What course would a donkey like me follow in order to "earn an equivalent to an undergraduate degree in gravity"?
Excepts from "A Degree in Gravity" by Tim Thompson and Ivan Viehoff
From Tim Thompson: "You simply cannot pick and choose which part of physics you want to study. It doesn't work. However, a personal study program does not need to match the same depth of preparation as one would find in a formal program. So one can, rather then study lots of prerequisites first, simply establish a light but broad foundation, and then concentrate on "prerequisite" topics as they come along. That way one does not lose patience by studying something that seems irrelevant; you only study what you need to study to handle the matter at hand.
Before I go on to describe how I think this personal study program should look, let me add that a strong understanding of basic algebra and calculus, up to and including differential equations is an absolute prerequisite for any "advanced" study of any topic in physics, where by "advanced" I mean anything roughly equal to a bachelor's degree. The better your foundation in mathematics is, the easier it will be to handle the topics in physics which make use of it.
"How to create your own "B.S. degree" in Gravity":
- Step 1 - "Mathematical Foundation
One does not need to spend time studying mathematics on its own as a prerequisite. Rather, the practical way to proceed is to be prepared to cover the topics needed as they come up, with the idea in mind that studying gravity specifically is the ultimate goal. So, here is my advice on a list of mathematical reference books one should have at hand which are most likely to support the primary goal.
Explorations in Mathematical Physics: The Concepts Behind an Elegant Language by Don Koks; Springer, 2006. This 529 page book should be the primary reference. It includes special emphasis on special & general relativity, along with differential geometry, tensor calculus, and specific tools for solving covariant field equations.
Mathematics for Physics & Physicists by Walter Appel; Princeton University Press, 2007 (originally published in French in 2001). This 626 page book covers the more general mathematics not necessarily found in Kok's book; i.e., complex analysis & conformal mapping, Laplace & Fourier transforms & etc. A good supplement to Kok's book.
Mathematical Perspectives on Theoretical Physics: A Journey from Black Holes to Superstrings by Nirmala Prakash; Tata McGraw Hill, 2000. Don't be put off by the emphasis on string theory in the title. This 802 page book includes advanced topics crucial to understanding gravity, such as conformal field theory, symmetry & supersymmetry, quantum theory, group theory, Yang-mills theory and special emphasis on general relativity & gravitation
- Step 2 - Physical Foundation
The Six Core Theories of Modern Physics by Charles F. Stevens; MIT Press, 1995 (I have the 6th printing from 2002). This 221 page book has brief review chapters on classical mechanics, electricity and magnetism, quantum mechanics, statistical physics, special relativity and quantum field theory. It does not replace a few years of concentrated study in specialized formal course, but it does get the points across quickly. It does establish an appreciation for what it important and what is not, an appreciation that will be valuable later on as a guide to where one should focus personal effort.
Obviously, one does not replace all of the basic foundations in physics by one book. To supplement this book, if one later needs more in depth treatment, I recommend:
The Feynman Lectures on Physics by Richard Feynman; Addison-Wesley, 2005 (originally published in 1963). Basic physics is basic physics, it's the same now as it was 5 years ago. Feynman's books really are both educational & relatively easy to read, and the 3 volume set covers all of the classical physics in Stevens' book, but in much more detail.
- Step 3 - Astrophysics
Theoretical Astrophysics: Volume 1, Astrophysical Processes
Theoretical Astrophysics: Volume 2, Stars and Stellar Systems
Theoretical Astrophysics: Volume 3, Galaxies and Cosmology
This 3 volume set is by Thanu Padmanabhan; Cambridge University Press, 2000, 2001, 2002. This is your primary reference for astrophysics. The set is extensive and covers the basics of any topic you are going to run in to. The final volume has worked its way to the primary topics of gravitational interest, galaxy formation and cosmology. If you successfully plow through all 3 volumes, you will know more about astrophysics that anyone you are likely to meet. Volume I includes a chapter in general relativity. Volume II concentrates on stars, you could go lightly or skip it, but there will be a hole in your knowledge if you do. After all, Volume III goes in to galaxies, and galaxies evolve because the stars in them do.
You should be able to get the physics you need from these books, and the mathematical references.
- Step 4 - Gravity
Gravitation by Misner, Thorne & Wheeler; W.H. Freeman, 1973. Don't be put off by the fact the book is 35 years old. There is a reason it is still in print. It remains the primary reference to the basic study of gravity and general relativity. You want to use this 1216 page book as a reference, rather than as a working text book, though there will doubtless be places you want to work through. This is where you will need the deepest mathematical preparation.
As a working textbook, I recommend:
Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll; Pearson Education & Addison-Wesley, 2004. This 499 page book, (I don't include index pages), covers general relativity and gravitation directly, and is a textbook with problems, exercises, etc. Chapters include special and general relativity, gravitation, manifolds, curvature, and gravitational radiation. This is your main goal. Work through this book, and you will know gravity.
You may find additional references to be useful here, aside from Carroll's text book, and the massive tome by Misner, Thorne & Wheeler. So, along those lines, I recommend:
Gravity: An Introduction to Einstein's General Relativity by James B. Hartle; Pearson Education & Addison-Wesley, 2003. This 567 page book could be used as the working text instead of Carroll. The goal is the same, but the layout is different. Hartle emphasizes geometry & geodesics in spacetime, and includes additional material on black holes & gravitational collapse.
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg; John Wiley & Sons, 1972. Another oldie still in print for good reasons. Like Misner, Thorne & Wheeler, it's age means the approach is different, and a lot of newer material found in Hartle & Caroll won't be there. Nevertheless, this is one of the classic introductions to the field. It's 631 pages concentrate on applications in cosmology, with very little astrophysics. So there is a much greater emphasis on geometry, curvature, and the mathematical basis of the field equations of pure general relativity.
Of course, I left out huge chunks of physics that somebody else might want to include. But the point was to create an abbreviated rather than comprehensive course of study. There is plenty of Newtonian gravity in Six Core Theories, Feynman's Lectures, Padmanabhan's trilogy, and Gravitation. The latter book includes extensive treatments of post-Newtonian formalism, as does Weinberg's book.
I am not such a big fan of the post-Newtonian formalism for understanding gravity or general relativity. Obviously, this is an opinion open to considerable debate. But I find the post-Newtonian formalism to be more confusing than enlightening. It is found in the older books, but not much in the newer books, and I suspect that's because others are of the same opinion. The modern methods are easier for me to understand.
Certainly, a deep understanding of Newtonian gravity can't hurt. I simply decided that it was not necessary to go deeply into orbital mechanics or central force problems as separate topics, because they are all treated well enough (I think) in the books I reference, for the task at hand. Remember, even in the case of a formal program of study, the "degree" is where you start studying, not where you stop studying. So my only goal is to get the "student" up to speed on the basics as quickly as reasonable. I assume the "student" will proceed to work on these ideas for the rest of their lives, leaving plenty of time to pick up the more arcane aspects of gravitational physics, or a broader understanding of applications like radiative transfer in collapsing giant molecular clouds.
Finally, I don't think studying the Principia itself is of any value in the context of the question at hand, namely a modern understanding of gravity in cosmology and astrophysics. I have several editions of Newton's Principia; its geometric formalism is so arcane it is almost unreadable. That's why Chandrasekhar decided to translate parts of the Principia into modern mathematical notation:
Newton's Principia for the Common Reader by Chandrasekhar. After seeing his book I can say I admire Chandrasekhar's notion of what constitutes a "common" reader. He translates Newtonian geometry into differential equations, which are much easier for modern readers to understand, assuming that the "common" reader commonly studies differential equations. But so much work has gone into classical mechanics and classical Newtonian gravity as to render the Principia itself of purely historical interest, in my opinion.
- Step 5 - Newtonian Gravity
Classical and Celestial Mechanics: The Recife Lectures , edited by Hilberto Cabral & Florin Daciu; Princeton University Press, 2002. This book concentrates on gravity in classical mechanics. It is a good reference I think.
Classical Dynamics: A Contemporary Approach by José & Saletan, Cambridge University Press, 1998 (reprint 2002). This 668 page book concentrates on Hamiltonian & Lagrangian mechanics. These methods are key to understanding the field theory approach to gravity.
The Classical Theory of Fields by Landau & Lifshitz; Elsevier, 1975 (first edition 1951). Volume 4 of their multi-volume set on theoretical physics. This is of course the classical text on classical field theory, which obviously does include a field theory discussion of both Newtonian gravity and general relativity.
If you want to get into Newtonian gravity specifically, there's three ways to do it.
Final Comments
All of the books I recommend here are books I have in my own library of a few thousand books on mathematics and physics. But this message is clearly biased by my own thinking about how I teach myself from my own resources. Others may find other paths to learning more efficient. Some people prefer to take formal courses, others do better on their own. And some of us are just too old and lazy to work to somebody else's formal class schedule. I decided to write this message because the more I thought about it, I was struck by the challenge. How does one go about self teaching a complex topic like this? This is the answer I came up with for myself." Tim Thompson
From Ivan Viehoff: "I wonder if you could shorten the reading list by buying this one:
The Road to Reality, A Complete Guide to the Laws of the Universe, by Roger Penrose. The first part of the book includes all the relevant mathematics you need. It also has all the quantum stuff, but I think it is arranged so that you can concentrate on gravity, etc.
Mathematics isn't just something you learn, it is something you understand. While I was doing my Mathematics degree, some of my colleagues reached a ceiling of understanding. They were very conscientious, hard-working students, but they had just reached their limit. I reached my own ceiling of understanding in one or two branches. You'd reach a point where I just no longer had a sufficient complete grasp of what was going on to understand how the next statement came out of the preceding, or how to move the argument on myself. One of these branches was classical mechanics.
My point is that a great number of students can read course material in a subject like gravity, without actually being able to truly grasp it, even though they may think they do.
Another important aspect of it involves the interpretation of a models. The mathematics in basic special relativity is quite simple. It is simple enough that I was taught it in high school. Understanding what it means, and knowing how to apply it, is very difficult." Ivan Viehoff
Donkey logged onto the forum and thanked all involved for their valuable insight and wisdom. I thank them again here.
I may not yet know exactly how to get where I want to go, but I am beginning to see how I might get there and how to know that I have arrived. There will be many obstacles, (donkeys aren't known to be very smart ya' know), but now Donkey has a road map ... Hee Haw!
Next blog entry: "Beginning at the Beginning": Back to Grade School
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