PAllen's latest activity

It seems obvious to me that the finite particle approach can be made to exceed any energy bound, based on extrapolation from my post...

And one more thing… the issue of transitioning to a continuum is a problem for the radial direction, but not the tangential. So the...

I think there is no nice way to allow a continuum model to capture the notion that there is no upper bound to particle energy before c...

One further way of looking at this is that transitioning from a ring of point particles to a smooth annulus adds width to the particles...

Also note that the constraint of constant density per a local, comoving observer, means adding particles to outer rings as they are sped...

I think I have the key notion for the finite to continuum transition that explains the counterintuitive continuum result. The uniform...

But the uniform COM frame density follows necessarily from any simple (i.e. not including stress energy of holding the disc together)...

I don’t think the improper integral is the issue at all. In some sense, you never need to get there. Intuition suggests that any chosen...

Note that if we change the mass assumption to be constant density in each element's local comoving frame, we get for an arbitrary...

To decouple an annulus from any hypothetical containing disc, we introduce ##\rho## which is simply ##M/\pi r^2## in post #1 convention...

This is a nice simplification for further discussion, that changes the main questions not at all. This would be computing the total...

If I am picking, for example, a finite set of approximating elements, I can certainly have the angular span of each be inversely...

I think this can be dispensed with. Just keep ##r d\phi## constant along a radial path. Also, @Ibix integral in the OP already keeps...

This would be the same as long as you have uniformly distributed rest mass per the COM inertial frame. To get any other result you have...

It’s possibly important to understand what M means. The construction of the integral means it summed from constituent rest masses.