PHYSICAL REVIEW D VOLUME 7, NUMBER 4 15 FEBRUARY 1973 Vector and Tensor Radiation from Schwarzschild Relativistic Circular Geodesics* R. A. Breuert University of Maryland, College Park, Maryland 20T42 R. Ruffini JosePh Henry Physical Laboratories, Princeton, Nese Jersey 08540 and J. Tiomno Institute fax Advanced Study, Princeton, New Jersey 08540 C. V. Vishveshwaraf. New York University, Net York, Nezo York 10003 (Received 8 June 1972) For the case of high multipoles we give an analytic form of the spectrum of gravitational and electromagnetic radiation produced by a particle in a highly relativistic orbit r p = (3 + D)M around a Schwarzschild black hole of mass M. The general dependence of the power spectrum on the frequency in all three spin cases (s = 0 for scalar, s= 1 for vector, and s = 2 for tensor fields) are summarized by power P o- cu ' exp(-2'/(A)~it). Although they have the common feature of an exponential cutoff above a certain frequency ~, ~t (4/~6)~p, where cop is the frequency of the orbit, the tensor case has a much broader frequency spectrum than scalar or vector radiation. I. INTRODUCTION Regge and Wheeler' developed techniques to treat the most general small perturbation in a Schwarzschild geometry. These were applied to the stability problem for the Schwarzschild met- ric by Regge, Wheeler, and Vishveshwara, ' First calculations of gravitational radiation from a highly relativistic source, based on an extended Regge-Wheeler formalism, were done by Thorne' and used for the analysis of gravitational radiation from neutron stars by Thorne and Campolattaro' and Price and Thorne. ' Radiation from material falling into black holes has been given by Zerilli' and by Davis and Ruffini, ' Davis, Ruffini, Press, and Price, ' Davis, Ruffini, and Tiomno, ' and Davis, Ruffini, Tiomno, and Zerilli. ' In this paper the method is applied to treat (in the fixed Schwarzschild background geometry) the radiation emitted by a test particle moving at a velocity close to the local speed of light in the neighbor- hood of a black hole. As usual the particle as well as the radiation is described by a perturba- tion of the background metric, which can be ex- panded in terms of a set of scalar, vector, or ten- sor harmonics with both (-l)' electric (even) par- ity and (-l)'" magnetic (odd) parity. This problem is of direct interest in relation to the possible en- hancement of synchrotronlike gravitational radia- tion effects as recently suggested by Misner. " As we are interested in the spectral distribution, we analyze only the Fourier transform of the per- turbation. The radial part of the expansion satis- fies a Schrodinger-type wave equation of the form where r*=x — 3My2Mln(r/M 2); M -is the mass of the Schwarzschild black hole; l and m are the angular and azimuthal quantum numbers of the multipole expansion. V,ff depends on the particu- lar spin of the field under examination. We can adapt the formalism to the study of scalar (spin-0), electromagnetic (spin-l), or gravitational (spin-2) radiation. In the limit l»1 the potential V,«ac- quires a standard form independent of the spin of the field The source terms will be different in the three cases, even for l» 1. The explicit forms both for the potential and for the source in the case of vector fields have been derived by Ruffini and 1002