Radiation emitted from a source rotating around a black hole

We analyze the radiation emitted from a scalar source rotating around a Schwarzschild black hole using the frame work of quantum field theory at the tree level. We show that for relativistic circular orbits the emitted power is about 20% to 30% smaller than what would be obtained in Minkowski spacetime. We also show that most of the emitted energy escapes to infinity. Our formalism can readily be adapted to investigate similar processes. 98.62.Mw, 98.62.Js, 95.30.Sf, 04.70.Dy, 04.62.+v Typeset using REVTEX 1 Observational confirmation of the existence of black holes is one of the most important challenges in astrophysics. Recently a number of compact objects in X-ray binary systems were identified as black holes since a careful analysis has shown that their masses are far beyond any limit accepted for dead stars in general relativity [1]. There also exists indirect evidence of the presence of supermassive black holes in the center of some galaxies [2]. Nevertheless, unambiguous confirmation of the existence of black holes would require the observation of effects due to the event horizon itself. This is expected to be achieved through precise measurements of the electromagnetic radiation emitted from black hole accretion disks [3,4], and from the gravitational radiation emitted from companion stars orbiting black holes [5,6]. Because radiation from sources orbiting black holes plays such a crucial role in modern astrophysics and because increasingly precise measurements are leading to the observation of relativistic effects occurring in the vicinity of the horizon [7], a comprehensive investigation of how radiation-emission processes are modified by the nontrivial curvature and topology of the black-hole spacetime would be necessary. In this paper we analyze the radiation emitted by a scalar source rotating around a black hole using the framework of quantum field theory at the tree level, and compare the results with those obtained in Newtonian gravity and in a theory associated with one-graviton exchange in flat spacetime. We show that these results coincide asymptotically as expected but considerably differ close to the last stable circular orbit. We also calculate the amount of emitted energy absorbed by the black hole. We use natural units c = h̄ = G = 1 and signature (+−−−) throughout this paper. A nonrotating black hole with mass M is described by the Schwarzschild line element ds = f(r)dt − f(r)dr − rdθ − r sin θdφ , (1) where f(r) = 1− 2M/r. Let us consider a circularly moving scalar source at r = RS with constant angular velocity Ω > 0 (as measured by asymptotic static observers) on the plane θ = π/2 described by j(x) = q √−g u0 δ(r − RS) δ(θ − π/2) δ(φ− Ωt) (2) with g ≡ det(gμν), where the constant q determines the magnitude of the source-field coupling. The four-velocity of this source is u(Ω, RS) = [(f(RS)−R SΩ), 0, 0,Ω/(f(RS)−R SΩ)] . We have normalized the source j by requiring that ∫ dσj(x) = q, where dσ is the proper three-volume element orthogonal to u. Let us now minimally couple j(x) to a massless scalar field Φ̂(x) so that the total Lagrangian density is L = √−g ( 1 2 ∇Φ̂∇μΦ̂ + jΦ̂ ) . The positive-frequency modes in spherically symmetric static spacetime can be given in the form uωlm(x ) = √ ω π ψωl(r) r Ylm(θ, φ)e −iωt (ω > 0) , (3)