Expressions for optical scalars and deflection angle at second order in terms of curvature scalars

Abstract

We present formal expressions for the optical scalars in terms of the curvature scalars in the weak gravitational lensing regime at second order in perturbations of a flat background without mentioning the extension of the lens or their shape. Also, by considering the thin lens approximation for static and axially symmetric configurations we obtain an expression for the second-order deflection angle which generalizes our previous result presented in [1]. As applications of these formulas we compute the optical scalars for some known family of metrics and we recover expressions for the deflection angle. In contrast to other works in the subject, our formalism allows a straightforward identification of how the different components of the curvature tensor contribute to the optical scalars and deflection angle. We also discuss in what sense the Schwarzschild solution can be thought as a true thin lens at second order.PACS numbers:

Gravitational deflection of relativistic massive particles by wormholes

“…This expression is the same to the result obtained by applying the GB theorem to the optical metric [15,16,57,59]. From Eq.…”

“…(2) is actually the same with a special optical metric related with massive particles in Ref. [57]. The general form for a static and spherically symmetric metric is written as…”

Higher order corrections to deflection angle of massive particles and light rays in plasma media for stationary spacetimes using the Gauss-Bonnet theorem

“…We want to point out that an explicit expression for the Finsler-Randers metric (26) was obtained for the particular case of photons propagating in a cold non-magnetized plasma over a Kerr spacetime by Perlick in [28]. Expressions (20) and (22) are their generalization to any stationary spacetime, and in fact even when to our knowledge they were not explicitly presented before in the literature, they are already implicit in [28]. On the other hand, it was recently reported in [43] an expression of the Finsler-Randers metric for massive particles propagating in a Kerr spacetime.…”

“…On the other hand, it was recently reported in [43] an expression of the Finsler-Randers metric for massive particles propagating in a Kerr spacetime. Expressions (20) and (22) also contain this particular case through the known correspondence between the motion of test massive particles in a given stationary axisymmetric spacetime and photons moving in a homogeneous plasma environment in the same background. We will use this correspondence later to discuss the motion of test massive particles.…”

“…In [18] it was established for the first time a correspondence between the motion of light rays in a particular non-homogeneous plasma and the one of relativistic test charged massive particles in vacuum. In the same work, the Gauss-Bonnet method was applied to obtain an expression for the deflection angle in terms of the components of the energy-momentum tensor in a plasma environment generalizing in this way previous works restricted to the pure gravity case [19][20][21][22].…”