The transitive property of thermal equilibrium is shown to be equivalent to the transitiveproperty of synchronization of rates of coordinate clocks. A necessary and sufficient condi-tion for the validity of the zeroth law of thermodynamics in Riemann space time is obtained.It is pointed out that the validation of the zeroth law is a necessary condition to construct"simultaneity surfaces", but not a sufficient condition. The reason why one has to use the"dragged system" in the study of Hawking radiation in Kerr spacetime has been explained.
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