Essential Math Topics for GR: Semi-Riemannian Geometry (Newman)

[Shirish]

I'm reading the book "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Newman. There are definitely other questions on math background needed for GR, but my aim is to find which topics from that particular book aren't essential so that self study is more efficient.

The ToC for the book is here: https://www.barnesandnoble.com/w/semi-riemannian-geometry-stephen-c-newman/1133040658

My purpose is to get an idea about the mathematical underpinnings of intermediate-level GR. It would be incredibly helpful to me if people familiar with GR here can give me an idea on which topics I can skip and which ones are critical.

For example, I know that ch 1-6, ch 9, ch 10, ch 14-15, ch 18-19 are essential. And I think that ch 11-13 can be skipped since their purpose seems to be to give a concrete foundation for more abstract intrinsic differential geometry concepts, but I'm comfortable with starting out from abstract concepts. About the rest of the chapters, I'm not sure. Would appreciate your views on this!

 

[PeroK]

I'm reading the book "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Newman. There are definitely other questions on math background needed for GR, but my aim is to find which topics from that particular book aren't essential so that self study is more efficient.

The ToC for the book is here: https://www.barnesandnoble.com/w/semi-riemannian-geometry-stephen-c-newman/1133040658

My purpose is to get an idea about the mathematical underpinnings of intermediate-level GR. It would be incredibly helpful to me if people familiar with GR here can give me an idea on which topics I can skip and which ones are critical.

For example, I know that ch 1-6, ch 9, ch 10, ch 14-15, ch 18-19 are essential. And I think that ch 11-13 can be skipped since their purpose seems to be to give a concrete foundation for more abstract intrinsic differential geometry concepts, but I'm comfortable with starting out from abstract concepts. About the rest of the chapters, I'm not sure. Would appreciate your views on this!

If your objective is to get a undergraduate level understanding of GR, then you can try Hartle's Gravity: An Introduction to GR. It's only if you want to take things further that you need heavy differential geometry.

Altenatively, there is a series of largely self-contained graduate lectures from MIT:

https://ocw.mit.edu/courses/physics/8-962-general-relativity-spring-2020/video-lectures/index.htm

That said, before you start GR you need to know Special Relativity to the level that you can explain it fully to others. Hartle gives a brief revision of SR and the MIT lectures begin with a geometric treatment of SR.

 

[Shirish]

If your objective is to get a undergraduate level understanding of GR, then you can try Hartle's Gravity: An Introduction to GR. It's only if you want to take things further that you need heavy differential geometry.

Altenatively, there is a series of largely self-contained graduate lectures from MIT:

https://ocw.mit.edu/courses/physics/8-962-general-relativity-spring-2020/video-lectures/index.htm

That said, before you start GR you need to know Special Relativity to the level that you can explain it fully to others. Hartle gives a brief revision of SR and the MIT lectures begin with a geometric treatment of SR.

Thank you! I'm aiming at a grad level understanding of GR. I completely agree with mastering SR first, but as of now my aim is only to get a good grasp on the math needed for grad-level GR. I won't touch GR and the physics behind it all without mastering SR. I hope that makes sense.

That said, the heavy DG that you mention for the same - that's the tricky bit. Because in the book there are all sorts of topics and I'm uncertain which ones aren't strictly necessary. Any kind of help in which ones are skippable and which ones aren't would be very useful.

 

[PeroK]

Thank you! I'm aiming at a grad level understanding of GR. I completely agree with mastering SR first, but as of now my aim is only to get a good grasp on the math needed for grad-level GR. I won't touch GR and the physics behind it all without mastering SR. I hope that makes sense.

That said, the heavy DG that you mention for the same - that's the tricky bit. Because in the book there are all sorts of topics and I'm uncertain which ones aren't strictly necessary. Any kind of help in which ones are skippable and which ones aren't would be very useful.

If you are studying Newman's book as a preliminary to learning SR followed by GR, then you are probably wasting much of your time. It doesn't necessarily help to get so far ahead in the mathematics department. You should aim to develop your physics and mathematics in parallel. For example, many people would say that classical electromagnetism is a pre-requisite for GR, because it's too big a step to tackle GR without having the experience of classical EM. The authors of graduate texts in GR will assume you have a working knowldege of classical EM in any case.

The sooner you start SR the better (the mathematical prerequisites are minimal). The first chapter of Morin's book is free online:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

Also, I particularly like Helliwell's book:

https://www.goodreads.com/book/show/6453378-special-relativity

That said, Newman's book looks excellent for grad students.

 

[PeroK]

The ToC for the book is here: https://www.barnesandnoble.com/w/semi-riemannian-geometry-stephen-c-newman/1133040658

@ergospherical do you have any thoughts on Newman's book? Looks good?

 

[ergospherical]

@ergospherical do you have any thoughts on Newman's book? Looks good?

Haven’t read it, sorry! Looks awfully expensive, though.

 

[martinbn]

@ergospherical do you have any thoughts on Newman's book? Looks good?

It looks good, but it is a geometry book first, and not aimed at just the parts relevant for physics. It is similar in spirit to O'Neill or Sternberg's books. If one wants to study geometry first and then GR, it is perhaps easier and faster to just look at a differential/riemannian geometry books. Say Lee's Introduction to Riemannian Manifolds.

edit: Sorry didn't realize this was not addressed to everyone.

 

[Shirish]

It looks good, but it is a geometry book first, and not aimed at just the parts relevant for physics. It is similar in spirit to O'Neill or Sternberg's books. If one wants to study geometry first and then GR, it is perhaps easier and faster to just look at a differential/riemannian geometry books. Say Lee's Introduction to Riemannian Manifolds.

edit: Sorry didn't realize this was not addressed to everyone.

Looking at the ToC, could you give some advice on which chapters can be potentially skipped (in the context of grad level GR prerequisites) if possible?

e.g. as I said in the OP I know that ch 1-6, ch 9, ch 10, ch 14-15, ch 18-19 are essential. And I think that ch 11-13 can be skipped. But not sure about the rest.

Sorry for the bother

 

[martinbn]

Looking at the ToC, could you give some advice on which chapters can be potentially skipped (in the context of grad level GR prerequisites) if possible?

e.g. as I said in the OP I know that ch 1-6, ch 9, ch 10, ch 14-15, ch 18-19 are essential. And I think that ch 11-13 can be skipped. But not sure about the rest.

Sorry for the bother

You can skip the rest on first reading.