How much math do I need to learn for general relativity?

[Simfish]

So in general, it seems that most physics courses don't really require anything beyond linear algebra, multivariable calc, and differential equations. All the other math is taught as the student learns Quantum Mechanics or Electromagnetism (this is how students learn how to do PDEs and such). As such, those three math courses are the only three needed for physics majors here (although they do take a math methods sequence, though this only extends the math toolkit to partial differential equations, complex variables, and Fourier transforms).

But general relativity seems a lot different. It has items from a math major's toolkit - abstract algebra, topology, differential geometry, and even possibly math at the graduate level. Do most physics students end up having to take those math courses when they learn general relativity, or are they still able to "learn math along the way of learning general relativity?" Does general relativity also require real analysis?

 

[union68]

Differential geometry is a big one for GR, and that's part of the reason I've never attempted to get into it, ha. I don't know a blooming thing about it. Don't be so fixated on one theory though, make sure you check out the other parts of physics along the way!

 

[JDGates]

Like other physics courses, a good GR course will teach you the necessary math along the way. Even folks I knew who did GR research in grad school generally didn't take extra math courses, though of course it's an option.

 

[fasterthanjoao]

Like other physics courses, a good GR course will teach you the necessary math along the way. Even folks I knew who did GR research in grad school generally didn't take extra math courses, though of course it's an option.

The courses I have completed in GR have been very much like this. The first half of the course is setting up the framework. The second half is the physics.

The reason I think this is a good way to approach GR is that the mathematics can be difficult to comprehend for a physicist - the way we're trained to think about things isn't conducive to the way you need to approach GR. Example: physicists like to visualise quantities and how problems exist - GR is all about tensors, something that, for almost all of us normal people, is impossible to comprehend. Then, topics that you'll be used to as a physicist even require a different way of thinking. For instance, in GR, a vector isn't defined in any way that you'll likely be used to. A vector isn't a line- it's anything that, when contracted with a one-form, results in a number.

So yes, the mathematics is very different from what you require in most other physics courses: but, in my opinion, any well designed GR course will incorporate the necessary mathematical training, afterall a physicist is a different animal from a mathematician and should be approached as such. Of course, if you do take extra mathematics courses in these topics you'll do yourself no harm but you will find that you'll end up learning far more than is required for physics.

 

[Fredrik]

As others have said, a course on GR will teach you what you need...at least what you need to be able to do general relativity. Abstract algebra isn't a part of it. Some topology is needed to fully understand the definition of a manifold, but I think most physics courses just skip that part. That actually makes sense, because knowing exactly what a Hausdorff and second countable topological space is, won't help you with the exercises. What you really need to know before you start is just the most basic stuff about vector spaces (linear independence, bases, linear operators and matrices), and the chain rule. I like to write it in the form $$(f\circ g)_{,i}(x)=f_{,j}(g(x))g^j{}_{,i}(x)$$.

So what math courses you should study depends on what you intend to do in the future. A person who intends to do research in a field like mathematically rigorous quantum field theory must absolutely study advanced analysis, measure and integration theory, topology, differential geometry, and functional analysis. A person who intends to do research in GR should definitely study differential geometry from a good set of books at some point, but you won't have to do it to pass an introductory class. (I recommend the books "Introduction to smooth manifolds", and "Riemannian manifolds: an introduction to curvature", both by John M. Lee).