Numerical investigation of the dynamics of linear spin $s$ fields on a Kerr background: Late-time tails of spin $s=\pm 1,\pm 2$ fields

The time evolution of linear fields of spin $s=\pm 1$ and $s=\pm 2$ on Kerr black hole spacetimes are investigated by solving the homogeneous Teukolsky equation numerically. The applied numerical setup is based on a combination of conformal compactification and the hyperbolic initial value problem. The evolved basic variables are expanded in terms of spin-weighted spherical harmonics, which allows us to evaluate all angular derivatives analytically, whereas the evolution of the expansion coefficients in the time-radial section is determined by applying the method of lines implemented in a fourth order accurate finite differencing stencil. Concerning the initialization, in all of our investigations, single mode excitations—either static or purely dynamical-type initial data—are applied. Within this setup the late-time tail behavior is investigated. Because of the applied conformal compactification, the asymptotic decay rates are determined at three characteristic locations—in the domain of outer communication, at the event horizon, and at future null infinity—simultaneously. A recently introduced new type of “energy” and “angular momentum” balance relations are also applied in order to demonstrate the feasibility and robustness of the developed numerical schema and also to verify the proper implementation of the underlying mathematical model.

Figure 1

For the specific choice of excitation with multipole parameters $s=-1$, ${\ell }^{\prime }=1$, $m=0$, the time dependence of the $\ell =1$ mode is depicted at the outer horizon $R=R_{+}$ at the $R=1$ line representing future null infinity and at some intermediate $R=\text{const}$ locations. At each fixed spatial location, the initial long-lasting quasinormal oscillatory phase is supplemented by a specific power-law decay corresponding to some integer rate $n$.

Figure 2

The $T$ dependence of the LPI of the $\ell =1$ excited mode is depicted for various resolutions. The multipole indices of applied static excitation were $s=1$, ${\ell }^{\prime }=1$, $m=1$. It is visible that the use of 512 spatial grid points does not even allow us to get a hint of the correct value of LPI. It is also clearly visible that by increasing the number of involved grid points the value of $\mu$ becomes more accurate.

Figure 3

The $R$ dependence of the LPI values is depicted for a mode with multipole indices $s=1$, $\ell =1$, $m=0$ that was yielded by applying a static initial excitation with ${\ell }^{\prime }=1$. The value of $\mu$ at the outer horizon $R=R_{+}$ is $-6$, while it is $-2$ at the $R=1$ line representing future null infinity. It is also clearly indicated that the closer a finite intermediate spatial location with $R_{+}<R<1$ is to the outer horizon $R=R_{+}$ or to the $R=1$ line representing future null infinity, the longer it takes to settle down to the pertinent shared LPI value $\mu =-5$.

Figure 4

The time dependence of the real and imaginary parts of the energy and angular momentum balance relations $\delta E$ and $\delta J$ are shown, respectively. This is done in each panel for three different resolutions with 1024,2048,4096 spatial grid points. In order to make it transparent that our numerical implementation, as desired, is of fourth order accurate, the numerical values of $\delta E$ and $\delta J$ relevant for the resolutions 2048 and 4096 are multiplied by 16 and 256, respectively. (a) The time dependence of the real part $\mathrm{Re}(\delta E)$ of energy balance relation $\delta E$ is plotted for three resolutions. The convergence rate is slightly better than 4. (b) The time dependence of the imaginary part $\mathrm{Im}(\delta E)$ of energy balance relation $\delta E$ is depicted for three resolutions. The convergence rate is exactly 4. (c) The time dependence of the real part $\mathrm{Re}(\delta J)$ of angular momentum balance relation $\delta J$ is shown. The convergence rate is seen to be slightly better than 4. (d) The time dependence of the real part $\mathrm{Im}(\delta J)$ of angular momentum balance relation $\delta J$ is shown. The convergence rate remains close to 4.

Figure 5

The time dependence of the real and imaginary parts of the total energy ${\mathcal{E}}_T$ and total angular momentum ${\mathcal{J}}_T$, as well as those of the corresponding flows ${\mathcal{F}}_E$ and ${\mathcal{F}}_J$ are depicted. Though only the initial parts of the time evolution are shown in these plots, they make transparent about 99% of the pertinent transport processes. (a) The time dependence of the real part $\mathrm{Re}({\mathcal{E}}_T)$ of the total energy ${\mathcal{E}}_T$, along with the real part $\mathrm{Re}({\mathcal{F}}_E)$ of the integrated energy flow ${\mathcal{F}}_E$, evaluated at $R_{+}$ and at ${\mathcal{I}}^{+}$, is plotted. It is visible that first about half of the energy stored by the initial data is leaving the domain of outer communication via the black hole event horizon. The other half leaves later through future null infinity ${\mathcal{I}}^{+}$. (b) The imaginary part $\mathrm{Im}({\mathcal{E}}_T)$ of the total energy ${\mathcal{E}}_T$, along with the imaginary part $\mathrm{Im}({\mathcal{F}}_E)$ of the integrated energy flow ${\mathcal{F}}_E$, evaluated at $R_{+}$ and at ${\mathcal{I}}^{+}$, is plotted against time. Two spikes are visible on this figure which occur exactly when the pulses go through the event horizon and through future null infinity. Apart from these spikes the transport processes here remain small scale. (c) The time dependence of the real part $\mathrm{Re}({\mathcal{J}}_T)$ of the total angular momentum ${\mathcal{J}}_T$, along with the real part $\mathrm{Re}({\mathcal{F}}_J)$ of the integrated energy flow ${\mathcal{F}}_J$, evaluated at $R_{+}$ and at ${\mathcal{I}}^{+}$, is plotted. The angular momentum drops significantly by the loss through the black hole event horizon which is followed by some negative flow through null infinity. Notably, there is more angular momentum in the system at $T=10$ than around $T=5$. (d) The imaginary part $\mathrm{Im}({\mathcal{J}}_T)$ of the total angular momentum ${\mathcal{J}}_T$, along with the real part $\mathrm{Im}({\mathcal{F}}_J)$ of the integrated energy flow ${\mathcal{F}}_J$, evaluated at $R_{+}$ and at ${\mathcal{I}}^{+}$, is plotted against time. The losses here are of opposite sign w.r.t. the real part, i.e. the flow through the black hole event horizon is negative whereas the flow through null infinity is positive. These losses almost emptying the imaginary part of the total angular momentum.