There are three reasons why I wanted to write this book. First, gravitational lensing is emerging as a powerful tool in many areas of astronomy, from exoplanets to cosmology. This breadth of applications emerging from a single phenomenon – the bending of light by curved space-time – is the second reason, for it is yet another example of why physics is so interesting: we can explain many things with a few simple laws.

The final reason for this book is that, while important for many different applications, lensing is not that hard. There have been elegant, beautiful papers and books exploring the foundations of lensing and many of its formal aspects. But most of the applications simply do not require all that much formalism. General relativity makes several appearances here, but even if you skip those few sections, you will still learn essentially everything that's here: multiple images, magnification, micro-lensing, shear, etc. And armed with that information, I hope you will be able to begin research on real problems that are opening up in so many areas of astronomy and be prepared to analyze the ever-improving datasets that are coming our way.

The goal then is that this book can serve as the text for an undergraduate or graduate course on gravitational lensing or be used for independent study by someone interested in jumping into research.

Thanks are due to colleagues who generously gave of their time to answer questions: Gary Bernstein, Daniel Fabrycky, Bhuvnesh Jain, Rachel Mandelbaum, and Ben Wandelt. I am very grateful to the students who took this course, especially in 2015, as they helped immensely by providing feedback on an early draft of this book. So thank you to these rising stars: Adam Anderson, Gourav Khullar, Meng- Xiang Lin, Monica Mocanu, Pavel Motloch, Andrew Neil, Zhaodi Pan, Jason Poh, Amy Tang, Rito Thakur, and Alexander Tolish.

I am extraordinarily grateful to the people who fund my research, especially Fermilab, the Office of High Energy Physics at the Department of Energy and, by extension, the citizens of the United States. I am so fortunate to live in a society that values basic research. Some of this work was carried out at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.

This chapter derives the general formulae that will be used throughout the book for the deflection of light. We start, though, with a relatively simple problem: the deflection of light by a single point mass.We will tackle this problem in two ways: first using Newtonian gravity and then using the formalism of general relativity. Both approaches find that the deflection angle is proportional to MG/bc 2 where b is the distance of closest approach, but the two theories return coefficients that differ by a factor of 2. The general relativity calculation is presented in §2.2 and §2.3. If you know general relativity, you will have no problem with those sections. If you have not yet encountered general relativity, you can skip those sections and pick up in §2.4 without missing a beat. The mathematical details in the general relativity sections, though, are designed so that even those without backgrounds in this field can work through them. The advantages of doing so are three-fold: it's always good to work through the math; you will get a glimpse of the details of many general relativity calculations and should be able to carry out others; and you might emerge with an appetite to understand the theory at a deeper level. The disadvantage is that the sections do not attempt to capture the beauty and profundity of the theory.

Armed with the result for the deflection angle, we will be able to calculate one effect exactly: the Shapiro time delay, the amount by which light is slowed down as it passes by a massive body. The rest of the chapter builds on the formula for the deflection angle and derives the lens equation, the underpinning of all gravitational lensing phenomena. Starting with this general version, where the deflection is expressed in terms of the gravitational potential, we then circle back to show how it recaptures the simple cases of a single lens and then the even simpler case of a single point mass lens.

Newtonian Deflection

Let us consider a point particle with small mass m passing by a compact object with much larger mass M.