Toward stable 3D numerical evolutions of black-hole spacetimes

Three dimensional (3D) numerical evolutions of static black holes with excision are presented. These evolutions extend to about 8000M, where M is the mass of the black hole. This degree of stability is achieved by using growth-rate estimates to guide the fine tuning of the parameters in a multi-parameter family of symmetric hyperbolic representations of the Einstein evolution equations. These evolutions were performed using a fixed gauge in order to separate the intrinsic stability of the evolution equations from the effects of stability-enhancing gauge choices.Comment: 4 pages, 5 figures. To appear in Phys. Rev. D. Minor additions to text for clarification. Added short paragraph about inner boundary dependenc

Application of energy and angular momentum balance to gravitational radiation reaction for binary systems with spin-orbit coupling

We study gravitational radiation reaction in the equations of motion for binary systems with spin-orbit coupling, at order (v/c)^7 beyond Newtonian gravity, or O(v/c)^2 beyond the leading radiation reaction effects for non-spinning bodies. We use expressions for the energy and angular momentum flux at infinity that include spin-orbit corrections, together with an assumption of energy and angular momentum balance, to derive equations of motion that are valid for general orbits and for a class of coordinate gauges. We show that the equations of motion are compatible with those derived earlier by a direct calculation.Comment: 12 pages, submitted to General Relativity and Gravitatio

Multipole particle in relativity

We discuss the motion of extended objects in a spacetime by considering a gravitational field created by these objects. We define multipole moments of the objects as a classification by Lie group SO(3). Then, we construct an energy-momentum tensor for the objects and derive equations of motion from it. As a result, we reproduce the Papapetrou equations for a spinning particle. Furthermore, we will show that we can obtain more simple equations than the Papapetrou equations by changing the center-of-mass.Comment: 22 pages, 2 figures. Accepted for publication in Phys. Rev.

Potential for ill-posedness in several 2nd-order formulations of the Einstein equations

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.Comment: v1:6 pages; v2:7 pages, discussion extended, to appear in Phys. Rev. D; v3: typos corrected, published versio

Numerically generated quasi-equilibrium orbits of black holes: Circular or eccentric?

We make a comparison between results from numerically generated, quasi-equilibrium configurations of compact binary systems of black holes in close orbits, and results from the post-Newtonian approximation. The post-Newtonian results are accurate through third PN order (O(v/c)^6 beyond Newtonian gravity), and include rotational and spin-orbit effects, but are generalized to permit orbits of non-zero eccentricity. Both treatments ignore gravitational radiation reaction. The energy E and angular momentum J of a given configuration are compared between the two methods as a function of the orbital angular frequency \Omega. For small \Omega, corresponding to orbital separations a factor of two larger than that of the innermost stable orbit, we find that, if the orbit is permitted to be slightly eccentric, with e ranging from \approx 0.03 to \approx 0.05, and with the two objects initially located at the orbital apocenter (maximum separation), our PN formulae give much better fits to the numerically generated data than do any circular-orbit PN methods, including various ``effective one-body'' resummation techniques. We speculate that the approximations made in solving the initial value equations of general relativity numerically may introduce a spurious eccentricity into the orbits.Comment: 6 pages, 4 figures, to be submitted to Phys. Rev.

3D simulations of linearized scalar fields in Kerr spacetime

We investigate the behavior of a dynamical scalar field on a fixed Kerr background in Kerr-Schild coordinates using a 3+1 dimensional spectral evolution code, and we measure the power-law tail decay that occurs at late times. We compare evolutions of initial data proportional to f(r) Y_lm(theta,phi) where Y_lm is a spherical harmonic and (r,theta,phi) are Kerr-Schild coordinates, to that of initial data proportional to f(r_BL) Y_lm(theta_BL,phi), where (r_BL,theta_BL) are Boyer-Lindquist coordinates. We find that although these two cases are initially almost identical, the evolution can be quite different at intermediate times; however, at late times the power-law decay rates are equal.Comment: 12 pages, 9 figures, revtex4. Major revision: added figures, added subsection on convergence, clarified discussion. To appear in Phys Rev

Innermost circular orbit of binary black holes at the third post-Newtonian approximation

The equations of motion of two point masses have recently been derived at the 3PN approximation of general relativity. From that work we determine the location of the innermost circular orbit or ICO, defined by the minimum of the binary's 3PN energy as a function of the orbital frequency for circular orbits. We find that the post-Newtonian series converges well for equal masses. Spin effects appropriate to corotational black-hole binaries are included. We compare the result with a recent numerical calculation of the ICO in the case of two black holes moving on exactly circular orbits (helical symmetry). The agreement is remarkably good, indicating that the 3PN approximation is adequate to locate the ICO of two black holes with comparable masses. This conclusion is reached with the post-Newtonian expansion expressed in the standard Taylor form, without using resummation techniques such as Pad\'e approximants and/or effective-one-body methods.Comment: 21 pages, to appear in Phys. Rev. D (spin effects appropriate to corotational black-hole binaries are included; discussion on the validity of the approximation is added

Searching for Gravitational Waves from the Inspiral of Precessing Binary Systems: New Hierarchical Scheme using "Spiky" Templates

In a recent investigation of the effects of precession on the anticipated detection of gravitational-wave inspiral signals from compact object binaries with moderate total masses, we found that (i) if precession is ignored, the inspiral detection rate can decrease by almost a factor of 10, and (ii) previously proposed ``mimic'' templates cannot improve the detection rate significantly (by more than a factor of 2). In this paper we propose a new family of templates that can improve the detection rate by factors of 5--6 in cases where precession is most important. Our proposed method for these new ``mimic'' templates involves a hierarchical scheme of efficient, two-parameter template searches that can account for a sequence of spikes that appear in the residual inspiral phase, after one corrects for the any oscillatory modification in the phase. We present our results for two cases of compact object masses (10 and 1.4 solar masses and 7 and 3 solar masses) as a function of spin properties. Although further work is needed to fully assess the computational efficiency of this newly proposed template family, we conclude that these ``spiky templates'' are good candidates for a family of precession templates used in realistic searches, that can improve detection rates of inspiral events.Comment: 17 pages, 22 figures, version accepted by PRD. Minor revision

Gravitational waves from inspiralling compact binaries: Parameter estimation using second-post-Newtonian waveforms

The parameters of inspiralling compact binaries can be estimated using matched filtering of gravitational-waveform templates against the output of laser-interferometric gravitational-wave detectors. Using a recently calculated formula, accurate to second post-Newtonian (2PN) order [order $(v/c)^4$, where $v$ is the orbital velocity], for the frequency sweep ($dF/dt$) induced by gravitational radiation damping, we study the statistical errors in the determination of such source parameters as the ``chirp mass'' $\cal M$, reduced mass $\mu$, and spin parameters $\beta$ and $\sigma$ (related to spin-orbit and spin-spin effects, respectively). We find that previous results using template phasing accurate to 1.5PN order actually underestimated the errors in $\cal M$, $\mu$, and $\beta$. For two inspiralling neutron stars, the measurement errors increase by less than 16 percent.Comment: 14 pages, ReVTe

3D simulations of linearized scalar fields in Kerr spacetime

We investigate the behavior of a dynamical scalar field on a fixed Kerr background in Kerr-Schild coordinates using a ~311!-dimensional spectral evolution code, and we measure the power-law tail decay that occurs at late times. We compare evolutions of initial data proportional to f (r)Y,m(u ,f), where Y,m is a spherical harmonic and (r,u ,f) are Kerr-Schild coordinates, to that of initial data proportional to f (rBL)Y,m(u BL ,f), where (rBL ,u BL) are Boyer-Lindquist coordinates. We find that although these two cases are initially almost identical, the evolution can be quite different at intermediate times; however, at late times the power-law decay rates are equal