Boosted Kerr Black Hole in the Presence of Plasma †

Optics in a Curved Space-Time

Our main goal is to study the effect of plasma on the gravitational lensing for a boosted Kerr black hole in the weak-field limit. For this reason, we consider a static spacetime metric which describes a weak gravitational field g_µν and has the form [6,8]
g^αβ = η^αβ − h^αβ and g_αβ = η_αβ + h_αβ, (1) where η_αβ is the Minkowski spacetime, h_α,β are small perturbation (h_αβ 1) and h_αβ → 0 for x^α → ∞
and h^αβ = h_αβ. The form of g_µν in Equation (1), in the context of general relativity, means that the
spacetime metric is nearly flat [25]. In this sense, our study is done far from the gravitational field of the source where gravity is "weak". Also, the assumption of a weak field, enable us to consider an approximation in the motion of photons: the null approximation.
Before studying the gravitational lensing for a boosted Ker black hole surrounded by plasma, it is necessary to obtain the equation of motion for photons in the presence of a gravitational field and include the plasma contribution into these equations. To do so, we follow the ideas of J.L. Synge [5] and the works of Bisnovatyi-Kogan et al. and V. S. Morozova et al. [6,8]. In reference [5], where the optics in a curved-spacetime was first developed, Synge starts by considering a single infinity 3-spaces. These single infinity 3-spaces, can be spacelike, null, or timelike and are called waves or phase-waves by the author (see Figure 1).