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

In this work, we study the weak and strong gravitational lensing in the presence of an accelerating black hole in a universe with a positive cosmological constant Lambda. First of all we derive new perturbative formulas 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 our 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 formulas for the deflection angle in the weak and strong gravitational lensing regimes.

Automated summary

This study explores weak and strong gravitational lensing in the presence of an accelerating black hole and positive cosmological constant Λ. New perturbative formulas for event and cosmological horizons are derived, revealing interesting properties of null circular orbits and the inability to distinguish different types of C-black holes based on Sachs optical scalars alone. Analytical formulas for deflection angles in weak and strong gravitational lensing regimes are established for observers and light rays belonging to specific invariant cones.