The extended power law as an intrinsic signature for a black hole

We analyze the exact general relativistic integrodifferential equation of radiative transfer describing the interaction of low-energy photons with a Maxwellian distribution of hot electrons in the gravitational field of a Schwarzschild black hole. We prove that, owing to Comptonization, an initial arbitrary spectrum of low-energy photons unavoidably results in spectra characterized by an extended power-law feature. We examine the spectral index by using both analytical and numerical methods for a variety of physical parameters as such the plasma temperature and the mass accretion rate. The presence of the event horizon as well as the behavior of the null geodesics in its vicinity largely determine the dependence of the spectral index on the flow parameters. We come to the conclusion that the bulk motion of a converging flow is more efficient in upscattering photons than thermal Comptonization, provided that the electron temperature in the flow is of order of a few kilo-electron volts or less. In this case, the spectrum observed at infinity consists of a soft component, which is produced by those input photons that escape after a few scatterings without any significant energy change, and a hard component (described by a power law), which is produced by the photons that underwent significant upscattering. The luminosity of the power-law component is relatively small compared to that of the soft component. For accretion into a black hole, the spectral energy index of the power law is always higher than 1 for plasma temperatures of order of a few kilo-electron volts. This result suggests that the bulk motion Comptonization might be responsible for the power-law spectra seen in the black hole X-ray sources.