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Global spectral representations of black hole spacetimes in the complex plane

Edited by Robert M. Wald, University of Chicago, Chicago, IL, and approved November 17, 2005 (received for review September 26, 2005)
Abstract
Binary black hole coalescence produces a finite burst of gravitational radiation, which propagates toward quiescent infinity. These spacetimes are analytic about infinity and contain a dimensionless coupling constant M/s, where M denotes the total massenergy and s is an imaginary distance. This work introduces a globally convergent Fourier series in a complex radial coordinate, allowing spectral representations of black hole spacetimes in all three dimensions. This method circumvents singularities associated with black holes and includes infinity, thereby avoiding the need to impose radiative boundary conditions. We illustrate this representation theory on Boyer–Lindquist initial data of two black holes and a scalar wave equation with signal recovery by Cauchy's integral formula.
Rapid progress in sensitivity in the broadband gravitationalwave observatories (1–4) is creating new opportunities for searches for the bursts of gravitationalwave emissions produced by coalescing black holes. Detection strategies will benefit greatly from a priori understanding of their gravitationalwave emissions, which includes matched filtering techniques against a catalog of signals precomputed by numerical relativity. It poses an interesting challenge of designing highly efficient and stable computational algorithms.
Numerical relativity is particularly challenging in calculating bursts and preceding chirps over many waveperiods in the presence of singularities associated with black hole spacetimes. Spectral methods provide an attractive approach. They are optimal in efficiency and accuracy, both in amplitude and phase, provided that the metric is smooth everywhere. Here, we focus on bursts of radiation produced by black hole coalescence, which propagate toward quiescent infinity. By causality, these spacetimes preserve quiescence at infinity for all finite time.
We can introduce globally smooth metrics for black hole spacetimes by using complex distances. To illustrate this approach, we note that the asymptotic structure of black hole spacetimes, which are asymptotically flat and quiescent at infinity, shares the same asymptotic properties as the Green`s function G of Minkowski spacetime. Consequently, these spacetimes carry a dimensionless coupling constant M/s, where M denotes the total massenergy of the space and s is a finite step into the complex plane. The Green's function of the Laplacian in flat spacetime, is a function of the distance D between an observer at location y and a source at location p. We can expand G in Legendre polynomials P _{l} upon expressing y and p in spherical coordinates (r, θ, φ) which shows that G is analytic in each coordinate away from the singularity at y = p with domain of convergence r > p. We now complexify the real nonnegative radial coordinate r ≥ 0 into a complex variable z by where the real variable x assumes both positive and negative values. The analytic continuation of Eq. 1 given by is globally convergent on the straight line –∞ < x < ∞ outside z = p and parallel to the real line s = 0. (The corresponding branch cut for the square root is (–∞, 0].) To be precise, for each θ Eq. 4 defines a Taylor series in 1/z, and, for each z, it defines a convergent expansion in Legendre functions. Thus, p/s is a running coupling constant, and it must be less than one for convergence. We shall refer to M/s as the fixed coupling constant, noting that p/s is decreasing in binary coalescences.
Analyticity of the metric in Im(z) ≥ p defines an optimal distribution of points on z = x + is for numerical implementation by the conformal transformation Here, w describes a circle of radius (2s)^{–1} about w _{0} = (2is)^{–1}, whose interior corresponds to Im(z) ≥ s. There exists a Taylor series in w about w _{0}, which is convergent up to and including the boundary C: w(ψ) = (2is)^{–1}(e^{i} ^{ψ} + 1). The corresponding Fourier series in ψ can be efficiently computed by using the fast Fourier transform on a uniform distribution of points ψ_{k} = 2^{–} ^{m} ^{+1}πk(k = 1, 2,..., 2 ^{m} ) on C. Their images on z = x + is produce a unique nonuniform distribution of points whose largescale cutoff is set by the proximity of the {ψ _{k} } to ±π.
We apply these observations to Boyer–Lindquist initial data for two black holes (5, 6), described by the potential for a pair of black holes with masses M _{1,2} at positions p and q along the polar axis of a spherical coordinate system (r, θ, φ). The Boyer–Lindquist data represent an exact solution to Laplace's equation, representing the solution of the Hamiltonian constraint for timesymmetric initial data. For this reason, the Boyer–Lindquist data are representative in general of the asymptotic properties of the Green's function in curved spacetime.
It will be appreciated that the singularities z = pe ^{±} ^{i} ^{θ} are essential in making Φ nontrivial: the alternative of an entire function that is analytic at infinity would be constant by Liouville's theorem. For complex z = x + is with s larger than both p and q, Φ(z, θ) is finite and pointwise analytic everywhere on –∞ + is < z < ∞ + is. Following Eqs. 4 and 6, we capture analyticity by the Fourier transform on the image w = M/z. It gives rise to an efficient and accurate functional representation in (z, θ) with spectral accuracy in derivatives by spectral differentiation. Fig. 1 shows the computational results on the alternative representations where the series are finite in our numerical implementations.
The application to black hole spacetimes in the complex plane can be considered in the lineelement for a threemetric h_{ij} . Working in the complex plane, we are at liberty to choose a lapse function N and vanishing shift functions in accord with the asymptotic Schwarzschild structure of spacetime at large distances. It applies without loss of generality to any binary black hole spacetime with total massenergy M as measured at infinity, defined by the residue For binary black hole coalescence, M is the sum M _{1} + M _{2} of the masses of the two black holes, plus the energy in gravitational radiation and minus the binding energy in the system. Thus, M is a constant. In contrast, M is a nonincreasing function of time on any finite computational domain with outgoing radiation boundary conditions. For large z, the lineelement (Eq. 9) satisfies up to order (M/z)^{2}, where dΣ^{2} = z ^{2} dθ^{2} + z ^{2} sin^{2}θdφ^{2} denotes the lineelement of the coordinate sphere of radius z. Accordingly, the threemetric h_{ij} is decomposed into diagonal and offdiagonal elements given by the matrix factorization In accord with the spectral representation of Eq. 4, the potential Eqs. 7 and 8, the coefficients (A_{i}, B_{i} ) satisfy the general expansion in spherical harmonics Together with a Fourier expansion in the 2π periodic variable ψ of w(ψ), Eqs. 14 and 15 define a spectral representation in all three coordinates. Fig. 2 shows the results in case of the axisymmetric Boyer–Lindquist initial data, including the computational error behavior in the scalar Ricci tensor. Here, we use the general expansions (Eq. 8) to probe independently the dependencies on the degree N of the Fourier series of w(ψ) and the maximal degree L of the spherical harmonics.
Timeevolution of h_{ij} on z = x + is gives rise to burst of radiation that propagates toward quiescent infinity, where the observer`s signal is defined on nonnegative radii z ≥ 0. This signal can be calculated by projection according to Cauchy's integral formula where u denotes a relevant metric component or tensor quantity. Timeevolution on z = x + is is stable with constant translation s, because it preserves reality of the dispersion relation. We can illustrate this on the linear waveequation for a scalar field u(t, r) in spherical coordinates (t, r, θ, φ) with timesymmetric initial data, given by By analytic continuation, the equivalent initial value problem K′ with complex radial coordinate on –∞ + is < z < ∞ + is is [In general, the analytic continuation of the initial data are the extension of data on (r, θ, φ) and (r, π – θ, φ + π).] We can implement K′ numerically using finitedifferencing and leapfrog timestepping. Fig. 3 shows a representative numerical result, which is verified to satisfy independence of s. The scalar waveequation with timesymmetric initial data shows the appearance of both left and right movers, the first propagating toward z = –∞ + is and the second propagating toward z =∞+ is. Only the right movers are projected onto the observer's nonnegative radii z ≥ 0. The left movers are ultimately dissipated “unseen” in their propagation to –∞ + is, while maintaining a contribution to the total massenergy M in Eq. 10. We emphasize that the projection onto real radii is only required at the observer, that is, LIGO/Virgo, and hence for large distances away from the source. (Projections are not needed to recover a real representation of the source itself for the purpose of LIGO/Virgo searches of gravitational radiation.)
To conclude, the complex plane offers a unique opportunity for circumventing spacetime singularities and enabling spectral representations in all three dimensions, consisting of spherical harmonics in the angular coordinates and Fourier series in the complex radial coordinate. This approach introduces a computational duality between weakly nonlinear spacetimes in the complex plane and strongly nonlinear spacetimes on the real line, by virtue of Cauchy's integral formula. It creates a most efficient representation of black hole spacetimes and promises a reduction in computational effort by orders of magnitude compared with standard finite differencing approaches. It will be of interest to take advantage of this approach in timedependent calculations described by the fully nonlinear equations for general relativity in any of its hyperbolic forms (e.g., ref. 7).
Acknowledgments
I thank G.'t Hooft for stimulating discussions and the referee for a detailed consideration of the manuscript. Part of this work was supported by the Laser Interferometer GravitationalWave Observatories, constructed by Caltech and the Massachusetts Institute of Technology with funding from National Science Foundation Cooperative Agreement PHY 9210038. The Laser Interferometer GravitationalWave Observatory Laboratory operates under cooperative agreement PHY0107417.
Footnotes

↵ † Email: mvp{at}ligo.mit.edu.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.
 Copyright © 2006, The National Academy of Sciences
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