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. We also show that 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. It follows that the least mass of any black hole with fixed conserved electric and magnetic charges is given by the mass of the double-extreme black hole with these charges. Our work allows us to interpret the previously known observation that for all extreme black holes the moduli fields at the horizon take a value φ = φfix(p, q) depending only on the electric and magnetic conserved charges: φfix(p, q) is such that the scalar charges Σ(φfix, (p, q)) = 0.