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Please can someone recommend some books on 'higher-level' (couldn't think of a better way to phrase...) books on GR? I've read over half of Wald (General Relativity) and I'm about to finish Carroll (Spacetime and Geometry). I didn't really enjoy reading Wald (sacrilege!), but I've really really enjoyed Spacetime and Geometry - I felt it was more modern, and almost like I was being lectured in the subject. For some reason, it didn't feel the same with Wald, most probably because I read Wald from a .pdf and Carroll from hardback.

I really enjoyed the Black Hole aspects of S&G, but I feel that I'm little weak on gravitational waves. I'd also like to see some more advanced differential geometry too, like spinors and tetrads (even though I don't know what they are yet, they sound important). If someone could recommend a book with the feel of Carroll but more advanced content, I'd be extremely grateful.

I feel the answer to my question is to go back and try Wald again, however I would like some other opinions!

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    $\begingroup$ This would be more appropriate for physics.stackexchange.com $\endgroup$ – Paichu Jul 25 '20 at 20:30
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    $\begingroup$ You're probably right, but I'd rather get an opinion from a mathematician's point of view than a physicist's. I'm still a little scarred from my first year university studies... $\endgroup$ – lloydy.99 Jul 25 '20 at 20:34
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    $\begingroup$ The best way to gain a better understanding of GR and SR, is to start studying differential geometry. What one comes to understand is that spacetime is a four-dimensional topological manifold with a smooth atlas carrying a torsion-free connection compatible with a Lorentzian metric and a time orientation satisfying the Einstein equations. When this statement is perfectly clear in all its components, studying GR will be easier. $\endgroup$ – Taylor Rendon Jul 25 '20 at 20:36
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    $\begingroup$ Perhaps finding a set of lecture notes on arxiv that speak to you would be a good idea (as long as they are from a reputable source) especially for topics such as gravitational waves. Or you could set a long term goal for yourself that would allow you to pick up tools along the way, such as understanding and working through the Kerr metric. Rotating frames around black holes are particularly instructive, demolishing previous incorrect intuition much like an introductory analysis course does for budding mathematicians. $\endgroup$ – Ninad Munshi Jul 25 '20 at 20:47
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    $\begingroup$ I like Misner, Thorne, Wheeler, Gravitation which contains also a thorough description of Differential Geometry. Another good basic one is Schulz Relativity which is a much shorter but also very nice book. $\endgroup$ – Logos Jul 25 '20 at 21:03
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I came up with the some suggestions to begin to even start looking at GR and SR in a way that a mathematician would.


Make a disciplined effort to get through Ted Shifrin's texbook (.pdf free to everyone) on differential geometry. This will give a good review of some much needed analytical geometry and introduce the notion of covariant differentiation, parallel translation, and geodesics - which are pinacle ideas used in GR and SR.


If the ideas in this book sit well and you are feeling comfortable getting through (most of) the exercises, then take a look into purchasing Tensors: The Mathematics of Relativity Theory and Continuum Mechanics, by Anadijiaban Das. This book is a relatively expensive, but worth every penny; especially if you're serious about learning the anatomy of GR and SR. It is a mathematicians approach to introducing the machinery needed to study problems in GR and SR - not to mentioning its rewarding rigor.

if you are looking for some broad lectures that give a good overview of the mathematics needed to start being a better physicist, take a look at Fredric Schuller's lectures on the geometrical anatomy of theoretical physics.. He is by far one of the most precise lecturers I have ever learned from. These lectures encompass a vast amount of topics that are pertinent in studying physics - especially Einstein gravity. Getting through these lectures from start to finish will definitely help understand my comment: spacetime is a four-dimensional topological manifold with a smooth atlas carrying a torsion-free connection compatible with a Lorentzian metric and a time orientation satisfying the Einstein equations.

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