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 Arnowitt-Desner-Misner mass is extremized at fixed A, J, (p,q) when the moduli fields take the fixed value φ_fix(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  =  φ_fix(p,q): φ_fix is such that the scalar charges vanish: Σ(φ_fix,(p,q))  =  0.