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

We show that under variation of moduli fields {phi} the first law of black hole thermodynamics becomes {ital dM}={kappa}{ital dA}/8{pi} +{Omega}{ital dJ}+{psi}{ital dq}+{chi}{ital dp}{minus}{Sigma}{ital d}{phi}, where {Sigma} are the scalar charges. Also the ADM mass is extremized at fixed {ital A}, {ital J}, ({ital p},{ital q}) when the moduli fields take the fixed value {phi}{sub fix}({ital p},{ital 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 {phi}={phi}{sub fix}({ital p},{ital q}): {phi}{submore »