Numerical investigation of the late-time Kerr tails

Abstract

The late-time behavior of a scalar field on fixed Kerr background is examined in a numerical framework incorporating the techniques of conformal compactification and hyperbolic initial value formulation. The applied code is 1+(1+2) as it is based on the use of the spectral method in the angular directions while in the time-radial section fourth order finite differencing, along with the method of lines, is applied. The evolution of various types of stationary and non-stationary pure multipole initial states are investigated. The asymptotic decay rates are determined not only in the domain of outer communication but along the event horizon and at future null infinity as well. The decay rates are found to be different for stationary and non-stationary initial data, and they also depend on the fall off properties of the initial data toward future null infinity. The energy and angular momentum transfers are found to show significantly different behavior in the initial phase of the time evolution. The quasinormal ringing phase and the tail phase are also investigated. In the tail phase, the decay exponents for the energy and angular momentum losses at I + are found to be smaller than at the horizon which is in accordance with the behavior of the field itself and it means that at late times the energy and angular momentum falling into the black hole become negligible in comparison with the energy and angular momentum radiated toward I + . The energy and angular momentum balances are used as additional verifications of the reliability of our numerical method. *

Axisymmetric fully spectral code for hyperbolic equations

“…In implicit methods, eqs. (27) represent for an ODE a system of algebraic equations for the K A (i) , whereas for PDEs in (1+d)-dimensions, they correspond to a system of spatial differential equations in d dimensions. By restricting ourselves to Diagonally Implicit Runge-Kutta methods (DIRK), in which a (i)(j) = 0 for (j) > (i), the system decouples with respect to the Runge-Kutta index (i), i.e., for each (i) = 1...s, eq.…”

“…For explicit equations, the formulations (25) and (27) are completely equivalent to (30) and (31). We now assume that the latter formulation can be carried over to the realm of general implicit ODEs of the form…”

Slicing black hole spacetimes

“…Recently this method has been successfully applied to the numerical investigation of tail decay rates in a Kerr spacetime by Racz and Toth (RT) [43] (see also Refs. [44,45,46]).…”

Null Infinity Waveforms from Extreme-Mass-Ratio Inspirals in Kerr Spacetime

“…The only numerical computations of black-hole perturbations using the hyperboloidal method without spherical symmetry deal with scalar perturbations [62][63][64][65]. One goal of this paper is to present the application of the hyperboloidal method to Teukolsky equations in Kerr spacetime for the calculation of gravitational waveforms at null infinity.…”

“…One is to use a single smooth surface avoiding the transition zone of [55]. This technique has recently been applied by Rácz and Tóth in a detailed study of polynomial decay rates of a scalar field in Kerr spacetime [64]. They construct the first smooth, horizon-penetrating, hyperboloidal foliation of Kerr spacetime for their study.…”