Evolving metric perturbations in dynamical Chern-Simons gravity

The stability of rotating black holes in dynamical Chern-Simons gravity (dCS) is an open question. To study this issue, we evolve the leading-order metric perturbation in order-reduced dynamical Chern-Simons gravity. The source is the leading-order dCS scalar field coupled to the spacetime curvature of a rotating black hole background. We use a well-posed, constraint-preserving scheme. We find that the leading-order metric perturbation numerically exhibits linear growth, but that the level of this growth converges to zero with numerical resolution. This analysis shows that spinning black holes in dCS gravity are numerically stable to leading-order perturbations in the metric.

Figure 1

Constraint damping functions ${\gamma }_0$ and ${\gamma }_2$ used to evolve metric perturbations on a Kerr background. The functions are largest where the metric perturbation source has the highest value, and exponentially decay to $R\to \infty$. While the functions extend to $R=0$, the computational domain terminates outside the apparent horizon inner boundary (here shown by the black dashed line at $R=2M$ in the case of Schwarzschild).

Figure 2

Behavior of the perturbed constraints given in Sec. 3e for a dCS perturbation on a Kerr background with $\chi =0.1$. For each constraint $\mathrm{\Delta }{C}_A$, we compute the L2 norm of the constraint over the entire computational domain ($\parallel \mathrm{\Delta }{C}_1\parallel$ for the 1-index constraint, for example), and divide by the L2 norm of its normalization factor ($\parallel N_A\parallel$) (cf. Sec. 3e). We see that the constraints remain constant in time and are exponentially convergent with resolution.

Figure 3

Metric perturbation $\mathrm{\Delta }{g}_{ab}$ on a Kerr background with $\chi =0.1$. We present the behavior at low, medium, and high resolutions, and find that we increase the numerical resolution, the level of linear growth in time decreases.

Figure 4

Similar to Fig. 3, but for spin of $\chi =0.6$. For each resolution, we use the initial data for $\mathrm{\Delta }{g}_{ab}$ we have solved for at that resolution, and hence $\mathrm{\Delta }{g}_{ab}$ has different initial values depending on resolution. We have checked that these initial values converge to the highest-resolution result.

Figure 5

Behavior of the derivative of the norm of the metric perturbation with time for a background with spin $\chi =0.1$. We plot ${\partial }_t\parallel \mathrm{\Delta }{g}_{ab}\parallel$, the time derivative of the norm of the metric perturbation. Each line corresponds to a different resolution. We see that after an initial period of junk radiation, the time derivative is convergent towards zero with increasing numerical resolution.

Figure 6

Similar to Fig. 5, but for spin $\chi =0.6$.

Figure 7

Similar to Fig. 5, but for spin $\chi =0.9$.

Figure 8

The structure of this figure is similar to that of Fig. 5. However, in this case, we solve for the initial data for $\mathrm{\Delta }{g}_{ab}$ purely at the “High” resolution. We interpolate this data onto the “Low” and “Medium” resolution grids to give initial data at these resolutions. We see that as the simulation progresses, the linear growth in $\mathrm{\Delta }{g}_{ab}$ remains at roughly the same level for all resolutions. This suggests that the zero-frequency mode in $\mathrm{\Delta }{g}_{ab}$ is present in and due to the resolution of the initial data, rather than spontaneously appearing during the evolution.

Figure 9

Constraints for evolution of a transformed multipolar wave perturbation on flat space, as described in Sec. pp2-s1. For each constraint $\mathrm{\Delta }{C}_A$, we compute the L2 norm of the constraint over the entire computational domain ($\parallel \mathrm{\Delta }{C}_1\parallel$ for the 1-index constraint, for example), and divide by the L2 norm of its normalization factor ($\parallel N_A\parallel$) (cf. Sec. 3e). We see that the constraints converge exponentially with numerical resolution.

Figure 10

Behavior of $\mathrm{\Delta }{g}_{ab}$ for the multipolar wave test described in Sec. pp2-s1 for low, medium, and high resolution. We see that the value of the metric perturbation decreases as the wave propagates toward $R\to \infty$ (and leaves the computational domain), and that with increasing resolution the behavior of the variables converges to the highest-resolution value. We additionally plot the analytical solution for the behavior of the multipolar wave, which sits on top of the highest-resolution result.

Figure 11

Behavior of $\mathrm{\Delta }{g}_{ab}$ for the small data on Schwarzschild test described in Sec. pp2-s2. We see that with increasing time, the field with initial magnitude of $\sim {10}^{-16}$ remains close to roundoff error.