Mass Formulae of 4-Dimensional Dilaton Black Holes

Abstract

Integral and differential mass formulae of 4-dimensional stationary and axisymmetric Einstein-Maxwell-dilaton systems are derived. The total mass (energy) of these systems are expressed in terms of other physical quantities such as electric charge of the black hole suitably modified due to the existence of the dilaton field. It is shown that when we vary slightly the fields (metric of the spacetime $g_{μν}$, $U(1)-$gauge potential $A_μ$, and dilaton $ϕ$) in such a way as they obey classical equations of motion, the variation of the dilaton does not contribute explicitly to the variation of the total mass, but contributes only through the variation of the electric charge of the black hole.