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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."
start with this pdf and then slog through the wikipedia articles on GR http://web.mit.edu/edbert/GR/gr1.pdf
For the interested.
You've made an admirable attempt to define your question clearly, but you didn't quite succeed. General relativity can be understood at a variety of mathematical levels, so saying you want to understand "the mathematics of general relativity" doesn't really pin it down.
The other issue is that you haven't defined your physics background. If you really want to understand GR, you need to be fairly sophisticated in physics.
The first thing I'd suggest is that you build a solid foundation of understanding in special relativity. The best intro to SR is Taylor and Wheeler, Spacetime Physics, and you already have the math background to understand that.
Physically, GR is a field theory. The first field theory was electromagnetism. E&M is a lot easier to understand than GR, because it takes place on a fixed background of flat spacetime, and it also connects directly to everyday experience. The more intuition and technical skill you can build up in the context of E&M, the better prepared you'll be for GR. For someone ambitious about going far in physics, the best intro to E&M is Purcell, Electricity and Magnetism. Purcell uses vector calculus, and he tries to teach you all the vector calc you need as he goes along. However, you will want some of the preparation provided by a second-semester calc course, and you will probably also have an easier time if you can also study from a separate book on vector calculus. Here is a free online calc book that I like, and here is a free vector calc book you could use. When you're learning second-semester calc, I'd suggest you skip the integration tricks that form the bulk of such a course; they're largely irrelevant to your goal, and nowadays you can use Maxima or integrals.com for that kind of thing.
With that background, you're more than prepared to start studying GR at the level of Exploring Black Holes, by Taylor and Wheeler.
If you want to go on after that and understand GR at a higher mathematical level, you could try an upper-division undergrad book such as Hartle or my own free book, and then maybe move on to a graduate-level texts. The mathematics used in graduate-level texts is typically introduced explicitly in the text itself; basically tensors and calculus on a manifold. You don't need any more math prerequisites than vector calculus before diving in. The classic graduate text is Misner, Thorne, and Wheeler. I would still recommend it wholeheartedly, except that it's now decades out of date. A more modern alternative is Carroll; there is a free online version, plus a more complete and up to date print version. Other GR books worth owning are General Relativity by Wald and The Large-Scale Structure of Space-Time by Hawking and Ellis.
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+1 on this and all related posts: Relativity is about physics, about beautiful physics, and is not about math.
There are bits of relativity for which Einstein had to go math-shopping: He knew what the physics must look like, he needed to know if the mathematicians had any tools that matched what he wanted to express (they did, Lorentz transformations being one of the most important).
Note: I have a physics degree, which means I have studied more math than anything else. The math is important to express the physics precisely, important to get useful answers to specific questions. But the physics come first. (There's the old trope of the physics prof saying "set C to 1 so you can see the physics happening.)
Read about and try to reproduce Einstein's thought experiments. Start with the one about travelling at the speed of light, and what you would see as you approached C (hint: if you travel at C, photons can only reach you from in front, from along your axis of travel). Think about the "falling in an elevator" experiment. These get you a long way to the principle of equivalence, the principle of relativity, etc.
Only once you have some idea of the physics should you attempt to tackle the math - and by that time, you'll be starting to get a good idea of what the math might look like.
Do not attempt to learn the math first and thereby get to the physics. There lies madness.
I'm here EdgeKeep Inc.
That's not really true. Dirac went looking to remove the square from E=mc^2 since it allowed for the possibility of negative matter and energy. Eventually he came up with a solution using matrices, which as it happened once again left the door wide open for negative matter and energy and ultimately lead to the prediction and subsequent discovery of antimatter. In this case the maths directly lead to a major advance in physics.
Without maths, how would physicists even theorise anything? All they would have is their intuition which is at best useless and at worst an active hindrance to the the discovery and understanding of major advances in physics of the 20th century and beyond.
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 "
Sorry, no. First, most physicists nowadays don't talk about mass increasing with speed. Mass in that sense is really just energy (divided by c^2). It's much more meaningful to talk about invariant mass (also called "rest mass", since it is unambiguously the mass the object has as measured in the frame of reference where it's at rest) in pretty much any context where mass, rather than energy, is relevant. But, even ignoring that, your math is wrong. Using the interpretation that mass increases with speed, an object traveling at .5% of c will have a mass increase of about .00125%. An object traveling at 50% of c will have a mass increase of about 15.5%, and an object traveling at 95% of c will have a mass increase of 220% (so, it will be 3.2 times heavier than at rest).
Furthermore, it takes no energy expenditure at all to continue moving at a constant speed. You only need to expend energy to change your speed (or direction).
But I think it boils down to not only can we not exceed C we can't go slower either. Everything moves at C and the axis of that motion we perceive as time. And everything else we call reality is the contortions required to make that so under all circumstances.
Sir –
I wouldn't quite describe it that way, from the perspective of the epiphany Einstein must have had. I don't think it's that complex, and in any case I think it's more beautiful than that. As a matter of interest, perhaps someone will find the following worth reading.
We have space, and it's where we live. This space is physical but can be represented by representations in our brains and on various media, which representations we call physics.
We make rules in physics to reflect what happens in our space, our reality. Some rules we can see, and they are generally intuitive. For example: Two points - places - are distinct when not the same position, and these points are indivisible (identity). Also, two lines added together make a third line, regardless of the order those lines are added in (commutativity). Three lines can be added in any order to equal the same distance (associativity). Two lines never meet (parallelism). This is the Euclidian space, and applying such to our universe is Newtonian physics (aka classical physics).
Suppose though that the physical world in which we live is not Euclidean, contrary to our observations and intuition. Suppose in this world parallel lines in our world meet at infinity. We can call this a Lobachevsky space (also known as a hyperbolic geometry), and its principles formed the essential breakthrough in general relativity.
Once one accepts as axiomatic that we live in a Lobachevskian space, the acceleration of mass becomes governed (for reasons beyond the scope of this note) - otherwise we would violate other rules (e.g. identity). Hence the perception of time slows in lieu of infinite acceleration (imagine two trains travelling at the speed of light towards each other; to each other they would appear to be travelling only at the speed of light - not, as one might expect, twice the speed of light - because time relative to each other slows; contrast a stationary that expects both to pass at the speed of light in opposite directions). This effect is observed and compensated for in our Global Positioning System.
All to say, by changing our perspective from representing our accepted physical world as a Euclidean geometry to something unintuitive, a Lobachevskian geometry, we arrive at the ability to represent and predict what happens in our physical world.
The consequences inherent to the axiomatic perspective of living in Lobachevskian space are commonly and collectively referred to as "general relativity", and they are non-trivial. The underlying premise that commenced that perspective is itself quite simple.
Even Einstein himself never claimed to understand the why of GR. GR is all about the math. The vague analogies sometimes bandied about aren't science. They are flights of fancy and completely unproven and were only ever used to try to explain the math to people who didn't understand the long tensor calculus equations. The math itself is the science. There is no way around the equations. GR cannot be explained with natural language. Only with mathematics.
Quite an experience to live in fear, isn't it? That's what it is to be a slave.
Unless you can work out on your own how to put numbers on it, your understanding is imperfect. Being able to run some numbers through an equation doesn't mean you understand it even as well as the guy who doesn't know any maths but knows where to stand to catch the ball.