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

We show that under variation of moduli fields $\ensuremath{\varphi}$ the first law of black hole thermodynamics becomes $dM\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{\ensuremath{\kappa}\mathrm{dA}}{8\ensuremath{\pi}}+\ensuremath{\Omega}dJ+\ensuremath{\psi}dq+\ensuremath{\chi}dp\ensuremath{-}\ensuremath{\Sigma}d\ensuremath{\varphi}$, where $\ensuremath{\Sigma}$ 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 ${\ensuremath{\varphi}}_{\mathrm{fix}}(p,q)$ 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 $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\varphi}}_{\mathrm{fix}}(p,q)$: ${\ensuremath{\varphi}}_{\mathrm{fix}}$ is such that the scalar charges vanish: $\ensuremath{\Sigma}({\ensuremath{\varphi}}_{\mathrm{fix}},(p,q))\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$.

Перевод пока недоступен