Extended two-dimensional characteristic framework to study nonrotating black holes

We develop a numerical solver that extends the computational framework considered in [Phys. Rev. D 65, 084016 (2002)] to include scalar perturbations of nonrotating black holes. The nonlinear Einstein-Klein-Gordon equations for a massless scalar field minimally coupled to gravity are solved in two spatial dimensions. The numerical procedure is based on the ingoing light-cone formulation for an axially and reflection-symmetric spacetime. The solver is second-order accurate and was validated in different ways. We use for calibration an auxiliary 1D solver with the same initial and boundary conditions and the same evolution algorithm. We reproduce the quasinormal modes for the massless scalar field harmonics $\ell =0$, 1, and 2. For these same harmonics, in the linear approximation, we calculate the balance of energy between the black hole and the world tube. As an example of nonlinear harmonic generation, we show the distortion of a marginally trapped two-surface approximated as a q-boundary and based upon the harmonic $\ell =2$. Additionally, we study the evolution of the $\ell =8$ harmonic in order to test the solver in a spacetime with a complex angular structure. Further applications and extensions are briefly discussed.

Figure 1

Spacetime diagram with the problem setup. The foliation is based on advanced time $v$; the geometry of the world tube ($\mathcal{W}$) is kept fixed at all times and is given by the Schwarzschild values. The ingoing light cones emanate from $\mathcal{W}$. The initial ingoing light cone $\mathcal{N}$ at $v_0$ is distorted with the specification of an arbitrary outgoing massless scalar field. In general, the evolution generates gravitational radiation that, together with scalar radiation, is scattered toward and away from the distorted black hole.

Figure 2

Ingoing marching algorithm for one particular angle. Initial data are set on the initial null cone $\mathcal{N}$; boundary conditions are set on the world tube $\mathcal{W}$. Successive levels of the advanced time $v$ are depicted (diagonal lines $n$ and $n+1$) along with the radial grid (dashed vertical lines $i-2$, $i-1$, $i$, $i+1$). The radial grid starts at $\mathcal{W}$ ($i=0$) and terminates inside the horizon $\mathcal{H}$, at the inner boundary $\mathcal{B}$ ($i=N_r$). The evolution equation for $\widehat{\gamma }$ relates the field values ${\widehat{\gamma }}_R$, ${\widehat{\gamma }}_S$, ${\widehat{\gamma }}_P$, ${\widehat{\gamma }}_Q$ to the value of ${\mathcal{H}}_C$. The same structure of the integration procedure applies to $\psi$.

Figure 3

Treatment of the first (left) and second (right) points for the ingoing marching algorithm.

Figure 4

Distribution of the grid points on the ingoing light cone using a pseudo-Cartesian space $X=r\cos \vartheta$, $Y=r\sin \vartheta$, where $\vartheta =\pi /2-\theta$. The points $X=0$ denote the axis of symmetry, whereas $Y=0$ denotes the equatorial plane. The grid is uniform in $r$ and $y$ but nonuniform in $\theta$.

Figure 5

Log of the absolute value of the function $r\psi$ at $r=10M$ as a function of Bondi time, showing the quasinormal mode regime oscillations for $\ell =0$. The solid line is the output for $N_y=90$ and $N_r=2000$; the dashed line is the quasinormal mode extracted from the data.

Figure 6

Log of the absolute value of the function $r\psi$ at $r=10M$ as a function of Bondi time, showing the quasinormal mode regime oscillations for $\ell =1$. The solid line is the output for $N_y=90$ and $N_r=2000$; the dashed line is the quasinormal mode extracted from the data.

Figure 7

Log of the absolute value of the function $r\psi$ at $r=10M$ as a function of Bondi time, showing the quasinormal mode regime oscillations for $\ell =2$. The solid line is the output for $N_y=90$ and $N_r=2000$; the dashed line is the quasinormal mode extracted from the data.

Figure 8

Energy conservation (multiplied by ${10}^8$) as a function of the Bondi time for $\ell =0$ (curve 0), $\ell =1$ (curve 1), NS $\ell =2$ (curve 2). This calculation was done using the same grid parameters as for Fig. 5. For each specific $\ell$, the descending curve corresponds to energy given by Eq. (24). The ascending curve corresponds to the algebraic sum of $E_{\text{in}}=\int P_{\text{in}}dv$ and $E_{\text{out}}=\int P_{\text{out}}dv$.

Figure 9

Percentage variation in $\mathrm{\Sigma }(v)$ with respect to $\mathrm{\Sigma }(0)$ as a function of Bondi time for $\ell =0$ (curve 0), $\ell =1$ (curve 1), and $\ell =2$ (curve 2). The graph shows that energy is conserved to within less than 4.5% of the energy content of the initial surface.

Figure 10

Radius of the $q$ boundary as a function of $y=-\cos (\theta )$ for $\ell =2$, $N_y=90$, $N_r=2000$, $\lambda =0.1$, and different Bondi times $v$: 0 (curve 1), 0.4 (curve 2), and 0.8 (curve 3). The MTS is estimated using the $q$ boundary method (see Ref. [23]) with one iteration.

Figure 11

Sequence of snapshots illustrating the evolution of finite-amplitude black hole oscillations via an ingoing light-cone approach. The upper-left panel shows the scalar field $r\psi$ (multiplied by ${10}^6$) at $v=0$, representing a localized finite-amplitude perturbation of a black hole. The plot coordinates are pseudo-Cartesian: $X=r\cos \vartheta$, $Y=r\sin \vartheta$, where $\vartheta =\pi /2-\theta$, hence placing the axis of symmetry along $X=0$ and the equator along $Y=0$. Shown are contour levels of the distortion of the light cone. In this case, the initial datum is an $l=8$ harmonic. The next snapshot (upper-right panel) shows the evolution of $r\psi$ at $v=2.5M$. The outward propagation of the data is visible, but also the change of phase near the horizon. The sequence proceeds with the lower-left panel, at time $v=5M$. The evolution continues in the lower-right panel ($v=7.5M$). The panel shows only the innermost region of the computational domain (about $1/12$ of the total radial extent). Despite the oscillatory nature of these first cycles, examination of the signal shows that only for later oscillations does the black hole spacetime approach the typical quasinormal mode ringing.