Amateur Lorentzian geometry, part 2.

A simple guide to getting to the research frontier of Lorentzian geometry.

It is my belief that Lorentzian geometry is sufficiently ill-developed that it is easy to get to the research frontier as an amateur. Of course, this statement is open to interpretation, YMMV.

The lingua franca for Lorentzian geometry is differential geometry. Differential geometry is one of the older subjects in modern mathematics and as such it was the genesis for much of the more abstract mathematics that research attention is focused on. Differential geometry has a reputation for being difficult to learn. Do not be intimidated. If you have done university level calculus and a little analysis you can pick up basic differential geometry. Just remember that what you are learning is new notation, not new concepts.

Differential Geometry

I recommend Tu and Lee for beginners in that order. If you're feeling adventurous and want to get to the point quickly open up Kobayashi and Nomizu. It is well worth tackling Kobayashi and Nomizu at some point, it is the classic go to for every differential geometer.

Lorentzian Geometry

With some differential geometry under your belt I suggest taking a look at either Hawking and Ellis or Penrose. Hawking and Ellis will give you a good back ground on the physically motivated results in Lorentzian geometry which is important to appreciate. The book also includes a review of Lorentzian geometry, something that you will not have seen when learning differential geometry. Penrose covers a collection of techniques which have played a limited, but very important role, in the development of the subject. These techniques are specific to global constructions in Lorentzian manifolds and should be something that you are comfortable with. Fortunately both Hawking and Ellis and Penrose do a good job of explaining them, though Penrose is more thorough. Fair warning though: at the time both books were written the notation and definitions of the objects involved in these techniques were not pinned down, so there are some differences between the two sources.

O'Neill is an excellent reference book for Lorentzian geometry. It is much drier that Hawking and Ellis and Penrose as those two books have specific goals (the singularity theorems) where as O'Neill is more like a description of common techniques in Riemannian geometry translated in to the Lorentzian world. This is actually very important as Riemannian results are usually false in Lorentzian geometry. Admittedly there are Lorentzian versions of Riemannian results, but they have a very different flavour. For example the Hopf Rinow Theorem is false in Lorentzian manifolds, but with certain assumptions some of the implications (with modifications) will hold (Beem, Erhlich and Easley discuss this). I recommend using O'Neill as a reference.

The book by Beem, Erhlich and Easley is an excellent and very readable account of the focus of research in Lorentzian geometry circa 1996. This book will absorb substantial time but the effort is worth it. Once you have read this, it's time to jump into the literature. There is no more modern reference, though there are books that cover specific areas.

Getting in touch with the literature

Once you feel comfortable with the books above it's time to move on to the literature. I suggest picking a topic from Beem, Erhlich and Easley looking up one of the references for that topic and following the references to find the most recent paper. Read it. Repeat until you have a question about something. Because of the nature of Lorentzian geometry this question is likely to be both unanswered and challenging. Find an answer to your question.

Congratulations you've got something to publish.

Other books worth looking at.

There are some specialist books in Lorentzian geometry, or related fields, that might be worth looking at - depending on your own interests. As I think of books I'll add them here.

nothing here so far Send me an email to include your favourite book.

A practice problem

I find the best way to learn something is to have a specific problem in mind. So... how about you explain the twin paradox using Lorentzian geometry.

  1. Get a coordinate system,
  2. Write down the equations for the paths of the two twins (one of the paths will have no change in spatial position - remember normal coordinates),
  3. Determine which twin has the longer path (according to the Lorentzian metric),
  4. Interpret the length as proper time experienced by the twins,
  5. The twin with the larger age stayed still.

The wikipedia page for the twin paradox focuses on special relativity and inertial frames. Inertial frames have no meaning in General Relativity and hence aren't a concept in Lorentzian geometry. What's important is that the time spent travelling for the two people are clearly different and this allows you to identify which twin travelled.

This problem will help to get you into Lorentzian geometry since you'll have to do several things which are very common:

  1. Draw diagrams and interpret them correctly,
  2. Determine equations within a coordinate system,
  3. Perform line integrals,
  4. Physically interpret the results.

Strictly speaking physical interpretation isn't part of Lorentzian geometry, but it's not a bad skill to work on.