Here's a "dilemma" that came up in another discussion:
"Suppose you have a planet at the center of a shell. Both are
blackbodies at the same temperature. Photons from the planet will be
red-shifted coming up, while photons coming down will be blue-shifted.
Thus there will be more energy going down to the planet than up from it.
Thus you have a net flow of energy even though the temperatures are the
same."
(Or make the outer object SLIGHTLY cooler. Then you could have a net
flow of energy from the cool shell to the warm planet, again in
violation of the 2nd law.)
I've never heard it discussed, but I conclude that "at the same
temperature" is also relative.
One line of reasoning looks at the clocks at the two locations. The
clock on the planet will run slower. If the planet was radiating 400
J/s/m^2, the shell "at the same temperature" would radiate 400 J/s/m^2
-- but according to its own clock. According to the clock on the
ground, it would be radiating slightly more than 400 J/s/m^2. So
observer travelling from one to the other would measure the same
temperatures in the proper frames. But an observer at either location
will see the planet as slightly cooler than the shell. Thus there
should be a net flow of thermal energy inward.
You could also consider the speeds of the atoms. In their own reference
frame, both would have the same distribution of speeds --> same
temperature. But the observer on the planet would see the atoms in the
shell moving faster --> higher temperature.
In either line of reasoning, there will be a net flow of energy from the
shell to the planet even though both are "the same temperature".
1) Have you ever run across this "dilemma" before?
2) Do my explanations seem right?