Yes, I heard. The sentence that you attempted to correct was "Kinetic energy is not relativistic mass".
I didn't deny that. I just said they were "closely related" since the
total energy and relativistic mass are proportional. I included the clarification "total
(kinetic + rest)" for a reason you know. Also in picking this nit you are missing the wider context I posted this "correction" in: the total energy is relevant in GR (more so than just the kinetic energy), and since the total energy and relativistic mass are proportional to one another, the relativistic mass is just as relevant in GR as the total energy is.
That is what I was correcting you on.
Also I'm not actually going to say you were
completely wrong. It's a point of view issue that's a bit vague because exactly what "relevant" means is a bit vague. As I explained to RJ the gravitational field in GR doesn't depend on just the energy/relativistic mass in a given coordinate system, but also on momentum flux and density. This is why it's possible in GR, as you yourself said earlier, that a mass won't turn into a black hole just because it started moving or you changed frames. In that sense you were perfectly correct when you said the relativistic mass was irrelevant (to the general characteristics of the gravitational field around a dense mass). I'm just saying that you can't deny that the relativistic mass is relevant in GR
in exactly the same way and
to exactly the same extent as the total energy is, because apart from a factor of
c[sup]2[/sup] they're exactly the same thing.
First off, I did it, so you are off base. I wasn't going to just write it down for him, I gave him the tools to write it himself.
But it's clear that the energy in the top and bottom halves isn't the same. The total energy in each half is given by an integral over the perimeter which goes something like
$$E = \int \mathrm{d}l \, \rho \, k(v)$$
where $$\rho$$ is the linear mass density and $$k(v)$$ is the energy per unit mass (either $$\gamma c^{2}$$ or $$(\gamma - 1) c^{2}$$ depending on whether you're interested in the total or just the kinetic energy). Since both these functions are higher throughout the top half of the wheel than in the bottom half (because the mass density is higher, and because the speed is higher and
k(
v) is a monotonically increasing function in
v), and the integrals are over domains of the same length (half the perimeter of the ellipse), the total energy of the top half is necessarily going to be higher than in the bottom half, and you've certainly made an error if you actually did the calculation and found otherwise.
So prove it, assertions don't count as proofs.
I did, later in the very same post.
True but there is nothing about rigidity in this problem.
So why did you bring it up?
I don't think this is a valid explanation, especially the bit about length contraction generating higher density in the upper half.
So specifically what's wrong with it? Also the bit about length contraction at the end wasn't the main explanation.
I had to explain a similar issue over a year ago to tsmid in some posts leading to and following [POST=2294731]this[/POST] one. The topic was different (the current in a wire loop), but the issues involved were similar. (Warning: tedium ensues, since the wire loop wasn't the only thing being discussed.)
A much simpler proof is that the midline in the ground frame is no longer horizontal but inclined (due to relativistic aberration), so it no longer coincides with the axis of symmetry of the ellipse
Come again?
EDIT: 2000[sup]th[/sup] (non-cesspooled) post. Woo.
