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I am an actuarial science graduate who also minored in physics. I have a physics background of around a year 3 undergraduate physics student, and I've been meaning to self-learn general relativity someday.

I've heard much about how difficult the math of GR is and I'd like to know more about what I should know first before I properly tackle this subject.

I'm a pretty mathy person and I'd like to understand GR from a more rigorous and mathematical standpoint. Here's a brief list of what I know so far:

Physics:

  1. Classical mechanics. Studied this subject up to the level of Morin's classical mechanics textbook. I'm fine with Lagrangians and all that.
  2. Electrodynamics. Made it through half of Griffiths, enough to cover all of Maxwell's equations.
  3. Quantum Mechanics. Not sure if this will relate to GR, but I've also made it halfway through Griffiths for this.
  4. Special relativity. I've learnt all my SR from Morin, and so far I'm comfortable with ideas like events and lorentz transformations, as well as all the fundamental effects. I still need to read about 4-vectors though.

Math

  1. Calculus. Quite familiar with all the computational algorithms for finding derivatives and integrals. Pretty much had all of those drilled into my brain since high school. This includes vector calculus as well.
  2. Analysis. I quite enjoy analysis, and so far I'm learning analysis for single variable calculus.
  3. Set theory. I've delved a bit into the foundations of mathematics from studying analysis, so I am comfortable with set theory and set notation. Don't think this is useful for physics at all, but I find it useful to ground all my mathematical knowledge on a rigorous foundation.
  4. Linear algebra. I think I understand this subject well enough. Things like vector space axioms, inner products, orthogonality. Had to learn these things for QM.

What I want to ask now is whether there's anything I should add to this list? Do I need concepts like topology? Or do you think I'm good to go?

I'd also like to know if there is a mathematically rigorous textbook I could use for GR. If I place Einstein on one end of a spectrum, for people who care more about physics than math, and Hilbert on the other end, for people who care more about math than physics, I'd say I lean pretty heavily towards Hilbert, so I would appreciate a textbook based more on rigour than intuition.

(Thanks in advance for taking the time to read through my question.)

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    $\begingroup$ Does this answer your question? Mathematically-oriented Treatment of General Relativity $\endgroup$
    – J. Murray
    Commented Nov 30, 2021 at 3:56
  • $\begingroup$ I suspect this question will be closed due to its being less about physics and more about a subjective evaluation of whether or not you have the conceptual prerequisites to learn a particular subject. That being said ... the main thing absent from your list is differential geometry, but the overwhelming majority of physicists learn differential geometry via an introductory text on GR (for better or worse). I'd say you may as well give it a shot - the worst case scenario is that you find out you need to patch a few conceptual holes before moving forward. $\endgroup$
    – J. Murray
    Commented Nov 30, 2021 at 4:00

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I've literally seen first-year undergrads tackling GR with a basic knowledge of Linear Algebra and Calculus, so I'd say you are quite good to go. Most books on GR will cover the basic math you need. If you want to dive deep on all this math before tackling GR (it will probably take a long time, but it is up to you to choose), the "basic math" consists of General Topology, Differential Geometry (you will need a little Diff. Topology for this, but most books on DG will cover it), some multilinear algebra (namely, tensors). This is pretty much the essentials.

You could take a look at the 1973 book by Hawking & Ellis' The Large Scale Structure of Space-time. It is a beautiful, mathematically-oriented treatise on General Relativity which also discusses a lot of Physics as well. It starts from the very beginning (for example, covering the notions you'll need about Topology, Differential Geometry, and everything else) and is a maths book (Proposition, proof, Lemma, proof, Lemma, proof, Theorem, proof). Physics students will usually take a look after learning the basics somewhere else, but since you are more interested in the formalism I think it could be interesting. This is far from being the only option, and there's a post with many other excellent suggestions as well.

I will also mention Wald's General Relativity. While it is definitely a Physics book, it usually has quite some care with maths, and might be easier to tackle than H&E or other Mathematical Relativity texts. Wald's is the go-to book for Relativists and it might come in handy if the other texts are too mathy. In my opinion it is quite clear from a mathematical standpoint, but it won't put every theorem in a box.

I should also mention the excellent lectures by Frederic Schuller at the WE Heraeus International Winter School on Gravity and Light (see this link for a link to the YouTube Channel and for some typed notes). Schuller's approach is quite careful, clear, and depends essentially on just a good understanding of multivariable calculus and linear algebra. Every time I've seen these lectures mentioned on Phys.SE or on Reddit, someone would comment on how amazing they are, and I've never seen anyone complaining about them. They would provide an excellent, mathematically clear overview of the theory. It might be everything that you wanted to learn about GR, or it might be a great starting point for you to know what you want to see next.

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