Smarr Mass formulas for BPS multicenter Black Holes

TL;DR

This paper derives mass formulas for multicenter BPS black holes in four dimensions, revealing a continuous spectrum of solutions and an effective force between centers, generalizing known laws and incorporating scalar moduli effects.

Contribution

It introduces new mass formulas relating ADM mass, intercenter distance, and charges for multicenter BPS black holes, including Smarr-like relations with an effective force term.

Findings

Mass formulas relate ADM mass, charges, and intercenter distance.

Existence of a continuous class of black hole solutions with varying intercenter distances.

Effective force between black hole centers is always attractive, resembling Newtonian gravity.

Abstract

Mass formulas for multicenter BPS 4D black holes are presented. For example, ADM mass for a two center BPS solution can be related to the intercencenter distance $r$, the angular momentum $J^2$, the dyonic charge vectors $q_i$ and the value of the scalar moduli at infinity ($z_\infty$)by $M_{ADM}^2 =A\left (1+ \alpha J^2\left(1+\frac{2M_{ADM}}{r}+\frac{A}{r^2}\right)\right)$ where $A(Q),\alpha(q_i)$ are symplectic invariant quantities ($Q$, the total charge vector) depending on the special geometry prepotential defining the theory. The formula predicts the existence of a continuos class, for fixed value of the charges, of BH's with interdistances $r\in (0,\infty)$ and $M_{ADM}\in (\infty,M_\infty)$. Smarr-like expressions incorporating the intercenter distance are obtained from it: $$ dM\equiv\Omega d J+\Phi_i d q_i+ F dr,$$ in addition to an effective angular velocity $\Omega$ and electromagnetic potentials $\Phi_i$, the equation allows to define an effective "force", $F$, acting between the centers. This effective force is always negative: at infinity we recover the familiar Newton law $F\sim 1/r^2$ while at short distances $F\sim f_0+f_1/r^2$. Similar results can be easily obtained for more general models and number of centers.