The math of General Relativity

[nasijen]

Hello,

As a soon-to-be Astro grad student, I was wondering if taking classes such as Vector & Tensor analysis as an undergrad, or any form of tensor analysis is necessary for studying GR in graduate school. It's pretty high on the math list here as it has a couple of prereqs (Real analysis, which I'd figure..) but I heard GR's math consists of a lot of tensor analysis.

Thanks

 

[jfy4]

Hello,

As a soon-to-be Astro grad student, I was wondering if taking classes such as Vector & Tensor analysis as an undergrad, or any form of tensor analysis is necessary for studying GR in graduate school. It's pretty high on the math list here as it has a couple of prereqs (Real analysis, which I'd figure..) but I heard GR's math consists of a lot of tensor analysis.

Thanks

If you got Dirac's General Theory of Relativity along with a cozy tensor book, that would prepare you a pretty good amount.

To answer you, I don't think it's necessary, but I would recommend it.

 

[bcrowell]

No, it's not necessary. They will teach you tensors in your graduate GR course. However, if you're planning to be a specialist in theory doing GR, studying tensors in advance could help you to get the most out of your GR course.

 

[Ikoro]

I am an undergrad i took a vector and tensor analysis class anyways...b4 i took Gen. Relativity my sophomore year {yes i am a rising junior with too many physics classes}

 

[nasijen]

Thanks for the replies.

I am an undergrad i took a vector and tensor analysis class anyways...b4 i took Gen. Relativity my sophomore year {yes i am a rising junior with too many physics classes}

Did you find taking vector & tensor analysis helpful when you took GR?

 

[Ikoro]

Yes it helped..the little tricks i learned helped me in Gen. Relativity so go for it..doesn't hurt, or you could just buy the book and study it on your own save you the money...give the economy and all that.

 

[nasijen]

I could do that but I still haven't taken any real analysis which I think is necessary to study tensors, :(

 

[Ikoro]

not really..i mean what is mathematical analysis anyways...jst diff. integ. analytic func...i mean its all what you probably know..go straight for it.

 

[nasijen]

Well I have done all of integral/differential calculus/equations. The tensor class at my school has Real Analysis as prerequisite. I wish I could just jump into tensor analysis, would save time xD

 

[jfy4]

Well I have done all of integral/differential calculus/equations. The tensor class at my school has Real Analysis as prerequisite. I wish I could just jump into tensor analysis, would save time xD

You can learn Tensor Calculus, without the analysis. The analysis is exactly what it says, its the proofs and discussions around the methods of tensor calculus. But to do well with general relativity, I would not say it's necessary. It might help eventually if you plan to move well past a superficial understanding of the mathematics, but for a first course, if you can do well with normal 3-vector calculus, tensors are just a step more.

 

[Daverz]

If you have the chance, I would take a "Differential Geometry of Curves and Surfaces" course. This will give you a better understanding of concepts like curvature. It's not absolutely necessary for learning GR, though. Also if you are lucky enough to have an opportunity: a "Calculus on Manifolds" course (if your school's "tensor" course has analysis as a prereq, then it probably is a calculus on manifolds course). Again, you don't need to know things in that much detail to learn GR.

I would prioritize these courses below any elective physics courses that my be useful for astrophysics (e.g. fluid or continuum mechanics).

 

[nasijen]

If you have the chance, I would take a "Differential Geometry of Curves and Surfaces" course. This will give you a better understanding of concepts like curvature. It's not absolutely necessary for learning GR, though. Also if you are lucky enough to have an opportunity: a "Calculus on Manifolds" course (if your school's "tensor" course has analysis as a prereq, then it probably is a calculus on manifolds course). Again, you don't need to know things in that much detail to learn GR.

I would prioritize these courses below any elective physics courses that my be useful for astrophysics (e.g. fluid or continuum mechanics).

Yeah, I agree with most of the stuff you said. We do have a math course called Differential Geometry. I'm not sure what they do in Real Analysis but I've been told they prove all of calculus by the end or something. Perhaps I'm confusing it with Real Variables...Anyways, Astro classes as an undergrad don't seem to involve much mathematics besides algebra..

The thing about the Tensor course here is that in the course description, it states that they will discuss applications to relativity theory. However, the prerequisites for this Tensor class seem a little useless for Astrophysics.

 

[Daverz]

Yeah, I agree with most of the stuff you said. We do have a math course called Differential Geometry.

The kind of course I'm talking about covers the local theory of curves and surfaces. Typically it's a 3rd year course. Otherwise it's a course that starts from the beginning with the abstract manifold approach.

I'm not sure what they do in Real Analysis but I've been told they prove all of calculus by the end or something. Perhaps I'm confusing it with Real Variables...Anyways, Astro classes as an undergrad don't seem to involve much mathematics besides algebra..

The thing about the Tensor course here is that in the course description, it states that they will discuss applications to relativity theory. However, the prerequisites for this Tensor class seem a little useless for Astrophysics.

Real analysis is very useful for later study in math (if your research interests require that you learn more mathematics). It reinforces the concepts of calculus (particularly if your calculus courses emphasized computation over theory) and will help you develop "mathematical maturity".

But for astrophysics you should probably prioritize taking all the undergrad physics you can.

 

[nasijen]

Applications of the calculus to the study of the shape and curvature of curves and surfaces; introduction to vector fields, differential forms on Euclidean spaces, and the method of moving frames for low- dimensional differential geometry.

^ That is the description of Diff Geometry at my school. It actually sounds really interesting in general. Ill add this course to my math-list along with Tensor & Vector analysis. With that being said, I think this suffice as undergraduate mathematics as far as Astro is concerned.

 

[haushofer]

I think the lecture notes by Sean Carroll are the perfect balance between mathematical rigor and physical applications and intuition. You don't need to study differential geometry beforehand; it's all in there. Also, the lecture notes by Gerard 't Hooft on Black Holes are a very nice introduction.

You could take a look to these notes to get an impression :) Have fun with it, it's a great field!

 

[Daverz]

Applications of the calculus to the study of the shape and curvature of curves and surfaces; introduction to vector fields, differential forms on Euclidean spaces, and the method of moving frames for low- dimensional differential geometry.

^ That is the description of Diff Geometry at my school. It actually sounds really interesting in general. Ill add this course to my math-list along with Tensor & Vector analysis. With that being said, I think this suffice as undergraduate mathematics as far as Astro is concerned.

That does sound like an excellent course.

It's hard to say what math will be most useful for Astrophysics as it's a huge subject area. If you end up studying dust clouds or something like that, the geometrical stuff may not be very helpful.

 

[nasijen]

I think the lecture notes by Sean Carroll are the perfect balance between mathematical rigor and physical applications and intuition. You don't need to study differential geometry beforehand; it's all in there. Also, the lecture notes by Gerard 't Hooft on Black Holes are a very nice introduction.

You could take a look to these notes to get an impression :) Have fun with it, it's a great field!

I will keep that in mind, and check out those references when I get the chance. Thanks, ill try to :).


I will be done with my Physics degree as an undergraduate fairly soon, and having taken a Stellar Astrophysics course along with an Intro to Cosmology, it's definitely the sub-field I want to go into as a graduate student. I just want to be ready for GR, and other advanced physics courses mathematics-wise.

It is hard to say what's required for Astro because it is a huge field. I do want to do theoretical work (physical cosmology, black holes, etc.) which why I am a bit worried about the mathematical preparation.

 

[MikeLizzi]

I sat in on a graduate course "Introduction to General Relativity" at my local university. I already knew SR. I listened very hard tried to follow the GR textbook. I didn't know what a tensor was when I started and I still don't.

Whenever someone writes that you can learn the math at the same time you are learning the physics I feel like committing homicide.

 

[nasijen]

I sat in on a graduate course "Introduction to General Relativity" at my local university. I already knew SR. I listened very hard tried to follow the GR textbook. I didn't know what a tensor was when I started and I still don't.

Whenever someone writes that you can learn the math at the same time you are learning the physics I feel like committing homicide.

Haha, I totally understand. Physicists teach physics and use a language you already should know. I take on the analogy that how can you teach English to a foreigner while reciting Shakespeare. It's one of the reasons I am afraid to start of going into something like GR without knowing its Math.

 

[LBloom]

I know a lot of people learn about tensors and the relavant math while taking GR or are taking Differential Geometry at the same time they're taking GR. With a lot of Physics classes you will be learning new math while you're taking the course and this will probably true in grad school too since you'll be learning graduate level math and beyond but may not be taking grad math courses (i don't know for sure, I'm an undergrad).

In terms of learning about Tensors for GR, I wouldn't worry too much. For an independent study project I read through a book called Tensor Geometry and it basically taught the relevant concepts of tensors and riemannian geometry for a physicist. It doesn't require too much background knowledge. At the time I was reading the book I hadn't taken my school's real analysis class although I was familiar with proofs from an introduction to logic and proofs class and from my linear algebra class.

It's always best to learn the math before you learn the physics but sometimes its best to learn math through a physics textbook.

For GR i'd recommend the Schutz, Sean Carroll's notes, and maybe a short course in general relativity.

 

[WannabeNewton]

It depends on how MUCH math you want to learn. You could learn a lot of tensor analysis and differential geometry by reading Caroll's text and Wald's text but if you read a math textbook on the subject(s) then IMO you would gather more insight when actually learning the physics because you can more easily see how the mathematics beautifully falls into place without cluttering your mind with the mathematics and physics at the same time.

 

[nasijen]

It depends on how MUCH math you want to learn. You could learn a lot of tensor analysis and differential geometry by reading Caroll's text and Wald's text but if you read a math textbook on the subject(s) then IMO you would gather more insight when actually learning the physics because you can more easily see how the mathematics beautifully falls into place without cluttering your mind with the mathematics and physics at the same time.

Yeah, I definitely want to learn my mathematics from a Math class. I need to see it get done formally first with all its details shown. When the Physics rolls around, I don't mind if they use it informally. This is usually the kind of mental frame I've had since I began studying Physics.