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/8 π + Ω dJ + ψdq + χdp - Σ dφ, 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.