Which text on differential geometry to supplement relativity

In summary: If the focus is relativity, I would chooseB. O'Neill: Semi-Riemannian Geometry with Applications to RelativityorF. de Felice & C.J.S. Clarke: Relativity on Curved Manifolds If the focus is relativity and other physical topics, I would chooseT. Frankel: The Geometry of Physics (who also has a small book called Gravitational Curvature )orP. Szekeres: A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry
  • #1
Kostik
58
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I am looking to pick up one of these texts, but I don't really want to buy all three. Is there a considered favorite? Thanks in advance.

B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity

T. Frankel: The Geometry of Physics

B. Schutz: Geometrical Methods of Mathematical Physics
 
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  • #3
It depends.
Those three focus on different topics and will appeal to different audiences. What kind of relativity course of study are you pursuing?
What relativity text are you using?
 
  • #4
My favorite differential geometry reference is the series of books by John Lee: https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20

Frankel is not a differential geometry text per se, but more of a grab bag of advanced geometry, topology, and algebra needed in several areas of mathematical physics. In this role, I recommend Nakahara over Frankel.
 
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  • #5
robphy said:
It depends.
Those three focus on different topics and will appeal to different audiences. What kind of relativity course of study are you pursuing?
What relativity text are you using?

robphy: I am refreshing my E&M / SR with Ohanian's E&M text. I want to study GR and have several books - Schutz (old edition - small green paperback), D'Inverno, Weinberg, Ohanian-Ruffini, Dray and Zee. I was planning to read Weinberg first to get "to the physics" as quickly as possible.

I have a PhD in math but in analysis and number theory. My background in geometry and topology is weak, but I am generally good with proofs and mathematical "maturity". I was considering getting a supplementary text as noted above, since I find I often sidetrack myself to put the math on solid footing if the physics text doesn't do so. Thanks.
 
  • #6
I'm partial to Nakahara, especially if you're comfortable with a more "mathy" treatment.
 
  • #7
For a supplement to relativity, I would prefer a book that made explicit connections with relativity.
"How can mathematics be used to model the physics?"

The three books you listed are written by mathematically-oriented relativists
It's difficult to pick one.
I offer my opinions as a physicist interested in geometrical formulations.

If the focus is relativity, I would choose
B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity
or
F. de Felice & C.J.S. Clarke: Relativity on Curved Manifolds

If the focus is relativity and other physical topics, I would choose
T. Frankel: The Geometry of Physics (who also has a small book called Gravitational Curvature )
or
P. Szekeres: A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

B. Schutz: Geometrical Methods of Mathematical Physics
would be good as an overview... but you might find yourself looking elsewhere for more details.

I admit that I'm not so comfortable with Nakahara.You might find these useful:
http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf
https://projecteuclid.org/euclid.bams/1183539848 (article by Sachs & Wu., who also have an old book "General Relativity for Mathematicians")
http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html
 
  • #8
I'm a Nakahara-fan :P But a text like Carroll also gives sufficient background info, I guess. Btw, I absolutely love Zee's book. It's big, but a fun read full of surprising insights and topics!
 
  • #9
haushofer said:
I'm a Nakahara-fan :P But a text like Carroll also gives sufficient background info, I guess. Btw, I absolutely love Zee's book. It's big, but a fun read full of surprising insights and topics!
Several of reviews on amazon say that Nakahara has too many typos. Are they very bothering?(I mean the typos!)
 
  • #10
Shyan said:
Several of reviews on amazon say that Nakahara has too many typos. Are they very bothering?(I mean the typos!)
Not really. I didn't have the impression is was so bad.
 
  • #11
Kostik said:
I am looking to pick up one of these texts, but I don't really want to buy all three. Is there a considered favorite? Thanks in advance.

B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity

T. Frankel: The Geometry of Physics

B. Schutz: Geometrical Methods of Mathematical Physics

I like Schutz, it's pretty basic, but well done. I also think it's better oriented towards physics. I think it's a little too basic to make it you're only text, but I guess it depends on what you're looking for as an end goal. I think Frankel is good, but I found it to be a bit idiosyncratic. As a math person you might like it better though. I don't know O'Neill, so can't comment on how that compares.
 

1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using methods from calculus and linear algebra. It provides a framework for understanding the geometry of spaces that vary smoothly and continuously.

2. How is differential geometry related to relativity?

Differential geometry is an essential tool for understanding and formulating the theory of relativity. It provides a mathematical framework for describing the curvature of spacetime, which is a fundamental concept in relativity.

3. What are some recommended texts for learning about differential geometry in relation to relativity?

Some popular texts for learning about differential geometry in relation to relativity include "Gravitation" by Charles Misner, Kip Thorne, and John Wheeler, "General Relativity" by Robert M. Wald, and "A First Course in General Relativity" by Bernard Schutz.

4. Are there any prerequisites for studying differential geometry in relation to relativity?

A solid understanding of calculus and linear algebra is essential for studying differential geometry in relation to relativity. Some familiarity with classical mechanics and electromagnetism may also be helpful.

5. How can differential geometry be applied to areas other than relativity?

Differential geometry has applications in various fields, including physics, engineering, and computer graphics. It is used to study the behavior of physical systems, design optimal structures, and model curved surfaces in computer graphics, among other things.

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