Observability of Zero-angular-momentum Sources Near Kerr Black Holes

Abstract: We revisit monochromatic and isotropic photon emissions from the zero-angular\linebreak-momentum sources (ZAMSs) near a Kerr black hole. We investigate the escape probability of the photons that can reach to infinity and study the energy shifts of these escaping photons, which could be expressed as the functions of the source radius and the black hole spin. We study the cases for generic source radius and black hole spin, but we pay special attention to the near-horizon (near-)extremal Kerr ((near-)NHEK) cases. We reproduce the relevant numerical results using a more efficient method and get new analytical results for (near-)extremal cases. The main non-trivial results are: in the NHEK region of a (near-)extremal Kerr black hole, the escape probability for a ZAMS tends to $\frac{7}{24}\approx 29.17\%$, independent of the NHEK radius; at the innermost of the photon shell (IPS) in the near-NHEK region, the escape probability for a ZAMS tends to \begin{equation} \frac{5}{12} -\frac{1}{\sqrt{7}} + \frac{2}{\sqrt{7}\pi}\arctan\frac{1}{\sqrt{7}}\approx12.57\% . \nonumber \end{equation} The results show that the photon escape probability remains a relatively large nonzero value even though the ZAMS is in the deepest region of a near-horizon throat of a high spin Kerr black hole, as long as the ZAMS is outside the IPS. The energies of the escaping photons at infinity, however, are all redshifted but still visible in principle.