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James,
You are absolutely correct that there is a component of the
stress-energy tensor that contains the temperature. It is rather messy
but I can try to put it up here or send it to you personally if you
wish. I wish I understood this really well, but I think that is limited
to only a couple of people who have a true intuitive feel for
relativity, one of which is John A. Wheeler (you are close though).
Sam Held