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My Background - I finished Master's in Mathematics with heavy emphasis on Topology and Differential Geometry.I have no training in Theoretical Physics.

What I want to learn - I want to learn the foundation of Physics leading to Cosmology.I don't need one book containing all but it could be a study plan leading to cosmology.I am not worried about how much time it would take.At least tell me how to start.

What kind of books do I want - 1) I want to know how physicists think the world works and what led them to think that(be it experiments or anything). 2) I want physical intuition behind everything that are being done in the book and I want mathematics to make it rigorous(more like giving those physical intuitions some mathematical framework). 3) Not necessary but it is preferable that the mathematics is Geometry and Topology heavy.

                          *I don't need the same book for physical intuition and mathematical 
                           framework.They can be different but support each other properly.

What kind of books I do not want - 1) I don't want pop-sci books. 2) Although I want mathematical rigor behind intuitions, I don't need chapters on mathematical preliminaries.

edit - I said books but it actually can be any study material.

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  • $\begingroup$ I want to know how physicists think the world works and what led them to think that --- Perhaps begin with The Feynmann Lectures in Physics (freely available "read only" format). If you really like these, I recommend getting the hardback versions for greater ease in reading/studying. FYI, I very carefully read all of Volume I, and less carefully parts of the other two volumes, in 1982 using hardback versions that I purchased then and still have. Of course, back then I think they only cost ~50 dollars. $\endgroup$ – Dave L. Renfro Apr 20 at 11:46
  • $\begingroup$ Skimmed some part of the lectures.Looks like it contains good physical intuition. $\endgroup$ – Sagnik Biswas Apr 21 at 10:32
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I am a physicist and I had specialized in Astrophysics back at university. Since you already have a background in differential geometry and topology, I would suggest the (one of the) two all-time classics about gravitation:

Note, however, that these are ancient books (mid 1970's). But at least at around 2000 they were still state-of-the-art in teaching. Maybe you can find more recent books. Of course, Gravitation theory as such has not changed since Einstein, but applications and probably some solution techniques probably will. And, of course, our understanding of the large scale structure of the universe has changed a lot since these times. However, dark matter and dark energy, which play a role in the latter are not at all understood, so as far as we know, gravitation theory/general relativity is still the same and only the matter that fills the universe has to be 'corrected' (presumably).

If you want to dive into supersymmetric/string theories and quantum gravitation, I cannot recommend any books because I am not into that. I would not recommend that you start with that because it would probably not give you much physical understanding (which seems to be what you're after), and it requires a lot of quantum field theoretic knowledge as well. On the other hand, you as a mathematician might enjoy the 'otherworldliness' of these theories very much. Be prepared that this has not (yet) much to do with observable physics.

If you want to understand star formation or the so-called 'big bang', there is a lot to understand from quantum field theory and especially nuclear physics. I think this is a huge topic, and you will either spend a lifetime on that alone, or you will end up reading more 'phenomenological' treatments of it.

A word of caution: physics, as it is taught, is a very different style compared to mathematics. I am seeing this from the opposite perspective than you do. While mathematicians lay out their assumptions very carefully and then prove everything, physicists often omit (presumably) self-evident assumptions and only sketch proofs. Sidenote: the most ironic thing about contemporary physics is that quantum field theory is the best working theory we ever had (predicting measurements up to 10 or so significant figures), but it still does not have a rigorous mathematical foundation. This is already painful for me, but I can hardly imagine how this must feel for a mathematician.

This is all for the purpose of getting straight to the point instead of (presumably) wasting time with trivialities. So compared to mathematics, you will need a very good memory of what was said on page 20 while you're at page 500. Most things will be referenced back, but many things will just be assumed that you remember them. And while imagination is kind of 'sugar coating' in mathematics, it is essential (with respect to fundamental physical experiments) in physics. I would even say this is rule #1: know the essential experiments/observations that led to the theories. For gravitation, this is just a handful (Basics of Electrodynamics and Michelson-Morley for special relativity, Kepler's Law, Free Fall, perihelion shift of mercury, Pound-Rebka, gravitational lensing, etc. for general relativity).

It will probably be pretty painful for you to read a physics book as it is for me to read a mathematics book. If I read a math book, I am always impatient with all the abundant notation, the lemmas and auxilliary theorems that are just building blocks for the central theorems. Physics tries to get to the point faster. You will probably find that the way theoretical physics is treated is unbearably sloppy.

Having said that, the book by Weinberg above is a little more rigorous mathematically, as far as I remember.

Of course, your mileage may vary. If you can let yourself go, theoretical physics will not be a problem for you.

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  • $\begingroup$ I know,theoretical physics is taught very differently than mathematics but if things "makes sense" but not rigorous,that is okay. And, You are right in saying,I am after physical understanding but then I may try to see if it is possible to look at that mathematically.Also,I do like the "otherworldliness" of the modern theories but I want to know physically, why they believe those otherworldly theories. Lastly,I like getting to the point and checking the consistency of the trivial cases later.Thank You. $\endgroup$ – Sagnik Biswas Apr 21 at 10:11
  • $\begingroup$ Also,what are the physical prerequisites of those books? $\endgroup$ – Sagnik Biswas Apr 21 at 10:11
  • $\begingroup$ You have to know classical mechanics, of course. Some essentials of electrodynamics are beneficial because they are the foundation of special relativity (which in turn is the foundation of general relativity, i.e. the metric), but not mandatory. Since GR is a field theory, you would benefit from ED being a field theory as well, but from the mathematical point of view, that's already covered by knowledge of partial differential equations. To reiterate: don't try to approach GR by understanding just the math. Pay very much attention to experiments or observations that are mentioned in the books. $\endgroup$ – oliver Apr 21 at 10:18
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So this is not a book but Frederich Schuller has a series of lectures on General Relativity leading upto cosmology and black holes at the end, he's quite rigorous with the math which is great. If you're familiar with the geometry and topology you can probably skip a good amount of lectures. In my opinion this guy is probably one of the greatest lecturers out there I would definitely recommend you take a look.

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    $\begingroup$ If I can follow that, it seems like it would be very efficient and suitable for me. But do you know how much physics prerequisite they assume?Thanks for finding it by the way. $\endgroup$ – Sagnik Biswas Apr 21 at 6:08
  • $\begingroup$ Not much if any, he motivates all the physical intuition and then uses the math to rigorously formulate it. $\endgroup$ – Basel J. Apr 21 at 6:42
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What about these?

Theoretical Physics

Mathematics for Physicists

Mathematics of Classical and Quantum Physics

Also, have a look at the "customers also bought this" as that is what you might be interested in too.

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