Exact solutions of Einstein's field equations

waitedavid137

Honorable
This is going to take a lot of explaining so I'm just going to post a bit of it now and then. So lets start looking at the Majumdar-Papapetrau solution
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If you have a static charge distribution of extremal charge which is the case where
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then that will be the exact solution to Einstein's field equations for the spacetime outside of it where
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corresponds to a Newtonian gravitational potential for the mass distribution.
Essentially what I've found in working out exact solutions to the field equations, which I'll get to demonstrating, is that general relativity is an already internally unified classical theory of gravity and electromagnetism where masses correspond to an extreme charge whose observed value is perturbed by the surrounding electromagnetic radiation field.
Consider the Riessner-Nordstrom solution
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This is the exact solution to Einstein's field equations for the spacetime outside a spherically symmetric charge.
The stress-energy tensor outside of the charge, list order
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, is according to these coordinates
gif.latex

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What coordinates you can express things in for general relativity are not unique. There are other coordinates that can be used to express
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Another choice of r and t coordinates yields
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In these coordinates, the relation to the Majumdar-Papapetrau solution becomes clear. The first couple of terms correspond to that solution exactly and is what you get in the case of
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where the stuff on the right is then made to vanish. The stuff on the right is a perturbance to that solution when the observed charge q is allowed to vary from the mass M. In this coordinate choice the expression for the stress-energy tensor transforms to
gif.latex

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more to come...
 

waitedavid137

Honorable
The next thing I want to introduce is Brinkman's plane polarized electromagnetic and gravitational radiation solution.
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h is a function of
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.
In these coordinates it is the exact solution to Einstein's field equations for
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The upper left term is the energy density of the electromagnetic radiation. When
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the metric still represents a speed of light wave traveling though it with spacetime curvature, but the stress energy tensor here goes to all zero and so it becomes then an exact solution to the field equations for gravitational plane waves.
Now I introduce a transformation to a light-like coordinate in place of this time coordinate.
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The solution is now expressed
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where h is now a function of
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and the expression for the stress-energy tensor is transformed to
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So in this kind of light like coordinate the time space cross term of the top row is minus the energy density of the electromagnetic radiation and its column tells you the direction of the momentum flow. In his case the flow of momentum is in the
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direction.
Yeah it struck me as crazy that general relativity can use light like coordinates when there is no valid light like frames in special relativity. But it works fine.
 
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Dejan Corovic

As above, so bellow
gif.latex


I can translate this equation to plain English, just to score Purple Heart :) :

ds^2 - is some kind of space dimensional change, maybe movement of a point in the observed cube of space.
(2GM)/(r c^2) * dct^2 - is slowing of the time due to presence of the large mass,
(kGq^2)/(r^2c^4) - is speeding of the time due to the electrical charge attached to the large mass,
dr^2/(1-(2GM/rc^2)) - is shrinking of the space due to the large gravitational mass M,
dr^2/(1-(kGq^2/r^2c^4) - is the expansion of the space due to electrical charge q.
r^2d-omega^2 - is a radial shrinking of some kind (?) in the elevation plane. A bit strange because it's not tied to any physical actor, like mass or charge.

So the overall equation is telling us: mass is shrinking and the charge is expanding space-time.

Am I an overconfident smart Alec :), who deserves to eat humble pie or not?
 
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waitedavid137

Honorable
Now I can move to introduce the charged Vaidya solution. The Vaidya solution is the classical general relativity exact solution to Einstein's field equations for electromagnetic radiation coming from or collapsing to a central spherically symmetric source. In other words you don't need quantum mechanical modification to a black hole solution to consider electromagnetic radiation exactly with only general relativity. Yes, this does in Hawking and so you won't hear about it any time soon from anyone but me. But you can find papers on it which media and public relations, which are immoral level of left politically motivated, won't tell you about. In one choice of coordinates this solution can be expressed
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where M is a function of
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and the electromagnetic radiation is in the
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direction. In these coordinates the stress energy tensor that this is the exact solution for is expressed
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where
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and
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Here u is a light like coordinate and we can transform the solution to be expressed in terms of that coordinate instead of ct.
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M is now expressed as a function of u.
According to this transformation the stress-energy tensor is now expressed
gif.latex
 
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waitedavid137

Honorable
gif.latex


I can translate this equation to plain English, just to score Purple Heart :) :

ds^2 - is some kind of space dimensional change, maybe movement of a point in the observed cube of space.
(2GM)/(r c^2) * dct^2 - is slowing of the time due to presence of the large mass,
(kGq^2)/(r^2c^4) - is speeding of the time due to the electrical charge attached to the large mass,
dr^2/(1-(2GM/rc^2)) - is shrinking of the space due to the large gravitational mass M,
dr^2/(1-(kGq^2/r^2c^4) - is the expansion of the space due to electrical charge q.
r^2d-omega^2 - is a radial shrinking of some kind (?) in the elevation plane. A bit strange because it's not tied to any physical actor, like mass or charge.

So the overall equation is telling us: mass is shrinking and the charge is expanding space-time.

Am I an overconfident smart Alec :), who deserves to eat humble pie or not?
I am no longer capable of speaking in terms that layman can understand. Please get your favorite physics guy to look at what I am saying. Ask him if he can translate. However few physics guys can touch this. I don't know what to do about that. Maybe one minute chunk at a time. Ask what one minor thing is and I'll explain and hope understandably.
 
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Dejan Corovic

As above, so bellow
I am no longer capable of speaking in terms that layman can understand. Please get your favorite physics guy to look at what I am saying. Ask him if he can translate. However few physics guys can touch this. I don't know what to do about that. Maybe one minute chunk at a time. Ask what one minor thing is and I'll explain and hope understandably.

I completely understand. That's how I felt when I finished university, nobody understood me and I understood nobody else :). Confusing side effect of overeducation. We are just simple animals that are only supposed to survive and reproduce.

Let's try "blue pill, red pill". I am only looking at what bundle of physical parameters is acting on which differential, space or time.

upload_2020-7-19_9-32-8.png

Let's take the "red" pill first.

In the equation above, there is a bundle of physical params marked in a shade of red. They are all neatly related to the physical mass. And joined up with "-" sign. We all know, from pop-sci magazines, that mass is slowing time, here represented as differential dct^2. By that, all things being equal, I guess the equation is telling us: "mass has a negative effect on time" or "mass will cause the time to slow down". Is that right interpretation?

upload_2020-7-19_9-26-39.png

Now the "blue" pill.

Anyway, I am mostly interested in the physical parameters bundle marked in blue, above. Because it is saying the opposite. There is "+" sign in front of the bundle of physical parameters, all related to electrostatic charge. That means that electrostatic charge has a positive effect on the time dilation and that will cause time to "speed up". Basically, mass is slowing down the time and electrostatic charge is speeding up the time. Am I getting it right here?

Physical parameters tied up with dr^2 can be interpreted in a similar way, but it will be easier to understand if we keep it short. One step at a time.

@Thomas R. Morrison what do you say about this my translation to plain English? Is there a chance it's on the right track?
 
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Dejan Corovic

As above, so bellow
Essentially what I've found in working out exact solutions to the field equations, which I'll get to demonstrating, is that general relativity is an already internally unified classical theory of gravity and electromagnetism where masses correspond to an extreme charge whose observed value is perturbed by the surrounding electromagnetic radiation field.

There is this video by Sabine Hossenfelder listing five most promising approaches to gravity and the fifth approach is Emerging gravity @5:58. The way she explained it is that these theories assume that gravity is an emerging property, very much like temperature and thus GR and QM can not be unified because there is nothing to unify.

That goes hand in hand with what you said, that GR is already internally unified theory of gravity and electromagnetism

 
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waitedavid137

Honorable
I completely understand. That's how I felt when I finished university, nobody understood me and I understood nobody else :). Confusing side effect of overeducation....
lol no I blew past leftist education brainwashing decades ago. Literally I can no longer communicate where I am at on the common level. Get physics friends to look at it and try to translate. BTW as a footnote
gif.latex

where M is still a function of u is the exact solution combining Reissner-Nordstrom and Vaidya and deSitter.
 

waitedavid137

Honorable
So I mentioned that the Riessner-Nordstrom solution
gif.latex

can be expressed in different coordinates as
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in which its stress-energy tensor is expressed as
gif.latex

where
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for which its relation to the Majumdar-Papapetrau solution
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becomes clear.
I also mentioned that the charged Vaidya solution for which it radiates can be expressed in different coordinates as
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for which the stress energy tensor is expressed as
gif.latex

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so the next step is to express the charged Vaidya solution in appropriate coordinates that it also shows its immediate relation to the Majumdar-Papapetrau solution. With a different choice of a radial like r coordinate the charged Vaidya solution can be expressed as
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where M is a function of u and in these coordinates, the stress-energy tensor is expressed
gif.latex

where E is now defined instead in terms of this r coordinate as
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Now for
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this immediately demonstrates its relation wherein for extreme charge
gif.latex
and constant M it immediately reduces to the Majumdar-Papapetrau solution.
 
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waitedavid137

Honorable
So as mentioned, the Reissner-Nordstrom solution can be written in coordinates so that it is expressed as
gif.latex

and in these coordinates the expression for the stress-energy tensor transforms to
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and we were motivated to do this in looking for how it relates to the Majumdar-Papapetrau solution. The next step then would be to start looking at other simple charge geometries and consider analogous coordinates such that the relation to it is clear.
So consider the stress-energy tensor
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where
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and
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This would represent the stress-energy for a uniform electric field off of a charged plane of charge per area
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and mass per area
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. My exact solution for that is
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where
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is a constant ultimately related to internal pressure in the sheet of charged mass.
 

waitedavid137

Honorable
so, what are your equations saying?

Is UFO propulsion possible or not?
This isn't directly about UFO propulsion. Solving the field equations along the lines I am going is about better understanding the connections between gravity and electromagnetism and given the unintuative level of increadibly nonlinear behavior from the field equations about better understanding something several claim to understand, but few do, strong gravitation.
 

waitedavid137

Honorable
So I've mentioned that the charged Vaidya solution can be in one choice of coordinates expressed as
gif.latex

where u was a light like coordinate and according to these coordinates the expression for the stress energy tensor becomes
gif.latex

and the motivation in going to these is that its connection to the Majumdar-Papapetrau solution is clear.
For simplicity I am going to drop absolute value signs and just look at the +z side of a charged radiating sheet. Consider the stress-energy tensor
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where
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and
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, again dropping absolute values, considering the +z side of the sheet, capital
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is the energy density of the light-like radiation, and in this context lowercase u will be a light-like coordinate given by
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, (radiation coming from the sheet corresponds to this choice of sign.
My exact solution to Einstein's field equations for this is
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Where capital U relates to this by
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The radiation is electromagnetic with the exception of regions in which
gif.latex

for which it is purely gravitational radiation.
 

waitedavid137

Honorable
So I've mentioned that the charged Vaidya solution can be in one choice of coordinates expressed as
gif.latex

where u was a light like coordinate and according to these coordinates the expression for the stress energy tensor becomes
gif.latex

and the motivation in going to these is that its connection to the Majumdar-Papapetrau solution is clear.
For simplicity I am going to drop absolute value signs and just look at the +z side of a charged radiating sheet. Consider the stress-energy tensor
gif.latex

where
gif.latex
and
gif.latex
, again dropping absolute values, considering the +z side of the sheet, capital
gif.latex
is the energy density of the light-like radiation, and in this context lowercase u will be a light-like coordinate given by
gif.latex
, (radiation coming from the sheet corresponds to this choice of sign.
My exact solution to Einstein's field equations for this is
gif.latex

Where capital U relates to this by
gif.latex

The radiation is electromagnetic with the exception of regions in which
gif.latex

for which it is purely gravitational radiation.
One sign typo. It should be
gif.latex
 

waitedavid137

Honorable
So we are making some progress in working out exact solutions to the field equations for electromagnetic fields and radiation which can be backchecked by ctensor within maxima. Again, I can walk anyone through doing that. I have mentioned that the Reissner-Nordstrom solution can be written in coordinates such that it is expressed
gif.latex

and in these coordinates the stress-energy tensor is expressed as
gif.latex

with the list order
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and I have mentioned that my exact solution to the field equations for
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list order
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with
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and
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is
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So the next simple geometry to look at would be a line charge.
The stress-energy tensor in analogous coordinates in relating to the Majumdar-Papapetrau solution will be
gif.latex

list order
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where r in this context is a radial like coordinate instead of a spherical like one and
gif.latex
and
gif.latex

gif.latex
is the charge per length and
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is the mass per length of the line. R is a constant.
My exact solution to Einstein's field equations for this is
gif.latex
 
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