Global spectral representations of black hole spacetimes in the complex plane Maurice H. P. M. van Putten† Massachusetts Institute of Technology–Laser Interferometer Gravitational-Wave Observatory, 175 Albary Street, NW17-161, Cambridge, MA 02139 Edited by Robert M. Wald, University of Chicago, Chicago, IL, and approved November 17, 2005 (received for review September 26, 2005) Binary black hole coalescence produces a finite burst of gravita- which shows that G is analytic in each coordinate away from the tional radiation, which propagates toward quiescent infinity. singularity at y ϭ p with domain of convergence r Ͼ p.Wenow These spacetimes are analytic about infinity and contain a dimen- complexify the real nonnegative radial coordinate r Ն 0 into a sionless coupling constant M͞s, where M denotes the total mass- complex variable z by energy and s is an imaginary distance. This work introduces a globally convergent Fourier series in a complex radial coordinate, z ϭ x ϩ is ͑s Ͼ p͒, [3] allowing spectral representations of black hole spacetimes in all three dimensions. This method circumvents singularities associ- where the real variable x assumes both positive and negative 1 ated with black holes and includes infinity, thereby avoiding the values. The analytic continuation of Eq. given by need to impose radiative boundary conditions. We illustrate this p lϩ1 ϭ Ϫ1 ͸ͩ ͪ ͑ ␪ representation theory on Boyer–Lindquist initial data of two black G p Pl cos ), [4] holes and a scalar wave equation with signal recovery by Cauchy’s z integral formula. is globally convergent on the straight line Ϫϱ Ͻ x Ͻϱoutside ͉z͉ ϭ p and parallel to the real line s ϭ 0. (The corresponding ͉ computation gravitation branch cut for the square root is (Ϫϱ, 0].) To be precise, for each ␪ Eq. 4 defines a Taylor series in 1͞z, and, for each z, it defines apid progress in sensitivity in the broadband gravitational- a convergent expansion in Legendre functions. Thus, p͞s is a ASTRONOMY Rwave observatories (1–4) is creating new opportunities for running coupling constant, and it must be less than one for searches for the bursts of gravitational-wave emissions produced convergence. We shall refer to M͞s as the fixed coupling by coalescing black holes. Detection strategies will benefit constant, noting that p͞s is decreasing in binary coalescences. greatly from a priori understanding of their gravitational-wave Analyticity of the metric in Im(z) Ն ͉p͉ defines an optimal emissions, which includes matched filtering techniques against a distribution of points on z ϭ x ϩ is for numerical implementation catalog of signals precomputed by numerical relativity. It poses by the conformal transformation an interesting challenge of designing highly efficient and stable computational algorithms. M w ϭ . [5] Numerical relativity is particularly challenging in calculating z bursts and preceding chirps over many wave-periods in the Ϫ1 Ϫ1 presence of singularities associated with black hole spacetimes. Here, w describes a circle of radius (2s) about w0 ϭ (2is) , Spectral methods provide an attractive approach. They are whose interior corresponds to Im(z) Ն s. There exists a Taylor optimal in efficiency and accuracy, both in amplitude and phase, series in w about w0, which is convergent up to and including the Ϫ ␺ provided that the metric is smooth everywhere. Here, we focus boundary C: w(␺) ϭ (2is) 1(ei ϩ 1). The corresponding Fourier on bursts of radiation produced by black hole coalescence, which series in ␺ can be efficiently computed by using the fast Fourier Ϫmϩ1 propagate toward quiescent infinity. By causality, these space- transform on a uniform distribution of points ␺k ϭ 2 ␲k(k ϭ times preserve quiescence at infinity for all finite time. 1,2,...,2m)onC. Their images on z ϭ x ϩ is produce a unique We can introduce globally smooth metrics for black hole nonuniform distribution of points spacetimes by using complex distances. To illustrate this ap- ϭ ͓ ␺ ͞ ͒ ϩ proach, we note that the asymptotic structure of black hole zk Ms tan( k 2 i], [6] spacetimes, which are asymptotically flat and quiescent at in- whose large-scale cut-off is set by the proximity of the {␺k}to finity, shares the same asymptotic properties as the Green‘s Ϯ␲. function G of Minkowski spacetime. Consequently, these space- ͞ We apply these observations to Boyer–Lindquist initial data times carry a dimensionless coupling constant M s, where M for two black holes (5, 6), described by the potential denotes the total mass-energy of the space and s is a finite step into the complex plane. The Green’s function of the Laplacian M1 M2 in flat spacetime, ⌽ ϭ 1 ϩ ϩ , ͱz2 ϩ p2 Ϫ 2pz cos ␪ ͱz2 ϩ q2 Ϫ 2qz cos ␪ 1 G͑y, p͒ ϭ , [1] [7] D͑y, p͒ for a pair of black holes with masses M1,2 at positions p and q is a function of the distance D between an observer at location along the polar axis of a spherical coordinate system (r, ␪, ␾). y and a source at location p. We can expand G in Legendre The Boyer–Lindquist data represent an exact solution to polynomials Pl upon expressing y and p in spherical coordinates Laplace’s equation, representing the solution of the Hamilto- (r, ␪, ␾) 1 p lϩ1 Conflict of interest statement: No conflicts declared. ϭ ϭ Ϫ1 ͸ͩ ͪ ͑ ␪ G p P cos ), This paper was submitted directly (Track II) to the PNAS office. ͱr2 ϩ p2 Ϫ 2pr cos ␪ r l †E-mail: [email protected]. [2] © 2006 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0508406103 PNAS ͉ January 17, 2006 ͉ vol. 103 ͉ no. 3 ͉ 519–522 Downloaded by guest on September 28, 2021 Fig. 1. The Boyer–Lindquist scalar field ⌽(z, ␪) possesses a Fourier–Legendre expansion in w ϭ M͞z, z ϭ x ϩ is (Upper) and cos␪ (Lower), here shown for a symmetric configuration consisting of two black holes of mass M1,2 ϭ 1 positioned at the north and south pole p ϭϮ1 of a spherical coordinate system (r, ␪, ␾). The Fourier coefficients show exponential decay by analyticity in Im(z) Ն s Ͼ s*(␪) ϭ psin␪ shown with cos␪ ϭ 0.25 and s* ϭ 0.9682. It gives rise to spectral accuracy in the first and second coordinate derivatives (Right Upper). The norm of the Legendre coefficients shows exponential decay when s is larger than p (Left). It gives rise to spectral accuracy in the first and second derivatives with respect to ␪ (Right Lower). nian constraint for time-symmetric initial data. For this rea- binary black hole spacetime with total mass-energy M as mea- son, the Boyer–Lindquist data are representative in general of sured at infinity, defined by the residue the asymptotic properties of the Green’s function in curved ϱϩis spacetime. i 1/4 Ϯ ␪ ϭ E ͵ ͑ Ϫ ͒ [10] ϭ i M R ͭ hzz 1 dzͮ . It will be appreciated that the singularities z pe are ␲ essential in making ⌽ nontrivial: the alternative of an entire Ϫϱϩis function that is analytic at infinity would be constant by Li- For binary black hole coalescence, M is the sum M ϩ M of the ouville’s theorem. For complex z ϭ x ϩ is with s larger than both 1 2 masses of the two black holes, plus the energy in gravitational p and q, ⌽(z, ␪) is finite and pointwise analytic everywhere on Ϫϱ ϩ Ͻ Ͻϱϩ radiation and minus the binding energy in the system. Thus, M is z is. Following Eqs. 4 and 6, we capture is a constant. In contrast, M is a nonincreasing function of time ϭ ͞ analyticity by the Fourier transform on the image w M z.It on any finite computational domain with outgoing radiation gives rise to an efficient and accurate functional representation boundary conditions. For large z, the line-element (Eq. 9) ␪ in (z, ) with spectral accuracy in derivatives by spectral differ- satisfies entiation. Fig. 1 shows the computational results on the alter- native representations M ds2 ϭ ϪN2dt2 ϩ ͩ 1 ϩ ͪ ͑dz2 ϩ d⌺2͒ ϩ ..., [11] 2 z ⌽͑ ␪͒ ϭ ͸N ͑ ␪ n ϭ ͸ L ͑ ͒ ͑ ␪ z, nϭ1an cos )w lϭ0bl z Pl cos ), [8] up to order (M͞z)2, where d⌺2 ϭ z2d␪2 ϩ z2 sin2␪d␾2 denotes the where the series are finite in our numerical implementations. line-element of the coordinate sphere of radius z. Accordingly, The application to black hole spacetimes in the complex plane the three-metric hij is decomposed into diagonal and off- can be considered in the line-element diagonal elements given by the matrix factorization 1 Ϫ M͞2 z 4 2 ϭ Ϫ 2 2 ϩ i j ͑ ͒ ϭ A1 B1 B2 ds N dt hij dx dx , N z ϩ ͞ , [9] 4 1 M 2 z h ϭ ␥ ͩB1 A2 B3ͪ␥. [12] ij 4 B2 B3 A3 for a three-metric hij. Working in the complex plane, we are at liberty to choose a lapse function N and vanishing shift functions In accord with the spectral representation of Eq. 4, the potential in accord with the asymptotic Schwarzschild structure of space- Eqs. 7 and 8, the coefficients (Ai, Bi) satisfy the general expan- time at large distances. It applies without loss of generality to any sion in spherical harmonics 520 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0508406103 van Putten Downloaded by guest on September 28, 2021 Fig. 3. Signal recovery of a wave propagating to quiescent infinity is shown Fig. 2.