At a given A, the differences in the values of M (A) of the three branches are much smaller than the differences in R. Therefore, one has F(A) > F* (A) and Funst*(A) > F*(A) under the integrals, and thus M*(A) < M(A). The proof is even simpler for a C\ C* transition shown in the left panel of Fig. 7.11.
The inequality M *(A) < M (A) is quite general. It means that the change in the EOS associated with the phase transition always lowers the total energy of the star. However, the reader should be warned that the energy excess AE = (M — M*)c2 is usually very small. Therefore, while calculating AE one has to be sure that the EOS satisfies the condition of thermodynamic consistency discussed in § 6.4.1. Otherwise, there is a danger of violating the strict inequality M*(A) < M(A), leading to an apparent paradox: a phase transition in a neutron star core is blocked by global energy conservation!
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