100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Black Holes 117⃗x 4 = −⃗x 3⃗x 1Rr 4r 1⃗x 2 = −⃗x 1⃗x 3Fig. 7. “Many Schwarzschild” initial data with four black holes. The initial data areexactly Schwarzschild within the four innermost circles and outside the outermost one.The free parameters are R, (⃗x 1 ,r 1 ,m 1 ), and (⃗x 3 ,r 3 ,m 3 ), with sufficiently small m a’s.We impose m 2 = m 1 , r 2 = r 1 , m 4 = m 3 and r 4 = r 3 .then there exists a numberand a C ∞ metric ĝ ⃗Mconstraint equationm =2N∑i=0m i + O(δ 2 )which is a solution of the time-symmetric vacuumR(ĝ ⃗M )=0,such that:(1) On the punctured balls B(⃗x i , 2r i )\{⃗x i }, i ≥ 1, ĝ ⃗M is the Schwarzschildmetric, centred at ⃗x i ,withmassm i ;(2) On R n \ B(0, 2R 2 ), ĝ ⃗M agrees with the Schwarzschild metric centred at0, with mass m;(3) If R 1 > 0, then ĝ ⃗M agrees on B(0, 2R 1 ) \{0} with the Schwarzschildmetric centred at 0, with mass m 0 .By point (1) above each of the spheres |⃗x − ⃗x i | = m i /2 is an apparenthorizon.A key feature of those initial data is that we have complete control of thespace-time metric within the domains of dependence of B(⃗x i , 2r i )\{⃗x i } andof R n \ B(0, 2R 2 ), where the space-time metric is a Schwarzschild metric.