Lapse function for Brill-Lindquist initial data

For binary black holes the lapse function corresponding to the Brill-Lindquist initial value solution is given in analytic form under the maximal slicing condition. In the limiting case of very small ratio of mass to separation between the black holes the surface defined by the zero value of the lapse function coincides with the minimal surfaces around the singularities. PACS number(s): 04.70.Bw, 04.20.Ex, 04.20.Fy In a recent paper [1], the authors re-obtained the Brill-Lindquist initial value solution for two black holes [2] using a point-mass model for the black holes followed by Hadamard’s “partie finie” regularization procedure in 3-dimensional space. (It can be shown that the same result appears by applying the dimensional regularization of Ref. [3].) In the present paper we supplement this result by the calculation of the initial lapse function. The result should be of importance for numerical head-on collision simulations of binary black holes under time-symmetric initial conditions. We use units in which 16πG = c = 1, where G is the Newtonian gravitational constant and c the velocity of light. In the paper all vectors and their lengths are defined in the 3-dimensional Euclidean space endowed with a standard Euclidean metric; x is the position of an arbitrary point in this space. In the generalized isotropic ADM gauge the 3-metric takes the form gij ≡ ( 1 + 1 8 φ 4 δij + h TT ij (1) and for the canonical conjugate π of the 3-metric the condition π ≡ 0 (2) holds. In case of time-symmetric initial conditions the black-hole linear momenta pai (a = 1, 2) and field momenta π , and thus the shift function too, are zero pai = π ij = 0. (3) Conformal flatness, i.e. gij = ( 1 + 1 8 φ )4 δij , (4) Email address: [email protected] Email address: [email protected]