One dimensional description of the gravitational perturbation in a Kerr background

The perturbation equation in a Kerr background is written as a coupled system of one dimensional equations for the different modes in the time domain. Numerical simulations show that the dominant mode in the gravitational response is the one corresponding to the mode of the initial perturbation, allowing us to conjecture that the coupling among the modes has a weak influence in our system of equations. We conclude that by neglecting the coupling terms it can be obtained as a one dimensional harmonic equation which indeed describes with good accuracy the gravitational response from the Kerr black hole with low spin, while only a few couplings are necessary to describe a high-spin one. This result may help us to understand the structure of test fields in a Kerr background and even to generate accurate waveforms for various cases in an efficient manner.

Figure 1

The waveform produced by an initial perturbation described by the $l=2$, $m=0$ mode, for two values of the rotational parameter of the black hole, $a=0.2$ at the top and $a=0.9$ at the bottom. The signal is measured by an observer located at $r=70M$.

Figure 2

As in the previous figure, but for an initial perturbation described by the $l=2$, $m=2$ mode.

Figure 3

At the top the signal produced by the axial code is displayed, measured by an observer located at $r=80M$ in the equatorial plane $\theta =\pi /2$. The rotational parameter of the black hole is $a=0.999$. At the bottom are displayed a superposition of the projection of the signal in the $Y_{20}$ and $Y_{30}$ mode and the ones obtained by the one dimensional harmonic decomposition.