Schwarzschild black holes as unipolar inductors: Expected electromagnetic power of a merger
Аннотация
The motion of a Schwarzschild black hole with velocity ${v}_{0}={\ensuremath{\beta}}_{0}c$ through a constant magnetic field ${B}_{0}$ in vacuum induces a component of the electric field along the magnetic field, generating a nonzero second Poincar\'e electromagnetic invariant $^{*}F\ifmmode\cdot\else\textperiodcentered\fi{}F\ensuremath{\ne}0$. This will produce (e.g., via radiative effects and vacuum breakdown) an electric charge density of the order of ${\ensuremath{\rho}}_{\mathrm{ind}}={B}_{0}{\ensuremath{\beta}}_{0}/(2\ensuremath{\pi}e{R}_{G})$, where ${R}_{G}=2GM/{c}^{2}$ is the Schwarzschild radius and $M$ is the mass of the black hole; the charge density ${\ensuremath{\rho}}_{\mathrm{ind}}$ is similar to the Goldreich-Julian density. The magnetospheres of moving black holes resemble in many respects the magnetospheres of rotationally-powered pulsars, with pair formation fronts and outer gaps, where the sign of the induced charge changes. As a result, the black hole will generate bipolar electromagnetic jets each consisting of two counter-aligned current flows (four current flows total), each carrying an electric current of the order $I\ensuremath{\approx}e{B}_{0}{R}_{G}{\ensuremath{\beta}}_{0}$. The electromagnetic power of the jets is $L\ensuremath{\approx}(GM{)}^{2}{B}_{0}^{2}{\ensuremath{\beta}}_{0}^{2}/{c}^{3}$; for a particular case of merging black holes the resulting Poynting power is $L\ensuremath{\approx}(GM{)}^{3}{B}_{0}^{2}/({c}^{5}R)$, where $R$ is the radius of the orbit. In addition, in limited regions near the horizon the first electromagnetic invariant changes sign, so that the induced electric field becomes larger than the magnetic field, $E>B$. As a result, there will be local dissipation of the magnetic field close to the horizon, within a region with the radial extent $\ensuremath{\Delta}R\ensuremath{\approx}{R}_{G}{\ensuremath{\beta}}_{0}$. The total energy loss from a system of merging black holes is a sum of two components with similar powers, one due to the rotation of space-time within the orbit, driven by the nonzero angular momentum in the system, and the other due to the linear motion of the black holes through the magnetic field. Since the resulting electrodynamics is in many respects similar to pulsars, merging black holes may generate coherent radio and high energy emission beamed approximately along the orbital normal. In addition, merging black holes may produce observable wind-driven cavities.