Electric charge rotating around a black hole

Abstract

We consider an electric charge rotating around a Schwarzschild black hole. We compute, using quantum fi eld theory in curved spacetime at the tree level, the power emitted by the rotating charge minimally coupled to the Maxwell fi eld. We also compute how much of the radiation emitted by the swirling charge is absorbed by the black hole.We are commemorating the fi rst centennial of Special Relativity. Despite how exotic Special and General Relativity may have appeared at fi rst glance, they have become fundamental tools for now-a-days precision astrophysics. It is impossible to understand the center of active galaxies, pulsars and probably gamma ray bursts without evoking objects as black holes and neutron stars. As astrophysics measurements become more and more accurate, the influence of the spacetime curvature in particle wave modes will also need to be considered to explain the experimental data properly. This can be naturally accomplished in the context of quantum fi eld theory in curved spacetimes.To illustrate this, we analyze the radiation emitted by an electric charge rotating around a Schwarzschild black hole (see Ref.[1] for the emission of scalar radiation from a rotating source around a Schwarzschild black hole). We assume natural units = c = G = 1 and metricLet us consider the line element of the Schwarzschild spacetime in the formwhere f (r) = 1 − 2M/r and the electromagnetic fi eld described by the Lagrangian density in the modifi ed Feynman gaugeThe corresponding Euler-Lagrange equations are, thus,which can be written in terms of components asHere a and b denote angular variables on the unit 2-sphere S 2 with metricη ab and inverse metricη ab [with signature (−−)], ∇ a is the associated covariant derivative on S 2 ,∇ a ≡η ab∇ b and∇ 2 ≡η ab∇ a∇b . We write the complete set of positive-frequency solutions of Eq. (3) with respect to the Killing fi eld ∂ t asThe index ε stands for the four different polarizations. The pure gauge modes, ε = G, are the ones which satisfy the gauge condition G = 0 and can be written as Awhere Λ is a scalar fi eld. The physical modes, ε = I, II, satisfy the gauge condition and are not pure gauge. The nonphysical modes, ε = NP, do not satisfy the gauge condition. The modes incoming from the past null infi nity J − are denoted by n =← and the modes incoming from the past event horizon H − are denoted by n =→. The l and m are the angular momentum quantum numbers.The physical modes can be written asandwith l 1 (since the gauge condition G = 0 is not satisfi ed for l = 0). The radial part of the physical modes satisfi es the differential equationwhere λ = I, II and V S = f l(l + 1)/r 2 is the Schwarzschild scattering potential. Y lm and Y lm a are scalar and vector spherical harmonics [2], respectively.The conjugate momenta associated with the fi eld modes are

Corrigendum: Scalar radiation emitted from a source rotating around a black hole (2000 Class. Quantum Grav. 17 19)

“…Therefore, equations ( The subsequent low-energy approximation for W S,obs needs to be corrected accordingly. We also note that similar corrections need to be made in [1][2][3][4][5][6], which used equations similar to (35) and (36) for estimating the portion of the radiation emitted to infinity.…”