On the Doppler effect for light from orbiting sources in Kerr-type metrics

Abstract

A formula is derived for the combined motional and gravitational Doppler effect in general stationary axisymmetric metrics for a photon emitted parallel or antiparallel to the assumed circular orbital motion of its source. The same formula is derived by both the eikonal approximation and Killing vector approaches to elucidate connections between observational astronomy and modern Relativity. The formula yields expected results in the limits of a moving or stationary source in the exterior Kerr and Schwarzschild metrics and is useful for broad range astrophysical analyses.

Light ring and the appearance of matter accreted by black holes

“…The late-time decay is the same as described in the main text, and corresponds to the "cooling" of the light ring, with the radiation that was fed onto it while the source was on circular motion. The spectral content of the late-time signal is dominated by radiation blueshifted to ωo 1.4ωe and redshifted down to ωe 0.5ωe, as expected from an analysis of the Doppler shift of radiation from such binaries [18,28,48].…”

Gravitational tuning forks and hierarchical triple systems

“…We removed the (linear) CM contribution, which only induces a low-frequency modulation. Observers on the equatorial plane see gravitational and Doppler-induced frequency shifts, consistent with analytical predictions [45,82] when the CM is moving towards the observer. The amplitude of the wave can vary by orders of magnitude because of relativistic beaming [43,50,51] and gravitational lensing [53,83].…”

“…Doppler effect induces frequency shifts, relativistic beaming and gravitational lensing modulations in the amplitude. The maximum blue-shift is well described by ωmax = ω 0 Υ (Υ + vCM)/(Υ − vCM), with Υ = 1 − 2M/R, M ω 0 = 1 the proper SB frequency and vCM is the CM velocity[45,82].…”

“…There are also attempts at modeling the effects of a third body directly into the waveform. These include Doppler shifts [45][46][47][48][49], relativistic beaming effects [50,51], gravitational lensing [52,53] and other dynamical effects in triple systems caused by the third-body [54][55][56].…”

Self-lensing flares from black hole binaries I: general-relativistic ray tracing of black hole binaries

“…Again, we plot the total combined light as well as the individual light curves for each BH. The individual BHs show again a modulation that we identify as Kerr-Doppler boosting [44] (which is not limited to spinning BHs only). Which is the Doppler effect caused by the movement of the BHs, causing the approaching BH to be blueshifted, while the receding BH is redshifted.…”