Gravitomagnetism and acceleration

[Xezlec]

Wow, what a cool web forum! Here's a question I've been wondering since I read about gravitomagnetism on Wikipedia.

I'm very familiar with Maxwell's equations (having majored in electrical engineering), and I know that self-inductance resists any attempt to accelerate a charge, and that the energy exerted in doing so is radiated as an EM wave.

When I flip the signs everywhere that "charge density" appears (as I am told we have to do in gravitomagnetism, for fairly clear reasons), and try to re-derive self-inductance, I end up with negative self-inductance. So general relativity actually... helps us accelerate our masses? Huh? Where does the extra energy come from?

Thanks. This has been driving me crazy.

 

[JustinLevy]

Yeah, this drives me nuts (I asked a similar question earlier).

First let's look at a simple electromagnetism problem. Consider a charged ring at rest. Now apply a magnetic field. The change in magnetic flux will cause the ring to start spinning, creating a magnetic field to reduce the enclosed flux.

Now look at the gravitoelectromagnetic equivalent. Consider a ring with a non-zero mass at rest. Now apply a gravitomagnetic field. The change in flux will cause the ring to start spining, BUT due to the different signs (in GEM as compared to Maxwell's eqs.), this will INCREASE the enclosed flux. And thus will create a run away solution.

...
Am I making a sign error? Or is there some reason that this simple thought experiment falls outside the "applicability regime" of this approximation? (If so, why? There aren't fast moving masses, nor is this a "strong field" problem.)

In my next post I seemed to have convinced myself I found the sign error (but the reasoning makes no sense now that I'm re-reading it after some sleep). So unfortunately I don't have the answers either.

It seems you've stumbled across the same "sign error problem" as me, and it's gnawing at me as well. Xezlec, if you figure it out later, please post the answer here. I'll start pondering it again as well.

 

[pervect]

I haven't looked at this issue very closely as of yet, but I did run across the following tidbit in http://www.arxiv.org/abs/gr-qc/0011014 that may be relevant

It is important to note that the [ed: linearized GEM] field energy density is negative and that there is a flux of energy that circulates around the mass in a direction opposite to its sense of rotation.

also

It is then tempting to construct the analog of the Maxwell tensor as
in electrodynamics,

However, the physical significance of this quantity is doubtful as first pointed
out by Maxwell [13] in his fundamental work on the dynamical theory of the
electromagnetic field. The basis for Maxwell’s considerations was the notion
that the attractive nature of gravity would lead to a negative energy density
for the field, while the electromagnetic analogy would imply a positive result.
In fact, in our approximation scheme we can use instead the standard pseudotensor
tμ of general relativity that gives a negative energy density. This is the subject of section 4.

 

[pmb_phy]

Wow, what a cool web forum!

Hi Xezlec. Welcome to the forum!

I hope you get as much out of as most of us do!

Best regards

Pete

 

[Xezlec]

pervect: thanks, but if that information is relevant to our question, I can't see how (I spent an hour or two thinking about it). It's interesting though, and might provoke another question.

 

[pervect]

The energy stored in the field of an inductor is .5 L I^2. If L is negative, the energy stored in the field must also be negative. (I^2 is always positive). The article above suggests that by the energy stored in the gravitational field around a rotating body is negative (* see note)

Therefore it tends to support your suggestion that the gravitational equivalent of L will be negative.

* Note that "energy stored in the gravitational field" is somewhat ambiguous. The approach taken in this article uses the Landadu-Lifschitz pseudotensor, this should be equivalent to the ADM energy. In some other recent posts, I use an approach based on the Komar energy (i.e. Komar mass) for example. The two approaches should be expected to give the same energy for an isolated system (if the necessary preconditions for both sorts of energy to exist are met), but should not be expected to agree on exactly how that energy is distributed.

 

[Xezlec]

OK, let me see if I understand this. The extra energy that helps me accelerate a mass comes from "nowhere", leaving a gravitational field around it with negative energy.

End result: objects accelerate slightly more than what I would expect by the amount of energy I put in. But that seems to be contrary to what I know of special relativity, which says that objects accelerate slightly less than I would expect based on the amount of energy I put in.

And there's another problem. I thought that GR explains why an object's mass is the quantity that determines its resistance to acceleration. Since GEM is one particular limit of GR, I would expect GEM to lead to something like F=ma. But that could only happen if self-inductance were positive.

 

[pervect]

OK, let me see if I understand this. The extra energy that helps me accelerate a mass comes from "nowhere", leaving a gravitational field around it with negative energy.

End result: objects accelerate slightly more than what I would expect by the amount of energy I put in. But that seems to be contrary to what I know of special relativity, which says that objects accelerate slightly less than I would expect based on the amount of energy I put in.

And there's another problem. I thought that GR explains why an object's mass is the quantity that determines its resistance to acceleration. Since GEM is one particular limit of GR, I would expect GEM to lead to something like F=ma. But that could only happen if self-inductance were positive.

I think the situation is rather that the simple SR defintion is an overestimate, and that the circulating "negative energy Poynting flux" that arises in some approaches such as those mentioned in my quotes brings this overestimate back to the correct value. (And I would add that I would expect that this correct value is still higher than the Newtonian value even if it is less than the simple SR estimate).

But I'm not positive about this - as you can see, the treatment in the literature is rather involved, I'm afraid I haven't taken the time to sort through it all.

I could go on a bit more about energy in the full theory at more length than I could about energy in the linearized theory - see for instance http://en.wikipedia.org/wiki/Mass_in_general_relativity - but that would probably be hijacking this thread, as you expressed an interest in "L", not an interest in energy. I pointed out the connection by bringing up .5 L I^2.

I will say that energy in GR depends on some gauge choices, so that there is not any single correct assignment of energy to a specific location. That's one reason that the paper I quoted mentions about 3 different ways of accounting for energy - and they even left one of my favorites out (the Komar approach).

 

[Xezlec]

*Sigh* well, I was also curious why it can't be said that energy is assigned a specific location, since you could theoretically measure the location of energy by observing its gravitational field, just like you can measure the location of mass that way. But after looking at that link, I'm getting the feeling that my assumption that I had a chance of understanding GEM or anything about relativity without knowing GR in its entirety was pretty naïve.

I guess I'll just have to stick with Newton, Maxwell, and friends.

Thanks for your help, though!

 

[pervect]

*Sigh* well, I was also curious why it can't be said that energy is assigned a specific location, since you could theoretically measure the location of energy by observing its gravitational field, just like you can measure the location of mass that way. But after looking at that link, I'm getting the feeling that my assumption that I had a chance of understanding GEM or anything about relativity without knowing GR in its entirety was pretty naïve.

I guess I'll just have to stick with Newton, Maxwell, and friends.

Thanks for your help, though!

If you are familiar with electromagnetism, then you have probably heard about the "gauge degree of freedom" that E&M has. Basically, you can add any constant to the electrostatic scalar potential $\Phi$, and the gradient of any function to the magnetic vector potential A. In 4-vector terms, you can add a 4-gradient to the 4-vector potential A.

Any such "gauge transformation" (adding a 4-gradient to the 4-potential) does not affect the electromagnetic fields, E and B, and in E&M it also does not affect the value of the energy of the field. This field energy can be computed by several means, the one that is most relevant to this topic is the computation of the energy from the Lagrangian of the field for a free field.

You may or may not have had Lagrangian mechanics yet, but it's not too much further beyond standard Newtonian physics to the Lagrangian formulation. Lagrangians of lumped systems (i.e. point masses, springs, etc) are the easiest, often derived from the principle of "virtual work", but the formalism can also be extended to cover fields (such as the electromagnetic field) this is discussed for instance near the end of Goldstein, "Classical mechanics". See for instance pg 583-584. Given that one wants the stress energy tensor to be symmetric, one winds up with a unique energy for the (free) electromagnetic field via the Lagrangian approach, this is actually worked out on these pages. (It also gives you the Poynting vector).What happens in GR is that one finds that the energy associated with the field is NOT independent of the gauge choice.

And one of the things Emily Noether proved (in physics: Noether's theorem) is basically that this result (the fact that the field energy depends on the gauge choice) is inherent in the way that GR is formulated as a diffeomorphism invariant theory.

Because of this fact, there just isn't any unique way to assign energy to the "gravitational field" as there is in electromagnetism.

 

[JustinLevy]

* Note that "energy stored in the gravitational field" is somewhat ambiguous.

While I realize the question of the "energy in the fields" is ambiguous, I'd feel extremely disheartenned if this is an excuse for why our "inductor" (a ring with a mass) would not only accelerate more, but actually create a runaway solution (to infinite kinetic energy) if any initial torque is applied. I really doubt (and hope) that is not the case.

The logic in steps
1. A mechanical torque is applied (for a finite period of time) to a ring with non-zero mass, causing it to start spinning
2. As the ring begins to spin, it creates an increasing gravitomagnetic field
3. This changing flux in the ring creates an "emf"
4. The "emf" provides a torque on the ring ... but due to the negative "inductance", this increases the rate of spin (instead of hindering it)
5. Even after the mechanical torque is stop being applied, the ring is now in a run-away solution because of the negative inductance.

This can not possibly be correct. What is wrong here? How can negative inductance be allowed?

Also, since that paper suggests there is negative energy in a gravitomagnetic field ... it seems like the vacuum is at a horribly unstable state, just waiting to collapse into a state with a huge gravitomagnetic field.

I just can't take this seriously. We must be making a sign error, as I just can't believe the theory predicts these effects.

 

[Jonathan Scott]

If you have a massive rotating ring, it does slightly rotate frames of reference within it in the same sense (by the "Lense-Thirring effect"). The effect is so slight that you would have to rotate everything in the universe around you to make the frame of reference rotate at around the same rate as the actual rotation.

If you had another ring within it at rest, the inner ring would therefore behave very very slightly as if it were rotating in the opposite direction, in that there would be very tiny centrifugal forces experienced at rest on the inner ring, and anything moving relative to the inner ring would experience slight coriolis forces. If the inner ring were set rotating at the induced rotation rate of the frame of reference, in the same direction, then these centrifugal and coriolis forces would vanish, as if it were not rotating.

As far as I know, a constant rotation rate would not induce any torque forces in a static object inside it, but would merely cause the frame of reference to appear to rotate. However, if the rotation rate is increased, I would expect that this would also cause a very slight torque in the same direction on the inner ring inside it, causing it to rotate a little, and that ring would equally cause a matching torque on the outer ring opposing the increased rotation. The fact that the inner object would also rotate in the same direction would increase the gravitomagnetic rotational flux inside it, but I assume that this increase is "paid for" by the energy expended in rotating it.

I don't believe there could be any build-up of gravitomagnetic flux. Although the outer ring causes the inner ring to speed up, the inner ring has a braking effect on the outer one, and I think that conservation of angular momentum would mean that the effective overall flux would not be affected by that type of interaction.

The above conclusions are based on what I remember from studying gravitomagnetism myself many years ago; I think they are correct but don't treat them as necessarily authoritative. For an authoritative reference, I personally like the treatment of the subject in the book "Gravitation and Inertia" by Ciufolini and Wheeler, but like most things relating to gravity it takes a lot of patience to get into.

 

[pervect]

While I realize the question of the "energy in the fields" is ambiguous, I'd feel extremely disheartenned if this is an excuse for why our "inductor" (a ring with a mass) would not only accelerate more, but actually create a runaway solution (to infinite kinetic energy) if any initial torque is applied. I really doubt (and hope) that is not the case.

I don't think that's the case either.

The logic in steps
1. A mechanical torque is applied (for a finite period of time) to a ring with non-zero mass, causing it to start spinning
2. As the ring begins to spin, it creates an increasing gravitomagnetic field
3. This changing flux in the ring creates an "emf"
4. The "emf" provides a torque on the ring ... but due to the negative "inductance", this increases the rate of spin (instead of hindering it)
5. Even after the mechanical torque is stop being applied, the ring is now in a run-away solution because of the negative inductance.

This can not possibly be correct. What is wrong here? How can negative inductance be allowed?

A negative inductor all by itself would be unstable. However, I don't believe that's what's happening here.

It might help if you expand your analogy and write the equations down.

If voltage is analogous to torque, and current is analagous to angular velocity, then V = L dI/dt becomes

torque = (constant) d (omega) / dt

Is that the correct analogy? You might have to throw in some unit conversions, but it seems generally right - the faster the disk rotates, the higher the "mass current" is, and torque is what drives a change in rotation.

Assuming this is what you had in mind, your inducatance represents an effective gravitomagnetic correction to the moment of inertia.

This correction term to the moment of inertia may be negative, but that doesn't make the complete moment of inertia negative.

In other words your negative inductor is in parallel with a positive inductor, and the net admittance is positive.

 

[Chris Hillman]

Sigh... since this is a fairly common misconception among newbies dropping into the gtr-scape from the terrain of electromagnetism, the very first thing which should have happened here, I think, would have been to refer the OP to http://math.ucr.edu/home/baez/RelWWW/HTML/grad.html#gem

 

[Xezlec]

Sigh... since this is a fairly common misconception among newbies dropping into the gtr-scape from the terrain of electromagnetism, the very first thing which should have happened here, I think, would have been to refer the OP to http://math.ucr.edu/home/baez/RelWWW/HTML/grad.html#gem

Thanks for the link, but having glanced at that, I stand by my assertion that the subject sounds a lot more involved than I had thought, and is beyond my mathematical abilities. I'm familiar with vector calculus and Maxwell's equations, and nothing further out than that. I doubt I can make much sense out of those papers.

 

[Chris Hillman]

OK, well, the textbooks by Carroll or D'Inverno or Schutz might be something to put on your reading list for the summer holiday then :-/

 

[pervect]

I've taken a quick look through the references Chris listed too. I have seen the first one by Marshoon before, some of the others are new to me. If they help answer the original question, it's unfortunately not apparent - though they are interesting.

We are actually getting back to that old question, the relativistic spinning disk or hoop. I still have to look up that article by Gron. The good news is that I'll have more time to do that, but it still involves a fairly long drive to the "right" library.

 

[pervect]

OK, well, the textbooks by Carroll or D'Inverno or Schutz might be something to put on your reading list for the summer holiday then :-/

I would suggest before even reading those to read or reread say, chapter 13 in Griffiths, i.e. a treatment of electrodynamics in relativity, including the Faraday tensor.

An alternative to Griffiths (not for the faint of heart) would be Jackson, "Classical Electrodynamics". One might even try both if they are both available.

At that point, one will be armed with both a better understanding of electromagnetism and how it relates to relativity plus an introduction to tensors.

This way one will have one's first introduction to tensors in a familiar context, that of E&M.

 

[pervect]

I think I see a way to provide some physical insight into some of the "negative field energy" results discussed in Marshoon.

If we look at http://www.arxiv.org/PS_cache/gr-qc/pdf/0011/0011014v1.pdf

eq 20

we see that using the LL pseudotensor approach, we have a field energy of order GM^2/r^4 per unit volume. (Different approaches still have different bookkeeping, I didn't see an expression for the energy density, t^00 for the GEM approach).

If we integrate this field energy over a volume element dr (r dtheta) (r sin theta dphi) we get a term of order GM^2/r

This is of the order of the gravitational binding energy of the system! See for instance the wikipedia article for the gravitational binding energy of a sphere http://en.wikipedia.org/wiki/Gravitational_binding_energy

This gives an idea of the order of magnitude of the possible effects. At issue is the "location" of the gravitational binding energy. This negative contribution to the energy of the system is being distributed differently (depending on one's method of bookeeping) and some of it is being assigned to empty space in this particular "set of books".

Physically, we can see that assigning a location to this negative energy is necessary to calculate the effect on the moment of inertia.

We can also see that the effect won't matter if we have a small relativistic rotating disk - we would need a disk big enough to have significant gravitational self-binding energy before it would be important.

 

[Chris Hillman]

Well, I know what I meant, but I concede that's not much help. As you know, this area of discussion presents extra-math/sci difficulties in PF sufficiently serious to preclude further participation by myself.