Smarr Mass formulas for BPS multicenter Black Holes

Mass formulas for multicenter BPS 4D black holes are presented. In the case of two center BPS solutions, the ADM mass can be related to the intercenter distance r , the angular momentum J2 , the dyonic charge vectors qi and the value of the scalar moduli at infinity ( z∞ )by the Smarr-like expression MADM2=A(1+αJ2(1+2MADM/r+A/r2)) where A(Q),α(qi) 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∈(0,∞) and MADM∈(∞,M∞) . First Law expressions incorporating the intercenter distance are obtained from it: dM≡ΩdJ+Φidqi+Fdr, in addition to an effective angular velocity Ω and electromagnetic potentials Φi , the equation allows to define an effective “force”, F , acting between the centers. This effective force is always negative: at infinity and at short distances we recover the familiar Newton law F∼1/r2 at the leading order. Similar results can be easily obtained for more general models and number of centers.