Collisions of charged black holes

We perform fully nonlinear numerical simulations of charged-black-hole collisions, described by the Einstein-Maxwell equations, and contrast the results against analytic expectations. We focus on head-on collisions of nonspinning black holes, starting from rest and with the same charge-to-mass ratio, $Q/M$. The addition of charge to black holes introduces a new interesting channel of radiation and dynamics, most of which seem to be captured by Newtonian dynamics and flat-space intuition. The waveforms can be qualitatively described in terms of three stages: (i) an infall phase prior to the formation of a common apparent horizon; (ii) a nonlinear merger phase that corresponds to a peak in gravitational and electromagnetic energy; (iii) the ringdown marked by an oscillatory pattern with exponentially decaying amplitude and characteristic frequencies that are in good agreement with perturbative predictions. We observe that the amount of gravitational-wave energy generated throughout the collision decreases by about 3 orders of magnitude as the charge-to-mass ratio $Q/M$ is increased from 0 to 0.98. We interpret this decrease as a consequence of the smaller accelerations present for larger values of the charge. In contrast, the ratio of energy carried by electromagnetic to gravitational radiation increases, reaching about 22% for the maximum $Q/M$ ratio explored, which is in good agreement with analytic predictions.

Figure 1

The numerical profiles for ${\gamma }_{zz}$ and $E^z$ (symbols) obtained in geodesic slicing at various times $\tau$ are compared with the semianalytic results (lines).

Figure 2

Convergence analysis for simulation d08q05 of Table  with resolutions $h_c=M/64$, $h_m=M/80$, and $h_f=M/96$. The panels show differences of the $(2,0)$ multipoles of the real parts of ${\Psi }_4$ (left) and ${\Phi }_2$ (right) extracted at $R_{\mathrm{ex}}=100M$; in each case, the high-resolution differences have been rescaled by a factor of 2.78 as expected for fourth-order convergence.

Figure 3

Monopole ${\varphi }_1^{00}$ (left) and quadrupole ${\varphi }_1^{20}$ (right) of the radial part of the electromagnetic field ${\Phi }_1$ extracted at $R_{\mathrm{ex}}=100M$ for simulation d08q05 of Table . The dashed curves show the predictions of Eqs. (4.3, 4.4) at $R=\infty$ in the static limit. For the monopole case, we also added the curves obtained by extrapolating the results to infinite extraction radius; these curves—dotted lines—essentially overlap with the predictions from Eq. (4.3).

Figure 4

Time for apparent horizon formation, rescaled by the factor $\sqrt{\mathcal{B}}$ and the apparent horizon formation time $t_0$ for an electrically neutral binary. We note that the change in the quantity we plot is only, at most, of 2%. The coordinate time itself, however, varies by a factor of 5 as one goes from $Q=0$ to $Q=0.98M$.

Figure 5

Real part of the $(2,0)$ mode of ${\Psi }_4$ (left) and ${\Phi }_2$ (right) extracted at $R_{\mathrm{ex}}=100M$.

Figure 6

Power spectrum for the gravitational (long dashed line) and electromagnetic (short dashed line) quadrupole extracted from simulation d08q03. Note that the spectrum peaks near the fundamental ringdown frequency of the gravitational mode; cf. Table .

Figure 7

Radiated fluxes for simulations d08q05, d08q09, and d08q00 of Table . We have aligned the curves in time such that their global maximum coincides with $t=0$. The inset shows the exact same plot with the $y$ axis in logarithmic units.

Figure 8

Energy radiated in the gravitational and electromagnetic quadrupoles as well as the ratio of the two as a function of $Q/M$.