Vacuum electromagnetic counterparts of binary black-hole mergers

As one step towards a systematic modeling of the electromagnetic (EM) emission from an inspiralling black hole binary we consider a simple scenario in which the binary moves in a uniform magnetic field anchored to a distant circumbinary disc. We study this system by solving the Einstein-Maxwell equations in which the EM fields are chosen with strengths consistent with the values expected astrophysically and treated as test fields. Our initial data consists of a series of binaries with spins aligned or antialigned with the orbital angular momentum and we study the dependence of gravitational and EM signals with different spin configurations. Overall we find that the EM radiation in the lowest $\ell =2$, $m=2$ multipole accurately reflects the gravitational one, with identical phase evolutions and amplitudes that differ only by a scaling factor. This is no longer true when considering higher $\ell$ modes, for which the amplitude evolution of the scaled EM emission is slightly larger, while the phase evolutions continue to agree. We also compute the efficiency of the energy emission in EM waves and find that it scales quadratically with the total spin and is given by $E_{\mathrm{EM}}^{\mathrm{rad}}/M\simeq {10}^{-15}(M/{10}^8{M}_{\odot }{)}^2(B/{10}^4\mathrm{G}{)}^2$, hence 13 orders of magnitude smaller than the gravitational energy for realistic magnetic fields. Although large in absolute terms, the corresponding luminosity is much smaller than the accretion luminosity if the system is accreting at near the Eddington rate. Most importantly, this EM emission is at frequencies of $\sim {10}^{-4}({10}^8{M}_{\odot }/M)\mathrm{Hz}$, which are well outside those accessible to astronomical radio observations. As a result, it is unlikely that the EM emission discussed here can be detected directly and simultaneously with the gravitational-wave one. However, indirect processes, driven by changes in the EM fields behavior could yield observable events. In particular we argue that if the accretion rate of the circumbinary disc is small and sufficiently stable over the timescale of the final inspiral, then the EM emission may be observable indirectly as it will alter the accretion rate through the magnetic torques exerted by the distorted magnetic field lines.

Figure 1

Recovery of Wald’s solution for an isolated Kerr black hole with dimensionless spin $a=0.7$. Shown in the top panel are the values of the electric field as measured at different distances from the origin; since $E{r}^2\sim B_{0}J$, the different lines should overlap at late times if the magnetic field is uniform which is evident in the figure. Shown in the bottom panel is the ratio of the electric and magnetic fields which is proportional to the black-hole spin only. Note the transitory state until $t\approx 70M$, when the solution reaches a stationary state.

Figure 2

Left panel: Magnetic (blue) and electric field (red/magenta) field lines at $t=200M$ for a Schwarzschild black hole. Central panel: the same as in the left panel but for a Kerr black hole with spin $a=J/M^2=0.7$ aligned with the magnetic field, i.e., $J^i=\{0,0,J\}$. Right panel: The same as in the center panel but for a Kerr black hole with spin $a=J/M^2=0.7$ which is orthogonal to the magnetic field, i.e., $J^i=\{J,0,0\}$. Indicated with black surfaces are the apparent horizons.

Figure 3

Left panel: Large-scale magnetic and electric field lines on the plane $y=0$ and at $t=200M$ for a Kerr black hole with spin $J/M^2=0.7$ aligned with the magnetic field, i.e., along the $z$ axis. Indicated with blue circles are the apparent horizons. Right panel: The same as on the left panel but on a smaller scale to highlight the fields structure in the vicinity of the black hole.

Figure 4

Left panel: Large-scale magnetic and electric-field lines on the plane $y=0$ and at $t=200M$ for a Kerr black hole with spin $J/M^2=0.7$ orthogonal to the magnetic field, i.e., along the $x$ axis. Indicated with blue circles are the apparent horizons. Right panel: The same as on the left panel but on a smaller scale to highlight the fields structure in the vicinity of the black hole.

Figure 5

Electric-field lines on the plane $y=0$ for the $r_0$, $s_0$, and $s_6$ configurations at $t=123$, 155, and $246M$, respectively. Left panel: Large-scale structure of the EM fields around the apparent horizons (blue circles). Right panel: The same as on the left but on a smaller scale to highlight the field structure in the vicinity of the black holes. Note that an additional magnification is applied to the black hole “on the right” so as to highlight the change of sign in the electric field near the horizon, i.e., at $x\simeq 3M$.

Figure 6

Electric (red/magenta) and magnetic-field lines (gray) in 3D for the $s_6$ binary during inspiral when both black holes are still far separated at time $t=328M$ (left panel), and after the merger at $t=690M$ (right panel).

Figure 7

Left panel: GWs as computed from the (2, 2) mode of ${\Psi }_4$ for the different binaries reported in Table . Right panel: The same as in the left panel but for the EM waveform as computed from ${\Phi }_2$.

Figure 8

Amplitude and phase evolution of the main $\ell =m=2$ modes for the Weyl scalar ${\Phi }_2$ and the first time integral of ${\Psi }_4$ (i.e., ${\tilde{\Psi }}_4$), relative to the $s_6$ configuration. The plots show the data in retarded time $t-r$ for a detector located at $r=100M$. While the $\ell =m=2$ modes show the same amplitude (up to a scale factor) and phase evolution, this does not apply to modes with higher $\ell$. For the $\ell =3$, $m=2$ and $\ell =4$, $m=2$ modes, the phase evolution is still identical but the amplitude no longer does not differ only by a constant scale factor.

Figure 9

The total energy flux per unit solid angle in terms of GW waves (left panel) and of EM waves (right panel); clearly they differ only up to a scaling factor. The different lines refer to the different binaries reported in Table .