Math of GR Exercises from Spacetime & Geometry by Sean Carroll

In summary, this conversation was about recommendations for books or lecture notes with practice problems on the topics of Manifolds and Curvature in the context of General Relativity. Some suggestions were given, including a book called "A Visual Introduction to Differential Forms and Calculus on Manifolds" and a book on Lie series. The conversation also briefly touched on the definition of Lie series and its application in solving differential equations.
  • #1
shinobi20
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I have been reading the book Spacetime and Geometry by Sean Carroll, especially Ch. 2 Manifolds and Ch. 3 Curvature. I'm just wondering are there any lecture notes or books with lots of practice problems (with solutions or at least answers the better) that is suitable for physicist?

To give an example, in section 2.3, the book talks about how the tangent space is defined and how tangent vectors are constructed; exercises might be of the form, given a coordinate transformation find this and that, or show that this and that are orthogonal, etc. I'm seeking for exercises that allows for practice using these concepts that are relevant to physicist. Some people might recommend just plain pure math references where you need to prove this and that, but that is not what I'm looking for.

The exercises should focus more on the "math used in GR" (but still tailored for physicist) as opposed to the physics of GR like, find the gravitational time dilation of..., compute the variation of the lagrangian and find the EOM, etc.

So in short, exercises that are relevant to Ch. 2 Manifolds and Ch. 3 Curvature of the book. I already know many GR resources like Zee, Nightingale, Schutz, Ohanian, Rindler, Blau, Tong, etc. but their exercises are either too few or have no immediate relevance to the topics mentioned above. I find that exercises related to the math of GR to be not abundant, at least in the context of GR books that I know of.
 
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  • #6
CJ2116 said:
You might want to look at A Visual Introduction to Differential Forms and Calculus on Manifolds. He covers in whole chapters what Carroll only discusses in a few paragraphs (i.e. pushfowards/pullbacks, deriving the covariant form of Maxwell's Equations etc.) So many examples as well. This is one of my favorite math books!
Very nice book, seems very promising! Didn't know there exist a book.
 
  • #7
Also, just wanted to ask if anybody knows of any notes that follow Carroll's book or at least expounds on his book? Specifically, Ch. 2 Manifolds and Ch. 3 Curvature. The book recommended by @CJ2116 is a very nice alternative look though.
 
  • #11
martinbn said:
What is Lie series?
After some web-searching it seems that Lie series are exponentials of operators.
 
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  • #12
martinbn said:
What is Lie series?
It's just a way to solve differential equations, essentially, it's just saying, given a differential equation, even simple ones like ##\frac{du}{dt} = -u^2, u_0 = 1## can be solved by using an exponential+series i.e ##u(t) = e^{-tu^2 \frac{d}{du}}u|_{u_0}##

In order to get the solution ##u(t) = \frac{1}{1+t}## you must expand out the exponential using a series.

I think, technically, it is using something with vector fields (hence why we have ##-u^2 \frac{d}{du}## in the exponential), but I wouldn't know the pure math details as I don't have the book near me, nor do I think it goes into that much detail, nor have I applied this technique that often to need to know the more grainy details.

EDIT: Here is a paper for those who want more details: https://www.sciencedirect.com/science/article/pii/0022247X8490057X
 
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1. What is the purpose of studying the math of General Relativity (GR)?

The purpose of studying the math of General Relativity is to understand the fundamental principles and equations that govern the behavior of gravity and the structure of the universe. It allows us to make predictions and explain phenomena such as the bending of light, the expansion of the universe, and the existence of black holes.

2. What are some key concepts and equations covered in the exercises from Spacetime & Geometry by Sean Carroll?

Some key concepts and equations covered in the exercises include the principle of equivalence, the geodesic equation, the Einstein field equations, and the Schwarzschild metric. These concepts and equations are essential for understanding the foundations of General Relativity and its applications.

3. Is prior knowledge of advanced mathematics necessary to understand the exercises?

Yes, a strong foundation in advanced mathematics, particularly calculus and linear algebra, is necessary to understand the exercises in Spacetime & Geometry by Sean Carroll. It is recommended to have a solid understanding of multivariable calculus, tensor calculus, and differential geometry.

4. How can practicing these exercises improve one's understanding of GR?

Practicing these exercises can improve one's understanding of GR by providing hands-on experience with applying the mathematical concepts to real-world problems. It also helps in developing problem-solving skills and gaining a deeper understanding of the underlying principles of General Relativity.

5. Are there any online resources available for further practice and learning?

Yes, there are several online resources available for further practice and learning of the math of General Relativity. Some recommended resources include online lectures and courses, interactive simulations, and online forums for discussion and problem-solving. It is also helpful to refer to other textbooks and research articles for a more in-depth understanding.

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