Three-dimensional Einstein-Klein-Gordon system in characteristic numerical relativity

We incorporate a massless scalar field into a three-dimensional code for the characteristic evolution of the gravitational field. The extended three-dimensional code for the Einstein-Klein-Gordon system is calibrated to be second-order convergent. It provides an accurate calculation of the gravitational and scalar radiation at infinity. As an application, we simulate the fully nonlinear evolution of an asymmetric scalar pulse of ingoing radiation propagating toward an interior Schwarzschild black hole and compute the backscattered scalar and gravitational outgoing radiation patterns. The amplitudes of the scalar and gravitational outgoing radiation modes exhibit the predicted power law scaling with respect to the amplitude of the initial data. For the scattering of an axisymmetric scalar field, the final ring down matches the complex frequency calculated perturbatively for the $\ell =2$ quasinormal mode.

Figure 1

The scalar field norm $\mathcal{Q}/{\lambda }^2$ is plotted vs retarded time $u$ for random initial scalar data with amplitude $\lambda ={10}^{-8}$ in a localized region between $r_a=3$ and $r_b=5$ on a grid of size $n_x=41$, $n_q=n_p=25$. The dashed line plots the rescaled norm $\mathcal{Q}/{\lambda }^2$ for the same initial random data but with $\lambda =2.5\times {10}^{-9}$ on a grid of size $n_x=81$, $n_q=n_p=45$.

Figure 2

The dashed line plots the waveform $r\varphi (u)$ (multiplied by ${10}^{11}$) radiated to ${\mathcal{I}}^{+}$ in the $q=p=0$ direction resulting from the backscattering of an $l=2$, $m=0$ pulse. The pulse parameters for the initial data are $r_a=3$; $r_b=5$, $\lambda ={10}^{-8}$. The grid size is $25\times 25\times 101$. The solid line is the analytic curve corresponding to the complex quasinormal mode frequency $\overline{\omega }=0.4875-i0.098$ obtained from a perturbation calculation.

Figure 3

The rescaled waveform $r\varphi (u)/\lambda$ is plotted for the $q=p=0$ direction at ${\mathcal{I}}^{+}$. The parameters of the initial data are $r_a=3$, $r_b=5$, $\ell =1$, $m=0$. The grid size is $25\times 25\times 101$. All the curves overlap for $\lambda ={10}^{-5},{10}^{-4},{10}^{-3}$.

Figure 4

The rescaled news function component $N_{+}(u)/{\lambda }^2$ is plotted for the $q=p=0.5$ direction at ${\mathcal{I}}^{+}$. The initial data parameters are $r_a=3$, $r_b=5$, $\ell =1$, $m=0$. The grid size is $25\times 25\times 101$. All the curves overlap for $\lambda ={10}^{-5},{10}^{-4},{10}^{-3}$.

Figure 5

Axisymmetric surface plots of $r\varphi$ on the North patch of ${\mathcal{I}}^{+}$ for $u=1,2,3,4$, in zig-zag order from top to bottom, resulting from the scattering of an axisymmetric pulse. The parameters of the initial data are $r_a=3$, $r_b=5$, $\lambda ={10}^{-1}$. The grid size is $89\times 89\times 101$.

Figure 6

Surface plots of the spin-weight invariant $J\overline{J}$ on the North patch of ${\mathcal{I}}^{+}$ for $u=1,2,3,4$, in zig-zag order from top to bottom, resulting from the scattering of an axisymmetric pulse with initial data parameters $r_a=3$, $r_b=5$, $\lambda ={10}^{-1}$. The grid size is $89\times 89\times 101$.

Figure 7

Global radial-angular view of $r\varphi$ on the North patch for $u=0,2,4,6,8,10$, in zig-zag order from top to bottom. We set $p=0$, $-1.2<q<1.2$ (from bottom to top), $0.5<x<1.0$ (from left to right). The initial data parameters are $r_a=3$, $r_b=5$, $\lambda ={10}^{-1}$. The grid size is $25\times 25\times 41$.

Figure 8

$r\varphi (u)/\lambda$ (multiplied by ${10}^5$) is plotted for the $q=p=0$ direction at ${\mathcal{I}}^{+}$; all the curves overlap for $\lambda ={10}^{-12},{10}^{-3},{10}^{-2},{10}^{-1}$.

Figure 9

$N_{+}(u)$ (multiplied by ${10}^2$) is plotted for the $q=p=0.5$ direction. The curves for $\lambda ={10}^{-3},{10}^{-2},{10}^{-1}$ (broken line) overlap. For $\lambda ={10}^{-12}$ (continuous line), numerical error breaks the expected quadratic scaling.

Figure 10

The rescaled coefficient $F_{20}(u)/\lambda$ (multiplied by ${10}^5$) is plotted for $\lambda ={10}^{-12},{10}^{-3},{10}^{-2},{10}^{-1}$; all the curves overlap.

Figure 11

The rescaled coefficient $F_{40}(u)/\lambda$ (multiplied by ${10}^5$) is plotted for $\lambda ={10}^{-12},{10}^{-3},{10}^{-2},{10}^{-1}$; all the curves overlap.

Figure 12

The rescaled coefficient $F_{60}(u)/\lambda$ (multiplied by ${10}^5$) is plotted for $\lambda ={10}^{-12},{10}^{-3},{10}^{-2},{10}^{-1}$; all the curves overlap.

Figure 13

Surface plots of $r\varphi$ on the South patch of ${\mathcal{I}}^{+}$ for $u=1,2,3,4$, in zig-zag order from top to bottom, resulting from the scattering of an asymmetric pulse. The initial data parameters are $r_a=3.5$, $r_b=12$, $\lambda ={10}^{-7}$, $\mu =0.03$, $q_s=0.2$, $p_s=0.3$. The grid size is $85\times 85\times 101$.

Figure 14

Surface plots of the gravitational energy flux $|N{|}^2=|N_{+}{|}^2+|N_{\times }{|}^2$ on the South patch of ${\mathcal{I}}^{+}$ for $u=1,2,3,4$, in zig-zag order from top to bottom. The initial data parameters are $r_a=3.5$, $r_b=12$, $\lambda ={10}^{-7}$, $\mu =0.03$, $q_s=0.2$, $p_s=0.3$. The grid size is $85\times 85\times 101$. The maximum value of $|N|$ is of order ${10}^{-15}$.