Carroll Sean M. Spacetime And Geometry An Introduction To General Relativity
Yazmín Bohon
Spacetime and Geometry: An Introduction to General Relativity provides a lucid and thoroughly modern introduction to general relativity for advanced undergraduates and graduate students. It introduces modern techniques and an accessible and lively writing style to what can often be a formal and intimidating subject. Readers are led from physics of flat spacetime (special relativity), through the intricacies of differential geometry and Einstein's equations, and on to exciting applications such as black holes, gravitational radiation, and cosmology. Subtle points are illuminated throughout the text by careful and entertaining exposition. A straightforward and lucid approach, balancing mathematical rigor and physical insight, are hallmarks of this important text.
Sean Carroll is an assistant professor in the Physics Department, Enrico Fermi Institute, and Center for Cosmological Physics at the University of Chicago. His research ranges over a number of topics in theoretical physics, focusing on cosmology, field theory, and gravitation. He received his Ph.D. from Harvard in 1993, and spent time as a postdoctoral researcher at the Center for Theoretical Physics at MIT and the Institute for Theoretical Physics at the University of California, Santa Barbara. He has been awarded fellowships from the Sloan and Packard foundations, as well as the MIT Graduate Student Council Teaching Award. For more information, see his Web site at http: //pancake.uchicago.edu/ carroll
The main focus of Sean Carroll's General Relativity textbook is to provide a comprehensive understanding of the theory of general relativity and its applications in modern physics. It covers topics such as the geometry of spacetime, Einstein's field equations, black holes, and cosmology.
While it is a comprehensive textbook, it is not recommended for beginners in the field of general relativity. It assumes a basic understanding of calculus, linear algebra, and classical mechanics. It is better suited for advanced undergraduate or graduate students in physics.
This textbook stands out for its clear and concise explanations, as well as its modern approach to the subject. It also includes more recent developments and applications of general relativity, such as gravitational waves and the expanding universe.
As mentioned before, a basic understanding of calculus, linear algebra, and classical mechanics is necessary. It is also helpful to have some knowledge of special relativity. Some prior exposure to differential geometry is also beneficial but not required.
Yes, this textbook can serve as a useful reference for researchers in the field of general relativity. It covers a wide range of topics and includes many exercises and examples for further understanding. However, it may not be as comprehensive as other specialized reference books on specific topics within general relativity.
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Spacetime and Geometry: An Introduction to General Relativity provides a lucid and thoroughly modern introduction to general relativity. With an accessible and lively writing style, it introduces modern techniques to what can often be a formal and intimidating subject. Readers are led from the physics of flat spacetime (special relativity), through the intricacies of differential geometry and Einstein's equations, and on to exciting applications such as black holes, gravitational radiation, and cosmology.
Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade.
It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by Book recommendations, but beyond that: What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail?
First general relativity is typically taught at a 4th year undergraduate level or sometimes even a graduate level, obviously this presumes a good undergraduate training in mathematics and physics. Personally, I'm more of the opinion that one should go and learn other physics before tackling general relativity. A solid background in classical mechanics with exposure to Hamiltonians, Lagrangians, and action principles at least. A course in electromagnetism (at the level of Griffiths) I think is also a good thing to have.
Mathematically, I think the pre-reqs are a bit higher and since the question asks about mathematical detail, I'll focus on that. I learnt relativity from a very differential geometry centric viewpoint (I was taught by a mathematician) and I found that my understanding of differential geometry was very helpful for understanding the physics. I've never been a fan of Hartle's book which I think is greatly lacking on the mathematical details but is good for physical intuition. However having worked in relativity for some time now I think it's better to teach from a more mathematical point of view so you can easily pick up the higher level concepts.
Additionally, I think you really need to understand what is going on mathematically to understand why we must construct things the way we do. I'm going to have to disagree with nibot here and say that you'll need more then just linear algebra and college calculus. Calculus you must have at least seen up to vector calculus and be familiar with it. Linear algebra is something you should have a very good understanding considering that we are dealing with vectors. A good course in more abstract algebra dealing with vector spaces, inner products/orthogonality, and that sort of thing is a must. To my knowledge this is normally taught in a second year linear algebra course and is typically kept out of first year courses. Obviously a course in differential equations is required and probably a course in partial differential equations is required as well.
I don't think a course in analysis is required, however since the question is more about the mathematical aspect, I'd say having a course in analysis up to topological spaces is a huge plus. That way if you're curious about the more mathematical nature of manifolds, you could pick up a book like Lee and be off to the races. If you want to study anything at a level higher, say Wald, then a course in analysis including topological spaces is a must. You could get away with it but I think it's better to have at the end of the day.
I'd also say a good course in classical differential geometry (2 and 3 dimensional things) is a good pre-req to build a geometrical idea of what is going on, albeit the methods used in those types of courses do not generalise.
Of course, there is also the whole bit about mathematical maturity. It's a funny thing that is impossible to quantify. I, despite having the right mathematical background, did not understand immediately the whole idea of introducing a tangent space on each point of a manifold and how $\\partial_i\$ form a basis for this vector space. It took me a bit longer to figure this out.
You can always skip all this and get away with just the physicists classical index gymnastics (tensors are things that transform this certain way) however I think if you want to be a serious student of relativity you'd learn the more mathematical point of view.
EDIT: On the suggestion of jdm, a course in classical field theory is good as well. There is a nice little Dover book appropriately titled Classical Field Theory that gets to general relativity right at the end. However I never took a course and I don't think many universities offer it anyway unfortunately. Also a good introduction if you want to go learn quantum field theory.
Some reponses here are close to "do a mathematics degree, then a physics degree". I think this is not what you expected. Learning all that subjects is factible and natural while you are at the university, but trying to adquire all that knowledge alone on your own in your free time, with no professors, no classes, no press to do exams in certain dates... that is almost impossible.
Having said that, I think I understand exactly what you want, because I have had the same question once. I am a physicist, but after my degree, I realized that my understanding of GR from a mathematical point of view was only superficial. So I researched on the question and finally designed a "step by step" bibliography that I am still following in my free time. Here you have it:
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