How does gravitational redshift not violate the Second Law of Thermodynamics?

Question

Suppose we had two black bodies at distinct heights in an uniform gravitational field. They are connected by an insulating tube that only allows the exchange of electromagnetic radiation. Since radiation from the body at the bottom gets redshifted when it reaches the top, it loses energy. Similarly, radiation from the top body gets blueshifted when it reaches the bottom. So, if the bodies were initially at equal temperatures, they exchange a net energy and one heats up while the other one cools down.

If we now bring the bodies together we can use the temperature difference to do some work, then restore the system and repeat this indefinitely.

Since it appears that we broke the Second Law of Thermodynamics, there must be something wrong with this reasoning. What am I missing?

Answer 1

This is really similar to a waterfall. You're exploiting the energy available by lowering the item (EM radiation in this case) down the gravitational field.

So the question becomes, how did that item get to that height in the first place? You had to raise it up there. In other words, the upper object after temperature equilibration gains energy (gains mass). So you have to put in more work to raise it in the field than you receive back by lowering it.

Answer 2

See gravitational redshift & thermodynamics applied (gives a fine structure constant inside structure by the way but ) it is the (new) result if you combine Einsteins Principles from GR with those from Thermodynamics which we have in a (older) textbooks already.