Is Gravitomagnetism Covariant and Are There Any Alternative Approaches?

[Quantum Immortal]

Gravitomagnetism is this thing
https://en.wikipedia.org/wiki/Gravitomagnetism
Like electromagnetism equations, but adapted for gravity

The equations just like that are manifestly problematic.
If we use four-momentum as the source( appropriate units), does the equation become covariant?
From the covariant electromagnetism equations, we replaced one covariant four vector with an other, plus other minor changes... It will be weird if they are not covarint.

the four-momentum should also include the energy/momentum from the gravitational field it self not just mater/electromagnetism.

if not. Can we use the stress energy tensor instead?
By grouping pressure, stress and energy density together, to form a new four vector?
00 would be the trace T plus the shear stress.
Pressure and stress aren't just part of the energy density? Pressure and stress aren't just macroscopic manifestations of a form of energy/momentum?

if not, can you see an other way to make it covariant. It can't be that hard, its almost there. Its basically the electromagnetism equations.

An other issue that is problematic, is that the hyper relativistic equations should be used instaid.
But it must be first covariant for that.
The level of precision is so high, that it doesn't matter if the fields are weak and with slow speeds. For example, the precession of mercury, is slow and weak field, but its an observation spanning 100 years.
If i understand corectly, the usuall lorentz transformations are no good, because we have constant accelaration, so the generalized lorentz transformations should be used instaid.

well...
at least it makes more testable predictions then string theory and loop gravity combined >:P

 

[PeterDonis]

Gravitomagnetism is this thing
https://en.wikipedia.org/wiki/Gravitomagnetism
Like electromagnetism equations, but adapted for gravity

While this analogy can be helpful, it can also be misleading.

The equations just like that are manifestly problematic.

Do you mean, because they're not manifestly covariant? Would you apply the same complaint to the EM counterparts given on the same page (which are just Maxwell's Equations)? Do you realize that Maxwell's Equations *are* covariant? (See below.)

If we use four-momentum as the source( appropriate units), does the equation become covariant?

The 4-momentum *is* the source; $\rho_g$ and $\vec{J}_g$ together form a 4-vector just as, in the EM case, $\rho$ and $\vec{J}$ together form a 4-vector. In fact, you realize this, because you go on to say:

From the covariant electromagnetism equations, we replaced one covariant four vector with an other

Then why are you asking what would change if 4-momentum were the source? As your statement just quoted indicates, and as I said above, it *is* the source.

the four-momentum should also include the energy/momentum from the gravitational field it self

No, it shouldn't, for two reasons:

(1) The source for Maxwell's Equations doesn't include energy and momentum in the E and B fields; it only includes the charge and current density. So by analogy, the source for the gravity equations should not either.

(2) There is no covariant way to capture "energy/momentum in the gravitational field". (This is one aspect in which the analogy with EM does *not* work: there is a local way of capturing the energy/momentum in the EM field.) This is ultimately a consequence of the equivalence principle.

Can we use the stress energy tensor instead?

The general formulation *does* use the SET. What is given on the Wiki page you linked to is, as the page notes, an approximation that is only valid for weak fields in an asymptotically flat spacetime.

By grouping pressure, stress and energy density together, to form a new four vector?

No, there's no such thing. You can only capture all these things in the full SET. In the weak field limit, pressure and stress are negligible compared to energy density, so they can be ignored.

Pressure and stress aren't just part of the energy density?

No.

Pressure and stress aren't just macroscopic manifestations of a form of energy/momentum?

Ultimately, in ordinary matter, yes, but not in a way that adds to energy density in the SET. If you're trying to model physics at a macroscopic level, you are ignoring the microscopic phenomena (like atoms and molecules interacting at short distances) that give rise to pressure and stress.

Also, thinking of the space-space components of the SET as "pressure and stress" works OK for ordinary matter, but there are other kinds of things that have an SET that aren't ordinary matter. For example, the SET of the electromagnetic field contains space-space components that appear the same way pressure and stress do for ordinary matter; but those components certainly don't arise as a result of short-distance interactions of atoms and molecules, since they can be present when no atoms or molecules are present, just the EM field itself.

An other issue that is problematic, is that the hyper relativistic equations should be used instaid.

What does this mean?

If i understand corectly, the usuall lorentz transformations are no good, because we have constant accelaration, so the generalized lorentz transformations should be used instaid.

I don't understand what this means either.

 

[maajdl]

A post by rpenner on this page can be helpful:

http://lofi.forum.physorg.com/Gravitation-And-Electromagnetism_27028.html

You can also find the original paper by Heaviside:

http://gsjournal.net/Science-Journals/Essays/View/2505

 

[Quantum Immortal]

Do you mean, because they're not manifestly covariant? Would you apply the same complaint to the EM counterparts given on the same page (which are just Maxwell's Equations)? Do you realize that Maxwell's Equations *are* covariant? (See below.)

Then why are you asking what would change if 4-momentum were the source? As your statement just quoted indicates, and as I said above, it *is* the source.

I just meant that it not covariant, nothing to do with manifestly covariant. Thats a big problem. The article says its not covariant. It doesn't use the actual 4 momentum apparently, i don't see why if it makes it covariant so simply as you say.

No, it shouldn't, for two reasons:

(1) The source for Maxwell's Equations doesn't include energy and momentum in the E and B fields; it only includes the charge and current density. So by analogy, the source for the gravity equations should not either.

(2) There is no covariant way to capture "energy/momentum in the gravitational field". (This is one aspect in which the analogy with EM does *not* work: there is a local way of capturing the energy/momentum in the EM field.) This is ultimately a consequence of the equivalence principle.

The source is energy momentum, its exact nature is not important, i dismiss 1 . So in the true analogy, the energy momentum of the gravitational field it self should be included one way or an other.

If you have E and B, you should be able to deduce how much energy/momentum the field contains. Gravitomagnetism, is a flat space model. Are you sure 2 still applies? Here, gravitation, is analogous to EM, by analogy it should be able to fit in the 4momentum, like his EM counterpart.
I guess this is a big difference with GR, what differences we'll see because of this?
In GR, gravity is the geometry of space time, here its just an other field inside flat space, obviously they will collide.
Note, that gravitational self energy effect on the gravitational field, is so minute, that this is a really tiny deviation. If I'm not mistaken, only in binary pulsars this is detectable observationally.

Ultimately, in ordinary matter, yes, but not in a way that adds to energy density in the SET. If you're trying to model physics at a macroscopic level, you are ignoring the microscopic phenomena (like atoms and molecules interacting at short distances) that give rise to pressure and stress.

Also, thinking of the space-space components of the SET as "pressure and stress" works OK for ordinary matter, but there are other kinds of things that have an SET that aren't ordinary matter. For example, the SET of the electromagnetic field contains space-space components that appear the same way pressure and stress do for ordinary matter; but those components certainly don't arise as a result of short-distance interactions of atoms and molecules, since they can be present when no atoms or molecules are present, just the EM field itself.

For matter, you can translate in an appropriate way, pressure and stress into energy/momentum.
Its a bit like temperature, temperature is a macroscopic thing, microscopically, particles have some average energy/momentum.
...You said yourself, ultimately yes... :P

I think you can do something similar with other SETs. Ultimately, they mean something in terms of energy/momentum. I don't see why you can't translate them into energy or density. Even if its not as simple as i originally proposed

but, the 4 momentum should still represent all the energy/momentum of the system. You don' really need the SET. In my opinion pressure and stress, shouldn't appear as such.
SET is just a bad choice for the source here i guess...

What does this mean?

I don't understand what this means either.

In electromagnetism, when a particle goes very fast, in its reference frame, it sees a different field. You can't just use electromagnetism equations just like that in that case. You need to use lorentz transformations. This is what i meant. The correct term is hyper relativistic limit?

In general, in textbooks, you get particle moving at constant speeds.
In general however, a particle is accelerated. You need to use the generalized lorentz transformation for arbitrary acceleration and rotation... I
https://www.physicsforums.com/showthread.php?t=60005

You need to use Lorentz transformation, even for low speeds, because the precision of observation is too high.

 

[Quantum Immortal]

A post by rpenner on this page can be helpful:

http://lofi.forum.physorg.com/Gravitation-And-Electromagnetism_27028.html

You can also find the original paper by Heaviside:

http://gsjournal.net/Science-Journals/Essays/View/2505

Heaviside's 1893 paper is not very relativistic :P
he uses normal matter and normal momentum.
Also a reference frame transformation is needed in there...

Still, its beater then pure Newton.

 

[WannabeNewton]

You're correct that the gravitoelectromagnetism equations are not manifestly covariant because we are explicitly introducing a 3+1 split of space-time and we're decomposing the derivative operator $\nabla_{\mu}$ into an "inertial" part and a "gravitational" part relative to a background global inertial frame.

We do a similar thing in electromagnetism with the 3+1 split of the Maxwell equations in terms of $\vec{E}$ and $\vec{B}$ relative to an inertial frame. With linearized gravity the only way to rewrite the gravitoeletromagnetism equations in a manifestly covariant manner is to bring back the metric perturbation $h_{\mu\nu}$ and the derivative operator $\nabla_{\mu}$ which gives us back Einstein's equation but then we lose the obvious connection to gravitomagnetism.

However there is a very elegant way in which we can talk about the "electric" and "magnetic" parts of gravity by looking at the electric and magnetic parts of the Riemann curvature tensor. Basically we can decompose $R_{\mu\nu\alpha\beta}$ into an electric part and a magnetic part by using the Hodge dual operator $\star$, or equivalently the levi-civita tensor $\epsilon_{\mu\nu\alpha\beta}$, and a time-like congruence with 4-velocity field $\xi^{\mu}$ by writing the electric part and magnetic part of $R_{\mu\nu\alpha\beta}$ relative to $\xi^{\mu}$ as $E_{\mu\nu} = R_{\mu\alpha\nu\beta}\xi^{\alpha}\xi^{\beta}$ and $B_{\mu\nu} = -\frac{1}{2}\epsilon_{\mu \nu}{}{}^{\alpha\beta}R_{\alpha\gamma\beta\delta}\xi^{ \gamma}\xi^{\delta}$.

This is in formal analogy with electromagnetism wherein we decompose the electromagnetic field $F_{\mu\nu}$ relative to $\xi^{\mu}$ into the electric field $E_{\mu} = F_{\mu\nu}\xi^{\nu}$ and magnetic field $B_{\mu} = -\frac{1}{2}\epsilon_{\mu \nu}{}{}^{\alpha\beta}F_{\alpha\beta}\xi^{\nu}$.

EDIT: See here for more: http://en.wikipedia.org/wiki/Bel_decomposition

For a much more detailed analysis of the electric and magnetic parts of the Riemann curvature tensor (including their physical interpretations), see chapter 7 of "Classical Measurements in Curved Space-Times"-Felice and Bini.

 

[PeterDonis]

I just meant that it not covariant

Then you're wrong; it is. It's just not manifestly covariant because it's written in 3+1 form instead of in explicit tensor form.

The article says its not covariant.

Does it? I couldn't find any such statement. (Not that I would necessarily believe whatever an Wikipedia article says anyway.)

It doesn't use the actual 4 momentum apparently

Not explicitly; but as I pointed out, the 3+1 formulation of Maxwell's equations given in the same article doesn't use an explicit 4-vector as a source either, yet it's still there. In the EM case, $(\rho, \vec{J})$ together form a 4-vector; in the gravity case, $(\rho_g, \vec{J}_g)$ together form a 4-vector. The only difference between that vector and the usual form of 4-momentum is units: $(\rho_g, \vec{J}_g)$ are in units of energy density and momentum density, whereas 4-momentum is in units of energy and momentum. But that's because 4-momentum as it's usually written idealizes objects as point particles, not continuous. It doesn't change the fact that $(\rho_g, \vec{J}_g)$ form a covariant 4-vector.

The source is energy momentum, its exact nature is not important, i dismiss

That doesn't make it correct to dismiss the difference; it just means you're making a mistake in doing so if keeping things covariant is so important to you. See below.

So in the true analogy, the energy momentum of the gravitational field it self should be included one way or an other.

There are ways to assign an energy and momentum to the gravitational field; there's just no way to do so covariantly. You have to pick a particular space-time split (and you have to be in a spacetime where you can do that and get a reasonable definition of the energy and momentum in the gravitational field).

If you have E and B, you should be able to deduce how much energy/momentum the field contains.

Have you tried doing this? The expression for field energy in terms of E and B in the EM case is simple. Have you tried evaluating the analogous expression for the gravity case to see what you get?

Gravitomagnetism, is a flat space model.

This is not correct as you state it. Gravitomagnetism in the general case is present when the fields are not weak. What is correct is that the particular *model* of gravitomagnetism given in the Wiki article is only valid when gravity is weak, i.e., when all the effects of gravity can be models as small perturbations of flat spacetime.

Are you sure 2 still applies?

Yes. The weak field, asymptotically flat case is one of the cases where you can pick a particular space-time split and get a useful definition of energy and momentum contained in the gravitational field; but it won't be covariant.

Here, gravitation, is analogous to EM, by analogy it should be able to fit in the 4momentum, like his EM counterpart.

I'm not sure what you mean here. The "source" in the EM case is charge and current, not energy and momentum. E and B fields carry no charge or current. So the only analogy I can see here, as I pointed out in #1 in my previous post, would say that the "source" in the gravity case should not include any energy and momentum in the fields.

In GR, gravity is the geometry of space time, here its just an other field inside flat space, obviously they will collide.

You do understand what "weak field approximation" means, right? It means you are ignoring all the aspects of gravity, as curvature of spacetime, that can't be modeled as a field on flat spacetime. So of course your model won't look the same as the full curved spacetime model.

Note, that gravitational self energy effect on the gravitational field

I assume you mean by this the nonlinearity of gravity; calling it "gravitational self-energy" is not really the best way to look at it. See below.

is so minute, that this is a really tiny deviation. If I'm not mistaken, only in binary pulsars this is detectable observationally.

Now you seem to be equating "gravitational self energy effect" with the emission of gravitational waves, which is something else again. Binary pulsar systems are indeed, to my knowledge, the only observations we have that definitely show emission of gravitational waves; but those waves are too weak to be nonlinear (i.e., they don't interact with each other). They do carry away energy from the binary pulsar system, but again, I'm not sure that "gravitational self-energy effect" is the best way to describe why that happens.

For matter, you can translate in an appropriate way, pressure and stress into energy/momentum.
Its a bit like temperature, temperature is a macroscopic thing, microscopically, particles have some average energy/momentum.
...You said yourself, ultimately yes... :P

Yes, I did. So what? As I said, if you're modeling matter at this microscopic level, you're not treating it as a continuous substance with a single stress-energy tensor; you're treating it like lots of little discrete objects interacting with each other, which is a very different theoretical model. It certainly isn't anything like somehow translating a tensor into a 4-vector. See below.

I think you can do something similar with other SETs. Ultimately, they mean something in terms of energy/momentum. I don't see why you can't translate them into energy or density.

Because a tensor has more independent degrees of freedom than a 4-vector, so trying to model something with a 4-vector when it should be modeled with a tensor is throwing information away. See below.

the 4 momentum should still represent all the energy/momentum of the system. You don' really need the SET.

This is nonsense. A symmetric 2nd-rank tensor has 10 independent components; a 4-vector only has 4. You can't model 10 degrees of freedom with 4.

You continue to fail to grasp that the model in the Wiki article is an *approximation*, only valid for weak fields in asymptotically flat spacetime. In that approximation, pressure and stress are negligible compared to energy and momentum and can be ignored. That's why the source in this model only needs a 4-vector as a source: all the other degrees of freedom in the full SET are negligible *in this approximation*.

In electromagnetism, when a particle goes very fast, in its reference frame, it sees a different field. You can't just use electromagnetism equations just like that in that case. You need to use lorentz transformations.

And once you Lorentz transform, the equations in the new frame have exactly the same form, in terms of the fields in the new frame, as the equations in the old frame had in terms of the fields in the old frame. That's what "Lorentz covariant" means. Try it!

In general, in textbooks, you get particle moving at constant speeds.

Obviously you haven't read enough textbooks.

In general however, a particle is accelerated. You need to use the generalized lorentz transformation for arbitrary acceleration and rotation...

You need to use Lorentz transformation, even for low speeds, because the precision of observation is too high.

Yes, all this is true. So what? What does it have to do with what we're discussing?

 

[Quantum Immortal]

Does it? I couldn't find any such statement. (Not that I would necessarily believe whatever an Wikipedia article says anyway.)

Its a big section with a clear explicit title about it.
This is why i use "covariant" in the title of the thread

I think the article really means 1893's heviside's gravitomagnetism. The article is a bit inconsistent with it self, because too many people cooked it.
I think you on the other hand just see gravitomagnetism as the week field approximation of GR
On the other hand i mean it as an analogy to EM. Just treat gravity like EM.

There are ways to assign an energy and momentum to the gravitational field; there's just no way to do so covariantly. You have to pick a particular space-time split (and you have to be in a spacetime where you can do that and get a reasonable definition of the energy and momentum in the gravitational field).

In the context of GM. Why the expression shouldn't be the same with the one of EM?
In GM, spacetime is flat... Gravity is just a field inside flat space.

This is not correct as you state it. Gravitomagnetism in the general case is present when the fields are not weak. What is correct is that the particular *model* of gravitomagnetism given in the Wiki article is only valid when gravity is weak, i.e., when all the effects of gravity can be models as small perturbations of flat spacetime.

I see it differently.
GM, is correct, when the fields are week. yes...
I'm seen in it like an analogy, not just as an approximation for week fields that must be imperatively dropped when the fields are strong, so that you get correct results all the time.
Are we on the same page now?

Now you seem to be equating "gravitational self energy effect" with the emission of gravitational waves, which is something else again. Binary pulsar systems are indeed, to my knowledge, the only observations we have that definitely show emission of gravitational waves; but those waves are too weak to be nonlinear (i.e., they don't interact with each other). They do carry away energy from the binary pulsar system, but again, I'm not sure that "gravitational self-energy effect" is the best way to describe why that happens.

I understood, that they detected more then just gravitational waves effects. I understood, they measured also the gravitational attraction between the pulsars contributed by the gravitational field it self, as there orbit decays because of the gravitational waves they radiate...
If i understood wrong, then non linear effects of gravity were never observed yet, right?

This is nonsense. A symmetric 2nd-rank tensor has 10 independent components; a 4-vector only has 4. You can't model 10 degrees of freedom with 4.

Purely mathematically yes. If however, the other six components are really just mangled in some complicated fashion momentums and energies, then i don't see why you can't re express it as a 4 vector, without losing information.
I think you said it your self, ultimately pressure and stress are just momentum and energy at the microscopic level.
If you prefer... the basis of the other 6 components, can be expressed in the basis of the first 4, in this special case. They aren't really 10 independent stuff, just 4, but obfuscated.
I guess, if done correctly, the normal SET, and the SET with explicit 4 independent components, should give exactly the same results in GR.

Pressure and stress are just composed of momentum and energy. They aren't fundamental. They are like temperature.

And once you Lorentz transform, the equations in the new frame have exactly the same form, in terms of the fields in the new frame, as the equations in the old frame had in terms of the fields in the old frame. That's what "Lorentz covariant" means. Try it!

I meant the fields, not the equation.
You can have just an electric field in one reference frame, and in an other you also have a magnetic field.
In the reference frame of the sun, you just have gravitostatics (ignore the rotations), in the reference frame of the Earth as it travels through the field, it also sees a gravitomagnetic field, add to that, that its not a uniform motion, but an accelerated one, you can't use the simple lorentz transformations but the general ones.
Covariance just let's you use the same equations in all reference frames, the results however are not valid in all reference frames.

Yes, all this is true. So what? What does it have to do with what we're discussing?

GR automatically contains SR. With GM, you removed SR.
In GR, you just use GR, and get directly the final result.
I'm interpreting, that what they really mean about the accuracy level of GM, is when they are used, just like that, like in the 19th century. Used just like that, in analogy with the way GR give the final result.
The article says, they are valid for slow moving objects.
In general, people in math/physics are very literal.
I'm interpreting, that if you use lorentz transforms on the moving objects, GM should be accurate also for fast moving objects.
EM equations are also valid for slow moving objects, if you go too fast, you need lorentz transforms, to get what the moving object is really seeing.
Again, i see this in the context of the analogy with EM.

You say GM are valid approximations for reasonably flat spacetime. If you go (reasonably) very fast, spacetime remains reasonably flat. Thus GM remain accurate even at high speeds, if used with SR, like EM (i think, this is called hyper relativistic).

 

[PeterDonis]

Its a big section with a clear explicit title about it.

Ah, I see, they use the term "lack of invariance". However, I note that there are no references given in this section; and I don't think the claim that "$(\rho_g, \vec{J}_g)$ do not form a four-vector (instead they are merely a part of the stress–energy tensor)" is correct. Given a stress-energy tensor $T^a{}_b$, one can simply pick a timelike 4-vector field $u^b$, representing the 4-velocity of the matter that is acting as a source of gravity, and write $J^a = T^a{}_b u^b$, giving a covariant 4-vector $J^a = (\rho_g, \vec{J}_g)$. My understanding is that that is how the equations given at the start of the article are derived. The only requirement is, as I said before, that pressure and stress have to be negligible compared to energy density, since contracting the full SET with a representative 4-velocity for the matter throws away the information about pressure and stress.

I think the article really means 1893's heviside's gravitomagnetism.

Reference, please? Heaviside formulated *electromagnetism* in a commonly used way; but AFAIK he never did any work formulating *gravity* in such a way.

I think you on the other hand just see gravitomagnetism as the week field approximation of GR

For purposes of this thread, as I've already made clear, I'm using the term "gravitomagnetism" (or more correctly "gravitoelectromagnetism", the term the Wiki article uses) to mean the same thing you are:

i mean it as an analogy to EM. Just treat gravity like EM.

My point about the weak field approximation is that "treating gravity like EM" in this way can only be done in the weak field approximation. With strong enough gravity you can't do it, because the things that have to be negligible for "treating gravity like EM" this way to work are no longer negligible.

As far as whether the usage of the term "gravitomagnetism" just described is "correct", see below.

Why the expression shouldn't be the same with the one of EM?

Once again: have you actually tried to do this? What did you get?

GM, is correct, when the fields are week. yes...

You continue to miss the point of the word "approximation".

"Gravitomagnetism" as a general term covers effects that occur in strong fields (such as formation of accretion disks and jets of plasma around rotating black holes) which cannot be modeled using the methods given in the Wiki article. There is, in general, no requirement that fields have to be weak for gravitomagnetism, in the general sense, to be "correct".

However, when the fields are weak, the effects of gravity, *including* gravitomagnetism (but not *just* that--you do realize that the Wiki article also includes "gravitoelectric" effects right?), can be modeled in a way that looks a lot like EM. But this is an approximation; it is not "correct" in the sense of being an exact theory, the way Maxwell's Equations are an exact classical theory of EM. And some of the effects which don't appear in this approximation, because they are too small if the fields are weak, are *also* "gravitomagnetism" (see above).

I'm seen in it like an analogy, not just as an approximation for week fields that must be imperatively dropped when the fields are strong, so that you get correct results all the time.
Are we on the same page now?

No, because I don't understand what this is supposed to mean.

I understood, that they detected more then just gravitational waves effects. I understood, they measured also the gravitational attraction between the pulsars contributed by the gravitational field it self, as there orbit decays because of the gravitational waves they radiate...
If i understood wrong, then non linear effects of gravity were never observed yet, right?

They measured the orbital parameters of the pulsars and how they changed over time. That showed that the total energy of the binary pulsar system, as evidenced by the orbital parameters, was decreasing, indicating that gravitational waves were being emitted by the system. As far as "gravitational attraction contributed by the gravitational field itself", I don't know what this means; does it just mean that the behavior of the orbits is not exactly what Newtonian gravity would predict? (For example, the periastron--point of minimum distance between the pulsars--precesses, just as the perihelion of Mercury about the Sun does, which is a non-Newtonian effect.) Does that count as a "nonlinear effect of gravity"? I think you need to be clearer about what sorts of effects you are referring to.

If however, the other six components are really just mangled in some complicated fashion momentums and energies

They aren't; at least, they aren't at the macroscopic level. You don't appear to understand the difference between macroscopic and microscopic models. See below.

i don't see why you can't re express it as a 4 vector, without losing information.
I think you said it your self, ultimately pressure and stress are just momentum and energy at the microscopic level.

You're kidding, right? At the microscopic level, the matter is composed of some huge number of atoms, something like 10^25 or more of them, each with its own energy-momentum 4-vector. So if you really insist on using only 4-vectors instead of tensors, you're going to have to deal with 10^25 of them, i.e., with 10^25 x 4 degrees of freedom, instead of just 10.

If you prefer... the basis of the other 6 components, can be expressed in the basis of the first 4, in this special case.

If you really believe this, then please show your work. You need to back this up with math instead of just waving your hands.

I guess, if done correctly, the normal SET, and the SET with explicit 4 independent components, should give exactly the same results in GR.

Again, please show your work. You appear to have significant misunderstandings about how GR as a tensor theory works.

Pressure and stress are just composed of momentum and energy. They aren't fundamental. They are like temperature.

Yep. And they're also a lot simpler to work with than 10^25 individual 4-vectors. See above.

I meant the fields, not the equation.

But that's not what "covariant" means. Covariance means the *laws* are covariant, i.e., the equations, not the fields.

Covariance just let's you use the same equations in all reference frames, the results however are not valid in all reference frames.

What? This is just wrong; if the results aren't valid in all frames, the equations aren't covariant. You appear to have a significant misunderstanding of what "covariant" means.

GR automatically contains SR. With GM, you removed SR.

Huh?

I'm interpreting, that if you use lorentz transforms on the moving objects, GM should be accurate also for fast moving objects.

Sure; but that means the equations given in the article are valid in the transformed frame of the fast-moving object. The *fields* will look different, as will the sources (since they transform too), but the *equation* satisfied by the fields and sources together will be the same.

Btw, EM works the same way, which you appear not to understand:

EM equations are also valid for slow moving objects, if you go too fast, you need lorentz transforms, to get what the moving object is really seeing.

Which, once again, means the fields and sources change, but the *equations* satisfied by them remain valid. Which is what "covariant" means.

You say GM are valid approximations for reasonably flat spacetime. If you go (reasonably) very fast, spacetime remains reasonably flat.

Yes, this is true.

Thus GM remain accurate even at high speeds, if used with SR, like EM

And the equations given in the Wiki article are covariant in this way (see above), so once again, I think the article is just wrong in its "lack of invariance" section.

(i think, this is called hyper relativistic).

I'm not familiar with this term, but I now understand how you were using it.

 

[WannabeNewton]

You know, I've said this to you (the OP) twice already but let me say it again: why don't you just access a textbook on general relativity and properly learn the theory so that you can avoid asking nonsensical questions prompted by skimming over wikipedia articles? That way Peter doesn't have to painfully respond to a slew of misunderstandings at each turn.

 

[Quantum Immortal]

My point about the weak field approximation is that "treating gravity like EM" in this way can only be done in the weak field approximation. With strong enough gravity you can't do it, because the things that have to be negligible for "treating gravity like EM" this way to work are no longer negligible.

Once again: have you actually tried to do this? What did you get?

You mean to use GR and deduce the energy?

You continue to miss the point of the word "approximation".

"Gravitomagnetism" as a general term covers effects that occur in strong fields (such as formation of accretion disks and jets of plasma around rotating black holes) which cannot be modeled using the methods given in the Wiki article. There is, in general, no requirement that fields have to be weak for gravitomagnetism, in the general sense, to be "correct".

However, when the fields are weak, the effects of gravity, *including* gravitomagnetism (but not *just* that--you do realize that the Wiki article also includes "gravitoelectric" effects right?), can be modeled in a way that looks a lot like EM. But this is an approximation; it is not "correct" in the sense of being an exact theory, the way Maxwell's Equations are an exact classical theory of EM. And some of the effects which don't appear in this approximation, because they are too small if the fields are weak, are *also* "gravitomagnetism" (see above).

when i say GM, i mean the EM-like equations, not the general term
If you want to use an other term for specifically the EM-like equations, just propose something.
I use "gravitomagnetism", because electrostatics looks like Newton, and just substitute the name.
I have to say every time "EM-like approximate gravity equations" or something?

They measured the orbital parameters of the pulsars and how they changed over time. That showed that the total energy of the binary pulsar system, as evidenced by the orbital parameters, was decreasing, indicating that gravitational waves were being emitted by the system. As far as "gravitational attraction contributed by the gravitational field its", Ielf don't know what this means; does it just mean that the behavior of the orbits is not exactly what Newtonian gravity would predict? (For example, the periastron--point of minimum distance between the pulsars--precesses, just as the perihelion of Mercury about the Sun does, which is a non-Newtonian effect.) Does that count as a "nonlinear effect of gravity"? I think you need to be clearer about what sorts of effects you are referring to.

I meant, that the gravitational field it self has a gravitational field. Making it non linear.
If i got it correctly, as you get closer to a mass, the mass seams to gravitate less then what it should, because as you get closer, you leave behind more and more of the energy of the gravitational field of the mass. You don't feel that attraction anymore, because its spherically symmetric ( Newtons shell theorem).
I meant, that i thought, they managed to actually detect that with the binary pulsars.
I'm not talking about the gravitational waves, I'm talking about the gravitational fields of the pulsars.
If you still didn't get what i meant, i guess i really misunderstood.
So i guess, we never observed what i meant. Is so minuscule anyway...

They aren't; at least, they aren't at the macroscopic level. You don't appear to understand the difference between macroscopic and microscopic models. See below.

You're kidding, right? At the microscopic level, the matter is composed of some huge number of atoms, something like 10^25 or more of them, each with its own energy-momentum 4-vector. So if you really insist on using only 4-vectors instead of tensors, you're going to have to deal with 10^25 of them, i.e., with 10^25 x 4 degrees of freedom, instead of just 10.

...
B[
what ever
i don't really care about this point anyway

Yes, this is true.

And the equations given in the Wiki article are covariant in this way (see above), so once again, I think the article is just wrong in its "lack of invariance" section.

Wait, you are saying, that the EM-like/maxwel-like approximate equations of gravity. Give wrong results only when the field is too strong? And that's it?
How you define strong field? Is the sun a strong field? You mean white dwarfs and above, right?

The treatment of the orbit of mercury, with the full special relativity combined with the aproximate EM-like equations of gravity (what i mean by hyper relativistic), will give what result?

 

[PeterDonis]

You mean to use GR and deduce the energy?

No, I mean do what you were talking about in the post that I quoted. You said that the expression for the "energy in the gravitational field" should look the same as the one for the energy in the EM field. The obvious thing to do to test this is to use the equations in the Wiki article to derive an expression for "energy in the gravitational field" by the same method as the expression for the energy in the EM field is derived. Have you done this?

when i say GM, i mean the EM-like equations, not the general term

But you appear to be claiming that the "EM-like equations" should be "correct", not just approximations that are only valid for weak fields. Is that what you're claiming? If not, what, exactly, are you claiming?

I meant, that the gravitational field it self has a gravitational field. Making it non linear.

It is true that the Einstein Field Equation is nonlinear, and this is one of the key *differences* between it and Maxwell's Equations (which are linear). However, this does not mean what you appear to think it means. See below.

If i got it correctly, as you get closer to a mass, the mass seams to gravitate less then what it should, because as you get closer, you leave behind more and more of the energy of the gravitational field of the mass.

I don't know where you're getting this from; GR doesn't say anything like this. (Something like this occurs with the energy stored in the *EM* field of a *charged* mass; but that only applies to the EM field energy. There is no analogous "gravity field energy" that works like this.)

I meant, that i thought, they managed to actually detect that with the binary pulsars.

No, that's *not* what they detected with the binary pulsars. I have described what they did detect several times now.

Also, even if such an effect as you describe exists, how could we measure it from light-years away? We would have to get close enough to the binary pulsars for a significant amount of their "gravitational field energy" to be above us rather than below us.

i guess i really misunderstood.

Yes, you did. See above.

So i guess, we never observed what i meant.

No, we haven't. See above.

i don't really care about this point anyway

So you are dropping your claim that the "source" of gravity can somehow, in the general case, be represented by a 4-vector instead of the full SET?

Wait, you are saying, that the EM-like/maxwel-like approximate equations of gravity. Give wrong results only when the field is too strong?

I would say "give results which are too inaccurate for practical use" instead of "wrong results" (after all, straight-up Newtonian gravity is "wrong", but it still gives results good enough for practical use in a lot of cases). But yes, the general "make gravity look like EM" (or "GEM") framework in the Wiki article is a good approximation as long as the field is not too strong.

How you define strong field? Is the sun a strong field? You mean white dwarfs and above, right?

Basically, yes. The sun's field is not a strong field in this sense; for example, you can get the correct answer for bending of light by the Sun using the GEM framework.

The treatment of the orbit of mercury, with the full special relativity combined with the aproximate EM-like equations of gravity (what i mean by hyper relativistic), will give what result?

AFAIK it gives the correct result for Mercury's orbit, including the perihelion precession. However, I don't think it will give the correct result for a binary pulsar system; in other words, I don't think the GEM framework can capture gravitational wave emission from such a system. (You can certainly model gravitational waves themselves in the GEM framework; they're just the analog of vacuum EM waves. But I don't think the framework can correctly give the relationship between GW emission by a binary pulsar system and the changes in the orbital parameters of the system.)

 

[PeterDonis]

AFAIK it gives the correct result for Mercury's orbit, including the perihelion precession.

Actually, thinking this over, I'm not sure about it; the GEM framework may not capture enough of the relativistic corrections to the Newtonian force to get the right result for Mercury's perihelion precession. I would have to see a reference that gives more explicit formulas for the gravitational $\vec{E}_g$ and $\vec{B}_g$ fields. It's even possible that different references use different formulas (based on different approximations) for the fields, which would explain why some of the statements in the Wiki article do not appear to be consistent with each other.

 

[PeterDonis]

you can get the correct answer for bending of light by the Sun using the GEM framework.

Looking again at the Wiki article, I'm no longer sure that even this is true; as with the case of Mercury's orbit, I would need to see a reference that gives more explicit formulas for the fields (and it may turn out that different references use different formulas).

The basic problem is that, as the gravitomagnetic field $\vec{B}_g$ is defined on the Wiki page, it requires *rotation* of the central mass, not just motion of the mass; but the Sun's rotation is so slow that it can be ignored for almost all calculations, and certainly does not contribute to the bending of light passing close to the Sun. So by the Wiki article's formula, the gravitational "Lorentz force" on a relativistic particle passing the Sun would only include the $\vec{E}_g$ term, which the Wiki article appears to assume is just the Newtonian force; and that alone does *not* give the right answer (it gives an answer half as large as the GR prediction, and the GR prediction matches observations). But other references might include relativistic corrections to $\vec{E}_g$, or might include terms due to general relative motion in $\vec{B}_g$ instead of just rotation.

 

[Bill_K]

Actually, thinking this over, I'm not sure about it; the GEM framework may not capture enough of the relativistic corrections to the Newtonian force to get the right result for Mercury's perihelion precession. I would have to see a reference that gives more explicit formulas for the gravitational $\vec{E}_g$ and $\vec{B}_g$ fields. It's even possible that different references use different formulas (based on different approximations) for the fields, which would explain why some of the statements in the Wiki article do not appear to be consistent with each other.

The validity of an approximation such as weak field or slow motion can only be stated as the smallness of a dimensionless quantity. In many cases this quantity is position dependent, and then we say the approximation is not uniformly valid. There is no such thing as a "weak black hole", for example. For as we approach the hole the field gets stronger without limit. The dimensionless parameter in this case is GM/rc², the ratio of the coordinate r to the Schwarzschild radius GM/c².

But this same quantity is also the ratio of the gravitational potential GMm/r of a planet to the planets rest energy mc². And for this reason, the weak field approximation and the nonrelativistic approximation are inextricably linked. To retain an ultrarelativistic particle in circular orbit requires a strong gravitational field. So it's impossible to talk about the precession of the orbit of an ultrarelativistic particle in the weak field approximation!

 

[PeterDonis]

it's impossible to talk about the precession of the orbit of an ultrarelativistic particle in the weak field approximation!

Yes, I see what you mean, and I agree. But I'm wondering if the version of the "weak field approximation" presented in the Wiki article is even a good enough approximation to capture the correct perihelion precession for a non-relativistic "particle" (like the planet Mercury), which can be held in orbit by a weakly gravitating source like the Sun. Any correct prediction of such a precession would have to include corrections to the Newtonian force, since the Newtonian force by itself produces closed orbits (i.e., zero precession).

 

[Bill_K]

Yes, I see what you mean, and I agree. But I'm wondering if the version of the "weak field approximation" presented in the Wiki article is even a good enough approximation to capture the correct perihelion precession for a non-relativistic "particle" (like the planet Mercury), which can be held in orbit by a weakly gravitating source like the Sun. Any correct prediction of such a precession would have to include corrections to the Newtonian force, since the Newtonian force by itself produces closed orbits (i.e., zero precession).

I think that's right. I've heard the perihelion precession touted as a test of the nonlinear theory. I think the linear theory predicts zero precession.

 

[PeterDonis]

I think the linear theory predicts zero precession.

This would make sense given the EM analogy; I believe the analogous situation for electromagnetism (a test body orbiting a much larger body of opposite charge) has zero precession (because the effective potential has the Coulomb term and the centrifugal term, but not the extra attractive term that arises in GR and produces the perihelion precession).

 

[Quantum Immortal]

No, I mean do what you were talking about in the post that I quoted. You said that the expression for the "energy in the gravitational field" should look the same as the one for the energy in the EM field. The obvious thing to do to test this is to use the equations in the Wiki article to derive an expression for "energy in the gravitational field" by the same method as the expression for the energy in the EM field is derived. Have you done this?

I'll see this later.
I don't even remember how you deduce the EM energy. You need to integrate with a test particle doing work... My math are a bit rusty.
Just using the formula already derived for EM should give a first order only answer.
In any case, this effect should be tiny... becoming important only with strong fields... oups...:D

But you appear to be claiming that the "EM-like equations" should be "correct", not just approximations that are only valid for weak fields. Is that what you're claiming? If not, what, exactly, are you claiming?

i meant an approximation
An approximation, that should be self consistent.

It is true that the Einstein Field Equation is nonlinear, and this is one of the key *differences* between it and Maxwell's Equations (which are linear). However, this does not mean what you appear to think it means. See below.

I don't know where you're getting this from; GR doesn't say anything like this. (Something like this occurs with the energy stored in the *EM* field of a *charged* mass; but that only applies to the EM field energy. There is no analogous "gravity field energy" that works like this.)

Like you said. Something like this occurs with the energy of the EM field.
If you don't follow the analogy, somehow (need to plug something as the energy of the gravitational field) .Then the approximate EM-like equations become inconsistent mathematically.
They should at least be self consistent.
The source is the 4 momentum, if the 4 momentum is not really all the 4 momentum of the system, then itself inconsistent.

No, that's *not* what they detected with the binary pulsars. I have described what they did detect several times now.

Also, even if such an effect as you describe exists, how could we measure it from light-years away? We would have to get close enough to the binary pulsars for a significant amount of their "gravitational field energy" to be above us rather than below us.

From the way the orbits of the pulsars decay? There mutual attraction should be nonlinear.

So you are dropping your claim that the "source" of gravity can somehow, in the general case, be represented by a 4-vector instead of the full SET?

No, i just got fed up in how literal you are.
A hot object has more gravitational pull then a cold one. Are we going to use some kind of relativistic thermodynamics, or just treat the object as a gabillion of particles?
Why we can't use thermodynamics here too, to deduce the pressure and stress, in terms of energy and momentums?

I would say "give results which are too inaccurate for practical use" instead of "wrong results" (after all, straight-up Newtonian gravity is "wrong", but it still gives results good enough for practical use in a lot of cases). But yes, the general "make gravity look like EM" (or "GEM") framework in the Wiki article is a good approximation as long as the field is not too strong.

This happens only in strong fields however?
Only when the nonlinear effects of gravity become detectable?

Is it a more correct statement to say. That the inaccuracies appear, only when nonlineair effects must be taken in too account?
The EM-like equations, deviate from observations, when nonlineair effects have to be taken in too account?

Basically, yes. The sun's field is not a strong field in this sense; for example, you can get the correct answer for bending of light by the Sun using the GEM framework.

AFAIK it gives the correct result for Mercury's orbit, including the perihelion precession. However, I don't think it will give the correct result for a binary pulsar system; in other words, I don't think the GEM framework can capture gravitational wave emission from such a system. (You can certainly model gravitational waves themselves in the GEM framework; they're just the analog of vacuum EM waves. But I don't think the framework can correctly give the relationship between GW emission by a binary pulsar system and the changes in the orbital parameters of the system.)

Actually, thinking this over, I'm not sure about it; the GEM framework may not capture enough of the relativistic corrections to the Newtonian force to get the right result for Mercury's perihelion precession. I would have to see a reference that gives more explicit formulas for the gravitational $\vec{E}_g$ and $\vec{B}_g$ fields. It's even possible that different references use different formulas (based on different approximations) for the fields, which would explain why some of the statements in the Wiki article do not appear to be consistent with each other.

From what i understood. They use the EM-like equations just like that. They don't also consider SR.
The speeds are slow, yes. But the precision of observation is so damn precise, that you need to consider SR also. The precession of mercury is on over 100 years. That looks precise enough to require SR corrections...

Looking again at the Wiki article, I'm no longer sure that even this is true; as with the case of Mercury's orbit, I would need to see a reference that gives more explicit formulas for the fields (and it may turn out that different references use different formulas).

The basic problem is that, as the gravitomagnetic field $\vec{B}_g$ is defined on the Wiki page, it requires *rotation* of the central mass, not just motion of the mass; but the Sun's rotation is so slow that it can be ignored for almost all calculations, and certainly does not contribute to the bending of light passing close to the Sun. So by the Wiki article's formula, the gravitational "Lorentz force" on a relativistic particle passing the Sun would only include the $\vec{E}_g$ term, which the Wiki article appears to assume is just the Newtonian force; and that alone does *not* give the right answer (it gives an answer half as large as the GR prediction, and the GR prediction matches observations). But other references might include relativistic corrections to $\vec{E}_g$, or might include terms due to general relative motion in $\vec{B}_g$ instead of just rotation.

You don't need rotation from the source. As mercury moves through the gravitostatic field of the sun, in its reference frame, it sees also a gravitomagnetic field.

For light, this effect is definitely worse, because it goes at the speed of light (i thought of this by my self :P)

I agree, that the effect of the rotation of the sun is probably negligible.

Again, i think that in general, they never consider SR in combination with the EM-like equations. Keep that in mind when you check your references...

To retain an ultrarelativistic particle in circular orbit requires a strong gravitational field. So it's impossible to talk about the precession of the orbit of an ultrarelativistic particle in the weak field approximation!

Its not an ultrarelativistic particle. Precision of observation is too grate. Its 40 arcseconds a century. Normal magnetism of electromagnetic nature is also a relativistic effect, detectable at low speeds, because electrostatic force in matter is incredibly strong. Magnetism, is just an imbalance of electric charge in the reference frame of the moving particle, due to the legnth/time contractions...
I guess you could say, that the bending of light involves an ultrarelativistic particle...

Any correct prediction of such a precession would have to include corrections to the Newtonian force, since the Newtonian force by itself produces closed orbits (i.e., zero precession).

You forget the "magnetic" field in the reference frame of mercury. That thing will definitely induce a rotation.

This would make sense given the EM analogy; I believe the analogous situation for electromagnetism (a test body orbiting a much larger body of opposite charge) has zero precession (because the effective potential has the Coulomb term and the centrifugal term, but not the extra attractive term that arises in GR and produces the perihelion precession).

apparent magnetic field in the moving charge's reference frame...
thomas rotation/precession...

 

[PeterDonis]

Just using the formula already derived for EM should give a first order only answer.

Yes, and the formula already derived for EM is easy to look up.

i meant an approximation
An approximation, that should be self consistent.

Ok, that helps to clarify where you're coming from.

If you don't follow the analogy

Oh, I follow the analogy. I just also understand that it's an analogy, which might work for certain aspects, but which is certainly not guaranteed to always work, even in the weak field approximation.

, somehow (need to plug something as the energy of the gravitational field) .Then the approximate EM-like equations become inconsistent mathematically.

Do you have a reference for this, or are you just assuming it happens without having actually tried to do the math?

The source is the 4 momentum, if the 4 momentum is not really all the 4 momentum of the system, then itself inconsistent.

But see, here's the thing: GR, the full version (i.e., *not* the weak field approximation), the Einstein Field Equation, does *not* include any "energy and momentum in the gravitational field" in the source term on its RHS, the stress-energy tensor; yet it *is* self-consistent.

Now suppose we do what I described in a previous post: we make the weak field approxmation, so that the pressure and stress terms in the SET are negligible compared to the energy and momentum (which does *not*, as I just pointed out, include any "energy and momentum in the gravitational field"), and we contract the standard EFE with a 4-vector field describing the motion of the matter described by the SET (which, as I noted before, amounts to throwing away the information about pressure and stress). That gives us a 4-vector source $J^a = ( \rho_g, \vec{J}_g )$ on the RHS, which does *not* include any "energy and momentum in the gravitational field". Yet the equation is still self-consistent, because I took a self-consistent, covariant equation and applied a self-consistent, covariant operation (contracting with a 4-vector field) to it.

I don't know that this is actually a way to derive the GEM equations given in the Wiki page; I suspect not (because I suspect there are a bunch more approximations that need to be made first). But it certainly does give a set of self-consistent equations with a 4-vector "source" on the RHS that includes no gravitational field energy-momentum. So I'm extremely skeptical that the GEM equations somehow fail to be self-consistent if the source does not include gravitational field energy-momentum.

From the way the orbits of the pulsars decay? There mutual attraction should be nonlinear.

So what? By your logic, we can only observe a change in the system's mass by being close enough to it for a significant amount of its "gravitational field potential energy" to be above us. So we would have to be *within* the binary pulsar system itself to measure what you were talking about.

To put it another way: in order to see the mass of a system decrease significantly by your logic, we ourselves would have to be gravitationally bound within the system. But we are certainly not gravitationally bound to any of the binary pulsar systems we observe; the gravity of those systems is negligible, by many orders of magnitude, to us here on Earth. Yet we *do* detect the masses of these systems decreasing (by, as you say, measuring the change in orbital parameters which shows the systems becoming more tightly bound). So whatever the reason for the observed decrease, it can't be what you're suggesting.

What I've just said also makes clear, btw, that it doesn't take binary pulsar observations to show that the mass of a system doesn't decrease as you get closer to it because of more "gravitational field energy" being above you. If that were the case, orbits of satellites and spacecraft near the Earth, and indeed anywhere in the solar system, would be substantially changed from what we observe. All of our observations confirm that, for example, the mass of the Earth is the same whether you measure it from just above its surface or from the distance of the Moon or further; it certainly doesn't get smaller as you get closer to the Earth, which is what your reasoning would imply.

A hot object has more gravitational pull then a cold one.

True.

Are we going to use some kind of relativistic thermodynamics, or just treat the object as a gabillion of particles?

As others have noted, relativistic thermodynamics is not a trivial exercise. But I'll agree that it beats treating the object as a collection of 10^25 individual particles, yes.

Why we can't use thermodynamics here too, to deduce the pressure and stress, in terms of energy and momentums?

I and others have already given you the reason: because at the macroscopic level, i.e., the level that thermodynamics deals with, there are too many degrees of freedom to be captured in a single 4-vector. You need a tensor, the stress-energy tensor, with 10 independent components. The fact that, at the microscopic level, the pressure and stress are ultimately due to the motion of atoms and the energy and momentum associated with that motion, does not change the fact that, when you do the thermodynamics, you can only reduce 10^25 degrees of freedom to 10, not 4.

If you ask *why* you can only reduce 10^25 degrees of freedom to 10, not 4, that becomes a question of statistical mechanics and probably deserves its own thread, since it's not even necessarily a question about relativity: after all, even in non-relativistic physics, stress in an object is described by a 3-dimensional symmetric tensor (with 6 independent components), not by a 3-momentum and a scalar for energy (3 components plus 1 number, for 4 numbers total).

Is it a more correct statement to say. That the inaccuracies appear, only when nonlineair effects must be taken in too account?

As a general statement, this seems OK to me (but see one caveat below). But "nonlinear effects" turns out to cover a pretty wide range, as the last few posts from me and Bill_K should indicate. For example, I now think that the GEM model (probably--see below) gives the wrong prediction for both light bending by the Sun and the precession of Mercury's perihelion.

Basically, if the GEM model is only correct for linear effects, the only non-Newtonian effect that I can see it predicting is frame dragging, the Lense-Thirring effect due to a nonzero $\vec{B}_g$ field arising from the rotation of the central body. However, that brings up the caveat I referred to: I'm not even sure that the L-T effect is considered a "linear" effect, from the viewpoint of GR. But the GEM equations, like Maxwell's Equations, are linear, and I think the discussion in the Wiki article is correct about them predicting frame dragging, so I think this is a minor point.

From what i understood. They use the EM-like equations just like that. They don't also consider SR.

"Not considering" SR is not the same as the equations not being covariant. They're obviously covariant just from looking at their form (the fact that they are formally the same as Maxwell's Equations, which are known to be covariant); they just aren't written in a way that makes the covariance obvious.

The speeds are slow, yes. But the precision of observation is so damn precise, that you need to consider SR also.

This is a valid point; even at the slow speeds of solar system objects we can make precise enough measurements to see effects that only arise at a fairly high power of $v / c$.

The precession of mercury is on over 100 years. That looks precise enough to require SR corrections...

No, the corrections needed to get the right precession for Mercury's perihelion are not "SR corrections"; they are "GR corrections". That's why the GEM model, even though it is covariant, doesn't (I believe--see below) get them right.

You don't need rotation from the source. As mercury moves through the gravitostatic field of the sun, in its reference frame, it sees also a gravitomagnetic field.

I agree that this ought to be true based on the analogy with EM; but the Wiki article's expression for $\vec{B}_g$ doesn't give any help. That's why I said I wanted to see more explicit expressions for the fields. (I would also like to see a similar computation for the straight EM case of, say, an electron with $v \rightarrow c$ flying by a proton or some other positively charged object and being deflected.)

For light, this effect is definitely worse

Not really; the effect on light should be similar to the effect on a particle like an electron with $v \rightarrow c$. (It certainly is in the full GR calculation.)

Again, i think that in general, they never consider SR in combination with the EM-like equations.

No, I think they just don't bother pointing out that, just like the EM equations, the GEM equations are covariant. I suspect this is because, if the GEM framework gets any practical use, it is for problems where none of the issues we've been discussing arise.

You forget the "magnetic" field in the reference frame of mercury. That thing will definitely induce a rotation.

Remember I said "the Newtonian force", which does not include the magnetic component, will produce zero precession. Before coming to a conclusion about the GEM prediction, I would want, as I said above, to see more explicit expressions for the fields. I would also, for this case, want to see a computation for the EM case of, say, an electron in an elliptical orbit about a proton or other positively charged object, to see if the EM equations predict perihelion precession. If Bill_K is right that a linear theory has to predict zero precession, then the EM equations will predict zero precession since they are linear. But the presence of the magnetic field is a significant difference between the EM case and the straight Newtonian (not GEM) case, so I think it's worth having a separate computation for the EM case.

 

[Quantum Immortal]

Yes, and the formula already derived for EM is easy to look up.

Ok, that helps to clarify where you're coming from.

for second order correction, you do the same for the gravitational field of the first order correction found earlier...

The energy we found earlier, has it self a gravity. We deduce its energy with the simple EM formula. Third order correction...

Of course, the new energy has it self a gravitational field... We add this again...
And again...

If i didn't messed up, the infinite series should converge.
If i didn't messed up, that's the full energy of the gravitomagnetic field, inside the self consistent framework of the EM-like equations.
Maybe I'm off by a scaling factor?

Were I'm wrong?

Do you have a reference for this, or are you just assuming it happens without having actually tried to do the math?

We defined the source, as ALL the energy/momentum.
So the source is all energy/momentum, except the one of the EM-like gravitational field?
Sounds weird... Why include ALL energy, of everything, every field, true or imaginary, but the gravitational field is excluded? Weird...
This is why i say, it will be inconsistent to just omit the contribution of the EM-like field of gravity.

But see, here's the thing: GR, the full version (i.e., *not* the weak field approximation), the Einstein Field Equation, does *not* include any "energy and momentum in the gravitational field" in the source term on its RHS, the stress-energy tensor; yet it *is* self-consistent.

I start to understand your complaint. You just say, that GR doesn't has the energy of the field as source, so why EM-like equations should? And, is this rigorously derived from GR?

The equations are of different nature.

There's that second term with the ricci scalar that isn't sure if it should be on the right or on the left of the equations. Is it really a curvature? Is it really part of the source?
Also, in the continuity equation of the tensor, you use the covariant divergence. Cristofel symbols all over the place...
Implicitly, space time curvature( gravity, in GR) is part of the source. This is why it get nonlinear.
In GR, you just need to take a covariant derivative, to take into account gravitation.
Obviously, you can't do that in flat space. When you pass to normal derivatives, you need to compensate for what you through out.

By analogy, the EM-like equations, should also have the field it self in its source, in some form.
Or they'll remain linear.
The most natural choice, is to use the energy momentum of the field. It sure has the right units, and can be plugged at the same place as the rest of the sources, and the equations don't get distorted, and become non lineair... The EM field gravitates, why the EM-like gravitational field is not aloud to gravitate?
You want to propose something else?

And no. Its not rigorously derived from GR...
As you can imagine, its a bit difficult.
Is it really important that its not derived rigorously? It's an approximation after all.
What you propose is the simplest way, to render the equations non linear?
Here, you just need to tweak the definition of the source a little...
If its complicated, the whole point of an approximation is pointless...

Now suppose we do what I described in a previous post: we make the weak field approxmation, so that the pressure and stress terms in the SET are negligible compared to the energy and

momentum (which does *not*, as I just pointed out, include any "energy and momentum in the gravitational field"), and we contract the standard EFE with a 4-vector field describing the motion of the matter described by the SET (which, as I noted before, amounts to throwing away the information about pressure and stress). That gives us a 4-vector source $J^a = ( \rho_g, \vec{J}_g )$ on the RHS, which does *not* include any "energy and momentum in the gravitational field". Yet the equation is still self-consistent, because I took a self-consistent, covariant equation and applied a self-consistent, covariant operation (contracting with a 4-vector field) to it.

I don't know that this is actually a way to derive the GEM equations given in the Wiki page; I suspect not (because I suspect there are a bunch more approximations that need to be made first). But it certainly does give a set of self-consistent equations with a 4-vector "source" on the RHS that includes no gravitational field energy-momentum. So I'm extremely skeptical that the GEM equations somehow fail to be self-consistent if the source does not include gravitational field energy-momentum.

GR has implicitly as source, the curvature of space time.
When you did the approximation that way, you just cut off this fact.
The most simple and natural way to have the EM-like gravity, be a source of it self, is to plug in its energy. You want to propose something else?

So what? By your logic, we can only observe a change in the system's mass by being close enough to it for a significant amount of its "gravitational field potential energy" to be above us. So we would have to be *within* the binary pulsar system itself to measure what you were talking about.

To put it another way: in order to see the mass of a system decrease significantly by your logic, we ourselves would have to be gravitationally bound within the system. But we are certainly not gravitationally bound to any of the binary pulsar systems we observe; the gravity of those systems is negligible, by many orders of magnitude, to us here on Earth. Yet we *do* detect the masses of these systems decreasing (by, as you say, measuring the change in orbital parameters which shows the systems becoming more tightly bound). So whatever the reason for the observed decrease, it can't be what you're suggesting.

What I've just said also makes clear, btw, that it doesn't take binary pulsar observations to show that the mass of a system doesn't decrease as you get closer to it because of more "gravitational field energy" being above you. If that were the case, orbits of satellites and spacecraft near the Earth, and indeed anywhere in the solar system, would be substantially changed from what we observe. All of our observations confirm that, for example, the mass of the Earth is the same whether you measure it from just above its surface or from the distance of the Moon or further; it certainly doesn't get smaller as you get closer to the Earth, which is what your reasoning would imply.

The pulsars are inside each other's gravitational fields. We can observe there orbits decaying. So, i assumed, we could detect how there mutual attractions with each other varies with distance. And from there deduce, if what i explained about was true or not. I didn't mean the gravitational pull with us on earth, i was trying to explain the phenomenon back then.

The effect is very very small. Thats why, we could in theory observe it with a binary system of pulsars orbiting each other, not with our moon...

too many degrees of freedom to be captured in a single 4-vector. You need a tensor, the stress-energy tensor, with 10 independent components.

Pressure and stress are macroscopic constructs.
Not the smartest way to do it. but...

you take the stress energy tensors of the gabilion of particles. They all have 0 in the space space components. You add them. You get the stress energy tensor of the whole system, with 0 in the space space components. Its not the smartest way to built the tensor of the system, but, here it is, nothing in the space space components, and you truly have all the energy of the system.

That proves, you can somehow re-express the space space components into the time components. Its not necessarily trivial in practice ...

No, the corrections needed to get the right precession for Mercury's perihelion are not "SR corrections"; they are "GR corrections". That's why the GEM model, even though it is covariant, doesn't (I believe--see below) get them right.

I said GEM + SR > for calculating the precession
You are saying ... GEM + GR, in the context of what i meant.
The hyper relativistic limit, the way i use the term...

I'm pretty sure that GEM is used just like that. They just put mercury in the equation, and that's it. They don't consider the apparent field mercury sees. Per bellow...

I agree that this ought to be true based on the analogy with EM; but the Wiki article's expression for $\vec{B}_g$ doesn't give any help. That's why I said I wanted to see more explicit expressions for the fields. (I would also like to see a similar computation for the straight EM case of, say, an electron with $v \rightarrow c$ flying by a proton or some other positively charged object and being deflected.)

If i had found that, i wouldn't be asking nonsense questions here...

Not really; the effect on light should be similar to the effect on a particle like an electron with $v \rightarrow c$. (It certainly is in the full GR calculation.)

We talk about GEM, not GR.
Light going at the speed of light, should be used with SR if considered in the context of GEM .

No, I think they just don't bother pointing out that, just like the EM equations, the GEM equations are covariant. I suspect this is because, if the GEM framework gets any practical use, it is for problems where none of the issues we've been discussing arise.

I don't think so. They usually are very literal. If they didn't, most likely, they didn't do it.
I think they just use GR instaid, not GEM/SR

Remember I said "the Newtonian force", which does not include the magnetic component, will produce zero precession. Before coming to a conclusion about the GEM prediction, I would want, as I said above, to see more explicit expressions for the fields. I would also, for this case, want to see a computation for the EM case of, say, an electron in an elliptical orbit about a proton or other positively charged object, to see if the EM equations predict perihelion precession. If Bill_K is right that a linear theory has to predict zero precession, then the EM equations will predict zero precession since they are linear. But the presence of the magnetic field is a significant difference between the EM case and the straight Newtonian (not GEM) case, so I think it's worth having a separate computation for the EM case.

I would like to see that too.I knocked at the right door i hope...

Bill is wrong. The "magnetic" part of the GEM field will induce rotations.
I just, don't know what it looks like...

 

[PeterDonis]

Were I'm wrong?

I can't tell because you're not actually showing me any math.

We defined the source, as ALL the energy/momentum.

No, "we" did not define the source as "ALL the energy/momentum". *You* did. *I* defined the source in the standard way it's defined in GR: the stress-energy tensor. Then I showed how you can get a covariant 4-vector from the SET, *if* pressure and stress are negligible. I notice that you have not addressed that point at all; you merely continue to wave your hands.

So the source is all energy/momentum, except the one of the EM-like gravitational field?

As I've said several times: there is no covariant way to define the energy and momentum in the gravitational field.

Sounds weird... Why include ALL energy, of everything, every field, true or imaginary, but the gravitational field is excluded?

Because, once again, there's no covariant way to define it.

Weird...

Perhaps, but that doesn't make it false.

This is why i say, it will be inconsistent to just omit the contribution of the EM-like field of gravity.

I understand this is your intuition, but that doesn't make it correct.

You just say, that GR doesn't has the energy of the field as source, so why EM-like equations should?

More or less, except that, unlike you, I'm not just waving my hands and saying it. I'm actually showing you the math.

And, is this rigorously derived from GR?

Yes.

There's that second term with the ricci scalar that isn't sure if it should be on the right or on the left of the equations. Is it really a curvature? Is it really part of the source?

It's part of the curvature. Here's how you can tell:

Also, in the continuity equation of the tensor, you use the covariant divergence.

Yes, because you're in a curved manifold, so ordinary partial derivatives do not give you the correct rate of change of quantities; you need to use covariant derivatives. So to properly express conservation of stress-energy, the *covariant* divergence (*not* the ordinary divergence) of the SET needs to be zero, which means the covariant divergence of both sides of the Einstein Field Equation needs to be zero. And the only way the Einstein Field Equation obeys the Bianchi identity, so that both sides have zero covariant divergence, is if the trace term $(1/2) g_{\mu \nu} R$ is on the LHS, with the curvature, instead of on the RHS, with the source.

Implicitly, space time curvature( gravity, in GR) is part of the source. This is why it get nonlinear.

No, gravity is nonlinear because the equation it obeys, the EFE, is nonlinear. And it's nonlinear regardless of which side you put the trace term on; do it the way I just described, so the Bianchi identity is obeyed, and the equation is still nonlinear.

In GR, you just need to take a covariant derivative, to take into account gravitation.

Yes, that's quite true, as I said above. What does that have to do with what the proper source term in the EFE is?

Obviously, you can't do that in flat space. When you pass to normal derivatives, you need to compensate for what you through out.

There are various ways of defining a "pseudo-tensor" describing the energy in the gravitational field, so that you can define a "total energy" pseudo-tensor whose *ordinary* divergence (not covariant divergence) is zero, yes. But that word "pseudo" should be a clue: doing this is *not* covariant. As I've already said several times, there is no covariant way to define the energy and momentum in the gravitational field.

By analogy, the EM-like equations, should also have the field it self in its source, in some form.
Or they'll remain linear.

They *are* linear, as I've already said several times. Just as Maxwell's Equations, which have the same form as the GEM equations, are linear. The GEM model is a *linear approximation* of gravity.

The most natural choice, is to use the energy momentum of the field. It sure has the right units, and can be plugged at the same place as the rest of the sources, and the equations don't get distorted, and become non lineair...

Can you show me some actual math? (Also, whatever you think you're describing here, it isn't the GEM model described in the Wiki article, which, as I've noted several times now, is *linear*.)

The EM field gravitates, why the EM-like gravitational field is not aloud to gravitate?

Because it can't in a linear model. This is one way in which the analogy breaks down. (Actually, it only breaks down if you fail to pay attention to what the analogy is actually telling you. What the analogy is actually telling you is that you need to match source to source. The source of the EM field is not energy-momentum, it's charge, and EM fields carry no charge. The source of gravity is energy-momentum, so by analogy with EM, gravity fields should carry no energy-momentum.)

And no. Its not rigorously derived from GR...
As you can imagine, its a bit difficult.

And yet you think you know how it's going to come out, without actually doing any math.

Is it really important that its not derived rigorously? It's an approximation after all.

As I said in a previous post, I don't actually know how the GEM equations are derived from the full EFE. All I described was the obvious way to get a 4-vector describing energy and momentum from the full SET, assuming pressure and stress were negligible. If you want to show me how the GEM model is actually derived, why don't you find some references that show that?

What you propose is the simplest way, to render the equations non linear?

I didn't propose any such thing. All I "proposed" was the obvious way to get an energy-momentum 4-vector from the SET; see above.

Here, you just need to tweak the definition of the source a little...

Then it should be easy to show me the actual math that does this.

GR has implicitly as source, the curvature of space time.

No, the "source" in GR is *not* the curvature of spacetime. Spacetime curvature is the *effect*; stress-energy is the source. You need to spend some time actually learning GR; you appear to have very basic misunderstandings about how it works.

The pulsars are inside each other's gravitational fields. We can observe there orbits decaying. So, i assumed, we could detect how there mutual attractions with each other varies with distance.

To the extent we can measure this, by measuring the orbital parameters, the masses of both pulsars are the same regardless of how far apart they are. But AFAIK the variation in the distance between them over an orbit is not very great, so this isn't much of a test.

The effect is very very small. Thats why, we could in theory observe it with a binary system of pulsars orbiting each other, not with our moon...

You're kidding, right? You postulate an effect that is very small, and you expect us to get a better measurement of it by observations of a binary pulsar system light-years away, than we could get by sending spacecraft with radio transponders between the Earth and the Moon? Weren't you the one that was saying that our measurements are so accurate in the solar system that we can detect even very small relativistic effects?

Pressure and stress are macroscopic constructs.

Which you don't appear to understand very well. See below.

you take the stress energy tensors of the gabilion of particles.

Individual particles don't have a stress-energy tensor. The SET is a macroscopic construct that is used when you are modeling matter as a continuous substance, by averaging the behavior of a large number of particles whose size and average separation are much smaller than the distance scale you are interested in.

That proves, you can somehow re-express the space space components into the time components.

No, it proves that you don't understand what an SET is. See above. You really need to look at some actual math.

I'm pretty sure that GEM is used just like that. They just put mercury in the equation, and that's it. They don't consider the apparent field mercury sees.

Once again, can you show me some actual math? Has somebody actually used the GEM model to calculate Mercury's orbit? Do you have any references? Or are you just waving your hands again?

Light going at the speed of light, should be used with SR if considered in the context of GEM.

So should an electron going at almost the speed of light, which was what I was saying.

Bill is wrong. The "magnetic" part of the GEM field will induce rotations.
I just, don't know what it looks like...

Have you actually tried to do the math? If not, how can you be so sure Bill_K is wrong?

 

[Quantum Immortal]

What you people think of this paper?
http://www.academia.edu/483186/Gravitomagnetism_a_novel_explanation_of_the_precession_of_planets_and_binary_pulsars

He uses GEM to calculate the precession of orbits of some planets and of some binary pulsars
example for mercury GR:42,98 GEM:43,02 observation:43,1

The gravitomagnetic field is apparent, in the reference frames of the planets moving through the gravitostatic field.
This apparent gravitomagnetic field, induces the rotations...
The results remain valid, even in the strong fields of the binary pulsars.

also he rewrite the Lorentz force and gets
F= (Newton) - (mv^2/r)(v/c)^2
(radial component of the force)
I think something like this can be readily compared with PPN

the binary pulsars are strong fields. They are the strongest fields i can think of, we can observe.
So why GEM is said not to be valid for strong fields? It seams to be doing reasonably well...
A reasonable guess, is that other strong field effects are reasonably predicted...

These calculations were possible since when the Lorentz transformations were derived (before SR).
Most likely, GEM covers correctly 100% of the observations of that time...
The precession of the orbit of mercury, was the most important...
The simplest possible way, of trying to build a relativistic theory of gravity, is to just adapt the EM equations to gravity. But GEM doesn't really show up in the history back then. Just an article from Heaviside of 1893... Why was it rejected back then?
...so what's going on here?

What are the observations GEM contradicts? Today and back then?
It seams the list is shorter, then what is usually claimed...

comments?

 

[Dale]

He uses GEM to calculate the precession of orbits of some planets and of some binary pulsars
example for mercury GR:42,98 GEM:43,02 observation:43,1

That is interesting. I remember reading that the original calculations of precession using GEM were wrong. I wonder what he changed.

What are the observations GEM contradicts? Today and back then?
It seams the list is shorter, then what is usually claimed...

comments?

To answer this question the best approach would be to determine the PPN parameters for GEM and see how they compare to experiment. I haven't seen any reference which did that, but I haven't particularly looked for one either.

Of course, one problem is that gravitomagnetism is usually considered a prediction of GR instead of its own independent theory as you seem to be attempting. So experimental confirmation of GEM is usually taken as confirmation of GR.

 

[Nugatory]

What you people think of this paper?
http://www.academia.edu/483186/Gravitomagnetism_a_novel_explanation_of_the_precession_of_planets_and_binary_pulsars

That paper basically demonstrates that gravitomagnetism is consistent with GR when stress and pressure is negligible. That's all well and good, but why would I ever prefer a non-covariant theory over the weak-field approximation of a covariant theory?

 

[PeterDonis]

What you people think of this paper?
http://www.academia.edu/483186/Gravitomagnetism_a_novel_explanation_of_the_precession_of_planets_and_binary_pulsars

I haven't checked the math in this paper, but I notice that the GEM equations it gives are different from the ones given in the Wikipedia article; this paper has the opposite sign on the RHS of the $\vec{\nabla} \cdot \vec{E}_g$ equation, and it's missing a factor of 4 on the RHS and a minus sign in the $\vec{J}_g$ term in the $\vec{\nabla} \times \vec{B}_g$ equation. AFAIK the equations as given in the Wiki article are the ones that can be derived from the full EFE; this paper's author refers to another paper of his (which I haven't looked at) where he derives his version of the equations "using quaternions". (The equation for gravitational Lorentz force in the paper is also different; it's missing a factor of $\gamma$.)

He uses GEM to calculate the precession of orbits of some planets and of some binary pulsars

I haven't checked the math here either; on the face of it I'm not sure I see how his formulas can give numbers that match up with the GR ones so closely, yet are not identical. If he's claiming his method can duplicate the GR equations at the given order of approximation, which is what it looks to me that he is claiming, then his numbers should be identical.

I also note that he does not calculate the rate of decay of the binary pulsar orbits due to gravitational wave emission; he only calculates the periastron precession. (He mentions the orbital decay only in a brief aside.)

 

[PeterDonis]

the GEM equations it gives are different from the ones given in the Wikipedia article; this paper has the opposite sign on the RHS of the $\vec{\nabla} \cdot \vec{E}_g$ equation, and it's missing a factor of 4 on the RHS and a minus sign in the $\vec{J}_g$ term in the $\vec{\nabla} \times \vec{B}_g$ equation. AFAIK the equations as given in the Wiki article are the ones that can be derived from the full EFE

On consideration, this observation kills the paper's theory outright. The reason the GEM equations derived from the EFE (the ones given in the Wiki article [Edit: assuming those are actually derived from the EFE, which we don't know for sure--see Jonathan Scott's post below and my response]) differ from what you would get by just putting $g$ subscripts on the quantities in Maxwell's Equations, which is what the equations given in this paper do, is that in GR, gravity is a spin-2 interaction, not a spin-1 interaction. Writing the equations the way the paper does amounts to modeling gravity as a spin-1 interaction, just like electromagnetism.

However, that has at least one obvious problem: for a spin-1 interaction, like charges repel! If you look at the $\vec{\nabla} \cdot \vec{E}_g$ equation given in the paper, with its positive sign, it's obvious that positive mass density should *repel* positive mass density, not attract it. You need a minus sign in that equation to make gravity attractive (and indeed the GEM equation in the Wiki article has a minus sign there). MTW has, IIRC, a whole set of exercises aimed at getting you to show this explicitly.

So even if the numbers given in the paper are correct for those particular predictions, the model it presents is obviously falsified. However, given what I just said above, I strongly doubt the numbers given in the paper are actually correct. The author does not actually prove that his model even leads to stable circular or elliptical orbits (note that in standard electromagnetism, you have to have charges of opposite sign to have stable orbits); he just assumes it and plugs in numbers based on that assumption. If he actually tried to calculate these "orbits" from first principles (by starting with some set of initial conditions and using his GEM equations to calculate the time evolution), he would find, based on the above, that the "orbits" were hyperbolas (as with like charges repelling in standard electromagnetism), not ellipses.

 

[Jonathan Scott]

Be cautious about the Wikipedia entry in this case.

There are multiple conventions for gravitomagnetism, differing for example in whether the "magnetic" field is treated as being equal to the angular velocity or differing by a factor of 2, like the "curl" of a velocity field, making it more like the electromagnetic field.

Back in 2008 I found that the Wikipedia article at that time mixed up different conventions in different places with the result that there were factor of 2 errors in various places, which I mentioned on the Talk page at the time, making it look spuriously more similar to electromagnetism than is in fact possible. I don't know to what extent that has now been fixed but after my comment I saw a lot of argument about which convention was best and it didn't seem to be heading for quick resolution.

 

[Dale]

There are multiple conventions for gravitomagnetism

I think that this is the key problem in comparing GEM to experiment. GEM is not a coherent theory of gravity, it is a loose term referring to several different sets of equations or even a term referring to different types of gravitational phenomena which are predicted by other theories.

I think that the OP needs to clarify exactly which theory he/she is interested in rather than using a generic moniker like GEM.

 

[PeterDonis]

MTW has, IIRC, a whole set of exercises aimed at getting you to show this explicitly.

After looking at MTW, I was partly right about what the exercise is aimed at showing. It is exercise 7.2, and it gives an action for a vector (i.e., spin-1) theory of gravity and then asks you to calculate the perihelion precession, the bending of light, and the energy carried by gravitational waves in this theory. It doesn't explicitly ask you to show that positive masses attract in this theory; but the action that is written down for the theory flips a key sign, compared to the analogous action for electromagnetism, so MTW were clearly aware that without flipping that sign, gravity would be repulsive in this theory and there would be no stable orbits. Since the paper QuantumImmortal linked to does *not* flip that key sign, what I said about the paper stands.

 

[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 teh 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.