Relevant scales for the $C$-metric with positive cosmological constant

TL;DR

This paper investigates gravitational lensing effects around accelerating black holes with a positive cosmological constant, deriving new formulas for horizons and lensing angles, and exploring their observational distinguishability.

Contribution

It introduces novel perturbative formulas for horizons and analytical expressions for lensing angles in the context of accelerating black holes with positive cosmological constant.

Findings

Null circular orbits are independent of the cosmological constant.

Black holes with and without $\Lambda$ are indistinguishable via Sachs scalars.

Derived analytical formulas for weak and strong lensing deflections.

Abstract

In this work, we study the weak and strong gravitational lensing in the presence of an accelerating black hole in a universe with positive cosmological constant $\Lambda$. First of all, we derive new perturbative formulae for the event and cosmological horizons in terms of the Schwarzschild, cosmological and acceleration scales. In agreement with previous results in the literature, we find that null circular orbits for certain families of orbital cones originating from a saddle point of the effective potential are allowed and they do not exhibit any dependence on the cosmological constant. They turn out to be Jacobi unstable. We also show that it is impossible to distinguish a $C$-black hole from a $C$-black hole with $\Lambda$ if we limit us to probe only into effects associated to the Sachs optical scalars. This motivates us to analyze the weak and strong gravitational lensing when both the observer and the light ray belong to the aforementioned family of invariant cones. In particular, we derive analytical formulae for the deflection angle in the weak and strong gravitational lensing regimes.