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I'm sticking with thermal equilibrium being isothermal, even for a
gas in the presence of the gravitational field.
In a curved spacetime, the thing that is the same everywhere is
Tolman's redshifted temperature T √(-g00).
In the weak-field limit, g00 = -(1 + 2Φ) where Φ is the depth of
the gravitational potential.
1) You can doggedly reject most of them them and cling to the idea
that ONLY one of them is "correct"
2) You can accept the fact that people simply WILL use many different
and not unreasonable definitions of work (usually without thinking
carefully enough about it) and then try to educate them about the
differences between those definitions and what energy changes they
turn out to be equal to.
Much of the previous [educational] research has involved processes
in which heat is the only means of energy transfer. Several of
these studies indicate that students tend to treat heat as a
substance residing in a body.