General relativity math self-study - what next?

[ibkev]

My goal is to develop an intuitive understanding of the math underlying general relativity and ultimately be able to take a book like Wald or Carroll and, as someone on these forums commented once, “be able to casually read it while sipping my morning coffee and listening to the news.” :)

So where am I now? Although I have an eng degree, I completed it years ago and I've been brushing up. Math-wise, I’ve worked through vector calculus (Stewart) and linear algebra (Poole.) Physics-wise, I've also brushed up on mechanics (Taylor), EM (Griffiths) and done some light conceptual reading on SR.

So now the question is where to go next? What I'd like to do is come up with a list of textbooks that would makes sense when used together. ie. not too much overlap and a progression towards grad-level texts that is reasonable for self-study. Also, rather than go with the math-for-physicists route, I’m interested in a broader understanding of the associated math. My hope is to find a path that offers the holy trinity of intuition, motivation AND rigour. That said, I am self-studying, so if a text is particularly good at developing intuition and offering motivation at the expense of rigour I would be fine with that.

I’ve done some skimming of book previews at amazon and recommendations here and come up with the list of topics below that seem important to my goals but I really don’t understand these terms well enough to know if my list makes any sense, or in many cases how these topics relate to one another. (For example, how do differential forms, differential geometry and manifolds relate?) Thus you can imagine my difficulty navigating a course through these topics and picking out associated textbooks for self-study.

• covariant/contravariant vectors
• tensors
• differential forms - I heard Lovelock’s book is pretty good, also Hubbard
  • https://www.amazon.com/dp/0486658406/?tag=pfamazon01-20
  • https://www.amazon.com/dp/0971576688/?tag=pfamazon01-20
• metrics
• coordinate free geometry
• differential geometry - I heard Keyszig is good but not “coordinate free” whatever that means
  • https://www.amazon.com/dp/0486667219/?tag=pfamazon01-20
• topology, manifolds- I hear John Lee’s series is very good, also Mendelson
  • https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
    
  • https://www.amazon.com/dp/0486663523/?tag=pfamazon01-20
• Riemannian manifolds - also Lee?

I have no particular love for the texts listed above, they're just my best guess right now.
Anyways, I would really welcome suggestions from this group - thanks!

 

[Orodruin]

All of the subjects you have listed are of importance to general relativity. However, many of them are related and you should be able to cover them with two or three good books at increasing difficulty levels.

 

[Markus Hanke]

As an amateur myself, I had struggled through a lot of texts over the years, but the one that stood out and really "did it" for me was Misner/Thorne/Wheeler "Gravitation". I have several notebooks full of material which I have condensed out of that text, and I cannot recommend it highly enough, even for other amateurs. Some of the information in it is of course outdated ( in particular with regards to available observational data etc. ), but the maths and physics behind GR are presented in a way that no other text I know of has managed to do.

In fairness though, you'd need a pretty good grounding in multivariate calculus, differential equations, and linear algebra, or else the text will probably not be of much use to you.

 

[haushofer]

I often recommend Nakahara's book on differential geometry. It's accessible when you know linear algebra, and makes you familiar with all the geometry and topology needed to understand GR, QFT and string theory. It is not a typical math-text however; some would say it is not rigorous enough, because sometimes proofs are omitted. For me that's just perfect; I'm more interested in the physics.

It helped me a lot. So I'd say: highly recommended! :)

 

[vanhees71]

I think it's best to first study books concentrating on the physics, using "old-fashioned" Ricci calculus. Among those I like most Landau&Lifshitz vol. II (Classical Fields) and S. Weinberg, Gravitation and Cosmology (1972). If you then like to go further with modern tensor calculus ("Cartan calculus"), I'd recommend Misner, Thorne, Wheeler. A good free online book is

http://www.blau.itp.unibe.ch/newlecturesGR.pdf