much as well. Every few years, a new one is being found, and it comes with a Nobel too, so here is the party.
But before that, we, humanity, discovered electrons and protons and neutrons. And the fact that electrons do not fall over the atomic core, does not bother anyone. Because they are held there by the electromagnetic force, the same one that sais that particles of inverse signs attract each other. Because so sais the physics, invented by us. So what if there is a little contradiction? And going further, protons and neutrons stay together in core, again against ‘’particles of identical sign repel one other’’, but they do so because there is the ‘’strong force’’, as so even if we have another controversy, we just solved it be adding another player to the table, even if is against logic and all that we, humans, physicists, have said before that. And if this goes for protons, that have the same charge, what about the neutrons? Why are they in the core? Why are they not with the electrons, for example? Again, no explanation to the why, just that so it is.
And protons became neutrons and neutrons became protons. So it is, because we can not explain that either, so we just invent another player, the weak force. And so on, and so on. We are really talented in finding another excuse to the unexplainable thing that we just explained. Cool, isn’t it?
And there is that thing, the wave-particle duality, and we kind of like it that too, because so the radiation interacts with particles, and we do not bother to explain why. And our trees are more and more and we just can’t find the forest any more.
Again please: gravity sais all attracts! All! That means matter. That means atoms. That means subatomic particles as well. Yes, but in the entire forest of particles, we still have not found the graviton. How comes that everything attracts, but no responsible one for this? One that we can point out and say – he is the one! We have not found him because it hides in plain sight. In the entire forest of particles (see one page above), it is really hard to imagine that a mistake was made, mainly because you got some sort of equations and explanations for all of the above.
What I mean to say, is that the entire universe is made from only one thing- energy. That is why E=mc². Before the 3 fatal seconds of Big Bang, all universe was in one point. Then it exploded and it cooled down. But still spins. And this spinning is continuous, it is another very important observation left aside, although it is in everything that surround us. And then, in those 3 seconds, particles have attracted each other, and so the first atoms came to life. And since then we have matter. Actually energy. The explanation resides in speed and direction of movement. Energy, as we know it, as electromagnetic radiation, in its elementary form, travels at speed of light and has no rest mass. So how comes matter? Well matter is another way to pack energy.
Otherwise you just can not logically explain all these. If there is something that God has made, are waves. And because He made them good, the entire Universe goes as we know it and it will so go on and on. A electromagnetic wave, as we know it, it goes only one way, and it will keep doing so, until something interacts with it. And interaction is something it gets every moment. Gravity interacts with them. Einstein was the first that proved that Earth’s gravity bends light waves by an angle of 2 seconds. Bigger stars can do more. So, returning to our Big Bang, if everything was concentrated in only one point, after it exploded started to spin in larger and larger spirals. If we ask ourselves why have they spin thereafter, it is because they were spinning before, as a Big Bang. The Big Bang was spinning too. It had its own spin, because it was made from waves. And not matter. And waves bend, oscillate and spin. This was demonstrated already a long time ago. They have sinusoidal patterns, a wave length and frequency. Yes, but how do we explain what we consider to be waves and particles? We know that waves interact with particles. So we say. Why? Because particles are waves too. And nearer is the frequency of the longitudinal wave to the one of those we call particles, more they interfere. The greater the frequency, smaller is the wave length, thus smaller the gap between 2 photons. And since they oscillate, bend and spin they can compact themselves. It is like a snake that bites his own tale. If you take the example of a snake, it can slipper in a certain direction with a given speed. If it bites his tale, but continues to move, he will move in a circle, with the same speed, but in regards to another that has not done so, it will remain practically still. Here is the difference between what we call waves and particles. A particle, analysed from a logical point of view, must be a wave that caught its own tale. It will continue to move in regard to itself with the speed of light, but in regard to one that has not done so, it will remain practically still, or at least very slowed down. The Big Bang was spinning. We know this, because thinking backwards or forwards, we always realise that inertia is something that we can not ignore. Thus, if things are spinning now, so did they before. If galaxies spin now, stars spin, planets spin, it is because the Big Bang was spinning. Electrons spin around the atomic core. Quarks spin too. Everything spins. Meaning it did so before. And when it blew, everything that blew apart was still spinning. And some waves caught their tale. And they made a loop, which in itself still moves with light speed, but in regard to every other that has not done so, it remained less mobile. That is why E=mc². Because energy is the product between light speed and unpacked energy. That that we call mass, is in fact a wave that travels at speed of light in a small loop, and when we manage to observe as it unpacks we can measure as how much it was in. The logical product of its unchanged internal speed, light speed, dissipated in all directions, that why square and all multiplied with the one that we use to call mass.
But let us go a little more with this. Once a wave managed to catch its tale, they do not remain only at this stage. They make small loops, small packs of energy, less or more active in regard to other packs. Because some of them attract, others reject. Explanation sits in their other possible range of movements. Now, please, on a separate piece of paper draw a sinusoidal curve. Now if you have drawn it on that piece of paper please cut it out and glue its ends. You should have now a small paper loop. If you will put it on a table, it will remain so. But please remember, unfolded, theoretically it travels at the speed of light. Folded as a loop, it is practically still. The speed of movement of the loop, taken as so, can be between 0 and speed of light in regard to an external observer. Now comes the other ranges of movement. One of them is internal (inner) rotation. If you take the loop on one of your fingers, and start to spin it, it will move, but in regard to your finger it will remain there. So is the situation with the ‘’particles’’ in an atom. They are packs of energy, that spin in regard to their geometric centre but move more or less in regard to the external world in our know universe.
And what happens if they do spin so? Well, circular forces appear, centrifuge and centripetal forces make their presences known. And starts to look like a small tornado.
Now, you also need to keep in mind, that electromagnetic waves are called so, because they always have 2 components, the electric and the magnetic one, placed at 90 degree angle in regard to the geometric centre of the unpacked wave. . If you look at it, unpacked, you can also spin it around its direction of movement, just like a drill. The electric and magnetic components will spin as well, but only together. So will they further spin, packed too, all together. Packed you will loose the independence of its direction of movement. That means, that taken as a whole package, it can accelerate, slow down and in rotate, but all at once. And so it comes to interactions at smaller speeds, that I will explain later.
But if internal rotation speed is small, meaning the part when you leave the loop sitting on a table, the loop is quite neutral. It does not do much. If internal rotation is big, turbulence appears, just like in a tornado or in a jumbo jet engine. And this external influence increases with the internal speed, just like on jumbo jets engine at full throttle. And just like that influence range is from a minimal, the dimensions of the engine when it does not run up to miles when it is in full throttle.
So, the possible range of movements of a wave that caught her tale are:
a) Spinning around its axes - meaning the rotation of the unpacked wave, just like a drill, around its geometrical axes, spinning that sill remains after packing.
b) Spinning around the geometrical centre of the loop, that can be independent to the previous one, but bought of them can coexist in the same time.
c) Spinning of the entire complex, would be that if you take the paper loop, that already moves as explained at a and b, and you move it so in your apartment.

These are the ones that I see. You already have seen, that each one can be independent of the other ones, and each one can have speeds between 0 and speed of light. The speed of original wave, linear or circular (when has caught its tale), remains the same and equal to the speed of light. This one will not change. Only the other 3 (described above at a, b and c). And here is the key in understanding why their behaviour and thus appearance is different for us.
If there is only one basic kind of wave, but the motion degrees are different, so will be the effects. Just like the cars. Some are green, some are yellow, other are red. Some stand still, some go slow, others fast. Some have small PS, others middle, race cars have a lot of PS. And so we get a lot of combinations of 3 from 3 in any direction we can imagine. And we have direction of movement too. Some are coming toward us, some are going away. And there is also the frequency of the original wave. This is also a parameter that can vary. Anyway, all these parameters above can explain the different way how a wave can appear to us and accordingly our perception of it.
Let’s see some examples. Let us say, that the original wave (a ring would be a appropriate picture of it) spins really fast. We know, from our life experience, how a tornado looks like. Or a jumbo jet engine. It usually spins one way. If you get in its ray of action, you can get sucked in. If it changes the way of its rotation, as pilots do with jet engines when they want to stop a plane at a very short runway, it will spit everything out (sort of say). A jet engine is only a few feet in diameter, bit its ray of action can be felt even miles away. Now replace the engine wit a wave. Our wave that caught her tale spins, at full power. That means, that its influence is much, much bigger that her diameter. And it attracts all it can attract. If you reverse the direction of rotation, it will reject everything and it will be very hard for something to get in her near. If someone or something will try to get in her near, it will probably remain at a certain distance, a balance point between the forces. Accordingly to this, if an observer would be in that point, it could see that the wave is there, active, spits or better said forces all other things to keep a distance, so it creates a own ‘’aura’’ that you can not surpass. You can not deny her existence, because you can see it and its effects, and as long it is ‘’on’’ and you do not have the force to go near it and touch it, it will just be there. If you have to waves, one the always attracts everything and one that rejects everything, the logic sais that if they somehow get in each others near, one will try to get the other as near as possible, even swallow it, the other one will do everything to reject this. So the ones force of attraction will keep the other in his near, at a certain distance, distance where the force of attraction and the force on rejection will balance themselves.
Now, from example to reality, the one that rejects all the time is the electron, little magic ‘’e’’. He can not touch anyone, so somehow it finds his balance and stays in the orbit of the one that attracts. The one that attracts is the positron. He spins too. With the same speed. But in opposite direction. A positron by itself will attract all that he can. In the first phase, it will catch all he can, mainly waves that are slower then he is. So, with the help of other smaller prisoners, he will get bigger. We call this gaining mass. And so he grows up and becomes a quark. Is as if our jet engine has caught on itself different things. And because he does not have wings to stop his spin, all these things spin with it. That why it is bigger. That is why we find more frequently quarks. Because if they come to life far enough from electrons, their tendency and actions will make them bigger as they were at birth. And because other slower waves are strapped to it, we can no longer see our disguised friend, and since then on we call it quark. Its charge is equal to the one of the electron. The charge of these two is equal, but opposite. And since we, humans, decided that attraction should be marked with ‘’+’’ and rejection with ‘’-‘’, since then we have ‘’eˉ’’ and the other one has a ‘’+’’. The one thing we call mass is and indicator of the number of smaller waves caught in the positive ones aura of influence. Now, if it has caught a lot of small waves, he will be slowed down, and so we will call it a ‘inferior quark’’. If he hasn’t caught to many and he keeps his speed, he will be a ‘’superior quark’’. If 2 of them come together , a middle class subatomic particle will come to life. If 3 of them come together, 2 superior ones and one inferior, we will have a proton.
The spin speed remains the same, only the range of action will increase. Strength grows in numbers.
And when you have 2 fast slick ones in one so named big particle, its range of influence (action) goes far, far more than its geometrical dimensions. And if an electron comes in its near? Well, that one rejects. Also said it wants its own private space. The proton, on the other way, he would gladly catch any time another fellow in his web. And so, they come to a balance. As if the electron would say ‘’I’m not really coming, but I will not go to far either’’. You can see me, salute me, but you can not touch me, ha, ha, ha! And as long this balance is kept, the electron will spin at a quite constant distance around the proton. And since the first proton has convinced the first electron to stay at a constant distance and orbit around him, since then we have our first atom. Hydrogen. He is made from 4 main waves. 3 ones that attract, 2 fast ones and one slower and one that rejects. And so we dissembled the electromagnetic force. Because she is not the one that keeps the electrons in orbit around the atomic core, it is positrons combined attraction force and their range of influence.
Further on, let us say that one of the main engines, a positron, due to a cause, as an interference with another wave, leaves the atomic core. Just the positron does that, because he is faster, and somehow, the other wave, maybe because it was still linear and not circular, could have found a way to get through the electronic cloud and impact directly with a quark. So he, the fastest one, gets separated. The first one that he will encounter will be the electron, because he is in the nearest vicinity of the core. Then instantly total annihilation. Or otherwise said, omnidirectional unpacking at the speed of light. That is why matter, (the thing that spins in one direction) annihilates itself with antimatter (actually the same thing that spins in the other way. That is why E=mc². Because they are the same. Matter, as we call it, is nothing more that circular packs of energy. Waves that bite on their own tales. An so we dissembled antimatter. Because is the same thing, just another direction of rotation.
And what remains behind after the engine is gone? Well the other fast one will be overcharged. If you have 2 horses and one dog that pull a big wagon with 10 miles on hour, and one horse gets loose, the other one and the dog will do their best. But his means that the speed will go down. The ‘’extra weight’’ will slow them down. It will probably be divided, accordind to each ones strength, but per general bought will be slowed down. Maybe every strong engine, as a positron or a electron actually have 3 overlapped ,fast, power rings, and when one positron leaves the core, the remaining ones, will rearrange themselves in 3 smaller engines with only two rings each. So the remaining slower waves will be further carried by these 3 smaller engines. Thus, the entire complex that remains will be slower. And if the charge sign, of +1 for positron and -1 for electron are only speed indicators, the logic makes us chose something between for the new slower complex. And what stays between -1 and +1? Well, you guessed right, it is 0. And since 0 takes no parts, then the new slower complex will be called neutron. Not because he is neutral, just because he is slower. Smaller engines, but same weight to carry. And if you try to catch a rabbit, and you use 2 grey hounds and one shi-tzu, you can guess ho will always get the rabbit. In our case the grey hounds represent the faster guys in the game, the positron and the electron.
So in this game of 3, whatever is the challenge, the neutron will never get there on time. So no results, no reason to call him otherwise than neutron. That is the only reason why he is neutral. Because he is always slower. And not many people until today have given a clear explanation why things are as they are with the neutron.
And here comes in place the weak force. Because, according to modern physics, she is the one responsible for these jumps from proton to neutron and vice versa. We will dissemble this one right away. There is another unclearly explained thing, the so magically called wave-particle duality. Well, since we have no particles and only waves, we do not have this thing any more. A wave, a linear one (unpacked), interacts with a compacted one (the so called particles). Why? Obviously, because they are bought waves. And if they have similar frequencies, they do interact. That is already known and proven until today. That is why there are no particles, just different ways how we perceive waves. If a wave, with a similar frequency as the one of an engine, goes through the electronic cloud and maybe hits at a certain angle, this wave can be caught and transformed in another engine, a bigger one, or can dissemble a bigger engine in smaller ones. So does the weak force work. And it actually all has to do with the spinning of the engine. The same force, no new one.
And when energy gets packed and slow down, we get the perception that they are 3D. And if at these low speed it encounters another similar pack, they interact. And somewhere, on this way, you can even get to touch it. And your hand that touches it is thought to be made from matter, but in fact it is energy, that touches other energy. We only get this perception because the interaction takes places at very small speeds in regard to the entire complex. And this is Newton’s physics and the chemistry. Wave interaction at very small and small speeds. We know that we are made from organs, that at their term are made from tissues, that are made from cells, that are made from cell organs, that are made from proteins and lipids and sugars, that are made from molecules, made from atoms, made from particles, that are in fact waves, small energy loops that interact with each other. So there is no duality, no wave-particle thing, because all are waves, just the speed differs. And because of this, because there is no matter, there can be no mass. An so another system errors are dissembled, and also a big hit to the 7 fundamental measuring units.
What remains? I just destroyed electromagnetic force, weak force, mass and wave-particle duality. We still have strong force and gravity. Until now we established that waves spin. And they spin and catch their tales, so they look like particles. And because some spin in a certain way, they attract all other particles. Slower and faster ones. With the spin in the same direction or opposite one. And those that attract get together. And they make bigger so named subnuclear particles. That at their term make nuclear particles. And they all spin together, on and on. And more of them get together. And they do attract themselves further on and on. And that is why they stay together in the atomic core. Because they spin in the same direction, the attraction direction. All dance together, in a big, happy circle. They have all the same charge sign, positive in this case for protons and 0 for neutrons, and unbelievable, they do not reject each other, as pretended from the electromagnetic force. Here being dissembled another system error, that particles with similar sign should reject each other. And so they dance on and on.
And the shy ones or the slick ones, they say they do not want to come in the dance circle, but dance at their own choice at a certain distance, without holding hands with the similar ones, so creating the electronic cloud. But those that are in the middle, hand in hand in the circle have are more than those that stay on the side. And their force is bigger and goes beyond the atoms diameter, and if some other atom comes in his near, they will attract themselves. Of course, it is not a perfect world. There are obviously more attracting engines, then repelling ones. And the form that they combine themselves is one in the favour of the attracting ones. Every attracting engine will try to get even a repelling one. Not all of them will succeed. So this is we have atoms with less electrons than protons. On a 3D model, that would mean that the electronic cloud is not balanced, there are holes that need to or can be filled. And so, when atoms with similar needs come together they bond in what we call a chemical reaction. As said before this is nothing more as the physics at low speed and temperatures. And still the attraction force reaches beyond the range of the molecule, because no molecule is sealed. The more balanced ones are ‘’chemically stable’’ as we call them, or less reactive. And even if they come together, they still move, they still spin. That is why Heisemberg’s principle. Because we try to see something moving using something else in movement. And when they do not form molecules, they just stay together. A crowd of them will attract another crowd. Bigger and bigger with every chance. And from really big crowds suns and planets will emerge. From planets and suns solar systems. From solar systems galaxies. And they all continue to spin. And this is gravity. Following the logic gravity is strong force. And strong force is gravity. And because gravity is made only from one word, strong force from two, and because gravity was observed long before the strong force was made up, I will call it from now on just gravity. And so we leave behind us the strong force, another system error.
And because protons and neutrons are the attracting ones, we will acknowledge that they have inside them them positive gravitons, ore g+. And the electrons, because they are negative, will be g-. And we found these ones too. The protons and the neutrons still need some dissection to find there core, but with the electrons the case is clear. Now that we have names for them is ok and we already know what they do. And how much they weigh. Still there is work to do. I still believe that there will be many that will still look for them from now on. But in all the huge family of particles is hard to realise that a system error was made, that only the hammer (mathematic) was used, logic was left aside, as follows all looked as nails and you will loose the sight of the forest behind the tree line and no one gets to say ‘’people, somewhere, along the way, we made a mistake, but look now how the things really are!’’.
Now let us demonstrate this from the other end. We will start again with electromagnetism. Because it was the next big found after gravity. According to it, the electrons jump from an atom to another. Why? Because, in the 3D model, if more g+ in the core have a certain influence, they will try to satisfy this need with whatever they find, meaning even g-. But g- guys do not want to bond with no one else. They want to run free in the big and broad universe. But when they do interact in some sort (impossible love), it is not forever. Every interference determines another action. And some guys, (not me), long ago, called it electromagnetic interaction. Actually it is gravitational interaction.
And if strong force is gravity, weak force is gravity, means that all we have is gravity. Made from waves with different speed and rotation axis. And that is why light is deviated with an angle of 2 seconds from Earth’s gravity. Because that is also gravitational interaction. :)

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Re: Uncommon GR derivation

Postby pogono » Tue Jul 23, 2013 5:48 pm

Ufff, generalization to General Relativity finished! :-)
In the way to arXiv.

I will appreciate your remarks and comments.

Title: Maxwell-like General Relativity formulation in the Planck limit

Abstract: We show that Geroch decomposition leads us to Maxwell-like representation of gravity in (3+1) metrics decomposition. For such decomposition we derive four-potential V^\mu and gravitational field tensor F^{\mu\nu} that may be associated with gravitational interaction. Next we introduce valid Lagrangian and equations of motion. Then we show that gravitational four-current J^\mu derived for introduced four-potential produce energy-stress tensor and generalize main General Relativity formula. At the end we introduce new approach to quantization of gravity that results in proper quantum values and is open to further generalization.

File: http://www.dilationasfield.net/gaf.pdf

I have also prepared brief explanation of DaF framework ended with GR generalization for gravitational field.


Thanks to Geroch's decomposition applied for Schwarzschild solution of the General Relativity one obtains metric in (3+1) decomposition (space + time). In this picture curved spacetime is equivalent to the flat space-like manifold minimally coupled to a scalar field \Phi, where \Phi is equal to inversed gravitational time dilation factor.

In obtained picture gravity is described as regular field with Maxwell-like equations on flat spacetime as follows.

We define scalar fields as follows:
\Phi=\frac{1}{\gamma_r}=\frac{d\tau_r}{dt}= \sqrt{1-\frac{r_s}{r}}
\Theta= r \cdot \beta_r = r\sqrt{\frac{r_s}{r}}
t is the time coordinate (measured by a stationary clock located infinitely far from the massive body)
\tau_r is the proper time of stationary observer located in distance r to the massive body
r is the radial coordinate
r_s is the Schwarzschild radius

As we know mass relation to Schwarzschild radius is:

Therefore (assuming c=1) one may easy calculate that the gradient of the scalar field \Phi is equal to proper gravitational acceleration "g" for Schwarzschild solution.
\nabla \Phi = \frac{r_s}{2r^2} \gamma_r = \frac{GM}{r^2} \gamma_r = g

Above "g" acceleration is measured for the reference frame of the stationary observer located at distance "r" to the massive body.

Using introduced above scalar fields one defines vector fields as below (we assume c=1 and \hat{e} as directional versors).
Vector field responsible for gravitational acceleration is denoted as “G”.

\vec{T} = \Phi \cdot \hat{e}_y
\vec{G} = -\nabla \Phi \times \hat{e}_y = -\nabla \times \vec{T}
\vec{V} = \nabla \Theta \times \hat{e}_x
\vec{B } = \nabla \times \vec{V}

Utilizing relations between above fields, one obtains Maxwell-like equations for gravitation:
\nabla \cdot \vec{G} = 0
\nabla \cdot \vec{B} = 0
\nabla \times \vec{G} = \gamma_{r} \cdot \frac{\partial \vec{B }}{\partial t}

\nabla \times \vec{B} = - \gamma_{r} \cdot \frac{\partial \vec{G}}{\partial t}

Therefore by analogy to electromagnetism we may introduce four-potential V in form of:


and related gravitational field tensor:

F_{\mu\nu}=\partial_\mu V_\nu - \partial_\nu V_\mu
After simple transformations one derives wave equation (d’Alembertian) as follows:

\gamma^2_r \cdot \frac{\partial ^{ 2}\vec{G} }{ \partial t^{2}} - \nabla^2 \vec{G} = 0

Above d’Alembertian describes the wave propagating in the flat spacetime with speed equal to:
v_{light}=c/\gamma_r=c \cdot \sqrt{1-\frac{r_s}{r}}

In result, in considered case curved spacetime is physically equivalent to the flat spacetime with variable speed of light, where refracting index for light is equal to

It should not surprise us, that in above picture, spacetime around the mass behaves as gravitational lens.

Lagrangian and Hamiltonian

Analyzing Einstein-Hilbert action for considered case one may derive proper Lagrangian and Hamiltonian for gravity on flat spacetime with refracting index for light speed.

For the stationary observer with rest mas 'm' that keeps his position against gravitation we obtain Lagrangian and Hamiltonian in the form of:
\mathcal{L}= mc^2 \frac{1}{\gamma_r}
\mathcal{H}= mc^2 \gamma_r

Thanks to superposition principle for some test body with rest mass 'm'' and four-velocity
we obtain proper Lagrangian and Hamiltonian in form of:
\mathcal{L}= mc^2 \frac {1}{\gamma_r} - mc^2 \frac {1}{\gamma}
\mathcal{H}= mc^2 \gamma - mc^2 \gamma_r

To comply with the Newtonian approximation:

\mathcal{H}=mc^2 (\gamma - 1) - V \left ( r \right )


V \left ( r \right ) =mc^2(\gamma_r-1)

Classic Newtonian approximation we obtain using Maclaurin expansion of above Hamiltonian, the same way that we do it for Kinetic energy approximation:
\mathcal{H}= mc^2 (\gamma-1) - mc^2 (\gamma_r-1) \approx mc^2 \frac{\beta^2}{2} - mc^2 \frac{\beta_r^2}{2} = \frac{mv^2}{2} - m \frac{c^2 r_s}{2r} =\frac{mv^2}{2} - G\frac{mM}{r}

If we consider above field V in the Planck limits, we obtain proper quanta equal to rest energy value.
For r_s << l_{P} we calculate:

\lim_{m \to m_P; r \to l_P} V \left ( r \right ) = m_{P}c^2 \left( \frac{1}{\sqrt{1-\frac{r_{s} }{l_P} } } -1\right) \approx m_{P} \cdot \frac{c^2 r_{s} }{2l_{P} } =\frac{c^{ 4}r_{s} }{2G}=Mc^2

Obtained quanta may be treated as some rest energy (some rest mass M) related to given radius r_s (Schwarzschild radius).

Equations of motion and relation to Newton-Cartan theory

The introduced Lagrangian locally satisfies the Euler-Lagrange condition:

\frac{d \frac{\partial \mathcal{L}}{\partial v}}{d\tau_r}= \frac{\partial \mathcal{L}}{\partial r}

what yields to:

\frac{d (mv\gamma)}{d\tau_r}= mc^2 \cdot \frac{r_s}{2r^2} \gamma_r

where LHS is relativistic force and RHS is just equivalent to "gravitational force" in Schwarzschild solution.
Using introduced scalar field \Phi we may rewrite it as:

RHS=mc^{2}\cdot \nabla \Phi

Considering above we see, that equations of motion should fulfill transitional condition (assuming c=1 to facilitate):

\frac{d(v\gamma)}{d\tau_r}= \nabla \Phi

Therefore equations of motion for considered case are in the form of:

\frac{d^2\vec{x}}{d\tau d\tau_r}= \nabla \Phi

\tau is the proper time of the test body
\tau_r is the proper time of stationary observer located in distance r to the massive body
\gamma=d\tau_r / d\tau

Above equations may be explained as relativistic form of Newton-Cartan theory equations of motion. In DaF, the proper times were taken in place of coordinate time "t".

The equations of motion in terms of four-velocity U and four-acceleration A may be rewritten for stationary observer reference frame as:

\frac{d U^\mu}{d\tau_r}= \partial_\mu \Phi

and for the test body reference frame as just:

A^\mu = \partial_\mu \Phi

General Relativity generalized for gravitational field

In the Lorentz gauge equation of motion may be rewritten as:
A^\mu=\frac{\partial V^\mu}{\partial \tau}

Above formula says, that gravitational acceleration is equal to derivative of the gravitational four-potential. Reversed it also says, that any move is the source of gravitational acceleration. It means, that any body with four velocity U^\mu is at the same time the source of gravitational potential V^\mu:

U^{\mu}= \left ( \gamma , \gamma \vec{v} \right ) \to V^{\mu}= \left ( \frac{1}{\gamma}, \vec{V} \right )

Four-acceleration A may be generalized to the tensor in the form of:
A^{\mu\nu}=\partial_\nu V^\mu

showing it is equal to the first element of introduced gravitational field tensor:
F_{\mu\nu}=\partial_\nu V_\mu - \partial_\mu V_\nu

where the second part is the acceleration for the other bodies.

If we introduce gravitation four-current by analogy to electromagnetism:
J^\mu=\partial_\nu F^{\mu\nu}=\frac{2\pi r_s}{V}\cdot(\gamma_r,\gamma_r\vec{v_r})

and generalize to the tensor multiplying by four-velocity U of the source of gravity:
J^{\mu\nu}=J^{\mu} \cdot U^{\nu}

we obtain stress-energy tensor T with respect to the constant
J^{\mu\nu}=\frac{4 \pi G}{c^4} T^{\mu\nu}

what drives to General Relativity main formula in the form of:
G_{\mu\nu}=2 \cdot J_{\mu\nu}

This way we have created General Relativity main equation equivalence as wave-based formulation that might help us with explaining the wave nature of mater.

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