Collisions of oppositely charged black holes

The first fully nonlinear numerical simulations of colliding charged black holes in $D=4$ Einstein-Maxwell theory were recently reported [Zilhão et al., Phys. Rev. D 85, 124062 (2012)]. These collisions were performed for black holes with equal charge-to-mass ratio, for which initial data can be found in closed analytic form. Here we generalize the study of collisions of charged black holes to the case of unequal charge-to-mass ratios. We focus on oppositely charged black holes, as to maximize acceleration-dependent effects. As $|Q|/M$ increases from 0 to 0.99, we observe that the gravitational radiation emitted increases by a factor of $\sim 2.7$; the electromagnetic radiation emission becomes dominant for $|Q|/M\gtrsim 0.37$ and at $|Q|/M=0.99$ is larger, by a factor of $\sim 5.8$, than its gravitational counterpart. We observe that these numerical results exhibit a precise and simple scaling with the charge. Furthermore, we show that the results from the numerical simulations are qualitatively captured by a simple analytic model that computes the electromagnetic dipolar radiation and the gravitational quadrupolar radiation of two nonrelativistic interacting particles in Minkowski spacetime.

Figure 1

Predictions from Eqs. (3.4) and (3.5) (black dotted lines) matched against our simulation results (models q+-070_d16_hf80 and q+-090_d32_hf192; see Table 1). Left plot shows results obtained from fitting a curve of the form $R_{\mathrm{ex}}^3{\varphi }_{1,2}^{10}=a_0+a_1/R_{\mathrm{ex}}$ to the numerically extracted ${\varphi }_{1,2}^{10}$ (for all time steps) using all available extraction radii. Right plot (showing only the $Q=\pm 0.9M$ case) depicts explicitly the extracted $R_{\mathrm{ex}}^3{\varphi }_1^{10}$ for all extraction radii. In this case, the curves from bottom to top correspond to $R_{\mathrm{ex}}=100M,\dots ,160M$ in steps of $10M$, with the uppermost being the result extrapolated to infinity. We note that for these cases, the BH merger happens roughly at around $t\sim 330M$.

Figure 2

The electric (left panel) and Hamiltonian (right panel) constraints along the collision axis at time $t=384M$ for model q+-090_d32. The solid (black) curves display the result obtained for lower resolution $h_f=M/160$ and the dashed (red) curves show that obtained for higher resolution $h_f=M/196$ and amplified by $1.2^4$ for the expected fourth-order convergence.

Figure 3

Real part of the electromagnetic ($l=1$, $m=0$ mode) and gravitational ($l=2$, $m=0$ mode) waveforms. These have been conveniently rescaled and shifted in time so that their peaks coincide.

Figure 4

Total energy radiated in the electromagnetic ($E_{\mathrm{rad}}^{\mathrm{EM}}$) and gravitational ($E_{\mathrm{rad}}^{\mathrm{GW}}$) channels. Solid lines show a fit to the numerical results of the form $E_{\mathrm{rad}}^{\mathrm{GW}}=8.53\times 10^{-5}+4.55\times 10^{-4}{\mathcal{B}}^{3/2}$, $E_{\mathrm{rad}}^{\mathrm{EM}}=4.00\times 10^{-3}{Q}^2\mathcal{B}$, in agreement with the scaling used in Fig. 3. Dotted lines show results from the analytic approximation taking $z_c/M=1.5$.