Gravitational Wavetrains in the Quasi-Equilibrium Approximation: A Model Problem in Scalar Gravitation

A quasi-equilibrium (QE) computational scheme was recently developed in general relativity to calculate the complete gravitational wavetrain emitted during the inspiral phase of compact binaries. The QE method exploits the fact that the the gravitational radiation inspiral timescale is much longer than the orbital period everywhere outside the ISCO. Here we demonstrate the validity and advantages of the QE scheme by solving a model problem in relativistic scalar gravitation theory. By adopting scalar gravitation, we are able to numerically track without approximation the damping of a simple, quasi-periodic radiating system (an oscillating spherical matter shell) to final equilibrium, and then use the exact numerical results to calibrate the QE approximation method. In particular, we calculate the emitted gravitational wavetrain three different ways: by integrating the exact coupled dynamical field and matter equations, by using the scalar-wave monopole approximation formula (corresponding to the quadrupole formula in general relativity), and by adopting the QE scheme. We find that the monopole formula works well for weak field cases, but fails when the fields become even moderately strong. By contrast, the QE scheme remains quite reliable for moderately strong fields, and begins to breakdown only for ultra-strong fields. The QE scheme thus provides a promising technique to construct the complete wavetrain from binary inspiral outside the ISCO, where the gravitational fields are strong, but where the computational resources required to follow the system for more than a few orbits by direct numerical integration of the exact equations are prohibitive.Comment: 15 pages, 14 figure

Computing the Complete Gravitational Wavetrain from Relativistic Binary Inspiral

We present a new method for generating the nonlinear gravitational wavetrain from the late inspiral (pre-coalescence) phase of a binary neutron star system by means of a numerical evolution calculation in full general relativity. In a prototype calculation, we produce 214 wave cycles from corotating polytropes, representing the final part of the inspiral phase prior to reaching the ISCO. Our method is based on the inequality that the orbital decay timescale due to gravitational radiation is much longer than an orbital period and the approximation that gravitational radiation has little effect on the structure of the stars. We employ quasi-equilibrium sequences of binaries in circular orbit for the matter source in our field evolution code. We compute the gravity-wave energy flux, and, from this, the inspiral rate, at a discrete set of binary separations. From these data, we construct the gravitational waveform as a continuous wavetrain. Finally, we discuss the limitations of our current calculation, planned improvements, and potential applications of our method to other inspiral scenarios.Comment: 4 pages, 4 figure

Implementing fully relativistic hydrodynamics in three dimensions

We report on our numerical implementation of fully relativistic hydrodynamics coupled to Einstein's field equations in three spatial dimensions. We briefly review several steps in our code development, including our recasting of Einstein's equations and several tests which demonstrate its advantages for numerical integrations. We outline our implementation of relativistic hydrodynamics, and present numerical results for the evolution of both stable and unstable Oppenheimer-Volkov equilibrium stars, which represent a very promising first test of our code.Comment: 5 Pages, 4 Figures, submitted to Proceedings of the 8th Canadian Conference on General Relativity and Relativistic Astrophysic

Analytical Representation of a Black Hole Puncture Solution

The ``moving puncture'' technique has led to dramatic advancements in the numerical simulations of binary black holes. Hannam et.al. have recently demonstrated that, for suitable gauge conditions commonly employed in moving puncture simulations, the evolution of a single black hole leads to a well-known time-independent, maximal slicing of Schwarzschild. They construct the corresponding solution in isotropic coordinates numerically and demonstrate its usefulness, for example for testing and calibrating numerical codes that employ moving puncture techniques. In this Brief Report we point out that this solution can also be constructed analytically, making it even more useful as a test case for numerical codes

The Moment of Inertia of the Binary Pulsar J0737-3039A: Constraining the Nuclear Equation of State

We construct numerical models of the newly discovered binary pulsar J0737-3039A, both with a fully relativistic, uniformly rotating, equilibrium code that handles arbitrary spins and in the relativistic, slow-rotation approximation. We compare results for a representative sample of viable nuclear equations of state (EOS) that span three, qualitatively different, classes of models for the description of nuclear matter. A future dynamical measurement of the neutron star's moment of inertia from pulsar timing data will impose significant constraints on the nuclear EOS. Even a moderately accurate measurement (<~ 10 %) may be able to rule out some of these competing classes. Using the measured mass, spin and moment of inertia to identify the optimal model computed from different EOSs, one can determine the pulsar's radius.Comment: 4 pages, ApJL in pres

Stability of coalescing binary stars against gravitational collapse: hydrodynamical simulations

We perform simulations of relativistic binary stars in post-Newtonian gravity to investigate their dynamical stability prior to merger against gravitational collapse in a tidal field. In general, our equations are only strictly accurate to first post-Newtonian order, but they recover full general relativity for spherical, static stars. We study both corotational and irrotational binary configurations of identical stars in circular orbits. We adopt a soft, adiabatic equation of state with $\Gamma = 1.4$, for which the onset of instability occurs at a sufficiently small value of the compaction $M/R$ that a post-Newtonian approximation is quite accurate. For such a soft equation of state there is no innermost stable circular orbit, so that we can study arbitrarily close binaries. This choice still allows us to study all the qualitative features exhibited by any adiabatic equation of state regarding stability against gravitational collapse. We demonstrate that, independent of the internal stellar velocity profile, the tidal field from a binary companion stabilizes a star against gravitational collapse.Comment: 13 pages, 10 figures, RevTex, to appear in Phys. Rev.

A Linear-Nonlinear Formulation of Einstein Equations for the Two-Body Problem in General Relativity

A formulation of Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions. The key idea behind this formulation is a separation of the dynamical variables into i) a fixed conformal 3-geometry, ii) a conformal factor possessing nonlinear dynamics and iii) transverse-traceless perturbations of the conformal 3-geometry.Comment: 7 pages, no figure

Evolving Einstein's Field Equations with Matter: The ``Hydro without Hydro'' Test

We include matter sources in Einstein's field equations and show that our recently proposed 3+1 evolution scheme can stably evolve strong-field solutions. We insert in our code known matter solutions, namely the Oppenheimer-Volkoff solution for a static star and the Oppenheimer-Snyder solution for homogeneous dust sphere collapse to a black hole, and evolve the gravitational field equations. We find that we can evolve stably static, strong-field stars for arbitrarily long times and can follow dust sphere collapse accurately well past black hole formation. These tests are useful diagnostics for fully self-consistent, stable hydrodynamical simulations in 3+1 general relativity. Moreover, they suggest a successive approximation scheme for determining gravitational waveforms from strong-field sources dominated by longitudinal fields, like binary neutron stars: approximate quasi-equilibrium models can serve as sources for the transverse field equations, which can be evolved without having to re-solve the hydrodynamical equations (``hydro without hydro'').Comment: 4 postscript figures. Submitted to Phys. Rev. D15 as a Brief Repor