Rapidly Spinning Mass Effects on GR & SR Gravitational Field

[alpha_wolf]

Does a rappidly spinning mass experience an effect in GR that is equivalent to mass dilation in SR? I so, would it have a stronger gravitational field than at rest?

 

[turin]

Does a rappidly spinning mass experience an effect in GR that is equivalent to mass dilation in SR?

No. There is a GR effect, but it is not equivalent to the SR effect. The GR effect is to induce cross terms in the gravitational force. SR is just a special case of GR; it does not account for graviational effects.

... would it have a stronger gravitational field than at rest?

I believe so. But also a different gravitational field (i.e. gravetomagnetic).

 

[rtharbaugh1]

I have recently been interested in this also, due to reading about Gravity Probe B and the tests for frame dragging. I would rewrite the question as follows: Does a rapidly spinning mass have greater gravitational attraction than an otherwise identical mass that is not spinning?

I imagine an experiment as follows: a pair of large massive disks in free space (no extraneous mass or charge fields) is set to rotate in the plane of the disks about an axis through the centers of mass, rather like spinning a pair of wheels on an axle. Does the attraction measured as a compressive force on the axis between the two disks vary with the rate of rotation of the disks? Does it make a difference if the disks are rotated in the same or in opposite directions?

It seems to me that if the disks are rotated they should show some attractive force along the axis. If frame dragging is measurable, there should be a difference between disks rotated in the same direction vesus disks rotated in opposite directions.

Perhaps someone more knowledgeable about these things will have better questions. Perhaps Gravity Probe B will have some answers.

THanks,

 

[alpha_wolf]

No. There is a GR effect, but it is not equivalent to the SR effect. The GR effect is to induce cross terms in the gravitational force. SR is just a special case of GR; it does not account for graviational effects.

I sould have written "similar", not "equivalent".. What I meant was an increase in mass or something like that due to angular velocity, similar to how mass increases due to linear velocity in SR.

I believe so. But also a different gravitational field (i.e. gravetomagnetic).

Hmm.. is there some simple formula that would relate the generated gravitaional field (in terms of force strengh - I don't care much about the resulting spacetime geometry..) to the rest mass, radius, and angular velocity of the body, assumming it some simple shape like an infnitely thin ring? Or is this one of those devilish calculations that leave one's lower jaw at approximately ground level?
Also, could you expand a bit on the cross terms in the gravitational field and the gravetomagnetic effect? I'm afraid I'm not very familiar with GR terminology...

rtharbaugh1:
Yes, that would be an interesting experiment indeed. :) But I imagine you'd have to spin the disks rather fast..

 

[turin]

What I meant was an increase in mass or something like that due to angular velocity, similar to how mass increases due to linear velocity in SR.

The SR effect is a GR effect, since GR contains SR. I thought that you were asking for something that GR would predict that SR would not. There is a relativistic mass increase that SR predicts for a spinning mass without appealing to GR. GR predicts an added magnetic-ype effect.

.. is there some simple formula that would relate the generated gravitaional field ... the rest mass, radius, and angular velocity of the body, assumming it some simple shape like an infnitely thin ring?

Assuming an "ordinary" situation like the Earth rotating, you use the weak field approximation. I haven't done the calculation, but I'm sure you can find it all over the place in textbooks and papers. Look for PN1 (1st order post Newtonian approximation). As an example, Einstein gives a brief treatment in The Meaning of Relativity that shows the emergence of a vector type potential.

...could you expand a bit on the cross terms in the gravitational field and the gravetomagnetic effect? I'm afraid I'm not very familiar with GR terminology...

This termonolgy isn't really GR incantation. If you know what magnetism is, then just take the straightforward analogy that

electric:magnetic::gravitational:gravetomagnetic

A word of caution though. The analogy is only an approximate analogy. It arrises from some low order approxiation of Einstein's equation. You can think of the cross terms as products of unlike vector components.

 

[alpha_wolf]

The SR effect is a GR effect, since GR contains SR. I thought that you were asking for something that GR would predict that SR would not. There is a relativistic mass increase that SR predicts for a spinning mass without appealing to GR. GR predicts an added magnetic-ype effect.

Oh.. I didn't know SR handles rotating bodies..

This termonolgy isn't really GR incantation. If you know what magnetism is, then just take the straightforward analogy that

electric:magnetic::gravitational:gravetomagnetic

A word of caution though. The analogy is only an approximate analogy. It arrises from some low order approxiation of Einstein's equation.

I understand the verbal meaning, but I don't quite get the physical meaning of the term.. how does this effect work?

You can think of the cross terms as products of unlike vector components.

You mean something like a vector cross product?

 

[turin]

Oh.. I didn't know SR handles rotating bodies...

Well, in some contexts SR has to be finagled quite a bit (a rotating frame of reference being one of them) in order to apply, so this can be misleading. It is usually not a good idea to rely on SR to consider non-inertial reference frames. If you are willing to suspend the curvature of space-time (i.e. in the absense of "modern" gravitation), then you can get away with treating a rotating body in SR by considering it from an inertial frame (i.e. the frame against which the body rotates). You actually get quite an interesting result: even using SR, you can realize that the 3-D geometry in the rest frame of the body is non-Euclidean (i.e. curved). To be more specific, the SR treatment results in the geometrical ratio of C/D > π in the rest frame of a rotating body (I think it's greater than).

I understand the verbal meaning, but I don't quite get the physical meaning of the term.. how does this effect work?

Do you understand how magnetism arrises from electrostatics? Around the beginning of the 20th century, the mathematical formulation of physics acquired a powerful criterion: covariance. In order for the low order (but not the lowest order) gravitational field equations to maintain their covariant nature, magnetic-type manifestations of the gravitational interaction must exist. This is quite similar to the reason why Maxwell's equations unite the electric and magnetic fields, and this similarity is why most people refer to the generalization as gravetomagnetism.

You mean something like a vector cross product?

Yes.

 

[alpha_wolf]

Do you understand how magnetism arrises from electrostatics? Around the beginning of the 20th century, the mathematical formulation of physics acquired a powerful criterion: covariance. In order for the low order (but not the lowest order) gravitational field equations to maintain their covariant nature, magnetic-type manifestations of the gravitational interaction must exist. This is quite similar to the reason why Maxwell's equations unite the electric and magnetic fields, and this similarity is why most people refer to the generalization as gravetomagnetism.

Ah, ok. I.e. gravitomagnetism relates to gravity like magnetism relates to electrostatics, and not that gravity somehow creates a megnetic force of sorts... Hmm, yea, the latter would mean that the grand unification theory has already been found, at least partially.

Ok, thanks!

 

[turin]

Beware

... not that gravity somehow creates a megnetic force of sorts...

Beware, some may try to obscure the meaning to just that.

Actually, I am not possitive that gravity does not create a literal magnetic field, but that is not what I was talking about, nor have I seen any theoretical or experimental evidence for it. In other words, you may have understood what I was saying, but what I have said I hold precipitately.

 

[pmb_phy]

Does a rappidly spinning mass experience an effect in GR that is equivalent to mass dilation in SR? I so, would it have a stronger gravitational field than at rest?

The faster a body spins the greater its gravitational field. This follows from the fact that the components of the metric are the gravitational potentials and the components of the metric are a function of the angular momentum of the body. How this compares to mass increase is another story. It'd be interesting to find the weight of a body which is at rest outside a rotating black hole and determine the weight as a function of angular momentum. It wouldn't surprise me if it did since relativistic mass plays a role in general relativity which is similar to the role that charge plays in electrodynamics. In fact in the weak field limit the equations can be placed in the same form. The active gravitational mass of a body plays the role of the charge density of a source of an EM field. So the faster a body rotates the faster the particles which make up the body move. The faster the particles move the greater the active gravitational mass. All this goes under the title "Gravitomagnetism" as others have mentioned above.

Pete

 

[Chris Hillman]

If we spun up the Earth, would we weigh more?

Hi, alpha_wolf, you asked a good question back in May 2004:

Does a rapidly spinning mass experience an effect in GR that is equivalent to mass dilation in SR? I so, would it have a stronger gravitational field than at rest?

No. There is a GR effect, but it is not equivalent to the SR effect. The GR effect is to induce cross terms in the gravitational force. SR is just a special case of GR; it does not account for graviational effects.

Actually, the best short answer is "yes", as I just said in another post in which I pointed out the alpha's question is analgous to the question of whether when we heat up an object, its effective gravitational mass increases.

I would rewrite the question as follows: Does a rapidly spinning mass have greater gravitational attraction than an otherwise identical mass that is not spinning?

Essentially, yes, although it can be quite difficult to speak of "otherwise identical objects" in two different spacetime models, or even at different times in the same spacetime model. For example, a radial coordinate may not have the same geometric meaning at all times.

I imagine an experiment as follows: a pair of large massive disks in free space (no extraneous mass or charge fields) is set to rotate in the plane of the disks about an axis through the centers of mass, rather like spinning a pair of wheels on an axle. Does the attraction measured as a compressive force on the axis between the two disks vary with the rate of rotation of the disks? Does it make a difference if the disks are rotated in the same or in opposite directions?

Interestingly enough, this raises a new point, the prediction of spin-spin forces in gtr. I just discussed the so-called "double Kerr solution" (two coaxial Kerr objects, with the same mass and with opposite but equal rotational angular momentum) in another post today (27 Nov 2006). It does make a difference whether they are spinning with the same sense or with opposite sense. See for example (9.35) in the textbook by Hans Stephani, General Relativity for more details.

It seems to me that if the disks are rotated they should show some attractive force along the axis.

You are right: if they are spinning in the same sense, their spins attract; if they are spinning in the opposite sense, they repel (although not enough to completely overcome the attraction due to their positive masses).

It is worth recalling here that in gtr, the source of the field is the full matter tensor, including momentum and stress as well as mass. The effect of a possibly complicated distribution of mass in an isolated object on its (asymptotically flat) gravitational field is described by the "mass multipoles", while the effect of a possibly complicated distribution of momentum is described by the "current multipoles". (Stephani's textbook offers an excellent treatment.)

Roughly speaking, the dominant gravitational interactions between objects is due to the interaction of their "mass monopoles", but the low order current multipoles can also be important.

Chris Hillman