Key points are not available for this paper at this time.
We show that under variation of moduli fields the first law of black hole thermodynamics becomes dM0ex{0ex}=0ex{0ex}8+++-, where 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 ₅₈ₗ (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 0ex{0ex}=0ex{0ex}₅₈ₗ (p, q): ₅₈ₗ is such that the scalar charges vanish: (₅₈ₗ, (p, q) ) 0ex{0ex}=0ex{0ex}0.
Gibbons et al. (1996) studied this question.