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The arguments for the formulas I put up were made on the basis
that heat is not just pure Energy but also carries some momentum. When
Einstein and Planck (actually Planck first) made their calculations
based on the first law of thermo, they required a "fake force" that
carried the momentum. However, the first law considers heat to be pure
energy.
D.L. Steinert "An evaluation of relativistic thermodynamics"
(Ph.D. thesis - 1969 - Mich. State)
Another reason for so much confusion is trying to correctly
define heat and work. Also, what do we consider in the system. Ott
only considered a gas, while Planck and Einstein a gas and a container
(where I believe their force came from) It is very hard to understand
so I will let you read up on the material.