Light bending in a two black hole metric

Abstract

We discuss the propagation of light in the C-metric. We discover that null geodesics admit circular orbits only for a certain family of orbital cones. Explicit analytic formulae are derived for the orbital radius and the corresponding opening angle fixing the cone. Furthermore, we prove that these orbits based on a saddle point in the effective potential are Jacobi unstable. This completes the stability analysis done in previous literature and allows us to probe into the light bending in a two black hole metric. More precisely, by constructing a suitable tetrad in the Newmann–Penrose formalism, we show that light propagation in this geometry is shear-free, irrotational, and a light beam passing by a C-black hole undergoes a focussing process. An exact analytic formula for the compression factor θ is derived and discussed. Furthermore, we study 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 obtain formulae allowing to calculate the deflection angle in the weak and strong gravitational lensing regimes.