Light trajectories and optical appearances in asymptotically Anti-de Sitter-Schwarzschild and black string space-times

The Event Horizon Telescope (EHT) imaging of the central objects in the M87 and Milky Way galaxies provide compelling evidence that these objects are consistent with (Kerr) black holes. In view of these observations and the future expectations of Very Long Baseline Interferometry (VLBI) on which the EHT observations are based, an intensive research work has been carried out in the literature to simulating light trajectories and reconstructing the corresponding optical appearance for a wide array of modified black holes and ultra-compact objects. The corresponding images are directly affected not only by the background space-time geometry but also by the physics of the accretion disk, whose combination yields a characteristic fingerprint. In this paper, we consider such a fingerprint for objects which are not asymptotically flat but instead approach a Anti-de Sitter space-time. This assumption significantly influences light trajectories and, consequently, the corresponding images of the objects as seen by an observer at some distance, which can be used in future VLBI observations for testing alternatives of this kind to the Kerr paradigm. We illustrate our considerations with the examples of a Schwarzschild-Anti-de Sitter black hole and a black string, discussing their most notable departures from canonical, asymptotically-flat black hole space-times.

💡 Research Summary

This paper investigates how a negative cosmological constant, which makes the spacetime asymptotically anti‑de Sitter (AdS), modifies light propagation and the resulting optical appearance of compact objects. The authors focus on two concrete geometries: the Schwarzschild‑AdS black hole and a static black string with cylindrical symmetry. Both metrics contain a term proportional to α²r² (α² = –Λ/3) that dominates at large radii, causing the spacetime to diverge from the usual asymptotically flat case.

Starting from the general static line element ds² = –A(r)dt² + A⁻¹(r)dr² + r²dΩ², the Schwarzschild‑AdS metric is A(r)=1–2M/r+α²r², while the neutral black string has A(r)=α²r²–β α r (β encodes the linear mass density). A charged string adds a γ‑dependent term, A(r)=α²r²–β α r+γ²α²r², introducing an inner Cauchy horizon for appropriate parameter choices. The authors first review the properties of these functions, emphasizing that the presence of the α²r² term pushes the photon sphere outward and raises the critical impact parameter b_c = r_c√A(r_c).

Null geodesics are treated using the conserved energy‑like quantity E and angular momentum‑like quantity L. The effective potential for photons is V_eff(r)=A(r)L²/r². Circular photon orbits satisfy V_eff = E² and V′_eff = 0, defining the photon sphere radius r_c. The impact parameter b = L/E determines the fate of a photon: b > b_c leads to scattering (direct image), b < b_c results in capture (shadow), and b ≈ b_c produces trajectories that loop around the compact object multiple times, giving rise to the hierarchy of photon rings indexed by the number of half‑turns n.

The ray‑tracing algorithm integrates the differential equation dφ/dr = ± b /