Weak Gravitational Lensing: A Brief Overview

TL;DR

This paper provides a comprehensive overview of gravitational lensing, covering fundamental concepts, relativistic formulations, photon trajectories in various spacetimes, and advanced geometrical methods for analyzing light deflection.

Contribution

It introduces a unified geometrical framework for light deflection in static and rotating spacetimes, extending previous methods with new analytical expressions and formalisms.

Findings

Derived analytical expressions for photon orbits and deflection angles.

Extended analysis to axisymmetric spacetimes using new formalisms.

Compared different methods like Rindler-Ishak and Gauss-Bonnet for bending angle calculation.

Abstract

Gravitational lensing constitutes one of the most direct observational manifestations of spacetime curvature and provides a powerful probe of compact astrophysical objects. In this work, we present a comprehensive analysis of the bending of light in curved spacetime, beginning with the fundamental aspects of gravitational lensing and the Newtonian approximation to light deflection. The relativistic formulation of lensing is then developed through the lens equation, lensing potential, and a geometrical interpretation of Fermat's principle using an effective refractive index in curved spacetime. Photon trajectories and light deflection are subsequently investigated in static, spherically symmetric geometries, followed by a detailed study of photon motion in the equatorial plane of the Kerr spacetime. Analytical expressions for the closest approach distance and critical parameters governing photon orbits are derived. Furthermore, the bending angle is examined using the Rindler-Ishak method and the Gauss-Bonnet theorem within the optical geometry. Finally, the analysis is extended to axisymmetric spacetimes using the OIA and GW-OIA formalisms, providing a unified geometrical framework for computing light deflection in both static and rotating gravitational fields.