Lensing of hot spots in Kerr space-time

An empirical relation for black hole spin estimation

¹ Department of Astrophysics/IMAPP, Radboud University P.O. Box 9010 6500 GL Nijmegen, The Netherlands
² Center for Astrophysics | Harvard & Smithsonian 60 Garden Street Cambridge MA 02138, USA
³ Black Hole Initiative at Harvard University 20 Garden Street Cambridge MA 02138, USA

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 23 April 2025
Accepted: 29 December 2025

Abstract

Context. Sagittarius A* (Sgr A*) exhibits frequent flaring activity across the electromagnetic spectrum that is often associated with a localized region of strong emission known as a hot spot.

Aims. We aim to establish an empirical relationship linking key parameters of this phenomenon – emission radius, inclination, and black hole spin – to the observed angle difference between the primary and secondary image (ΔPA) that an interferometric array could resolve.

Methods. Using the numerical radiative transfer code ipole, we generated a library of more than 900 models with varying system parameters and computed the position angle difference on the sky between the primary and secondary images of the hot spot. The key assumptions are equatorial and circular orbits.

Results. For these models we find that the average ΔPA over a full period is insensitive to inclination. This result significantly simplifies potential spin measurements, which might otherwise depend strongly on inclination. Additionally, we derived a relation connecting spin to ΔPA, given the period and emission radius of the hot spot, with an accuracy of better than 5° in most cases. Finally, we present a mock observation to showcase the potential of this relation for spin inference.

Conclusions. Our results provide a novel approach for black hole spin measurements using high-resolution observations, such as future movies of Sgr A* obtained with the Event Horizon Telescope, the next-generation Event Horizon Telescope, and the Black Hole Explorer. Although discrepancies will likely arise in the general case (lifting model assumptions), the methodology is sound, and easily extendable.