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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."
Save yourself some trouble and get Relativity; The Special and the General Theory by Einstein himself. In his words "The work presumes a standard of education corresponding to that of a university matriculation examination..." however note those words
were written in 1916 and education standards are somewhat lower now. What used to be required for admission are often not
learned during university at all.
I know I have read it several times now and when I finish and sit and think a bit I'll almost 'get it' before retreating from the gates of madness. Think Cthulhu.
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.
Democrat delenda est
start with this pdf and then slog through the wikipedia articles on GR http://web.mit.edu/edbert/GR/gr1.pdf
Linear Algebra, Differential Equations, Advanced Calculus, Partial Differential Equations, Electromagnetism, Waves, Introduction to Astronomy, Special Relativity, Differential Geometry
Feynman lectures are fun to read
What do you really want to do ? (My guess is that you are not sure.)
If you want to be able to write down and solve Einstein equations for some case, you need vector and tensor algebra, geometry and calculus. Many people who work in GR never do this (for others, it's all they do). If you are interested in some more particular case (black holes or gravitational radiation, say), you need to understand Einstein's equations at some level, plus whatever approximations or simplifications are used in that area (transverse traceless gauge or post-Newtonian approximations, for example). Also, you should get to where you understand Lorentz transforms in your sleep. If you can't do and understand Lorentz transforms, the actual GR math will likely be beyond you.
What I would recommend is to buy Misner, Thorne and Wheeler, and read and follow "track 1." I would allocate 1 year for that.
the one you draw, assuming one the |x> is one glyph means semi-direct product. http://en.wikipedia.org/wiki/Semidirect_product
if you meant |${SOME_NAMES}> it is the bra-ket notation : http://en.wikipedia.org/wiki/Bra_vector
for more help with the notations, wikipedia is your friend @ http://en.wikipedia.org/wiki/List_of_mathematical_symbols
Jehovah be praised, Oracle was not selected
The actual math needed to understand the basics of relativity[1] is actually quite simple. If you've had calculus, you have more than you need.
The hard part is wrapping your brain around the concepts and the fact that the rules you use to interact with the world around you are a subset of the rules of the universe.
A book I have recommended several times for people who want to start learning about physics is 'Asimov on Physics'. Dr. Asimov was a master of explaining difficult science in a way that laymen could understand.
[1] Going beyond the basic, or getting into odder corners of general relativity, is another matter.
Can't really understand it without the math, but over the decades innumerable "popular science" authors have attempted to write about general relativity for the "common man", with no math beyond maybe pythagoras.
Its kind of like having a verbal understanding of ohms law, without actually knowing how to divide. "So you increase the resistance and the current drops, assuming constant voltage, ok?". On a small scale its easier to understand the little bits, but its hard to grasp the entire thing.
One thing to look out for is relativity was "cool" some decades ago, so anything with a tenuous connection, will have GR on the cover and some pictorial representation of an elderly Einstein. Kaufman has a famous book for beginners "cosmic frontiers of general relativity" but note that only a few chapters talk about G.R., the rest is 40 year old black hole research. A better title would have been "black hole physics in the 70s, and related topics.". Its a perfectly good book, just not quite what you're asking for.
Another oddity is no one every provides a pix of Einstein when he did his famous work as a young man, only pictured as an elderly dude. Other scientists don't get that treatment; Feynman's "popular press photos" are all from his middle age when he was earning his 2nd Nobel, Tesla is usually portrayed as a steampunk vampire young goth man...
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
You could have left off the first paragraph and provided an informative response. I was going to post something about MIT's online courseware, too. But you had to preface a useful bit of information with a put-down. Welcome to slashdot where innocent questions are met with derision and insults.
it's = "it is"; its = possessive. E.g., it's flapping its wings.
From the preface:
This is a textbook on gravitation physics (Einstein's "general relativity" or "geometrodynamics"). It supplies two tracks through the subject. The first track is focused on the key physical ideas. It assumes, as a mathematical prerequisite, only vector analysis and simple partial-differential equations.
It is a really fun book to read at the first track level; especially if you are not on the hook for the homework.
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.
Find free books.
Leonard Susskind's Modern Physics lectures on the Stanford University's channel on youtube are excellent.
http://www.youtube.com/watch?v=hbmf0bB38h0
Gravity, by Hartle. It's the textbook we used in the undergrad GR course, so geared towards those with some math, without being too difficult, abstract, or esoteric. If you know college calculus and vectors, I think it does a good job of explaining any of the other math you need along the way. And if you have any questions, a bit of web searching will fill in any holes.
I didn't follow Bra-Ket notation at all until I read up on the history of it. For me, it helped a lot to know Dirac invented it, and that it was needed because it applied to Hilbert spaces, and that Hilbert developed that concept a few years before Dirac got started, and that John von Neumann was the guy who actually named Hilbert's concept "Hilbert Spaces". Why did those things matter?
1. Hilbert was discussing infinities, and he was familiar with Cantor's work (and liked it) so he was using the modern definition of infinities (plural), where there are multiple trans-finites possible. His math was meant to cover all that, and the use of it for QM was a limited case. Some events can be described using a quite limited number of spatial dimensions and the results will be understandable with a little calculus or even trig if you just understand how to take the notation used and put it into actual equations. For example, there's a Hilbert for a three dimensional Euclidean space. Other (particularly in QM) events need many spatial dimensions to describe, sometimes even an infinite number.
2. The Ket part of the notation is about those vectors in a Hilbert space. You could represent that Euclidean space I mentioned with just a Ket notation, for example. Since Hilbert spaces can have either a finite number of dimensions or an infinite number, and can entail complex numbers, the Bra part becomes needed when the Hilbert space has complex numbers involved. The Bra and Ket together are a short way of writing a formula for a complex conjugate, and the whole can be expressed just as a complex number. These can be mathematically manipulated by partial differential equations. Any person with a fair knowledge of Linear Algebra can derive information from them, secure that the treatment is mathematically both complete and rigorous. That seems to be the real point of the notation, it gets results into a form where the rest of the process uses math that's regarded as rock solid.
3. Dirac invented other math for areas where the completeness condition of all Hilbert Spaces didn't apply. He called some of these "rigged Hilbert Spaces" . He proved people could use the Bra-Ket system and similar operations to describe those QM events, but the results won't technically be proven to be correct in an absolute mathematical sense. many working physicists do it anyway.
4. People tend to refer to Feynman for a good source to understand all this and not mention von Neumann as much, but it looks like von N. was historically quite involved in it. Maybe some of what he wrote on QM could clarify Bra-Ket notation better for you than the standard modern textbooks.
Who is John Cabal?
Many introductory general relativity books give you some of the math background you need. A very good one in that regard is Bernard Schutz: A First Course in General Relativity, Cambridge University Press, ISBN 0-521-27703-5. It begins with a very good introduction to special relativity, and then develops the math needed for basic GR. I would avoid Misner, Thorne, and Wheeler. The 2 track approach is confusing, and the math is thrown at you in bits and pieces as you need it, making it hard to see the big picture.
If you are interested in math courses to take, multi-variable calculus, then differential geometry are good choices. If there are separate courses on tensor calculus or tensor analysis, they are good, but that material is often just taught as part of differential geometry. For really advanced stuff, like cosmology, you might need some topology as well.
If I can be modded down for being a troll, can I be modded up for being an orc, or a balrog?
First off, you should pick up an undergraduate text on "Modern Physics," which should include a really basic intro to both special and general relativity. Any text will do, but I own the one by Tipler/Llewellyn. This kind of text will be fairly light on the math, but will include some. This will also get you started with some really basic problems which should show that while you may not fully understand General Relativity (GR), you can do some really basic problems (e.g. gravitational redshift).
I. Calculus. Sounds like you already know some.
II. Differential Equations
A. Ordinary
B. Partial
III. Linear Algebra (Some texts teach ordinary differential equations and linear algebra together)
IV. Math Methods for Physicists (Arfken and Weber). Use this more for reference than for learning. Any math you need beyond the above set will be fairly specialized, so you can study by topic.
V. The best intro to relativity is in David J. Griffiths "Intro to Electrodynamics", a widely used textbooks for undergraduate physics majors. This only covers special relativity, but it's probably a really good place to start. For the graduate level, refer to Jackson's "Classical Electrodynamics," or possibly an easier equivalent.
VI. Another text by Griffiths is "Introduction to Elementary Particles", which includes some really useful stuff on relativity at the undergraduate level but for physics majors.
VII. (admission: I haven't studied General Relativity because I'm in another area of physics (CM), but I've harbored a secret desire to study it and maybe someday will steel away and do it.) A really common book is "Spacetime and Geometry: An Introduction to General Relativity" by Sean Carroll. I've flipped through this and it looks extremely well written, so when I do go ahead with my study, this is probably the book I'll select. Another good one is "A First Course in General Relativity" by Bernard Schutz. These are both graduate level texts, and I can't imagine there being an undergraduate level text.
This may take a long time and will be occasionally difficult, but it is certainly doable. Good luck.
The Geometry of Physics, Theodore Frankel; An excellent introduction to differential geometry and its application not just to GR but to other areas of physics as well. Highly recommended.
A First Course in General Relativity, Bernard Schutz; I found this book helpful in some specific areas -- notably understanding the notions of the stress-energy tensor.
Gravitation, Charles Misner, Kip Thorne, & John Wheeler; This is the classic text, and is comprehensive and comprehensible. I like Wheeler's way of thinking about physics, and it shows through here. There is the standard joke, that this is a text which not only discusses gravitation, but also attempts to demonstrate it by its high mass.
I found Ray d'Inverno's Introducing Einstein's Relativity a good place to start and very well presented (a much 'lighter' introduction than others, although goes in less depth, but if you have to start somewhere ...).
Here's the Amazon link if you are interested (although your university library may have it, mine did which is where I discovered this gem): http://www.amazon.com/Introducing-Einsteins-Relativity-R-dInverno/dp/0198596863
It is quite surprising how limited the mathematical arsenal needed for general relativity is. Considering it is one of the giant theories of physics, the amount of math background needed for general relativity can be learnt in a short time (2-3 months) (in comparison to other theories like String theory which require mind boggling amount of 20th century mathematics and can require several years of learning) . This is provided you have studied math at college level. All you need to know is vectors, tensor and tensor calculus and Reimannian geometry. Pick a good text book of relativity. Lot of books teach the math needed for relativity. I just started working through A Short Course in General Relativity by Foster and Nightingale and it is a very good book. Another good book is 'A First Course in General Relativity' by Schultz but it uses modern index less approach which is more concise and beautiful but also more abstract so harder to grasp for beginners. I found Nightingale much easier to understand.
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.
Several of the preceding responses have covered much of what you'll need.
If you've not had any exposure to tensor analysis, I'd recommend a gentle introduction called: A Brief on Tensor Analysis by James Simmonds.
If you're still needing a grounding in vector calculus Div, Grad, Curl and All That. is a good overview of it.
At least one has recommended Wald as a text. I'd recommend Gravitation by Misner, Thorne and Wheeler. Which one you prefer will become apparent pretty quickly.
And definitely, you will need a quite solid grounding in Special Relativity.
For doing the tensor manipulations with a computer program, GRtensorII for Maple was one I've used.
My instructor in it, Dan Finley at UNM has a page for the class he teaches on it at: http://panda.unm.edu/Courses/Finley/p570.html
One warning, Dan is not one to "spare the rod" when it comes to the mathematics. (Which to me, is a good thing.)
It's a worthy goal, but one that will take a lot of determination, work and preparation. Unfortunately, I had to drop out of Finley's class due to my full time job boiling over (we lost two other employees, and I had to cover). It's been 15 years, but someday I still intend to get back to it.
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 "
If its just a hobby I don't understand why you would want to know the in-depth details since you probably wont be playing with equations most of the time.
On the contrary, if it's a hobby he's probably interested in reading and playing with the various speculative equations for warp drive and time travel - for example, the Alcubierre Drive, or Kip Thorne's wormholes. Which has nothing to do with everyday physics, but everything to do with science fiction worldbuilding and geeky entertainment. Certainly that's what I would do if I understood enough of GR to get to the "test the equations" stage.
You are not a brain: http://books.google.com/books?id=2oV61CeDx-YC
Hey great thread! I can confidently state that I'm in lower percentile of the posters here regarding physics and math (I'm just above the random trolls and bellow everyone else). I found Penrose - The Road to Reality a great overview starting with math I already understood, educating me about some concepts I didn't get before and ending up with today's physics of which I understood, charitably ... uh ... 10 percent ... cough ... I already had a tourist knowledge of higher math but my actual arithmetic is a disgrace and I found Penrose kept me on the horse longer than other texts.
And I've been flamed for recommending this book for reasons I didn't understand in the past so YMMV.
Physics is like sex: sure, it may give some practical results, but that's not why we do it.