OK -- a little more follow up. I'm not super strong on relativity, but I was continuing this discussion a bit off-list and was trying to make sure I was understanding the physics involved.
Someone else (off-list) said:
" Strictly speaking, it is not possible to measure the temperature other than in the rest frame of each object. One can infer the temperature by observing the emitted radiation, but to do so one has to make certain assumptions – such as that the radiation one observes is thermal and has not been doppler shifted – that are not always valid. The temperature difference in my example is a real or absolute one. We can see this by imagining that the two objects are made of a single material that has a melting point above the temperature of the shell but below that of the planet. The shell will remain solid; the planet will melt."
I tend to disagree with most of this.
I would tend to say:
* "The" temperature can be measured in any frame. But in different frames the values are different. There is no "real or absolute one". Values measured in other frames are not simply "inferred" -- they are perfectly legitimate values.
(Furthermore, since we are explicitly including gravitational effects, neither the shell nor the planet are "rest frames". So "strictly speaking" none of the measurements would have been made in a rest frame.)
* The shell could be solid and the planet melted. Suppose the observer on the planet ("P") says the temperature is T (for both objects). Then the observer on the shell ("S") would say the temperature of both is T - dT. (And if either carried a thermometer from one to the other, they would agree the temperature reading changes.) So can we conclude the melting point is suppressed where gravity is stronger = where acceleration is greater?
The discussion went on to:
"... it demonstrates how an object at one temperature can heat another to a higher temperature; so this cannot be what the second law prohibits."
* This also seem incorrect, based on the discussion above. In equilibrium, the temperatures are the same as seen from any reference frame -- there is no "heat to a higher temperature" when properly dealing with relativity.
Any efforts to set either one of us straight would be appreciated. :-)