What Math Books Should I Read to Understand General Relativity?

[Sunnyocean]

Hi,

I started reading General Relativity but concepts such as Lorentz transformations, rotations, tensors etc. are, at least in my opinion, poorly explained. Or perhaps the authors assume that the readers are already familiar with such maths?

At any rate, I would very much like to read some good books on the above math topics. Could anyone recommend me some books / textbooks?

I really need to understand it well, from the basics to the very difficult concepts.

The more detailed, the better (and preferably with exercises and solutions, if possible, so that I can also practice the maths (and check my solutions), not just read the book).

Just a few words on difficulty. It seems to me that there are two meanings for "difficult". One of them is "too much detail". I don't mind that kind of "difficult". So yes, please do tell me about "the epsilon and the delta" of Ricci calculus etc. On the other hand, if "difficult" means that the book takes you through the whole history of mathematics before actually introducing the concept, then I am afraid that's not what I am looking for. Although if the latter are the only kind of books you would recommend, then they are obviously better than nothing.

So the short of it is: I am not afraid of difficult, and in fact I really want to understand the topic THOROUGHLY, but I don't want to waste my time with unnecessary details either.

 

[micromass]

What physics and math do you already know?

 

[Daverz]

What is your mathematics and physics background?

Which GR books did you try?

 

[Fredrik]

The best place to get a good introduction to SR and tensors is "A first course in general relativity", by Schutz. Read the first three chapters.

The best place to learn the differential geometry is the books by John M. Lee. You will need both "Introduction to smooth manifolds" and "Riemannian manifolds: An introduction to curvature". (Because the "smooth" book doesn't cover connections, parallel transport, geodesics, curvature). If you haven't already studied topology, you need to either skip some stuff in those books, or get a book on topology too. (It is possible to proceed without fully understanding the topological stuff. For a physicist, that's a reasonable compromise between following a dumbed down approach and getting really deep into the mathematics). I don't know what the best book on topology is, but it's a safe bet that Lee's book "Introduction to topological manifolds" covers what you need. (I haven't read it myself).

As for the actual general relativity, I'm not sure what to recommend. I know that Wald is a pretty good standard book that takes the math seriously, and it probably was the best choice when I studied GR, but it's possible that some newer book is better.

 

[Sunnyocean]

Fredrik, thank you very much.

I have not studied topology but I would prefer NOT to skip anything.

I am aware that it will take a lot of effort but I would prefer to avoid the dumbed down approach. I really don't like making compromises when it comes to understanding stuff.

So, could you please recommend me some good books on topology as well (I will check Lee's book, but you seem to know the topic pretty well - at any rate much better than me - so I would prefer books you did read, if that's ok).

I like books which take maths seriously very much :)

 

[WannabeNewton]

I can assure you that learning all this extra math will not only fail to be of any use in better understanding GR, it will also be a ridiculous time sink especially given the lengths of Lee's books. But if you have the time to spare then all the power to you, learn it if it interests you. Just know that physics books which overemphasize math tend to be quite bad in my opinion. Wald for example is great for learning the basic mathematical structure of aspects of GR e.g. causality, but you will not learn a word of actual physics from the book.

 

[Sunnyocean]

Oh, and in order to answer the previous questions, I know the math that the "average" physics student knows at the end of the second year - maybe a bit more. I was avoiding to say that because I wouldn't like people to say to me things like "oh, in that case don't read General Relativity, it's too much for you!" <--- Please don't give me that kind of reply :p

 

[Sunnyocean]

WannabeNewton, maybe you are right and maybe I will arrive at the same conclusion as you after reading all the stuff, but at the very least I need to check it for myself. I know I may well be wrong, but somehow I feel that I really need to know the math in detail.

 

[Fredrik]

I can assure you that learning all this extra math will not only fail to be of any use in better understanding GR,
[...]
but you will not learn a word of actual physics from the book.

That first statement is probably only about 75% wrong, but you certainly reached 100% with that last one.

 

[WannabeNewton]

That first statement is probably only about 75% wrong, but you certainly reached 100% with that last one.

Well geez looks like I must have just plucked the statements out of thin air without any experience whatsoever. Why don't we put a rain check on this while I survey the GR post-docs in my research group on their opinions regarding the matter after the next group meeting and get back to you on that.

 

[Fredrik]

So, could you please recommend me some good books on topology as well (I will check Lee's book, but you seem to know the topic pretty well - at any rate much better than me - so I would prefer books you did read, if that's ok).

I have picked up pieces of topology here and there. A little bit about metric spaces from Rudin's "Principles of mathematical analysis" a long time ago. The basics about topological spaces from Friedman's "Foundations of modern analysis", also a long time ago. A few years ago, I read the appendix on topology in Sunder's "Functional analysis: spectral theory", and supplemented it with Munkres's "Topology" when I needed more. This brought me up to the OK level, but I don't think this is the best path for you.

Micromass is much better at topology than I am, and he's much more familiar with the books on topology than I am. So you should check out what he has to say about it. If he doesn't answer here, you should be able to find recommendations he has made in other threads.

 

[Fredrik]

Well geez looks like I must have just plucked the statements out of thin air without any experience whatsoever. Why don't we put a rain check on this while I survey the GR post-docs in my research group on their opinions regarding the matter after the next group meeting and get back to you on that.

Good luck finding a person who thinks that there's no physics whatsoever in Wald. Or a person who thinks that understanding tensor fields or parallel transport is of no use in understanding GR.

 

[Mr-R]

Good luck finding a person who thinks that there's no physics whatsoever in Wald. Or a person who thinks that understanding tensor fields or parallel transport is of no use in understanding GR.

Dear Fredrik,

I don't think that when wannabeNewton said "extra math" meant parallel transport and tensor field. These stuff are standard and is covered in every intro to GR books like Schutz and D'inverno. I think there is some truth in saying that you don't need that much of maths to understand the Basics of GR. For example I think D'inverno relied too much on explaining computational stuff rather than physical concepts like Schutz.

I am just a beginner in this so I will just leave it at that.

 

[micromass]

Oh, and in order to answer the previous questions, I know the math that the "average" physics student knows at the end of the second year - maybe a bit more. I was avoiding to say that because I wouldn't like people to say to me things like "oh, in that case don't read General Relativity, it's too much for you!" <--- Please don't give me that kind of reply :p

We will obviously not give you this kind of reply. But I think you need to be a bit more specific with your knowledge. It's really difficult to give suitable recommendations if we don't know more or less precisely what you know and don't know. If anything, just give the names of books you worked through, we can figure it out from there too.

With respect to the debate between Fredrik and WBN, I think both have good points and it really depends on what kind of person the OP is. I certainly agree with WBN that you don't necessarily need any math books in order to start with GR, and that learning the underlying math takes a huge effort and takes a lot of time (we are talking about graduate level mathematics here!). However, knowing the math rigorously does sometimes have its benefits. Math texts make concepts much clearer and are sometimes easier to read because of vague explanations in the physics texts.

I guess it depends on the OP. Does he just want to learn physics or does he want to learn the foundations and the mathematics too. Both are valid approaches.

If you are interested in the mathematical approach, then I think you might want to start with linear algebra, a good book (despite its name) is Linear Algebra done wrong: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
Then you should learn analysis of metric spaces and topology. Good books (but maybe far too much) are the analysis book by Zorich, these will also cover manifolds in part II. Lee's books are a good alternative (but they're also very large!), if you already know the basics of metric spaces and proofs.

Even if you're following the mathematical approach, don't wait to start with GR until you have covered all math. Try to read Schutz now already (maybe together with the math books).

 

[Sunnyocean]

Mentor,

Thank you very much; your reply is really very useful.

I really don't remember all the math books I've read, but I will give you some EXAMPLES of what I know:

Calculus - integration, differentiation, both one-variable and multivariate (however, I know the "epsilon and delta" only for one-variable calculus). As for multivariate calculus, I have been taught to "just accept" the theories, something which I do not like very much to be honest.

With respect to linear algebra, I know matrix multiplication, determinants, eigenvalues etc.

I also know some vector field calculus, curl theorem, divergence theorem etc.

I have studied the basics of what groups are (what is an abelian group etc.)

Geometry-wise, I know Euclidean geometry. (And of course trigonometric functions etc.)

I hope this is detailed enough. The above is of course an outline, not an exhaustive list of the maths I know.

With respect to what a "huge" effort is, in my experience different people mean very different things by "huge effort". I can tell you that in high school my maths teacher used to say "If you haven't sat at the table at least three hours per day doing maths, you are not really studying maths". This, to me, is "normal" effort (NOT big, let alone "huge"). Hopefully this will serve as a description for what "normal effort", "huge effort" etc. mean. Oh, and by "sitting at the table" I mean really doing maths - no tv, no watching out the window, no facebook, no family members / friends coming to do "small talk", cell phone switched off, no snacks (just some water), no headphones (the room should be as quiet as possible, earplugs are not a bad idea) etc. Going to toilet is allowed!

I hope the above is detailed enough. If it is not, please do ask in case I need to write in more detail :)

If you think I know too little maths, please do let me know.

 

[Sunnyocean]

Oh, and I really AM *very* grateful to ALL the people who took the time to reply. Especially to the people who wrote detailed replies. I am writing this because the system told me that I cannot give any more reputation (I am a beginner on Physics Forums so I don't know well how this works). But it seems to me that I should be able to thank everyone who takes even some seconds from their precious time to answer my posts, and there should not be a limit to "how many kilograms of gratefulness" I am allowed to express.

Anyway, since I am not able to thank everyone individually, I would at least like to thank you all by writing this reply! :)

 

[Fredrik]

If you have to apply that normal effort for more than a year, then it adds up to a pretty huge effort. It will take a very long time, probably more than a year of normal effort, to first learn topology well and then learn differential geometry well. It's probably better to do something like this:

Start with the first three chapters of Schutz (SR + tensors in the context of multilinear algebra). Then take a look (in some book on differential geometry) at the definitions of "smooth manifold" and "tangent space", and the proof that if x is a coordinate system (=chart) that covers a region that contains a point p, then $\big\{\frac{\partial}{\partial x^i}\big|_p\big\}_{i=1}^n$ is a basis for the tangent space at p. You will see that some topology is used. So then you know what topics from topology you will have to study to understand manifolds and tangent spaces. At this point you study those things in topology (the basics of metric and topological spaces, and a little more), and then give those definitions and theorems in differential geometry another shot. Then you make sure that you understand vector fields and tensor fields, in particular the metric tensor. Then you study GR until you find something that you can't understand without consulting Lee.

 

[WannabeNewton]

Good luck finding a person who thinks that there's no physics whatsoever in Wald.

I wouldn't have said it if I didn't already know the results It's not as if we (the people in the group) haven't discussed this before. And I can say with 100% confidence that going through all of Wald and doing all the problems in it will help to no extent in solving GR problems that are actually of interest. Hell it wouldn't even help solve the physics problems in Lightman et al. If your main goal is mathematical GR then all the power to you, Wald is probably the best book to get started on that. But mathematical physics isn't physics.

Or a person who thinks that understanding tensor fields or parallel transport is of no use in understanding GR.

Mr-R said everything I have to say with regards to this. No one needs to go through Lee or a more advanced book in order to understand such material. It is very easy to understand as presented in GR texts.

 

[Sunnyocean]

If your main goal is mathematical GR then all the power to you, Wald is probably the best book to get started on that. But mathematical physics isn't physics.

WannabeNewton, since I am just starting it is probable that I won't understand what you mean, but can you tell me more about what you mean by "mathematical physics isn't physics"? Could you perhaps give an example?

While I do want to study everything in detail, I certainly don't think that mathematics is everything.

 

[WannabeNewton]

Could you perhaps give an example?

Sure no problem. I think the most demonstrative class of examples come from Penrose's grand sea of contributions to GR and Geroch's equally vast contributions to the subject. Both of them worked on aspects of GR that are termed mathematical GR but they also worked on aspects that one would align more with physics. For example Penrose (with Hawking) is famous for the proofs of the singularity theorems. The study of these theorems, their background, and proofs are entirely mathematical in that they really are an (elegant) use of relatively advanced differential geometry. There exist as a result entire projects on studying the kind of Jacobi fields that arise in the singularity theorems and how they classify certain manifolds.

Another example would be the contributions of Choquet-Bruhat to the field of mathematical GR. She is most famous for her work on the study of Einstein's equations in the context of PDE theory. Robert Geroch, as well as David Malament, have some extremely interesting work on the relationships between topology and causality of general space-time manifolds (Lorentzian manifolds that need not satisfy Einstein's equations) and pathological effects of causality in space-times with interesting topologies. Geroch also has papers on generating solutions to Einstein's equations for lineariztions over a curved background.

These are all instances of mathematical physics and chapters 7-10 of Wald cover the basics of everything I've mentioned. Chapter 11 is on the mathematically rigorous definition of asymptotic flatness for the most part and has a small section on the notion of energy-momentum of space-times. The latter (ADM energy-momentum of space-times) is actually an example of physics instead of mathematical physics and is still an active area of research for physicists working in GR but I didn't count it in Wald's book because he covers it so briefly and quickly that you can't learn anything from it unless you consult other books or literature (c.f. the very detailed review by Jaramillo and Gourgoulhon on arxiv).

Contrast this for example with Rindler's papers on rotation in GR which are definitely more on the side of physics. His most famous is probably his paper on using the gravitomagnetic potential in the corotating frame of observers in circular orbits in stationary axisymmetric space-times to calculate the Thomas, Geodetic, and Lense-Thirring precessions of gyroscopes at rest in the frame in an extremely elegant manner. This is work that is actually of interest to physicists who explore precessional and orbital effects in GR that can be verified by experiment (e.g. Gravity Probe B).

EDIT: Just for the record, Wald is my most favorite GR book. It is an extremely elegant and careful development of the mathematical structure of the theory and is what really made me fall in love with the subject. Since then I've worked through multiple other GR books and what I'm saying is by way of comparison. Wald's book simply does not go into any detail on calculations important for physical problems in GR (both research and textbook), it doesn't even teach you how to really think about approaching a GR problem and calculations therein, and it doesn't explore any of the interesting physics in physically relevant space-times that can eventually be compared with experiment. Wald is really something you want to look at after having seen/acquired all these things through e.g. MTW, Hartle, Straumann, Schutz, Hobson et al, Padmanabhan to name a few.

Best of luck!

 

[Sunnyocean]

Thank you very much, as I said, for the moment I am happy I can read the words in your reply :p

But I did make a copy of your reply for future reference, when I hope to be more able to understand what you wrote :)

 

[WannabeNewton]

But I did make a copy of your reply for future reference, when I hope to be more able to understand what you wrote :)

Haha well again best of luck!

BTW, I'm not trying to come off as a devil's advocate here or a downer on math. I'm just trying to save you time and effort. I did exactly what you are aiming to do now in this thread and I can tell you, while it helped me greatly in the short run in quickly grasping the basic mathematical structure of GR in more depth than that which is provided by GR books, in the long run it was not useful at all. It simply took precious time away from my learning the physics of GR more deeply, working on more involved GR calculations, and moving into more specialized arenas of GR.

I have yet to come across a GR paper in the literature relevant to the stuff I normally deal with either at school or in my own time when I'm reading as a hobby that required any deep knowledge of the material in e.g. Lee's topological or smooth manifolds books. Any unfamiliar math that did come up could easily be looked up or understood from a vastly superior math intellect than oneself (for me it was micromass ).

That being said, working through Lee's topological manifolds with micromass was some of the most fun I've ever had learning so if you want to learn it for the sake of learning it you will not be disappointed. It's extremely enjoyable.

 

[Sunnyocean]

EDIT: Just for the record, Wald is my most favorite GR book. It is an extremely elegant and careful development of the mathematical structure of the theory and is what really made me fall in love with the subject. Since then I've worked through multiple other GR books and what I'm saying is by way of comparison. Wald's book simply does not go into any detail on calculations important for physical problems in GR (both research and textbook), it doesn't even teach you how to really think about approaching a GR problem and calculations therein, and it doesn't explore any of the interesting physics in physically relevant space-times that can eventually be compared with experiment. Wald is really something you want to look at after having seen/acquired all these things through e.g. MTW, Hartle, Straumann, Schutz, Hobson et al, Padmanabhan to name a few.

Since we are talking about it, could you please give me the exact titles of the books by Hartle, Straumann etc.? Not only for future reference but also I might just want to take a look through them, to (try to?) get a glimpse of what awaits me :)

Also, I really don't know what "MTW" means...:p

 

[WannabeNewton]

Since we are talking about it, could you please give me the exact titles of the books by Hartle, Straumann etc.? Not only for future reference but also I might just want to take a look through them, to (try to?) get a glimpse of what awaits me :)

Also, I really don't know what "MTW" means...:p

Sorry that's entirely my fault. I'm just so used to giving only the author's last names. They're all included in a list I made in an older thread: https://www.physicsforums.com/showpost.php?p=4717552&postcount=7

MTW = "Gravitation" by Misner, Thorne, and Wheeler for future reference since almost everyone just calls it MTW.

 

[Sunnyocean]

Thank you very much :)

 

[micromass]

Face it wbn, topology problems were the most fun problems you've ever done aside from maybe kleppner :tongue:

But I do agree. I tell my students every time during topology that the topology exercises we are going to do are very fun, help you to grow mathematicaly and mentaly, and are extremely elegant. But they are also very useless. There are very few topology problems that I have done (and I have done quite a few!) that appear in later mathematics, let alone physics or other applied fields. Furthermore, a lot of the theory you'll learn in a topology book will also be not so useful for you unless you specialize in something topological. It does broaden your mind quite a bit and it offers you foundations to handle a lot of other mathematics.

But anyway, I think it's very fun to see two very different physicists on this thread. Both WBN as Fredrik are of course theoretically inclined and know the mathematics pretty well, but have a very different perspective on mathematics and physics. Both perspectives are equally valid in my opinion and both have their pro's and cons. The OP should decide himself what kind of physicist he is.

 

[WannabeNewton]

Face it wbn, topology problems were the most fun problems you've ever done aside from maybe kleppner :tongue:

Haha that's definitely true without a doubt. No question I've had way more fun doing the problems in Lee than doing the problems in my favorite GR books, and certainly they were infinitely more fun than the banal problems in the majority of Wald.

I just think it's important to draw the line between what's learned for the sake of pure enjoyment and what should be learned out of necessity.

 

[micromass]

the banal problems in the majority of Wald.

I certainly didn't hear you complain when you talked about the joy if index gymnastics...

I just think it's important to draw the line between what's learned for the sake of pure enjoyment and what should be learned out of necessity.

Agreed completely. But then again, some people find things necessary that other people don't find necessary. For example, I can imagine very well that there are people who are totally incapable of reading a physics text unless they know all of the associated mathematics first. This is of course extremely inefficient, but those kind of people get a mental block otherwise. On the other hand, other people are just happy with the physics text their explanations.

Finding the right math/physics text is a lot like finding the right partner (god, I sound pathetic now). But it's true: some people find some text very good while other people hate the text. It's really very personal.

 

[Sunnyocean]

For example, I can imagine very well that there are people who are totally incapable of reading a physics text unless they know all of the associated mathematics first. This is of course extremely inefficient, but those kind of people get a mental block otherwise.

That is exactly why I think that learning the mathematics would be useful, useless as it may be! (contradiction intended!)

As long as you don't know everything in your own mind, you are more or less like that person who needs to be explained the associated mathematics first, before understanding the textbook. There are many levels of "not understanding", and I don't want to be on any of them.

Besides, who knows what insights may spring out from the explanation of some forgotten mathematical formulae which most people take for granted...

Benjamin Franklin reportedly answered, rhetorically, "what is the use of a newly born baby?" when he discovered electricity. I think it is also useful to answer, rhetorically, "what is the use of an old man?" when learning that "useless", time-consuming mathematical background behind the physical theories.

 

[PeroK]

But then again, some people find things necessary that other people don't find necessary. For example, I can imagine very well that there are people who are totally incapable of reading a physics text unless they know all of the associated mathematics first. This is of course extremely inefficient, but those kind of people get a mental block otherwise. On the other hand, other people are just happy with the physics text their explanations.

Finding the right math/physics text is a lot like finding the right partner (god, I sound pathetic now). But it's true: some people find some text very good while other people hate the text. It's really very personal.

I've recently retired and I would like to learn GR. I did a Pure Maths undergraduate degree 30 years ago and have been revising: Linear Algebra, Vector Calculus, Differential Equations and Real Analysis (although I remembered most of that!). I've worked through SR (T. M. Helliwell) and classical EM (Maxwell's equations). This is in the last 12 months.

From this thread and a previous one by WBN, there seem to be three options:

Schutz
Hartle
Carroll

Get started with one of those and digress into Differential Geometry, Topology or whatever other maths I'm missing as required. Any thoughts on the pick of these for someone from a pure maths background?

 

[whyevengothere]

This is kinda good: Koks, Don Explorations in mathematical physics. The concepts behind an elegant language

 

[haushofer]

Nakahara. The golden road between math.rigour and physical applications.

 

[atyy]

I liked Crampin and Pirani https://www.amazon.com/dp/0521231906/?tag=pfamazon01-20 and Fecko https://www.amazon.com/dp/0521187966/?tag=pfamazon01-20 (very weird, but very fun and good explanations). My background is a biologist reading for fun, so I think these are pretty approachable like Nakahara, which I like also.

A great free set of notes on GR is Blau's http://www.blau.itp.unibe.ch/GRLecturenotes.html.

And another free set on more advanced topics is Winitzki's https://sites.google.com/site/winitzki/index/topics-in-general-relativity.

 

[Sunnyocean]

Thank you very much atyy :)

 

[bolbteppa]

Even ignoring manifolds and point set topology you guys can't really ignore field theory & action principles, i.e. classical field theory (electromagnetism) & the calculus of variations, the way you can ignore manifolds.

If you just jump into GR with a modest background you could ruin the subject for yourself - if you see an action or the EM Lagrangian you can only hope the 5 pages of preparation (if at all) they spent on these was enough, also it'd be pretty awful to start dealing with field tensors without action principles & the theory as a whole would seem pretty divorced from the rest of physics what with all the Riemann's & Christoffel's taking up your time... In other words, the fun naive feeling you have in your head as to what GR is could be totally shattered, you could be left with that awful feeling of confusion, possibly be tricked into thinking that not knowing manifolds and dual spaces is the reason why you're having problems, or just generally have feeling that you can't understand something because you skipped something else that you should have done.

There is a nice way to actually study the structure of GR by studying the prerequisites properly. Look at the table of contents of Landau vol. 2: chapter's 1 - 9 are SR & EM (special relativity and electromagnetism) done absolutely from scratch using the (multi-variable) calculus of variations, 10 - 14 are general relativity. If you look closely you'll notice:
1, 2, 3 ~ 10
4 ~ 11,
5 ~ 12,
6, 7 ~ 13 (& bits of 9)
That is, chapters 1, 2 & 3 are an easy version of chapter 10, i.e. the structure of GR in chapter 10 closely parallel's what you already did in 1, 2 & 3 taking special relativity and electromagnetism as your example, what you do in chapter 4 is analogous to what you do in chapter 11, etc...

By studying this book you are already studying general relativity on chapter 1, just an easy version of one part of it in such a way that when you get to chapter 10 you'll really appreciate how radical a change GR is in context, you are doing it honestly in a way you won't have to un-learn, developing the necessary prerequisites, & you see the absolute unity of physics at your feet as a by product. No manifolds, no stupid vector calculus EM (which you don't need to know, better not to tbh), no memorizing Maxwell's equations as if they're god-given or rely on their heuristic derivations, and no need to resort to books like these.

The only potential roadblocks standing in your way of being able to pick this book up are the (multi-variable) calculus of variations and classical mechanics (based on single variable calculus of variations), & Landau's hard so you'll probably have to look in other sources at times, but it's always worth more to struggle towards understanding what he's saying than running away, though you might need tools in other books to see it in a nice way, e.g. Noether in Gelfand, Tensor's in Borisenko, remembering some crazy EM derivations via differential forms in MTW, fun things from Padmanabhan. Do whatever you have to do but "http://www.researchgate.net/post/Any_recommended_reading_for_physics_undergraduate_student"

Following the above approach is to think of GR as a natural extension of the calculus you know and love, but it is a local approach (local means you always assume a coordinate system and basis, though arbitrary). Taking the manifolds approach to GR is basically taking the point set topology (think Kelley) and Lebesgue measure approach to calculus, or at least the metric space & Riemann-Stieltjes approach to real analysis (still paying lip service to the Kelley & Lebesgue approach) so that you can take a global approach to GR (global means you don't impose any coordinate system or vector space basis, you do things before laying these out). The benefits are generality, nice global geometric intuition, and more... The cost is that you'll instantly be told to work with a special kind of second-countable Hausdorff topological space out of the blue & to impose conditions that are obvious in, or naturally fall out of, the local approach - if we're going to get so pedantic why not go back to axiomatic set theory and do it properly, taking your time to first do single & multi-variable calculus, using these as practice with your tools so that you'll have some intuition by the time you get to GR? It's chancy to take unfamiliar tools and apply them to something as notoriously difficult as GR when you can do GR using tools you have or are relatively easily accessible. If you didn't do it for calculus, why arbitrarily do it for GR & ignore doing GR in a way you can do based on what you've already done?

To learn GR directly through the language of manifolds without properly developing those tools is to learn a subject unintuitively paying lip service to special cases you have little to no appreciation for while using tools you have little/no intuition for leaving yourself open to committing some basic basic topological error should you stray too far from the path your book sets out. A hint at what you're doing is that you're shifting the emphasis from thinking of your metric as $g_{\mu \nu}$ to thinking of it as $g_{\mu \nu} = g(\hat{e}_{\mu},\hat{e}_{\nu})$ and giving a set-theoretical & topological foundation to the notion of vectors on a curved surface, what a surface itself is, what it means to define $g$ over the whole surface and then playing a lot of necessary games to do what is very natural using the coordinate approach. There's a lot of value to this approach, but if you didn't have the urge to go back to point set topology in your study of calculus (or axiomatic set theory like I did ) don't think it's absolutely necessary in order to understand GR when it's not.