General relativistic radiative transfer: formulation and emission from structured tori around black holes

Aims. We aim to construct a general relativistic radiative transfer formulation, applicable to particles with or without mass in astrophysical settings, wherein ray-tracing calculations can be performed for arbitrary geodesics for a given space-time geometry. Methods. The relativistic radiative transfer formulation is derived from first principles: conserving particle number and phase-space density. The formulation is covariant, and transfer calculations are conducted along particle geodesics connecting the emitters and the observer. The geodesics are determined through the space-time metric, which is specified beforehand. Absorption and emission in the radiative transfer calculations are treated explicitly. The particle-medium interaction is evaluated in the local inertial frame, co-moving with the medium. Relativistic, geometrical and optical depth effects are treated self-consistently within an integral covariant framework. Results. We present a self-consistent general relativistic radiative transfer formulation with explicit treatment of emission and absorption. The formulation is general and is applicable to both particles with mass and without mass. The presence of particles has two major effects: firstly the particle bundle ray is no longer along the null geodesic, and secondly the intensity variation along the particle bundle ray is reduced by an aberration factor. The radiative transfer formulation can handle 3D geometrical settings and structured objects with variations and gradients in the optical depths across the objects and along the line-of-sight. Such scenarios are applicable in calculations of photon emission from complex structured accretion flows around black holes and neutrino emission from remnant neutron tori in neutron-star mergers. We apply the formulation and demonstrate radiation transfer calculations for emission from accretion tori around rotating black holes. We consider two cases: idealised optically thick tori that have a sharply defined emission boundary surface, and structured tori that allow variations in the absorption coefficient and emissivity within the tori. We show intensity images and emission spectra of the tori obtained in our calculations. Our findings in the radiative transfer calculations are summarised as follows. (i) Geometrical effects, such as lensing-induced self-occulation and multiple-image contribution are much more significant in accretion tori than geometrically thin accretion disks. (ii) Optically thin accretion tori show emission line profiles distinguishable from the profiles of lines from optically thick accretion tori and lines from optically thick geometrically thin accretion disks. (iii) The line profiles of the optically thin accretion tori have a weaker dependence on the viewing inclination angle than those of the optically thick accretion tori or accretion disks, especially at high viewing inclination angles. (iv) Limb effects are present in accretion tori with finite optical depths, due to density and temperature stratification within the tori. We note that in accretion flows onto relativistic compact objects, gravitationally induced line resonance can occur. This resonance occurs easily in 3D flows, but not in 2D flows, such as a thin accretion disk around a black hole.