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Covariant Gravitomagnetism

  1. Jan 7, 2014 #1
    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 wierd 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
     
  2. jcsd
  3. Jan 8, 2014 #2

    PeterDonis

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

    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.)

    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:

    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.

    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 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.

    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.

    No.

    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.

    What does this mean?

    I don't understand what this means either.
     
  4. Jan 8, 2014 #3

    maajdl

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    A post by rpenner on this page can be helpful:

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

    You can also find the original paper by Heaviside:

    http://gsjournal.net/Science-Journals/Essays/View/2505
     
    Last edited by a moderator: May 6, 2017
  5. Jan 8, 2014 #4

    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.



    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.

    For matter, you can translate in an appropriate way, pressure and stress in to 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 in to 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...


    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 text books, 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.
     
  6. Jan 8, 2014 #5
    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.
     
    Last edited by a moderator: May 6, 2017
  7. Jan 8, 2014 #6

    WannabeNewton

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    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.
     
    Last edited: Jan 8, 2014
  8. Jan 8, 2014 #7

    PeterDonis

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    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.

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

    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.

    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.

    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).

    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?

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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*.

    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!

    Obviously you haven't read enough textbooks.

    Yes, all this is true. So what? What does it have to do with what we're discussing?
     
  9. Jan 8, 2014 #8
    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.


    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.


    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?



    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?



    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, with out 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.



    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 lets you use the same equations in all reference frames, the results however are not valid in all reference frames.

    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).
     
  10. Jan 8, 2014 #9

    PeterDonis

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    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.

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

    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:

    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.

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

    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).

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

    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.

    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.

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

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

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

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

    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.

    Huh?

    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:

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

    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.

    I'm not familiar with this term, but I now understand how you were using it.
     
  11. Jan 8, 2014 #10

    WannabeNewton

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    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.
     
  12. Jan 9, 2014 #11
    You mean to use GR and deduce the energy?

    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?


    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...


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

    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 thats 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?
     
  13. Jan 9, 2014 #12

    PeterDonis

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    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?

    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?

    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.)

    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.

    Yes, you did. See above.

    No, we haven't. See above.

    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?

    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.

    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.)
     
  14. Jan 9, 2014 #13

    PeterDonis

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    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.
     
  15. Jan 9, 2014 #14

    PeterDonis

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    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.
     
  16. Jan 9, 2014 #15

    Bill_K

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    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!
     
  17. Jan 9, 2014 #16

    PeterDonis

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    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).
     
  18. Jan 9, 2014 #17

    Bill_K

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    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.
     
  19. Jan 9, 2014 #18

    PeterDonis

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    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).
     
  20. Jan 9, 2014 #19
    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


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

    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 its self inconsistent.


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


    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????


    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????


    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....

    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.....

    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...


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

    apparent magnetic field in the moving charge's reference frame....
    thomas rotation/precession...
     
  21. Jan 10, 2014 #20

    PeterDonis

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

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

    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.

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

    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.

    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.

    True.

    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.

    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).

    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.

    "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.

    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##.

    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 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.)

    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.)

    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.

    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.
     
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