High School Can a Spinning Object Increase its Mass through Acceleration?

[PAllen]

That is quite surprising. To my knowledge trajectories around a BH depend on its angular momentum. In order to make this happen without affecting the trajectory of the BH itself, the differences in energy and momentum would need to be emitted as gravitational waves. Is that the case?

Another point is the relativistic velovity dependence of the gravitational force. The gravitational mass is different from rest mass for relativistic velocities. In "Measuring the active gravitational mass of a moving object" [American Journal of Physics 53, 661 (1985)] DW Olson and RC Guarino derived a factor of $\gamma \cdot \left( {1 + \beta ^2 } \right)$ for hyperbolic trajectories (just to give an example). Keeping the same weight for rotating and non-rotating bodies would require that this effect cancels always out over all parts of a rotating body (at least with center of mass at rest) - independend from geometry and angular velocity. Is that the case?

This raises a number of question I have given some thought to.

To understand these issues, we need to separate notions:

1) A body's mass (within domain of GR where one can talk of a body with adequately defined mass), however it arises (rotating or not, to any degree). We don't care how internal components may add non-linearly to constitute mass, just whether we can speak of a reasonably isolated body whose mass can be measured. Then we can define effective passive gravitational mass by how it responds to the background metric of a much more massive body (of any nature, rotating or not). This same body's effects on much tinier test particles, far away, can be used to define an active gravitational mass. Far away, because we want the net influence of the body, not issues of energy/pressure distribution within it producing complex near field. One may also similarly define inertial mass.

2) Velocity dependent interaction (and even acceleration dependent interactions). The latter are discussed in Steve Carlip's well known paper "Aberration and the Speed of Gravity", where in he shows that (analyzed from a force point of view), a moving body generates both velocity and acceleration dependent gravitational forces, this being how the 'apparent speed' of gravity may appear to be near infinite. That a moving charge produces velocity dependent force on another test charge in no way is said to change the charge of the moving charge. Similarly, that a moving body produces velocity and acceleration dependent gravitational interactions with a test body doesn't change what the effective mass of the moving body is - it is the mass as in (1) measured in a quasilocal rest frame of the body. Guarino is discussing the velocity dependent effect.

3) Then, the question of how one accumulates component information to explain empirical mass (1), is a separate question. In the case of pressure-less, non interacting, uncharged, dust body, with insignificant self gravitation, then even if internal motions are relativistic, the invariant mass (= sum of particle total energies in the center of momentum frame) will be equal to the mass in (1). Beyond this, you would need to use either a quasilocal mass integrating the stress energy tensor in a nontrivial way (for example, Bartnik mass), or, more commonly, treat the body as embedded in isolation in asymptotically flat spacetime and compute its ADM mass. Another option is komar mass if the body may be treated as producing a stationary metric, which is true for a uniformly spinning disk in equilibrium. Generally, whenever several of these are valid, they come out the same.

A further issue is that the question of orbits near a Kerr BH is probing near field of an extended body. Nothing about the equivalence principle says probing the near field of different same mass bodies with different SET distributions will fail to detect differences. While a BH is all vaccuum, it may be considered the remnant of different collapses for different BH parameters (angular momentum, for our purposes), and its having different near fields is irrelevant to the EP.

The EP would come into play when we ask, even for the near field of an extreme Kerr BH, if the motion of a tiny (very low mass) non-rotating BH would be different from the motion of a tiny Kerr BH. Per numerous claims (see for example, Clifford Will's living review article on testing GR), GR is the unique theory that predicts no violations of even the strong equivelence principle, ever. Experiment so far is consistent with this, but coverage of extreme regimes is very limited. Note that a charged body is not considered valid for the EP because it is never isolated from its own field. However, collections of balanced extreme charges, with extreme motion, such there is no external field, are perfectly ok for the EP (this, of course, is ordinary matter).

Thus, I return to may claim that there is no sense in which the weight (passive gravitational mas) and mass (inertial or active gravitational) will differ, when they are definable at all. As for computing the mass of a spinning body, integrating relativistic energy in COM frame will only be valid when pressure and stress (in natural units) and self gravity are all insignificant. But that is a separate issue from weight and mass being different.

 

[PeterDonis]

To my knowledge trajectories around a BH depend on its angular momentum. In order to make this happen without affecting the trajectory of the BH itself

The trajectories in question are trajectories of test bodies, which by definition contribute no mass, angular momentum, stress-energy, etc. to the system as a whole. Obviously this is an idealization, but the point is that the EP, strictly speaking, only applies to test bodies in this idealized limit. So gravitational spin-spin coupling between two bodies, for example, does not violate the EP, because in order for it to be present at all both bodies must have non-negligible mass and angular momentum, hence neither one is a test body.

In practical terms, a body can be treated as a test body to a good approximation if its mass, angular momentum, stress-energy, etc. are sufficiently small compared to those of the BH (or whatever other central body is producing the overall spacetime geometry that determines the test body trajectories). For example, this approximation works very well for computing orbits in the solar system, even for planets which themselves obviously have non-negligible mass, angular momentum, etc. But it's still an approximation.

the differences in energy and momentum would need to be emitted as gravitational waves. Is that the case?

GR certainly predicts that this is the case if we drop the idealization of one body being a test body. For example, GR calculations of GW emissions from binary pulsar systems match observations very well. Pulsars spin rapidly, so the GR calculations would in principle include effects of spin-spin coupling between them, and any resulting effects on GW emission and changes in orbital parameters. (I suspect that these effects are very small compared to the primary source of GW emission, which is simply the orbital motion of the pulsars, but I have not looked up the calculations to see.) But in this case neither body is being treated as a test body.

The gravitational mass is different from rest mass for relativistic velocities. In "Measuring the active gravitational mass of a moving object" [American Journal of Physics 53, 661 (1985)] DW Olson and RC Guarino derived a factor of $\gamma \cdot \left( {1 + \beta ^2 } \right)$ for hyperbolic trajectories (just to give an example).

These calculations all involve the entire orbit, not just a small patch of spacetime, so they are irrelevant to the EP. The EP only applies within a small patch of spacetime. In such a patch, the velocity dependent effects you mention are not observable. They are only observable when you look at the entire trajectory, which cannot be covered by a single small patch.

For this reason, I think the language "gravitational mass different from rest mass" is misleading, although unfortunately it does appear to be commonly used in the literature.

 

[DrStupid]

weight (passive gravitational mas)

It seems we are talking cross purposis. I am talking about this: https://en.wikipedia.org/wiki/Weight

 

[PeterDonis]

I am talking about this

In a GR context, this definition is already problematic, because in GR gravity is not a force. Even the operational definition given in that article is technically incomplete, since it doesn't specify the state of motion of the object being weighed relative to the source of gravity.

The best quick definition I can come up with would be something like: the force that must be applied to an object to make it follow a worldline that is an integral curve of the timelike Killing vector field of the spacetime.

 

[DrStupid]

The trajectories in question are trajectories of test bodies, which by definition contribute no mass, angular momentum, stress-energy, etc. to the system as a whole.

This is not very helpful because the weight in the gravitational fiel of a massless body is always zero.

These calculations all involve the entire orbit, not just a small patch of spacetime, so they are irrelevant to the EP.

But they are relevant fort the weight. In a locally free falling frame of reference, there is no weight at all. That's the basis for GR.

 

[PeterDonis]

they are relevant fort the weight

If by that you mean that the force required to hold a body on its trajectory depends on the trajectory, of course that is true. But calling that force a "weight" might not be justified for all possible trajectories, at least not if you are trying to connect the "weight" with some property of the source of gravity. That's why I suggested a definition of "weight" that involves the timelike Killing vector field of the spacetime.

 

[PAllen]

It seems we are talking cross purposis. I am talking about this: https://en.wikipedia.org/wiki/Weight

So am I. If weight for a spinning body (to the extent that it can be treated as body in a background field) differed from inertial mass, you would have violation of WEP, which does not exist in GR.

The only aspect of your points I am questioning is the claim that adding energy to a body in the form of spin can possibly produce a result where the inertial mass is different from the gravitational mass; as long as that is not so, the weight will be the same as the inertial mass in appropriate units.

 

[DrStupid]

In a GR context, this definition is already problematic, because in GR gravity is not a force.

Yes, of course. That's why it is problematic to make claims about weight in general relativity. It's a classical concept and needs to be translated in some way - e.g. by considering space time around a source of gravity as flat and than calculating the force acting on a body in an orbit or on a hyperbolic trajectory. The result will always depend on the previous defined translation.

 

[DrStupid]

If weight for a spinning body differed from inertial mass, you would have violation of WEP, which does not exist in GR.

How do you want to test this violation? You can't have the same initial conditions for the spinning and non-spinning body. All parts of the non-spinning body are moving with the same initial velocity, but not the parts of the spinning body.

 

[PAllen]

How do you want to test this violation? You can't have the same initial conditions for the spinning and non-spinning body. All parts of the non-spinning body are moving with the same initial velocity, but not the parts of the spinning body.

I already described a way to do this for the most extreme case. Compare the predicted trajectory of a small nonrotating BH with Kerr BH, in a given background metric. In the case ordinary bodies, compare free fall of non-spinning gyroscope with spinning gyroscope. Check if their COM has the same trajectory. If the trajectories are the same, then weight is proportional to inertial mass, with the same porportionality.

 

[PeterDonis]

Compare the predicted trajectory of a small nonrotating BH with Kerr BH, in a given background metric.

This would be problematic because you can't consider a BH to be a test body; a test body can't have any effect on the spacetime geometry, but a BH does. Even a BH of very small mass has a horizon and a singularity inside.

In the case ordinary bodies, compare free fall of non-spinning gyroscope with spinning gyroscope. Check if their COM has the same trajectory.

This is the comparison that I would focus on.

 

[DrStupid]

I already described a way to do this for the most extreme case. Compare the predicted trajectory of a small nonrotating BH with Kerr BH, in a given background metric. In the case ordinary bodies, compare free fall of non-spinning gyroscope with spinning gyroscope. Check if their COM has the same trajectory.

And how does that help you? Different trajectories would'd violate the WEP because the initial conditions are different.

 

[PeterDonis]

This is the comparison that I would focus on.

On consideration, though, there is a caveat to this: I'm not sure how you model a test body in GR as having spin--more precisely, as having any angular momentum that is not orbital angular momentum, which is what a spinning gyroscope would have to have. Of course you can just say that the gyroscope has a vector or tensor attached to it that describes its spin and gets Fermi-Walker transported along its worldline, but that already assumes that the COM trajectory is not affected, since you have to know the COM trajectory in order to compute how things are Fermi-Walker transported.

 

[PAllen]

This would be problematic because you can't consider a BH to be a test body; a test body can't have any effect on the spacetime geometry, but a BH does. Even a BH of very small mass has a horizon and a singularity inside.

.

Well, being reminded of the series of small body papers by Gralla and Wald, something I got out of them was simply that if you put an envelope of influence around a body, that is small enough such that the body doesn't probe tidal gravity, then if the body is assumed to follow a timelike path, it must be a timelike geodesic of the background geometry. I don't see a part of the proof method that would exclude a sequence of ever smaller Kerr BH, maintaining constant spin parameter.

(A separate issue, addressed only by footnote in one of these papers, is what if you want to prove that a timelike trajectory must be followed direclty from the EFE; that is surprisingly delicate and independent of arguments for geodesic motion. However, Gralla-Wald side step this by simply assuming it. It is normally a noncontroversial assumption.)

 

[PAllen]

And how does that help you? Different trajectories would'd violate the WEP because the initial conditions are different.

Not true. I claim that if you posit the same COM initial motion, you get the same trajectory, irrespective of spin of a body.

 

[DrStupid]

Not true. I claim that if you posit the same COM initial motion, you get the same trajectory, irrespective of spin of a body.

For what reason?

 

[PAllen]

For what reason?

Because it would otherwise violate the EP to the extent the body's mass is (however extreme the spin) is small (compared to all other sources), and the extent of the body is small enough not to probe tidal gravity effects.

 

[DrStupid]

Because it would otherwise violate the EP to the extent the body's mass is (however extreme the spin) is small, and the extent of the body is small enough not to probe tidal gravity effects.

I do not see such a violation. Please explain.

 

[PAllen]

I do not see such a violation. Please explain.

Imagine putting a box around a super spinning gyroscope versus a non-spinning gyroscope. WEP says (with the technical caveat I explained before) they must follow the same trajectory. The internal state of a body does not influence is free fall trajectory is exactly what the WEP states.

 

[DrStupid]

Imagine putting a box around a super spinning gyroscope versus a non-spinning gyroscope. WEP says (with the technical caveat I explained before) they must follow the same trajectory.

It is not sufficient to repeat that claim. You need to prove or at least explain it sufficiently.

The internal state of a body does not influence is free fall trajectory.

Counter example: The trajectory of a bar-bell shaped body depends on it's initial orientation - even in classical mechanics. That's no violation of the WEP because it doesn't require identical trajectories of the COM for different initial states of a macroscopic object.

 

[PAllen]

It is not sufficient to repeat that claim. You need to prove or at least explain it sufficiently.
Counter example: The trajectory of a bar-bell shaped body depends on it's initial orientation - even in classical mechanics. That's no violation of the WEP because it doesn't require identical trajectories of the COM for different initial states of a macroscopic object.

The barbell case requires the barbell probe tidal gravity. If the size of the barbell is sufficiently small, or the field is sufficiently uniform over the region in space and time of the observation, there is no such effect.

 

[PeterDonis]

if you put an envelope of influence around a body, that is small enough such that the body doesn't probe tidal gravity, then if the body is assumed to follow a timelike path, it must be a timelike geodesic of the background geometry. I don't see a part of the proof method that would exclude a sequence of ever smaller Kerr BH, maintaining constant spin parameter.

I know the papers you refer to but haven't read them in a while. The key point I would want to look at is if there is any restriction on the topology of the region in the interior of the envelope of influence. Its surface is the surface of a "world tube" in the exterior geometry, which would have topology of $\mathbb{S}^2 \times \mathbb{R}$; but does the topology of the interior of that "world tube" have to be $\mathbb{R}^4$ for the proofs to work?

 

[DrStupid]

If the size of the barbell is sufficiently small, or the field is sufficiently uniform over the region in space and time of the observation, there is no such effect.

Does that mean that your claim is limited to point sized objects? That would be quite trivial and not very helpful.

 

[PAllen]

Does that mean that your claim is limited to point sized objects? That would be quite trivial and not very helpful.

The EP is always meant to cover behavior "up to when tidal gravity is detectable". How big this is depends; Jupiter in the field of the sun is fine to very high precision. But sensitivity of the experiment to tidal gravity comes into play, not just size.

 

[PAllen]

I know the papers you refer to but haven't read them in a while. The key point I would want to look at is if there is any restriction on the topology of the region in the interior of the envelope of influence. Its surface is the surface of a "world tube" in the exterior geometry, which would have topology of $\mathbb{S}^2 \times \mathbb{R}$; but does the topology of the interior of that "world tube" have to be $\mathbb{R}^4$ for the proofs to work?

Good question. Without reviewing them I cannot answer this.

 

[DrStupid]

The EP is always meant to cover behavior "up to when tidal gravity is detectable".

And therefore you can't refer to it in case of a macroscopic object like a disc. With sufficiently high precision you will always detect tidal gravity.

 

[PeterDonis]

With sufficiently high precision you will always detect tidal gravity.

But you can still ask the question: given two objects, one rotating and the other not, will there be any differences in their trajectories that are not due to the coupling of some difference in their internal states to tidal gravity? @PAllen is simply saying that the answer to that question is no.

 

[DrStupid]

But you can still ask the question: given two objects, one rotating and the other not, will there be any differences in their trajectories that are not due to the coupling of some difference in their internal states to tidal gravity?

Yes, of course you can ask this question - as long as you don't overgeneralise the answer. You must not conclude that rotating and non-rotating discs must have the same trajectory. That would require another argumentation.

 

[PeterDonis]

You must not conclude that rotating and non-rotating discs must have the same trajectory.

No, but you can conclude that, if the only differences in trajectory are due to coupling to tidal gravity, and if the effects of tidal gravity are not measurable, then any differences in trajectory are also not measurable. Which is highly relevant in a practical sense since this is exactly how we calculate the orbits of objects in the solar system: we assume they are all non-spinning test objects moving on geodesics in a background Schwarzschild geometry. And it works, to within our current accuracy of measurement.

 

[DrStupid]

And it works, to within our current accuracy of measurement.

This thread is about effects outside our current accuracy of measurement.