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Ask Slashdot: Math Curriculum To Understand General Relativity?
First time accepted submitter sjwaste writes "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."
For the interested.
I work in cosmology and use general relativity extensively in my day to day work. I have also fielded similar questions from friends and undergraduates, so I can provide you with advice based on my experience.
What approach you use depends on how well you want to understand. I am going to assume that you want to understand the equations and how to manipulate them --- that when asked about the anomalous procession of Mars, you could sit down with a pencil and graphing calculator for an hour and tell them that GR accounts for ~40 arcseconds/century. To get there, you will need to cover a series of courses: Classical Mechanics, Linear Algebra, Special Relativity, Multivariable Calculus, and then General Relativity. If you also study Electromagnetism and Differential Equations, you will get a bit more out of it, but those subjects are not necessary.
Classical Mechanics (prereqs: none): You don't need anything beyond an AP physics level understanding of mechanics, but you do need that. MIT has all of the 8.01 (classical mechanics) lectures online.
Linear Algebra (prereqs: none): You need to understand what a vector is, what a matrix is, what a linear transformation is, and what traces and determinants are. You probably have this knowledge from stats. If not, trys Jacob or any similar text.
Multivariable Calculus (prereqs: Linear Algebra): A standard undergrad book is fine. You need to know how to transform variables and use multivariable differential operators. A standard course is online.
Special Relativity (prereqs: Classical Mechanics, Linear Algebra): Special Relativity is essential for understanding General Relativity. Of particular importance is the 4-vector notation and the Lorentz transformation. A. P. French is one of the classic textbooks.
General Relativity (prereqs: Special Relativity, Multivariable Calculus): The nice thing about introductory Physics texts is that they teach you all the differential geometry you need to understand. The unfortunate thing is they tend to be aimed at Physics graduate students. There are a few undergrad textbooks, but they are not as rigorous and not as worthwhile to read. The classic General Relativity textbook is Misner, Wheeler, Thorne, but MWT is better as a reference text than as a first course. Better textbooks would be Wald, General Relativity, and Carroll, Spacetime and Geometry . Of the two, I would recommend the latter.
You should keep in mind that the texts will be hard and the learning curve will be steep. The best way to understand the material is to do most of the problems in the undergraduate books or all the problems in the graduate texts, and ideally, have someone read over your problem sets. It will, however, be rewarding.
There are some physical and mathematical fields that should be looked at first before a serious attempt to dig into general relativity.
On the physics side, I recommend looking at classical mechanics, special relativity, and the history of physics research (theory and experiment) during this critical time. I think it's important to know not just the results, but why they came around to that line of thinking. The history is also something you can do for entertainment or inspiration while you're building up the considerable list of prerequisites for the general theory.
The math side is very hard. As I see it, most of the math is under a vague title, "differential geometry". There are three main parts: differentiation and integration in multiple variables (generally, you're working in "3+1" variables for general relativity and dealing with partial differential equations in this space); manifold theory; and Riemannian geometry (which manifests in general relativity as the very similar Minkowski geometry). I mention partial differential equations above. They're nice to know, but not essential for the theory.
The first can be found in the end of college calculus books. Such treatments generally suffer from ignoring differential forms. I have a specific recommendation here. While you are going through that calculus book, also read "Differential Forms with Applications to the Physical Sciences" by Harvey Flanders. It is a smallish Dover book with a good treatment of differential forms (and their use in multi-variable differentiation, integration, and differential equations).
Manifold theory is one of the more interesting contributions of mathematics to the world. The idea is that you have an object, called a "manifold", that looks, locally like a fixed dimension Euclidean space at each point of the manifold. The dimension of the Euclidean space is in turn the dimension of the full manifold. For example, the surface of the Earth crudely looks like a plane with wrinkles (ignoring holes like arches and tunnels and whether you consider the top or bottom of oceans as "surface"). But it's sort of ball-shaped while a plane is infinite in extent.
On a plane, you can label the entire plane with a pair of coordinates so that each point of the plane has a unique coordinate and vice versa. Not so with the surface of Earth. However, you can map local pieces of the Earth's surface to a plane one-to-one and onto. That is typical behavior for a manifold.
The fundamental concept is that a manifold has local behavior and description provided by a particular set of "coordinate charts" which lead to global behavior and descriptions over the entire manifold. How that's done is hard to understand, but powerful in application. There are consistency conditions on that set of coordinate charts that allow for various structures (such as the subsequent "Reimannian metric") defined in terms of one coordinate chart to be converted via some change of variables algorithm to become in terms of another coordinate chart which happens to overlap with the first.
Finally, there's Riemannian geometry and its analogue, Minkowski geometry for general relativity. The idea here is that you have a manifold with an additional structure, a "metric" which defines a sort of inner product on the tangent vector fields of the manifold as well as a distance between points on the manifold. The Minkowski metric is no longer a true metric. One of the coordinates has become "time-like" resulted in a single dimension with negative length. You can't measure distance any more with the metric, but you still have the inner product property on the tangent vectors, which are now called phase vectors and can be used to describe velocity and momentum in a system with several space-like and one time-like coordinates.
And that's enough to describe general relativity, as a physical system operating on a manifold with a Minkowski metric which has three space-like coordinates and one time-like coordinate (dimension "