A note on methods for computing the critical curve of Kerr-like black holes

TL;DR

This paper compares different definitions of the black hole shadow for Kerr-like black holes, revealing their agreement in vacuum at large distances and divergence at finite distances or with plasma, and analyzes the impact of tetrad choice.

Contribution

It systematically compares celestial coordinate definitions of black hole shadows, clarifies their differences, and explores the effects of plasma and tetrad choices on the critical curve.

Findings

All three definitions agree in vacuum at large distances.

Definitions diverge at finite distances and with plasma.

Tetrad changes cause only horizontal shifts in the critical curve.

Abstract

This study systematically compares Bardeen's, de Vries's, and Grenzebach et al.'s celestial coordinate definitions of the critical curve ("shadow") of Kerr-like black holes. We find that all three definitions agree for black holes in vacuum or surrounded by inhomogeneous plasma observed from large distances. However, they diverge for observers located at a finite distance: Bardeen's definition yields the smallest critical curve, while de Vries's yields the largest. When homogeneous plasma is considered, critical curve computed using Bardeen's definition deviates from the other two even at large distances and contracts compared to the vacuum case with increasing plasma density. This is in clear contradiction with the behaviour predicted by de Vries's, Grenzebach et al.'s definitions, and previous gravitational lensing studies. We derive de Vries's definition assuming a critical curve on the observer's sky plane and explain its discrepancy with Grenzebach et al.'s definition. We further explore the effect of the change of tetrad on the critical curve. Using Bardeen and Carter tetrads, we plot the critical curve for Schwarzschild and Kerr black holes in the presence of plasma, highlighting that tetrad changes introduce only a horizontal shift in the critical curve.