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 [Phys.Rev.D 83:083007 (2011)]. 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.
Highlights
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
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]
In [1], we presented a new approach to the study of gravitational lensing in the weak field regime, which proved to be very useful since gives explicit gauge invariant expressions for the optical scalars and to the deflection angle
Summary
The phenomenon of gravitational lensing has become an active area of research at least from the beginning of the 1980’s when the observation of the first astrophysical gravitational lens was announced[2]. In [1], we presented a new approach to the study of gravitational lensing in the weak field regime, which proved to be very useful since gives explicit gauge invariant expressions for the optical scalars and to the deflection angle. In all these works the different expressions for the optical scalars or the deflection angle are generally written in a particular gauge or a given family of gauges All these studies are very useful and relevant in their respective range of validity, it is our intention to extend some of the results presented in [1]. Some auxiliary but relevant relations have been included in the four Appendixes
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