Magnetospheres of black hole systems in force-free plasma

The interaction of black holes with ambient magnetic fields is important for a variety of highly energetic astrophysical phenomena. We study this interaction within the force-free approximation in which a tenuous plasma is assumed to have zero inertia. Blandford and Znajek used this approach to demonstrate the conversion of some of the black hole’s energy into electromagnetic Poynting flux in stationary and axisymmetric single black hole systems. We adopt this approach and extend it to examine asymmetric and, most importantly, dynamical systems by implementing the fully nonlinear field equations of general relativity coupled to Maxwell’s equations. For single black holes, we study, in particular, the dependence of the Poynting flux and show that, even for misalignments between the black hole spin and the direction of the asymptotic magnetic field, a Poynting flux is generated with a luminosity dependent on such misalignment. For binary black hole systems, we show both in the head-on and orbiting cases that the moving black holes generate a Poynting flux.

Figure 1

Monopole test: Left: Magnetic field lines in the $z=0$ plane at $t=100M$ for a single spinning black hole with $a=0.1$. Right: The rotation frequency ${\Omega }_F$ of the magnetic field along the axial and azimuthal angles about the origin. The known perturbative solution is also shown for comparison purposes.

Figure 2

“Wald solution”: Late-time ($t=100M$) behavior for a single spinning black hole with $a=0.7$. Left: Contour lines of the rotation frequency ${\Omega }_F/{\Omega }_H$ between 0.01 and 0.9, together with the apparent horizon (green inner circle) and the ergosphere (blue outer ellipse) Right: The rotation frequency interpolated onto the half-circle in a meridional plane.

Figure 3

Comparison of electrovacuum and force-free solutions: Snapshot at $t=100M$ of a single spinning black hole with $a=0.7$ on the $y=0$ plane with the apparent horizon (green) and the ergosphere (magenta). For $x<0$, we show the force-free solution while for $x>0$ the electrovacuum solution is shown. Top: Magnetic field lines. Bottom: Electric field lines.

Figure 4

Dependence of luminosity on spin and orientation for a black hole with mass $M={10}^8{M}_{\odot }$ and initial magnetic field of $B_0={10}^4\mathrm{G}$. Top: The luminosity is shown as a function of the black hole spin $a$ for the aligned case ${\theta }_0=0$. Also shown (red, solid line) is a fit to $L_{\mathrm{EM}}\propto {\Omega }_H^2$. Bottom: Luminosity as a function of the angle ${\theta }_0$ between the asymptotic magnetic field and the spin angular momentum of the black hole for the cases $a=0.7$ and $a=0.1$. Also shown (red, solid line) is the curve $A_o[1+\cos ({\theta }_o{)}^2]$ where $A_o$ is a constant obtained by matching just the point at ${\theta }_o=0$.

Figure 5

Charge density and currents at $t=100M$ for a single spinning black hole with $a=0.7$. Left: An aligned black hole (${\theta }_o=0$). Right: A misaligned black hole with ${\theta }_o=\pi /2$.

Figure 6

Cross sections of the magnetic field structure at $z=4M$ for the aligned and 90 degree cases. Left: A clockwise toroidal $B$ field is generated in the aligned case. Right: We find two offset and counter-rotating toroidal fields for ${\theta }_o=\pi /2$.

Figure 7

Density of energy flux $F_{\mathrm{EM}}$ at $t=100M$ for a single spinning black hole with $a=0.99$, together with the magnetic field lines. Left: An aligned black hole (${\theta }_o=0$). Right: A misaligned black hole with ${\theta }_o=\pi /4$.

Figure 8

Head-on collision of two black holes: Snapshots at times $t=20M$, $40M$ and $60M$. Left: The electromagnetic energy flux (shaded) and representative magnetic field lines (solid blue lines). Right: The magnetic field lines in detail on the $x=0$ plane, showing only field lines close to the black holes are being perturbed, and that, after the BH passes, they recover their original shape due to the magnetic tension.

Figure 9

Head-on collision of two black holes: The charge density before and at the merger on the equatorial plane.

Figure 10

Orbiting binary black holes: The electromagnetic energy flux at different stages of the evolution: early when the black holes are separated; shortly after they merge. The final state is similar to the single spinning black hole studied previously. Notice that the “clumps” seen in the left frame are artifacts from the initial data.

Figure 11

Radiated power in the gravitational and electromagnetic band corresponding to the binary black hole case. Three electromagnetic luminosities; the collimated part (solid black), the corresponding to the $m=2$ and so induced by the orbital motion (short dashed red) and the expected for the final spinning black hole with $a=0.7$ (constant long dashed green). The isotropic part can be estimated by the subtracting the collimated to the orbital one. The gravitational power (long dashed blue) has been rescaled by a factor ${10}^{-13}$ to make it appear clearly in the plot.