Abstract We present two covariant radiative transfer formulations, and apply them to calculate emission from relativistic accretion flows around black holes. The first formulation is for situations with only line-of-sight absorption and emission, while the second formulation is for situations where scattering is important. We use the first formulation to calculate emissions from opaque accretion disks and tori around black holes absorbed by high-velocity cloudlets. Our calculations show the importance of effects due to space-time curvature, relativistic motions of the emitters and absorbers, and external line-of-sight absorption. We find that external absorption tends to take away line fluxes at energies red-ward of the line-centre energy. In its presence, emission lines from geometrically thin opaque relativistic accretion disks may not have broad asymmetric double-peak profiles: in some situations the lines may appear to be narrow, sharp and blue-shifted. We also find that geometric effects are important for thick accretion disks (accretion tori). When viewed at high inclination angles, the inner surface of an accretion torus can be self-occulted. As the most highly red-shifted and blue-shifted emissions are blocked, emission lines from an opaque accretion torus would suffer less broadening and the line intensities are less boosted. These lines may have single-peak profiles, but their line centres are slightly red-shifted. We apply the radiative transfer formulation to calculate emission lines from optically thin and semi-opaque accretion tori, and generalise it to calculate continua emission, such as the reflection spectra, of accretion disks. The second convariant radiative transfer formulation, which is based on a moment method, is used to calculate the emissions from accretion tori with the opacity being dominated by electron scattering. We demonstrate that the formulation is applicable for a wide range of optical depths appropriate for accretion tori around black holes.
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