First law for the Kerr-Newman family
= First law for the Kerr-Newman family
{title2=$dM=\kappa\,dA/(8\pi)+\Omega_H\,dJ+\Phi_H\,dQ$}
Differentiate $M=(r_+^2+a^2+Q^2)/(2r_+)$, $J=Ma$ and $A=4\pi(r_+^2+a^2)$. Eliminating $dr_+$ and $da$ gives the displayed relation, with $\Omega_H=a/(r_+^2+a^2)$ and $\Phi_H=Qr_+/(r_+^2+a^2)$. Matching <Hawking temperature> to this <First law of black-hole mechanics> fixes the <Bekenstein-Hawking entropy> as $A/4$.