Friday, 14 October 2011

General Relativity a reading list

Prompted by a discussion on the Maths Choice Forum I'm afraid I couldn't stop myself so here is a reading list for budding General Relativists. The question was what is a good mathematical approach to General Relativity. One of the contributors seemed a bit sniffy about the first book of the physics course The Relativistic Universe. Ok it's not the most mathematical approach but the basics are there namely introduction to relativity and the main solutions namely the Schwarzschild metric and the Robertson Walker metric. So a good introduction for those wanting more here is a brief survey.  

"Of the books I'm familiar with the most mathematical is Hawking and Elliss The large scale structure of space time but you do need a background in Topology and differential geometry before it becomes even remotely comprehensible. It deals with the singularity theorems that Hawking and Penrose discovered. There is little direct application to astrophysics.
And although there is a chapter on cosmological models you would have to spend a lot of time filling in the gaps

The monster which does justice to both the mathematics and astrophysical applications is Misner Thorne and Wheeler. Although some might find it a bit long winded more for reference than reading through
A reasonable compromise between concrete applications and abstract mathematics is Wald
All of these books are probably a bit dated an uptodate  version but more geared to applications than mathematics is
A book on mathematical physics which includes the background to differential geometry and also functional analysis for those interested in quantum physics is
Finally one of the first books on Geometric Algebra and its application to physics  is
Coauthored by the same Lasenby who wrote General Relativity an Introduction for Physicists.

Insertion added 15th October
Forgot to add these two classics
Chandresekhar on Black Holes

This has the encouraging paragraph at the end of chapter 9

"Every effort has been taken to present the mathematical developments in this chapter in a comprehensible
logical sequence. But the nature of the development simply does not allow a presentation that can be followed in detail, with modest effort:. The reductions that are necessary to go from one step to another are often very elaborate and so on occasion may require as many as ten, twenty or even fifty pages !! (my exclamation marks) "

Doesn't exactly make you want to read it does it.

And Weinberg's First book on General Relativity

All of these books have been sitting on my shelves and I occasionally take a peek sometimes even work through a chapter or two.

I'm in two minds about the usefulness of a purely abstract approach to General Relativity. Undoubtedly the Hawking and Penrose theorems regarding singularities are of great significance but the Topological Approach seems a bit like existence theorems in Pure maths. Yes they have been shown to exist, but what causes them or what are the consequences for that you need the detailed mathematical modelling that Astrophysicists use to construct what happens when stars collapse or work out the implications of a given cosmological model for say elemental abundances. For that the abstract mathematics is an elegant means to an end, be it differential forms or now geometric algebra, namely the components of the Riemann Tensor. Once you have those and solve the Einstein Field equations you end up with differential equations and the standard methods of solving them.
I'm currently trying to understand  the applications of variational methods to the standard solutions to general relativity especially the big bang. This seems a reasonable compromise between physical insight and the abstract methods certainly beats the usual slog of computing 128 Christoffel symbols only to find out most are zero.
I think to get the full picture you need both approaches but whether it is possible for a person to get the full picture in the typical lifetime of an undergraduate or a 1 year MSc course is another question.

Anyway hope to get back to my studies on the Variational method as applied to the Robertson Walker metric sorted by Christmas.

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