General Relativity and Gravitation, Vol. I9, No. 2, 1987 Nonexistence of Multiple Black Holes in Asymptotically Euclidean Static Vacuum Space-Time Gary L. Bunting and A. K. M. Masood-ul-Alam Received February 25, 1986 Using the positive mass theorem, we show that the connectivity assumption of the black hole boundary is not necessary for proving the spherical symmetry of asymptotically Euclidean static vacuum space-time; thus there exist no asymptotically Euclidean static vacuum space-times with multiple black holes. INTRODUCTION In this paper we use the positive mass theorem to prove that the exterior Schwarzschild geometry is the only maximally extended, static, asymp- totically Euclidean vacuum space-time with a smooth event horizon under the assumption that the intersection of the event horizon with the closure of a t = constant hypersurface may have more than one connected com- ponent. For such a space-time containing a single black-hole, a simple proof of the above assertion is given [1, 2, 3]. Nonexistence of static equilibrium configuration involving more than one black hole in an "axisymmetric" asymptotically euclidean, vacuum space-time was shown [4, 5] (see also [6]). We thus show the nonexistence of multiple black holes in an asymptotically euclidean static vacuum space-time, not necessarily axisymmetric. The relevance of the positive mass theorem for proving the uniqueness of complete asymptotically euclidean static space- times is discussed [7]. However, the situation here differs because of the black-hole-type boundary. For a general discussion on the static vacuum space-time we refer to [1, 2, and 3]. The space-time metric 4g exterior to event horizon is given by 4g=g~l~(x~)dx~dxl~-V2(xT)dt2 c~= 1,2, 3 (1) 147 OOOl-77Ol/87/o2oo-o1475o5.oo/o (, 1987Plenum Publishing Corporation