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 $dM{{0ex}{0ex}}={{0ex}{0ex}}{{κ}dA}{8{π}}+{Ω}dJ+{ψ}dq+{χ}dp{-}{Σ}d{φ}$, 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 φ0ex0ex=0ex0exφfix(p,q): φfix is such that the scalar charges vanish: Σ(φfix,(p,q))0ex0ex=0ex0ex0.