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

TL;DR

This paper extends the first law of black hole thermodynamics to include scalar charges and shows that the ADM mass is extremized at fixed charges when moduli fields are at specific fixed values, linking scalar charges to black hole extremality.

Contribution

It introduces a generalized first law incorporating scalar charges and demonstrates the extremization of ADM mass at fixed charges with moduli fields at specific fixed points.

Findings

The first law includes scalar charges as conjugate variables.

ADM mass is extremized at fixed charges when moduli are at specific values.

Scalar charges vanish at the fixed moduli values for extremal black holes.

Abstract

We show that under variation of moduli fields $\phi$ the first law of black hole thermodynamics becomes $dM = {\kappa dA\over 8\pi} + \Omega dJ + \psi dq + \chi dp - \Sigma d\phi$, where $\Sigma$ are the scalar charges. We also show that the ADM mass is extremized at fixed $A$, $J$, $(p,q)$ when the moduli fields take the fixed value $\phi_{\rm fix}(p,q)$ which depend only on electric and magnetic charges. It follows that the least mass of any black hole with fixed conserved electric and magnetic charges is given by the mass of the double-extreme black hole with these charges. Our work allows us to interpret the previously established result that for all extreme black holes the moduli fields at the horizon take a value $\phi= \phi_{\rm fix}(p,q)$ depending only on the electric and magnetic conserved charges: $ \phi_{\rm fix}(p,q)$ is such that the scalar charges $\Sigma ( \phi_{\rm fix}, (p,q))=0$.