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

[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 :-)

 

[mpolo]

Perhaps it would have helped more if the spinning disc experiment was done in deep space away from any other mass or gravitational field. Now this should eliminate any thing else affecting the spinning disc. In that case would the spinning disk's gravitational field increase. Would it emit a stronger gravitational field as its spin approaches the speed of light? If anything else this is teaching me to be more precise in how to ask a question.

 

[PAllen]

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

In GR, mass is anything but well defined in the nonlinear regime. Further, the measurement of energy for the same apparatus is affected by gravity (this is the basis of Komar mass). Thus, when you enter the regime where the box becomes a significant source of gravity, and try to compare two different scenarios where you have added the same energy as measured by a circuit outside, you have the problem that the measurement made by the circuit outside is no longer independent of what is going on inside the box.

 

[PAllen]

Perhaps it would have helped more if the spinning disc experiment was done in deep space away from any other mass or gravitational field. Now this should eliminate any thing else affecting the spinning disc. In that case would the spinning disk's gravitational field increase. Would it emit a stronger gravitational field as its spin approaches the speed of light? If anything else this is teaching me to be more precise in how to ask a question.

You can probe the speed of the rim approaching the speed of light without reaching the domain where mass and weight become complicated. If your hyper rigid disc weighs 1 gram, you can reach the regime of the rim being near c without the weight/mass of the disc getting beyond a few grams. At this point (accepting the idealization that such a disc can hold together), its mass would be increase over its starting point by just E/c², with E being the energy added (in this case, e.g. the yield of a couple of Hiroshima nuclear bombs). At this point, there would be no difference in mass or weight of the box whether the inside were a supercapacitor or a disc being spun up by a supermotor. You would need many orders of magnitude of energy beyond this to have my earlier caveat come into play.

 

[mpolo]

I being confused by the reference to a box. I am not using a box or an apparatus of any kind. There is just a disc and nothing else. It is magically spinning on its own near the speed of light. No need to worry about this disk flying apart. Its made of some super rigid material. There is nothing to be considered but the spinning disk. As it accelerates toward the speed of light does it gain in mass? Yes or No. That's all I want to know.

 

[DrStupid]

As it accelerates toward the speed of light does it gain in mass? Yes or No.

Yes, of course it does. But the question is how much exactly.

 

[PAllen]

I being confused by the reference to a box. I am not using a box or an apparatus of any kind. There is just a disc and nothing else. It is magically spinning on its own near the speed of light. No need to worry about this disk flying apart. Its made of some super rigid material. There is nothing to be considered but the spinning disk. As it accelerates toward the speed of light does it gain in mass? Yes or No. That's all I want to know.

ok, I was responding to Nugatory's variation of your scenario.

You question is actually very basic, and I thought well answered earlier.The mass of the disc increases with any amount of spin up. With the rim first reaching near c, you would be talking about e.g. doubling or tripling the mass compared to the stationary disc. If one is talking about a disc whose starting mass is grams or kilograms, then all caveats are irrelevant - its weight increase would be the same as its mass increase, with weight being defined by suspending it from a spring balance via a magic thread to its center.

Where Nugatory's suggestion is helpful is sidestepping computation of the mass increase in terms of dynamics of the disc. Just measure energy put into the disc. Then, well into near light speed rim speed, all you need do is take the energy input over c² to predict the increase in weight or mass.

[edit: as to why it is very helpful to use Nugatory's suggestion: modeling spinning up a disk to near c rim speed is extremely complex even in SR due to the fact that maintaining rigidity while spinning up a disk is impossible not just in an engineering sense, but in the sense the FTL motion is impossible. You can google the Ehrenfest paradox for more on this. Thus, you would have to allow the disc to behave like a deformable fluid, settling down to a disc at a final spin rate.]

 

[mpolo]

Yes, excellent. I agree determining how much it does is very important.

 

[Nugatory]

I being confused by the reference to a box. I am not using a box or an apparatus of any kind. There is just a disc and nothing else. It is magically spinning on its own near the speed of light. No need to worry about this disk flying apart. Its made of some super rigid material. There is nothing to be considered but the spinning disk. As it accelerates toward the speed of light does it gain in mass? Yes or No. That's all I want to know.

Who can say? The laws of physics don't work well in situations like the one you're describing, in which a disk made of some "super-rigid" (which is to say, physically impossible) material is spinning "magically" (which is to say, in violation of the laws of physics). You might as well be asking a mathematician to answer a question about the factors of a prime number, or the long side of a circle.

The point of the box with the two electrical terminals is that it allows us to define and measure the way the weight of whatever is in the box (in this case, a spinning disk because that's you're interested in) changes as the energy increases. That gives us both a question with a clear answer: yes, the weight increases (as we said back in post #6 of this thread); and tells us by how much: $E/c^2$, from $E=mc^2$ and a bit of algebra. It also makes it clear that the details of the non-magical method by which the disk spins up and its non-magical construction do not affect the basic physics; those moment-of-inertia formulas are an unnecessary distraction here.

 

[PAllen]

I computed something a bit surprising to me. Under the simplification of uniform density, integrating relativistic total energy over the disc, leads to the fact that the maximum amount you can increase mass over the rest state is to double the rest mass. This occurs in the limit as rim velocity approaches c.

This means the most energy you can store in 1 gram spinning disc of unobtainium is the approximate yield of the Nagasaki nuclear bomb.

[edit: I should clarify that the constant assumed density is invariant mass of a volume element divided by volume in COM frame of the disc. This means the mass and volume are measured in different frames. Since nothing about discs with near light speed rims is realistic, I use this density definition in the interests of computational tractability.]

 

[PAllen]

I computed something a bit surprising to me. Under the simplification of uniform density, integrating relativistic total energy over the disc, leads to the fact that the maximum amount you can increase mass over the rest state is to double the rest mass. This occurs in the limit as rim velocity approaches c.

This means the most energy you can store in 1 gram spinning disc of unobtainium is the approximate yield of the Nagasaki nuclear bomb.

A sample result of the fully relativistic formula for this ideal case is that a rim speed of 90% light speed produces only a 39% mass increase. A rim speed of 10% c produces only .25% mass increase.

These results also mean that you never ever have to worry about GR ambiguities in mass or weight for an ordinary rest mass disc, even up to rim speed approacing c.

 

[timmdeeg]

A sample result of the fully relativistic formula for this ideal case is that a rim speed of 90% light speed produces only a 39% mass increase.

Would the energy loss due to radiation of gravitational waves be negligible here?

 

[PAllen]

Would the energy loss due to radiation of gravitational waves be negligible here?

Once equilibrium is reached - constant angular speed - there would be no GW. So just repose the problem as doing whatever is necessary to end with a desired final state. Also, for starting mass of e.g. a gram or kilogram, GW would be utterly negligible - GW power goes as the fifth power of ( mass / c).

 

[PAllen]

So, the formula I get in a form nice for the OP, is comparing two discs of the same density (defined, as noted, as invariant mass per volume element measured in the COM frame), constant throughout the disc, with one stationary and the other spinning, then the mass increase factor ( 1 means 100% increase or doubling of mass) is given by:

(2 (1 - √ (1 - β²))/β²) - 1

where β is rim speed / c. The limit as β goes to 1 is 1, and of course is zero as β goes to 0.

Up to β = .1, a good approximation is simply β²/4, which is derived using the first two terms of Taylor expansion for square root (you get zero if you only use one term of the Taylor expansion).

Another useful figure is you would need a rim speed of about 20% c to get a mass increase of 1%.