Can a Spinning Object Increase its Mass through Acceleration?

[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%.