# List of required branches of mathematics to study GR

## Main Question or Discussion Point

Can you help me on making such list?
"list of required branches of mathematics to study GR"
I'm a high-school student, and I need to know what exactly I must study first to begin studying GR.

Can you help me on making such list?
"list of required branches of mathematics to study GR"
I'm a high-school student, and I need to know what exactly I must study first to begin studying GR.
First of all, to receive a good motivation for this stuff, start with the thin book "Physics in Spacetime" by Benjamin Schumacher. After you are though with it, the second thing to do is going to a library and picking d'inverno's book on GR "A first course in General Relativity" and coming back to home and then you should start with the section A concerning special relativity. After doing so, you have to go to some bookstore and buy a book named "Schaum's Tensor Calculus" by David C. Kay and do every single one of its excercises to get ready for stepping into GR. Then read the rest of d'inverno's book and whenever you hit a problem, we are here to help you out!

AB

The most important is differentiable manifolds/geometry. Then also calculus for functions of one and several variables. Of course, solving of differential equations is very important too. Otherwise you won't be able to find the solutions of Einstein's equations...

It is not necessary to study differentiable manifolds in the most rigorous mathematical way to understand and be able to work with GR. That's why physics courses on GR usually contain all the practical aspects of differentiable geometry that is needed for GR. After than, any physicist can study the differentiable geometry deeper by going to mathematics courses/textbooks if they would like to gain som better understanding.

There are lots of other mathematical subjects that are applied by people working on GR. But that is not required to study GR as a beginner.

EDIT: Oh, and by the way. A good understanding of special relativity is required of course...

Torquil

Fredrik
Staff Emeritus
Gold Member
You guys forgot linear algebra.

From single-variable calculus, you need to understand basic stuff about functions, everything about derivatives, and a little about integrals and limits of sequences.

From many-variable calculus, you need to understand everthing about partial derivatives (the chain rule for functions of many variables is especially important).

From linear algebra, you need to understand these things very well: vector spaces, linear operators, linear independence, bases, inner products, orthonormal bases, matrix multiplication, the relationship between linear operators and matrices. (This stuff, plus complex numbers, eigenvalues and eigenvectors, is even more important for quantum mechanics).

You will need that linear algebra stuff when you start studying tensors, and you'll need that calculus stuff when you study tensors in the context of differential geometry. The mathematics of GR is differential geometry, but Altabeh is right that you don't need a very solid understanding of differential geometry to pass a GR course. I agree with torquil that a solid understanding of special relativity is important too.

Some book suggestions: "Black holes and time warps: Einstein's outrageous legacy" by Kip Thorne is the best book about GR without math, so you can start reading it right now. "Linear algebra done right" by Sheldon Axler is my favorite linear algebra book. (I don't have any good recommendations for calculus...the books I studied were in Swedish). "A first course in general relativity" by Schutz has pretty good introductions to both SR and tensors.

I also like John (/Jack) Lee's books about differential geometry, but they are far too advanced for you right now. The names of those books are "Introduction to smooth manifolds" and "Riemannian manifolds". The former can teach you the basics about manifolds and tensors in the context of differential geometry, and the latter can teach you about geodesics (the curves that describe the motion of free-falling objects), and curvature, but don't bother with these books until you've studied those other things . Any GR book will cover those topics too, but it really helps to have a good differential geometry book where you can read about the things that the GR book doesn't explain well enough.

bcrowell
Staff Emeritus
Gold Member
I think some of the previous responses may have missed the part where the OP said he/she was a high school student. There are many different mathematical levels at which one can study GR. IMO what is most difficult about GR is not the math. GR requires a highly developed level of *physical* understanding.

In direct reply to the subject line of the OP ("list of required branches of mathematics to study GR"), my list is:
-high school geometry
-high school algebra

I would suggest starting with nonmathematical treatments of relativity. One good one is the one in Hewitt, Conceptual Physics. There is very little math in that book, but anyone who really deeply understands the relativity presented in it knows a lot of relativity. For example, it has a very good treatment of a twin paradox, which gets posted about endlessly here on PF. Another good nonmathematical book is Gardner, Relativity Simply Explained. (It's a little out of date, though.) Also: The First Three Minutes.

With nothing more than calculus, one can do the treatment of GR in the Feynman Lectures, and also Exploring Black Holes by Taylor and Wheeler, and Spacetime Physics.

Thanks a lot to all of you
Fredrik, Altabeh, torquil, bcrowell

I've already started studying special relativity, differential equations and linear algebra
so I suppose the next step would be differential geometry
and I thought I'll spend long time to find good books
but you told me everything!
thanks again :)

Susskind has a GR lecture course on youtube which includes both the physics and mathematics of GR.

Thanks a lot to all of you
Fredrik, Altabeh, torquil, bcrowell

I've already started studying special relativity, differential equations and linear algebra
so I suppose the next step would be differential geometry
and I thought I'll spend long time to find good books
but you told me everything!
thanks again :)
That's quite impressive, TrifidBlue! But also don't forget the basic topics that act as a foundation for all of physics. For example, nonrelativistic mechanics is very important, both in the Newtonian framework, and Lagrange/Hamilton framework. Also, try to apply your newfound mathematical knowledge to physical problems. I believe this can also lead to an improved mathematical understanding.

However, in math it is also sometimes necessary to let go of the apparent physical restriction of our world, in order to study the more exotic objects. Consider e.g. the flat two-dimensional torus surface.

So a good balance between the abstract and the concrete is in order for a theoretical physicist.

Torquil

This post falls into the question I have about the flow and direction of the math needed. I don't want to re-invent the wheel, I've got a smattering of math thru self study of physics from applied to GR and SR. I understand the concept behind Feynmans many body diagrams and can also follow the books I have acquired over the yrs in the various branch's of physics.

I'm very aware of my lack of a formalized mathematics background and would like to resolve that. I've got about a half dozen comp books I've filled with my own notes and abstracted math I made up to 'test' my ideas and try to put a 'proof' to my thoughts. Recently I decided to actually teach myself the math needed as I was reading Penroses' latest book 'the road to reality' the math intro was great and I realized I need to merge 'own' math with what everyone else is using.

So what comes first?
linear algebra ----> calculus ----> geometry (flavor?) -----> ???

I've got background geometry, trig and algebra. I also have a strong intuitive compass on Cartesian coordinates, polar and linear. I've been working with 3D software for 20yrs, it was a godsend over the borco table of old! finally I could work as I see it in my head. I'm an internally visual thinker, I see pictures and graphs to equations and numbers. the abstract is easy, the rigorous math can draw a blank.

Basically I don't want to study in circles, If I study out of turn then I have to back track. is there a generally accepted direction on the math flow?

I'm at the point where I'm trying to further my knowledge and I see that my lack in this area is holding my progress back. I can formalize a theory but I'm unable to show the math behind it. I'll dig the net and this forum, find that I was right and then be able to see the math and with some backtracking of the equations I can see the picture and it confirms the process I visually went thru to get there.

My apologies for the long post.

In the case of GR, I think the order goes something like this:

1. Multivariable Calculus
2. Linear Algebra
2b. Extension to Tensor Algebra
2c. Tensor Calculus
3. Differential Geometry