Can a Spinning Object Increase its Mass through Acceleration?

[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.

 

[PeterDonis]

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

You said we shouldn't overgeneralize the answer; what counts as "overgeneralizing" depends on how accurate your measurements are, which determines how small the effects are that you can detect. I gave a practical example of that. The details will of course differ for different specific cases.

 

[Nugatory]

The discussion in this thread has moved well beyond the B level of the original question. This is not a bad thing - communicating to the original poster that there's more going on here than just the simple answer is part of a good answer.

However, it might be possible to word the original question more precisely, so as to be able to give it a simple answer. Looking for the question implied by #5 of this thread ("has anyone tried weighing a spinning disk?"):

I am at sitting on the surface of a non-rotating earth, a situation which we will idealize as maintaining constant $r$, $\theta$, and $\phi$ coordinates in Schwarzschild spacetime. In front of me is a spring scale, and on the spring scale is a black box. The black box has two electrical terminals on the outside. I apply a potential difference (hook up a standard automotive battery charger?) to these terminals and find that there is a current flow. So I am doing work on whatever is in the box, and I can calculate the amount from the voltage drop and current flow. The reading on the spring scale will increase. The increase, for a given amount of work, can be calculated from $E=mc^2$ if the mechanism inside the box is a resistor dissipating heat. Will the increase be different if instead the mechanism is an electric motor spinning up a flywheel?

 

[PAllen]

The discussion in this thread has moved well beyond the B level of the original question. This is not a bad thing - communicating to the original poster that there's more going on here than just the simple answer is part of a good answer.

However, it might be possible to word the original question more precisely, so as to be able to give it a simple answer. Looking for the question implied by #5 of this thread ("has anyone tried weighing a spinning disk?"):

I am at sitting on the surface of a non-rotating earth, a situation which we will idealize as maintaining constant $r$, $\theta$, and $\phi$ coordinates in Schwarzschild spacetime. In front of me is a spring scale, and on the spring scale is a black box. The black box has two electrical terminals on the outside. I apply a potential difference (hook up a standard automotive battery charger?) to these terminals and find that there is a current flow. So I am doing work on whatever is in the box, and I can calculate the amount from the voltage drop and current flow. The reading on the spring scale will increase. The increase, for a given amount of work, can be calculated from $E=mc^2$ if the mechanism inside the box is a resistor dissipating heat. Will the increase be different if instead the mechanism is an electric motor spinning up a flywheel?

And the answer is there will be no difference, even if the flywheel is somehow able to reach relativistic speeds (e.g. by the flywheel's being super light and super rigid). This is true up to the inordinate precision needed to detect different couplings to tidal gravity over the size of said box. In the limit of infinite planetary radius at surface maintained at standard g, there will be no difference whatsoever, even at infinite precision.

[edit: and now for the inevitable caveat. In comparing different scenarios for what is in the box, in each case measuring the energy contributed in a standard way. locally, outside the box, suppose the energy added is such that the box becomes a significant source of tidal gravity. Now the scale can couple to this and be sensitive to different arrangements of stress/energy inside the box producing slightly different tidal gravity outside the box. Thus, if energy added is e.g. the energy equivalent of an asteroid, then the scale will give slightly different readings depending on how the energy is stored withing the box. To avoid this issue in a limiting way, you have to take limit of smaller and smaller boxes, but there is a lower limit on this due to BH formation. As compared to enlarging the planetary mass, and lowering density, so as to maintain a 1-g surface, which has no in principle limitations. ]

 

[mpolo]

I have been monitoring and enjoying the conversation. I am going to assume that the equation supplied earlier is still the best equation to explain the spinning disk problem. All this discussion has started me wondering. Would the initial equation above also work for a spinning sphere? I am guessing probably not. What would that equation look like? I am very interested in seeing how the structure of the equation changes in response to the change in the physical geometric structure of the object that is spinning.

 

[Nugatory]

IWould the initial equation above also work for a spinning sphere? I am guessing probably not. What would that equation look like?

If we're defining the weight of the object as I did above then the $E=mc^2$ calculation will work for a spinning sphere just as it does for a spinning disk (with $E$ being the amount of energy added to the box, calculated from the voltage and current flow). The details of what is going inside the box may be different (for example, in general the RPM of the sphere will be different for the same amount of added energy than the RPM of the disk) but the weight increase will be the same.

The caveat @PAllen provided above is important. We have to assume that the density of the planet we're standing on is sufficiently low and the radius is sufficiently large. If we don't, we no longer have a well-defined notion of weight and your original question ends up stuck somewhere between meaningless and hopelessly ambiguous (which is how a seemingly straightforward B-level question can generate four pages of posts by experts in the field). However, we can always find values for the radius and density that allow for an unambiguous definition of "weight".

 

[mpolo]

Would the question have been less ambiguous if I used the word mass instead of weight?

You have given the general form of the equation but I am interested in what Dr Stupid did and that was to modify the equation to account for the spinning disk geometry. What modification would need to be made to properly describe a spinning sphere. That equation he did was great. What do you call that when somebody modifies an equation like that? It is a very exciting thing to behold.

 

[DrStupid]

Would the question have been less ambiguous if I used the word mass was used instead of weight?

Definitely!

What modification would need to be made to properly describe a spinning sphere.

Here you can find the moments of inertia for different geometries:

https://en.wikipedia.org/wiki/List_of_moments_of_inertia

 

[PAllen]

Would the question have been less ambiguous if I used the word mass was used instead of weight?

No, it would not really help for the extreme situation of the caveat. This is because the question of the local, outside the box, energy measurement starts to be affected by gravitational influence of the box, so the question of mass itself becomes non-trivial. We can't exactly abstract from what is going on inside the box, even for determining how much energy is going in. Note, we are talking about adding an amount of energy more than the yield of the world's nuclear arsenal to make the box be a substantial source of tidal gravity, while its density is becoming like solar core material. This is really, really, absurd in practice.

 

[DrStupid]

No, it would not really help for the extreme situation of the caveat.

Mass is well defined in contrast to weigt. If we know total energy and momentum of the system than we also know its mass.

 

[jbriggs444]

Mass is well defined in contrast to weigt. If we know total energy and momentum of the system than we also know its mass.

But total energy is ill defined in GR.

Edit: stepping in deep waters where my feet do not touch bottom :-)