Is Gravitomagnetism Covariant and Are There Any Alternative Approaches?

[PeterDonis]

Be cautious about the Wikipedia entry in this case.

I agree this is always good advice. The equations as I'm seeing them on the page currently are (in units where $G = c = 1$):

$$
\vec{\nabla} \cdot \vec{E}_g = - 4 \pi \rho_g
$$

$$
\vec{\nabla} \cdot \vec{B}_g = 0
$$

$$
\vec{\nabla} \times \vec{E}_g = - \frac{\partial \vec{B}_g}{\partial t}
$$

$$
\vec{\nabla} \times \vec{B}_g = 4 \left( - 4 \pi \vec{J}_g + \frac{\partial \vec{E}_g}{\partial t} \right)
$$

Some of the numerical factors may be due to definitions, but the key points where the equations should differ from Maxwell's Equations--the minus sign on the RHS of the first equation and in the first term on the RHS of the fourth equation--appear to me to be "correct", in the sense that they make sense given that gravity is always attractive. However, as I noted earlier in the thread, I have not seen an explicit derivation of these equations from the EFE using a suitable approximation.

 

[Jonathan Scott]

Last time I was studying gravitomagnetism, I worked through the derivation in this paper:

Gravitoelectromagnetism: A Brief Review - Bahram Mashhoon
arXiv:gr-qc/0311030v2 17 Apr 2008

 

[Bill_K]

Some of the numerical factors may be due to definitions.

And then again, some are not. It's hard to put much faith in a set of equations in which gravitational waves propagate at one-half the speed of light.

 

[PeterDonis]

And then again, some are not. It's hard to put much faith in a set of equations in which gravitational waves propagate at one-half the speed of light.

Hm, good point. I note, btw, that in the paper Jonathan Scott linked to, the extra factor of 4 in the last equation is not present.

 

[Quantum Immortal]

Some of the differences are just different definitions. Others are not
The GEM that is meant here, is the ones that look the same with attractive EM. Period!

He defines E=-a

gravitational waves propagate at one-half the speed of light.

I just noticed that 4 in the wiki article. It also has a "factual accuracy is disputed" tag...
I think the 4 is at the wrong place in the wiki.
If you derive them from Newtons law with SR, the same way you derive EM from Coulombs law, there is no 4.
The paper has no 4. Its just straight out attractive EM-like...

I feel i need to comment about some attitudes here that i start to dislike a lot.
Why some people here are ready to believe, that the author and springer were so incompetent, as to use a repulsive model of gravity with half the speed of light as propagation speed?
A good high school student could notice these...
And if the wiki article says something supportive, we are extremely critical, "don't trust the wiki". But if it says something negative, we take it at face value...
How you spell "bigoted" and "preconceived"?
B[

The GEM that is meant here, is the ones that look the same with attractive EM. Period!

Can some one, with the skill needed, and without being preconceived against GEM. Tell me if the calculations of the article look correct? For now, i trust springer. Do i need not to trust springer?

Again. These calculations, could have been made over 100 years ago. If I'm correct, back at a time that only the orbit of mercury was an observable problem. So, if the paper is correct, it would mean that GEM would have been a complete theory back then, before GR, before modern SR even. What is done in GEM is so simple, that its impossible that no one thought about it. So, what's going on?
Some one wants to conjecture on this? I'm missing some of the history?

Can some one find, or already know, other examples, of comparing straight out EM-like gravity equations with observations?
I think, that frame dragging is an other example...
A straight comparison with PPN should be very informative.

 

[PeterDonis]

The GEM that is meant here, is the ones that look the same with attractive EM. Period!

Not if the equations say differently. See below.

He defines E=-a

Doesn't matter; his equation for $\vec{\nabla} \cdot \vec{E}_g$ is what makes positive masses repel other positive masses. See below.

I think the 4 is at the wrong place in the wiki.

I think it doesn't belong there at all, since, as I noted, the paper Jonathan Scott linked to, which actually goes through the derivation of GEM equations in the weak field, linear approximation from the EFE, does not have the 4. (Neither does the paper you linked to, but it has other problems; see below.)

The paper has no 4. Its just straight out attractive EM-like...

Not having the factor of 4 doesn't make gravity attractive in that model. The sign in its equation for $\vec{\nabla} \cdot \vec{E}_g$ is the opposite of that in the paper Jonathan Scott linked to. That makes a *big* difference; see below.

Why some people here are ready to believe, that the author and springer were so incompetent, as to use a repulsive model of gravity

Well, that's what the equations in the paper you linked to say; see below. As for why the paper passed review, lots of papers pass review that are later found to have errors in them. Since the author of the paper does not explicitly compute whether or not gravity is attractive in his model (he just assumes it), and since his main point was to compute perihelion precession, it may simply be that none of the reviewers thought to check; they assumed that he was correct about gravity being attractive, and just checked his computations of the precession.

with half the speed of light as propagation speed?

I believe Bill_K was referring to the Wiki page, not the paper you linked to. The factor of 4 in the $\vec{\nabla} \times \vec{B}_g$ equation on the Wiki page is what makes gravitational waves propagate at half the speed of light. As noted above, neither the paper you linked to nor the paper Jonathan Scott linked to have that factor of 4, so it looks to me like gravity waves in both of those papers would propagate at the speed of light, as expected.

And if the wiki article says something supportive, we are extremely critical, "don't trust the wiki".

"Don't trust the wiki" is, as I noted to Jonathan Scott, *always* good advice.

But if it says something negative, we take it at face value...

I don't think anyone in this thread has taken anything the Wiki page says at face value; I know that I've repeatedly said I want to see other derivations before accepting the equations it gives as correct (and now, of course, other derivations have shown its equations to be wrong in having the factor of 4 in the last one). So I don't know what you're complaining about here.

The GEM that is meant here, is the ones that look the same with attractive EM. Period!

You might have "meant" that, and so might the author of the paper you linked to, but you don't get to just declare it by fiat. You have to actually look at the math.

Look at the paper Jonathan Scott linked to; the sign difference in the equations that I described above is obvious. Now look at Maxwell's Equations, which are known to predict like charges repelling, not attracting. Which paper matches the sign in Maxwell's Equations?

These calculations, could have been made over 100 years ago. If I'm correct, back at a time that only the orbit of mercury was an observable problem. So, if the paper is correct, it would mean that GEM would have been a complete theory back then, before GR, before modern SR even. What is done in GEM is so simple, that its impossible that no one thought about it. So, what's going on?

I can't say why the Heaviside paper wasn't picked up on, but one hypothesis is that the pathway he was suggesting to modifying Newton's Laws just wasn't one that enough physicists were ready to follow. The pathway that Einstein used--first show that the kinematics of Newton's theory have to be modified, with SR, before tackling the dynamics, with GR--was apparently one that worked better.

 

[Dale]

A straight comparison with PPN should be very informative.

I agree. I think comparisons of the PPN parameters are the best way to understand the physical differences between gravitational theories and to compare them to experiments. Unfortunately, I don't know of anyone having done that.

 

[PeterDonis]

He defines E=-a

After looking at the paper again, I want to expand on this a bit more.

First, the paper never says $\vec{E}_g = - \vec{a}$. In some places it says the opposite: $\vec{E}_g = \vec{a}$ (for example, the beginning of section 2.1 where it talks about the gravitomagnetic field produced by a moving particle, and the beginning of section 4 where it gives the formula for the gravitational Lorentz force). In one place (the top of p. 64) it does say $\vec{E}_g = - \partial \vec{v} / \partial t$; but, aside from the fact that this contradicts the other places, it also is followed up almost immediately by this:

"Equations similar to these equations can be derived from the equations of general relativity in the limit of a weak field. However, these equations come with a negative sign of the density term (Mashhoon and Will 1989; Behera and Naik 2004)."

In other words, the paper's author explicitly *intends* for the sign of the RHS of his $\vec{\nabla} \cdot \vec{E}_g$ equation to be the opposite of the one derived from GR (which is the one given in the paper Jonathan Scott linked to); it's not just an accident or a mistake about the sign convention. In other words, the paper's author intends for this theory to look exactly like electromagnetism, in which the standard definition of $\vec{E}$ has no minus sign in front of $\vec{a}$.

In fact, as I've already noted, the paper's GEM equations are exactly the same as Maxwell's Equations, except for the subscript $g$ on things; so the actual definition of $\vec{E}_g$ that he is using must indeed be the same as the standard definition of $\vec{E}$ in the EM case, which means a repulsive force for like charges. And his equation for the gravitational Lorentz force is also exactly the same as the corresponding EM equation, which makes it clear that the standard definition of $\vec{E}$ is $\vec{E} = \vec{a}$ (no minus sign), since $\vec{E}$ appears in the Lorentz force equation with no minus sign. So gravity between like masses (i.e., positive masses) must be repulsive in this theory, just as the EM force between like charges is repulsive.

So, in summary: the paper's statement at the top of p. 64 that $\vec{E}_g = - \partial \vec{v} / \partial t$ is clearly inconsistent with the rest of the paper (unless he's adopting some weird sign convention for $\vec{v}$); and the math given in the paper as a whole makes clear that gravity between positive masses in the theory given is repulsive (which the author apparently doesn't realize), just as the EM force between positive charges is repulsive.

 

[Quantum Immortal]

I think he never uses one definition. He just uses what's more convenient every time, so that he doesn't have to bother with the signs. He never forgets that gravitation is attractive.

... O_O GEM is actually, implicitly non linear
The E field, is just the acceleration (F/m=ma/m=dv/dt). It doesn't care what it acts on. Its not like the electric field, that acts on charges only(F/Q).
So, the E field, also acts on light. Light been an object moving at the speed of light, and with no rest mass.
Similarly, it acts on gravitational waves and static gravitational fields, as objects with no rest mass, going at the speed of light.Also: I think i understood better the space space components of the stress energy tensor. The shear stress corresponds to angular momentum inside matter. The flow of angular momentum? Normal EM equations, don't explicitly bother with angular momentum. That would give a source, with 7 components.
I don't quite get what pressure is equivalent too... The flow of momentum?
But that's become important only inside very dense stars.
And the energy part, already represent all the non gravitational energy of the system.
So the space space parts are something to do with the flow of momentum, i don't clearly see what that means.
Maybe, this part could be derived, the same way we derive Ampere's law from Coulomb's law?

 

[Dale]

I think he never uses one definition. He just uses what's more convenient every time, so that he doesn't have to bother with the signs.

You cannot make a self consistent theory that way.

 

[PeterDonis]

He never forgets that gravitation is attractive.

He can't make gravity attractive in his theory just by saying so or believing that it is. As I've shown, the math he presents makes gravity in his theory repulsive, regardless of what he believes.

O_O GEM is actually, implicitly non linear

There's no such thing. "Linear" is a precise mathematical term; it's easy to check whether an equation is linear or not. There's nothing implicit about it.

The E field, is just the acceleration (F/m=ma/m=dv/dt). It doesn't care what it acts on. Its not like the electric field, that acts on charges only(F/Q).
So, the E field, also acts on light. Light been an object moving at the speed of light, and with no rest mass.
Similarly, it acts on gravitational waves and static gravitational fields, as objects with no rest mass, going at the speed of light.

None of which has anything to do with whether or not the equations are linear.

The shear stress corresponds to angular momentum inside matter. The flow of angular momentum?

This makes no sense. You can have shear stress in a material that isn't spinning, and in which none of the atoms are spinning or orbiting around each other, so there's no angular momentum anywhere.

Normal EM equations, don't explicitly bother with angular momentum.

That's because the source of electromagnetism is charge. The analogue of "angular momentum" for charge would be current flowing in a loop, which Maxwell's Equations certainly do address.

That would give a source, with 7 components.

In the GEM equations, the angular momentum of the source shows up in the $\vec{B}_g$ field, which is tied to rotating matter, i.e., the matter current is flowing in a loop, just like charge flowing in a loop creates an ordinary magnetic field. In other words, the GEM equations already contain angular momentum in the source. What they leave out is pressure and stress, which, as I noted above, have nothing to do with angular momentum.

I don't quite get what pressure is equivalent too... The flow of momentum?

No; momentum *is* a "flow", there's no such thing as "flow of momentum" other than momentum itself. And momentum is a different part of the stress-energy tensor from pressure.

I would recommend looking at a text about condensed matter physics that talks about how the ordinary stress tensor is derived from the microphysics of atoms. You appear to have some basic misconceptions about that which have nothing to do with relativity. Or you could ask about it in a separate post in the solid state physics forum here on PF; there are people there who can explain the details much better than I can.

 

[Bill_K]

I would recommend looking at a text about condensed matter physics that talks about how the ordinary stress tensor is derived from the microphysics of atoms. You appear to have some basic misconceptions about that which have nothing to do with relativity. Or you could ask about it in a separate post in the solid state physics forum here on PF; there are people there who can explain the details much better than I can.

I agree with this advice, which has been given to you already several times in this thread. You're just grasping at straws, and this is no way to learn something. There's nothing mysterious about the stress energy tensor, and no purpose in trying to reinvent it (and getting it wrong).

 

[PAllen]

No; momentum *is* a "flow", there's no such thing as "flow of momentum" other than momentum itself. And momentum is a different part of the stress-energy tensor from pressure.

Well, John Baez seems to disagree with you. From http://math.ucr.edu/home/baez/gr/stress.energy.html :

Tab is the flow in the a direction of momentum in the b direction!

To understand this, remember that a,b=0,1,2,3 correspond to t,x,y, and z respectively. Also, remember that "energy" is the same as "momentum in the time direction", and that "density" is the same as "flow in the time direction". Thus the top row of the stress-energy tensor keeps track of the density of energy --- that's T00 --- and the density of momentum in the x,y, and z directions --- those are T01, T02, and T03 respectively. On the other hand, T10, T20, and T30 represent the flow of energy in the x, y and z directions, respectively. The other entries keep track of the flow of spatial momentum in various spatial directions. For example, T12 keeps track of the flow in the x direction of momentum in the y direction.

 

[PeterDonis]

Well, John Baez seems to disagree with you. From http://math.ucr.edu/home/baez/gr/stress.energy.html :

Tab is the flow in the a direction of momentum in the b direction!

Baez is right, of course, given the sense in which he is using the word "momentum". But this is a different sense of the term "momentum" (and of "density", for that matter) than the usual lay person's sense. In this terminology, energy density is the flow in the time direction of "momentum" in the time direction. So I don't think this was the sense in which QuantumImmortal was using the term "momentum".

 

[Dale]

Quantum Immortal, I also concur with the advice by PeterDonis and Bill_K. Just learn about the stress-energy tensor. It is not that difficult of a concept to master and it will be well worth your while.

There really isn't any confusion about the various terms of the stress energy tensor. In a local inertial frame, the time-time component is energy density, the space time components are momentum density, and the space-space components are stress. There is nothing confusing or strange about any of that, they are perfectly ordinary quantities.

The only strange thing is that they are so intimately related that they are actually all part of the same tensor. This is perhaps a little surprising, but conceptually it is no different than finding out the relationship between other four-vector or tensor components. I.e. it is a surprise, but we have been surprised in exactly the same way so many times that we should recover quite quickly.

 

[Quantum Immortal]

You cannot make a self consistent theory that way.

Its just a local redefnition.The correct equations he uses, are when he explicitly states all 4 of them, and defines for them E=-dv/dt . Thats the only place, i think, that he uses dv/dt for the acceleration. He seam very formal and precise at that place.
He does that to clearly show its acceleration, we could have expected some force there instaid, and he puts the correct sigh there too.

It seams, that in the rest of the paper, he uses locally the convention that is more convenient, while at the same time, he takes into account that gravity is attractive. The only place that this could have messed up the final result, is when he deduces the magnetic field, that thing needing to be in the right direction in relation to E and v . I assume, he just plugged the equivalent result for attractive charges, and considered the details too trivial and unimportant.
The paper is not about elementary physics. The intended audience, is capable of reading through that.

People stop been annal about the freaking sign.
I'm not here to hear a lecture on elementary physics.
Does anyone else has something nontrivial to comment on the paper?
You can mention the signs, only if he messes up the direction of B in relation to E.
Or other wise renders the final rezult meaningless.

Nobody cares, that he seams to get the right numeric values?
Nobody cares, that GEM should give the wrong ones?
Where is the truth?
What you think of his method?

I start to be really annoyed. I assumed that people here were competent...
Instaid you are splitting hair about trivialities...
Don't mention the sign again, unless that messes up the final result...

There's no such thing. "Linear" is a precise mathematical term; it's easy to check whether an equation is linear or not. There's nothing implicit about it.

None of which has anything to do with whether or not the equations are linear.

You can't add two solutions to make a third one, because they normally would deform each other. The true "addition" is more then just adding them up.
Corollary of this, the field acts on it self. This is what nonlinear really mean.
"Can't add two solution" is just a mathematical statement, it doesn't mean anything by it self.
The electromagnetic field doesn't bend it self, it just adds up = linear.

There really isn't any confusion about the various terms of the stress energy tensor. In a local inertial frame, the time-time component is energy density, the space time components are momentum density, and the space-space components are stress. There is nothing confusing or strange about any of that, they are perfectly ordinary quantities.

The only strange thing is that they are so intimately related that they are actually all part of the same tensor. This is perhaps a little surprising, but conceptually it is no different than finding out the relationship between other four-vector or tensor components. I.e. it is a surprise, but we have been surprised in exactly the same way so many times that we should recover quite quickly.

There is understanding and understanding.
You can just state the Heisenberg uncertainty relation. Or you can say, that its the diffraction of the wave.
You can say that non linear is not been able to construct a solution by adding two others. Or you can say that the field acts on it self.
Just stating a mathematical truth, is not enough.

I think i really understood (my way) the stress energy tensor now. It's just all the conservations laws of mechanics, presented in an obscured way to the reader.
its the energy and momentum in the time components.
Shear stress is about conservation of angular momentum
Pressure is just about forces adding to 0.
The divergence of this, its simply all of mechanics...
Some one wants to disagree?

So the true analogy in GEM would be...
to also take the equations for rotating and accelerated charges.
Adapt correctly for gravity, in a mathematically correct way (not really sure about this). And Use the stress energy tensor as the source in GEM.
And, then GEM is consistent with it self.

Are you aware of any gravitational theory doing this?

The same happens in EM, where the currents are just moving charges, and the equations for moving charges, are just the lorentz transform of comlombs law, from a reference frame, were its actually exact. Because comlombs law is not covariant, the lorentz transform, messes it up, then you add the messed up comlombs law to the old one, as a system of equations, and define a new element in the source. The new construct has no other choice then been lorentz covariant.
True content of EM, SR and comlobs law. True content of GEM, SR and Newton.
This trick looks kind of lame if you ask me...

any comments?
Preferably, not about splitting hair again... B[

 

[pervect]

Quantum Immortal, I also concur with the advice by PeterDonis and Bill_K. Just learn about the stress-energy tensor. It is not that difficult of a concept to master and it will be well worth your while.

I have some space-time diagrams that illustrate some aspects of the 2 dimensional stress energy tensor (1 space dimension, 1 time dimension) that gives some examples of the flow of "stuff", and it is only necessary to replace "stuff" with energy, and the x components of momentum to get the 2d stress energy tensor. Of course one needs to add in y and z momenta to get the full tensor.

By replacing "stuff" with "particle density" one gets as a bonus the number-flux vector.

But to understand how the stress-energy tensor transforms, in addition to these diagrams, one would need to understand the following facts:

1a) A general understanding of space-time diagrams

1b) An understanding that the T' direction on a space-time diagram is different from the T direction (similar remarks apply to the X' direction being different from the X direction). Here we consider the primed coordinates to be the coordinates of a moving system derived from the coordinates of a reference system via the Lorentz transform.

2) How energy and momentum transform between frames

I could maybe get around 2) by appealing to the relativistic equations for energy and momentum in terms of velocity. But point #1b is crucial. And hard to explain to someone who isn't already familiar with the fact - at least it seems hard to explain in a manner that "gets through" to the target audience.

If QuauntumImmortal can tell me that he understands point #1a and #1b, I'll try to expidite the diagrams (and skimp a bit on the quality control process, perhaps, in the process). But if point #1a and #1b are big mysteries to him, it seems to me that I might as well not be in a huge rush, that the diagrams won't turn out to be useful to him.

 

[Quantum Immortal]

I have some space-time diagrams that illustrate some aspects of the 2 dimensional stress energy tensor (1 space dimension, 1 time dimension) that gives some examples of the flow of "stuff", and it is only necessary to replace "stuff" with energy, and the x components of momentum to get the 2d stress energy tensor. Of course one needs to add in y and z momenta to get the full tensor.

You didn't read my last post didn't you? :P

I think i really understood (my way) the stress energy tensor now. It's just all the conservations laws of mechanics, presented in an obscured way to the reader.
its the energy and momentum in the time components.
Shear stress is about conservation of angular momentum
Pressure is just about forces adding to 0.
The divergence of this, its simply all of mechanics...
Some one wants to disagree?

So the true analogy in GEM would be...
...
[

now, i think, that its really tork, not angular momentum...
right?

I understand the twin paradox, and how you fall in a black hole, and the experiment with the barn doors ...
Yea sure, i would like to see your diagrams
They are on the Internet?
There's xx element of P too?
These diagram show 'the truth' according to two coordinate systems.
I didn't bother to understand the diagrams of tachyons... too theoretical

If you want to be really really helpful. Can you tell me what you think of the paper I'm linking
here it is again
http://www.academia.edu/483186/Gravitomagnetism_a_novel_explanation_of_the_precession_of_planets_and_binary_pulsars
Please don't obsess about the sings...

What you think of the idea of how to get covariant GEM in my previous post?
About the non linearity in GEM? (in GEM, gravity attracts gravity...)

 

[Dale]

People stop been annal about the freaking sign.
I'm not here to hear a lecture on elementary physics.
Does anyone else has something nontrivial to comment on the paper?
...
I start to be really annoyed. I assumed that people here were competent...
Instaid you are splitting hair about trivialities...
Don't mention the sign again, unless that messes up the final result...
...
This trick looks kind of lame if you ask me...

any comments?
Preferably, not about splitting hair again... B[

Please don't obsess about the sings...

With that little rant the discussion is done. Whether you like it or not the objections to the paper you posted are legitimate scientific objections. Scientific theories require self-consistency and changing your sign convention on a whim is not self consistent.

Furthermore, sign errors are not trivial, they are hugely important. They are the difference between stability and instability, recession and expansion, nuclear decay and chain reaction, and many other critical distinctions.

In the end, it is up to the proponents of alternative theories of gravity to provide clear and self-consistent descriptions of their theories and to do things like calculate the PPN parameters so that their theories can be compared to experiment. Many other theoretical physicists have done just that. I don't know why the GEM people have not.

If you find a GEM reference with PPN parameters then we can discuss that in a different thread.