$$KE=m_0c^2(\gamma-1)$$

$$m_{relativistic}=m_0 \gamma$$

$$\frac{KE}{m_{relativistic}}=c^2 \frac{\gamma-1}{\gamma}$$

They obviously aren't the same thing, nor do they differ by a constant factor.

The main point was that gravity doesn't depend on relativistic mass.

Er, did you read what I said? The

*total* energy (which is really more relevant in GR than just the kinetic energy) and relativistic mass are proportional to one another. The

*total* energy is given by $$E = \gamma m_{0} c^{2}$$, and the relativistic mass by $$m = \gamma m_{0}$$, so you find that total energy and relativistic mass are just related by $$E = m c^{2}$$.

That is a good point, I did this approach as a (mistaken) backup since I couldn't get RJBerry to calculate the total energy of the upper and lower halves of the circle via a simple integral.

First, don't you think it's a bit silly complaining that RJ didn't calculate the integral when you've just admitted you couldn't be bothered doing it yourself?

Second, it's clear even without doing any calculation that RJ is actually right. The reason is that the speeds of wheel elements in the top and bottom halves of the wheel aren't just

*different*: they are consistently

*higher* in the top half than in the bottom half. Also as stated previously, most of the mass is also concentrated in the top half, which will only further increase the energy in the top half.

What makes you say that? The wheel is rigid

I think one of the nice things illustrated by the relativistic rolling wheel is the generally well known fact that perfectly rigid systems are impossible in relativity.

so why would a frame change move matter between the two halves?

It's just a manifestation of the relativity of simultaneity effect: imagine two wheel elements

*A* and

*B* on opposite sides of the wheel in the frame comoving with the axle. When wheel element

*A* crosses up into the top half, element

*B* on the opposite side simultaneously crosses down into the bottom half. In the frame where the wheel is rolling and the ground is at rest, these events are no longer simultaneous. If you think about it or work it out, you find in that frame that element

*A* crosses up into the top half

*before* *B* crosses down into the bottom half. The result is that wheel elements tend to lag in the top half. This also plays well with the fact that you expect the mass density to be higher in the top half anyway, since it's moving faster and length contraction is locally more pronounced there.