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

Probing Space-Time Through Numerical Simulations 161solutions were astrophysically relevant or not. A series of simplifying assumptionswere introduced to facilitate the task. Specifically, if one assumesinitial data with maximal embedding and a conformally flat 3-geometry,York’s version of the constraint equations take the simple form8 ˆ∆φ = − ij  ij φ −7 (11)̂∇ j  ij = 0 , (12)where φ is the conformal factor and Âij the trace-free, conformal extrinsiccurvature. Notice that in this form, one can solve first independently themomentum constraint (12) for Âij and then use this solution to solve thenon-linear constraint (11) for the conformal factor φ.Bowen and York showed 18,71 that a solution to equation (12) for Nblack holes, each with linear momentum Pi A and angular momentum Si A,is given by ij = 3 2N∑{ 1A=1r 2 A[PAi n A j + Pj A n A i − (η ij − n A i n A j )Pk A n k ]A+ 1 [rA3 ɛilk SA l nk A nA j + ɛ jlkSA l ] }nk A nA i , (13)where n A i is the unit normal at the throat of the A-th black hole, r A thedistance to the center of the A-th hole, and η ij is the flat-metric.The Bowen-York solution (13) has been extremely useful. For instance,Brandt and Brügmann used the Bowen-York solution to introduce the socalled puncture approach. 19 The puncture method solves (9) based on adecomposition of the conformal factor φ of the formφ = u + 1 p(14)such that1N p = ∑A=1(15)M Ar A, (16)with M A the “bare” masses of the black holes. The advantage of using thedecomposition (14) is that Eq. (9) reduces toˆ∆u = − 1 8ÂijÂij (1/p + u) −7 . (17)It is not difficult to show that the r.h.s. of this equation is regular everywhere,thus also the solution u. There is no need to excise the black holes.