|Jan29-13, 06:01 PM||#18|
Point masses versus infintesimal densities, is there a difference?
I'm going to go against what I said in my last post. In SR, rest mass is not additive in the discrete or continuous formalisms. In the discrete case, if we have several point masses, (with generally different velocities), then their rest mass is definitely not additive. And in the continuous case, if we are integrating over the density of some fluid, then over some region, the fluid may be at rest, so we are calculating its rest mass in this region. But generally, the fluid will have a non-uniform velocity field, so generally we will be calculating the relativistic mass, not the rest mass.
And the thing I said about vanishingly small difference in velocities: I don't think I was quite right. If we consider a continuous distribution, and imagine making its volume very small, then different parts of it can still have different velocities. So any angular motion of that tiny distribution will be very small, but as (I think) Studiot was saying, it will still be some non-zero value, which therefore means there is an associated degree of freedom.
It is only when we take some special limit, where we assume that the relative velocities literally go to zero, that the angular motion disappears. And this is the discrete (i.e. point-like) formalism (as Studiot was saying). So it is only when we pass over to the discrete formalism, that the degrees of freedom associated with the physical angular momentum of the distribution disappear. I think that how this limit is taken in quantum mechanics has something to do with renormalisation theory. I don't know, as I haven't learned that kind of stuff. But it seems pretty interesting.
Edit: sorry to start talking about quantum mechanics at the end there, we had been talking about classical physics. quantum is a different kettle of fish, or some similar idiom.
|Jan31-13, 05:26 PM||#19|
Thanks Studiot, I'm now convinced that infintesimal mass densities are not physically/metaphysically identical to point masses. Perhaps the best way to see this is as you say: point masses have six degrees of freedom where as mass densities, even when they are infintesimal, have three extra degrees of rotational freedom.
That seems right BruceW. The non-additivity of mass in SR should not matter to the physical differences or otherwise of point masses and infintesimal mass densities.
Incidently, I have some related questions regarding some mathematical differences/similarities between point masses and infintesimal mass densities. But since they are changing the topic slightly from the above I decided to create a new post. If you're interested...
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