Gravitational wave trains in the quasiequilibrium approximation: A model problem in scalar gravitation

A quasiequilibrium (QE) computational scheme was recently developed in general relativity to calculate the complete'gravitational wave train emitted during the inspiral phase of compact binaries. The QE method exploits the fact that the gravitational radiation inspiral time scale 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, quasiperiodic 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 wave train 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 ultrastrong fields. The QE scheme thus provides a promising technique to construct the complete wave train from binary inspirai 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.