Gravitational Lenses with Circular Symmetry

We have seen the beauty and power of gravitational lensing in both the local and distant universe, but exploiting this rich phenomenon requires a quantitative treatment. Many of the most important concepts in lensing can be gleaned from the axisymmetric case, where the mass distribution of the lens has either spherical or cylindrical symmetry in three dimensions, corresponding to circular symmetry in two dimensions. Strictly speaking, we must work within the confines of general relativity, but a Newtonian derivation of the deflection angle is nevertheless instructive (Sect. 2.1) and requires only a slight mathematical tweak to bring the result in line with relativity. Because the radial extent of a lens is much smaller than either the distance from the observer to the lens or from the lens to the source, we can project along the line of sight to arrive at a two-dimensional problem (Sect. 2.2). Using this “thin lens” approximation, we derive the equation that a deflected light ray must obey and discuss how to determine the positions and magnifications of lensed images. We then use this framework to analyze a few simple yet representative lens models (Sect. 2.3). The final three sections derive conditions for multiple imaging by axisymmetric lenses and discuss how to extend our analysis to non-axisymmetric lenses.