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

TL;DR

This paper analyzes photon escape probabilities and energy shifts from zero-angular-momentum sources near Kerr black holes, providing new analytical results especially for near-extremal cases and demonstrating significant photon escape even near the horizon.

Contribution

It introduces a more efficient method for calculating photon escape probabilities and derives new analytical results for near-extremal Kerr black holes, expanding understanding of photon behavior near black hole horizons.

Findings

Escape probability in NHEK region tends to 7/24 (~29.17%)

Escape probability at the innermost photon shell tends to approximately 12.57%

Photon energies are redshifted but still potentially observable at infinity.

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.