# Curvature radius and Kerr black hole shadow

## TL;DR

This paper introduces three innovative methods to determine the spin and inclination of Kerr black holes using the curvature radius of their shadows, enhancing accuracy and efficiency in black hole parameter estimation.

## Contribution

It proposes three new approaches utilizing curvature radius measurements at characteristic points to accurately infer black hole spin and inclination angle.

## Key findings

- Three new methods for spin and inclination estimation.
- Modeling of M87*'s shadow to calculate curvature radii.
- Identification of novel properties linking curvature radius and shadow features.

## Abstract

We consider applications of the curvature radius of a Kerr black hole shadow and propose three new approaches to simultaneously determine the black hole spin and inclination angle of the observer. The first one uses only two symmetric characteristic points, i.e., the top and the bottom points of the shadow, and is the smallest amount of data employed to extract information about spin and inclination angle amongst all current treatments. The second approach shows that only measuring the curvature radius at the characteristic points can also yield the black hole spin and the inclination angle. The observables used in the third approach have large changes to the spin and the inclination angle, which may give us a more accurate way to determine these parameters. Moreover, by modeling the supermassive black hole M87* with a Kerr black hole, we calculate the angular size for these curvature radii of the shadow. Some novel properties are found and analyzed. The results may shine new light on the relationship between the curvature radius and the black hole shadow, and provide several different approaches to test the nature of the black hole through the shadow.