Hybrid approach to black hole perturbations from extended matter sources

We present a new method for the calculation of black hole perturbations induced by extended sources in which the solution of the nonlinear hydrodynamics equations is coupled to a perturbative method based on Regge-Wheeler/Zerilli and Bardeen-Press-Teukolsky equations when these are solved in the frequency domain. In contrast to alternative methods in the time domain which may be unstable for rotating black hole spacetimes, this approach is expected to be stable as long as an accurate evolution of the matter sources is possible. Hence, it could be used under generic conditions and also with sources coming from three-dimensional numerical relativity codes. As an application of this method we compute the gravitational radiation from an oscillating high-density torus orbiting around a Schwarzschild black hole and show that our method is remarkably accurate, capturing both the basic quadrupolar emission of the torus and the excited emission of the black hole.

Figure 1

Harmonic behavior of an oscillating torus. Both panels refer to model (c) of Table  which was perturbed with a radial velocity having $\eta =0.05$. The left panel shows the time variation of the maximum rest-mass density normalized to its initial value for a representative portion of the timeseries. The right panel shows the power spectral density (PSD) calculated over 100 orbital timescales, with the eigenfrequency of the $f$ mode indicated with a vertical dashed line at 227 Hz.

Figure 2

Gravitational-wave energy spectrum produced by an oscillating, high-density torus orbiting around a $2.5M_{\odot }$ Schwarzschild black hole. The radiation has been computed for the $\ell =2$ mode using Eq. (2.13) within the Newtonian quadrupole approximation.

Figure 3

Same as Fig. 2 but derived integrating the RWZ equations in the frequency domain for $\ell =2$ and using Eq. (2.6).

Figure 4

Same as Fig. 2 but obtained integrating the BPT equation in the frequency domain for $\ell =2$ and using Eq. (2.11).