Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics

We show that under variation of moduli fields φ the first law of black hole thermodynamics becomes dM=κdA/8π +ΩdJ+ψdq+χdp-Σdφ, where Σ are the scalar charges. Also the ADM mass is extremized at fixed A, J, (p,q) when the moduli fields take the fixed value φfix(p,q) which depend only on electric and magnetic charges. Thus the double-extreme black hole minimizes the mass for fixed conserved charges. We can now explain the fact that extreme black holes fix the moduli fields at the horizon φ=φfix(p,q): φfix is such that the scalar charges vanish: Σ(φfix,(p,q))=0. copyright 1996 The American Physical Society