Gravitational searchlight and its astrophysical applications

Abstract

We wish to point out an interesting effect associated with light emitted in the forward direction by a source circulating in an equatorial plane around a highly collapsed object or a black hole of mass M. The essential feature of the emission is that provided the radius a of the orbit in Schwarzschild's coordinates exceeds 3GM/c², the light emitted is blueshifted when received by a distant observer. More specifically, if ν₀ is the frequency of light at emission and ν, the frequency at reception at a radial coordinate R≫R_S=2GM/c², we have ν/ν₀=√[(2−3ξ)(1−ξ)]/√[2(1−ξ)]−√ξ (1) where ξ=R_s/a (2). While ν>ν₀ for all values of ξ<⅔, it can be shown easily that as ξ→⅔, the blueshift diverges. Writing ξ=(⅔)(1−ξ), ν⋍(⅔)ν₀ε^−½ as ε→0 (3). Thus very high blueshifts are obtained as the orbit of the particle tends to the so called unstable circular orbit. Qualitatively, we can understand the result in terms of the competition between the Doppler blueshift of forward light emission and the gravitational redshift due to the central mass. The former increases much more rapidly than the latter as ε→0. We are interested here with orbits with very small ε, that is, orbits close to the unstable circular orbit.