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Proof of entropy principle in Einstein-Maxwell theory

We consider a static self-gravitating charged perfect fluid system in the Einstein-Maxwell theory. Assume Maxwell's equation and the Einstein constraint equation are satisfied, and the temperature of the fluid obeys Tolman's law. Then we prove that the total entropy of the fluid achieves an extremum implies other components of Einstein's equation for any variations of metric and electrical potential with fixed boundary values. Conversely, if Einstein's equation and Maxwell's equations hold, the total entropy achieves an extremum. Our work suggests that the maximum entropy principle is consistent with Einstein's equation when electric field is taken into account.