Black hole electrodynamics and the Carter tetrad

Abstract

The orthonomal frame first found by Carter (1968) is the natural frame to use when interpreting Newman–Penrose formalism outside the event horizon of a Kerr black hole. Observers at rest in this frame rotate round the hole at constant r and θ with angular velocity $$d\phi/dt=a/({r}^{2}+{a}^{2})$$.A simple set of Maxwell equations satisfied by field components in the Carter frame is obtained. Boundary conditions for a force-free black hole magnetosphere are found. Expressions for the rate of loss of energy and angular momentum by a black hole to a electromagnetic field are derived in terms of the toroidal components of the field at the event horizon. They are evaluated for a field that is slowly rotating with angular frequency ω and is approximately uniform at radii satisfying M ⪡ r ⪡ 1/ω. If B_⟘ is the magnitude of the component of the field perpendicular to the axis of rotation of the hole, the rates of loss of energy and angular momentum are $$(2/3)\omega B_{\perp }^{2}a{M}^{2}\,\text{and}\,B_{\perp }^{2}a{M}^{2}$$ respectively in geometrical units.