Will a mathematician understand GR if he knows differential geometry well?

[Arian.D]

The title says everything. Can a mathematician do fruitful research in general relativity if he masters differential geometry and manifolds?

 

[Dickfore]

Will knowing those mathematical concepts explain to you what a stress-energy tensor means?

 

[nonequilibrium]

No need to be snappy Dickfore, of course Arian.D implied he'd read up on the basic concepts of the theory, but he's asking whether he needs to go deep into physics books (as in: taking considerable time) to learn things he hasn't on the math side of things.

 

[Alexrey]

Well I recently finished a semester course in GR and funnily enough it was a course given by my maths department, not the physics department. I had no problems with GR despite having only done first year physics and no differential geometry. GR as I was taught it is heavily maths based so if you are good at maths and work consistently, GR won't be too bad at all (and yes, I'm sure having done DG would be a great leg up).

 

[homeomorphic]

You'd have to master more than just differential geometry, but yeah, it would help. I don't know enough GR to say to what extent it's worth pursuing, but it's definitely worth it to learn the basics, I think. I took GR from the physics dept. They didn't even use manifolds explicitly, but it probably would have helped for some things.

 

[deRham]

I think if you had a strong knowledge of both PDE and differential geometry (certainly subjects which are linked, but by the latter I understand emphasizing properties of the global objects, whereas PDE emphasizes actually understanding the differential equations), it would be possible to read and look up things as you're going. But I mean, of course you should actually read a book on GR first, so you know the particular kinds of questions addressed there. I think it would help if you have access to physics people to talk about things so it's not all you looking things up.

I'd like to learn something about GR myself, so I'm not quite qualified to answer...the above is what I gathered when I thought about learning some.

 

[mathwonk]

answer no. example me. unfortunately those few times i have attended meetings where physicists were invited to speak on the topic, all they talked about was differential geometry. thus i learned nothing from them. i suppose they were talking about the part they thought was mysterious, but i knew the manifold theory and wanted to learn the physics. the one time a physicist started to say something useful he was immediately interrupted by another physicist in the audience who stopped him by saying he was explaining it wrong.

'so another odd habit physicists seem to have that mathematicians do not have so much, is that physicists like to argue over who is right. mathematicians just prove what they are saying is true and that's pretty much it.

 

[Arian.D]

Well, I definitely didn't mean that I would like to do research in general relativity without knowing anything about physics. I know some Newtonian mechanics that they teach in a usual Physics I course to math students, I have passed Physics II, so I know about electric fields and magnetic fields, etc. I think I'd like to study special relativity on my own in full details, that's pretty easy, I understood special relativity easily when I was in high school from this book:
https://www.amazon.com/dp/0471717258/?tag=pfamazon01-20

I've also read Einstein's 'The meaning of relativity' and had no difficulty understanding the concept of simultaneity and space-time.
Now, my weak points are electrodynamics. I find the subject very unintuitive and absurd and that worries me. Moreover, I don't want to spend hours solving physics problems as I'm a math major, not a physics major. Having said all of these things, do you think I could finally understand GR and write research papers about it? For example, if I succeed in reading 'Gravitation' written by Misner, Thorne, Wheeler, will I be able to do research in GR like an expert?

I'm sorry if I sounded too vague at the beginning.

 

[mathwonk]

come on man, einstein's book is aimed at high school students, not grad students in physics.

 

[Nano-Passion]

come on man, einstein's book is aimed at high school students, not grad students in physics.

I wouldn't say it is aimed for high school students. Just skimming through it really quick, I see linear algebra, integrals, and PDEs.

https://www.amazon.com/dp/0691023522/?tag=pfamazon01-20

 

[Arian.D]

come on man, einstein's book is aimed at high school students, not grad students in physics.

Is Resnick's Introduction to special relativity aimed at high school students too? Actually SR is easily understandable by high school students if they don't introduce 4-vectors and Mikowski space in it I think.

 

[nonequilibrium]

mathwonk was maybe thinking of Relativity:The Special and General Theory

 

[homeomorphic]

Now, my weak points are electrodynamics. I find the subject very unintuitive and absurd and that worries me. Moreover, I don't want to spend hours solving physics problems as I'm a math major, not a physics major.

Electrostatics should be pretty intuitive. If I hadn't studied electrical engineering and physics, particularly electromagnetism, I might not really understand vector calculus to this day, which would be kind of embarrassing. I know math grad students, even gifted ones, who are, indeed, in exactly that embarrassing situation.

However, if you're more of a math person, eventually, you would probably feel better using differential forms, rather than the old classical vector calculus. Probably, my favorite treatment of forms is in V I Arnold's Mathematical Methods of Classical Mechanics. He doesn't cover E and M, but E and M is also good motivation for using forms. So, try to express Maxwell's equations in terms of forms instead of vector fields, first in 3-d. Then, maybe try 4-d. I sort of figured out how to do it myself, my only reference being Penrose's Road to Reality, which tells you the result, but not how to get there. Eventually, I want to write up my version of it and put it online. You can find more of a traditional physicist's version of this in Jackson. The next thing is to see if you can get a wave equation out of that.

Another very nice place to get a bit of intuition for E and M is in Feynman's Lectures on Physics, volume 2. The two things here that are really nice are the example of setting up a plane wave by accelerating an infinite sheet of charge and calculating the speed of light by using a plane wave. This gives you really good intuition for why waves will result and why they travel at a constant speed, regardless of reference frame (which ominously foreshadows the craziness of relativity--this was actually Einstein's main inspiration, rather than Michelson and Morley's results).

Finally, it helps to understand the sort of general idea of Green's function. This is a general technique in PDE, which also happens to apply to electrodynamics. Some very intuitive material discussing this general topic can be found in the last chapters of Visual Complex Analysis and in V I Arnold's book on PDE.

I'm still sorting out the intuition for electrodynamics, myself, and, like I said, when I have it all figured out, I want to put it all online--the most intuitive possible explanation of the whole story.

 

[nonequilibrium]

Homeomorphic, please send me a PM when you've written it.

 

[homeomorphic]

Homeomorphic, please send me a PM when you've written it.

For all I know, it might take me another 5 years, since life is throwing other obstacles in my way. But eventually, I will get around to it, I'm sure.

John Baez once quipped that it sometimes seemed like his life was a series of distractions that was set up to prevent him from writing his internet column, "This Week's Finds in Mathematical Physics."

I enjoyed that comment. I know what he means firsthand.