Slashdot Mirror
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
Mathematics of Relativity
As you can see, the first two hits are to Wiki with a very nice synopsis of the math subjects required.
Like the OP I do some pop science reading now and then, like the /. articles.
One thing I usually don't get are the QM articles. Can someone point out a good online resource for those things? Like, for instance, what the hell those |x> bra-ket things are suppossed to mean. Wikipedia is generally only good for those kind of things if you know the stuff already and just need a quick reference.
Who said he was still at uni?
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
If you haven't got anything useful to say, keep your fucking mouth shut.
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.
Troll doesn't read / understand summary.
Troll writes trollish thing about summary.
Troll gets modded Troll.
And the world will be just.
The ''problem'' with General Relativity is that it's differential geometry, so one should understand geometry of manifolds first...
Or just use the book Einstein wrote...
Then each of the math problems he has becomes an exercise to learn the math around it... He starts off with fairly simple math and works his way up more advanced stuff.
To understand some of it, a little of differential forms, tensors, differential equations should be enough (i assume analysis and linear algebra to be present already) - maybe 2 or 3 months for the basics.
To understand it fully and make own calculations at the state of the art - the same subjects and all related math fields. Think about something like 1-2years if you have a talent for it.
The Road to Reality : A Complete Guide to the Laws of the Universe
by Roger Penrose
http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679454438
Likely the most serious math book you will find in a retail, consumer bookstore. An excellent read and essential to truly understanding modern physics.
Favorite
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.
I am a physics major, about to get my BA at the end of this semester. I'd say it possible to understand and use every major concept in physics if you understand every thing up to vector calculus and throw in some linear algebra and diff eq (under stand 2nd order should be adequate). Obviously the more math you know the better, but up to this level should be enough to understand most of the material. Just make sure you chose the right physics text books that will hold your hand through the first few chapters and you'll be fine. Honestly I learned most of my advance math skills from my physics text books.
I seem to recall a physicist (I think it was Hawking) that said something along the lines of "if you think you understand [General Relativity], you don't." If you want a good place to start with the mathematics (without even needing more than Trig), pick up the book "Six Ideas that Shaped Physics, Unit R: The Laws of Physics are Frame-Independent" by Thomas A. Moore (ISBN-13: 978-0-07-239714-7, ISBN-10: 0-07-239714-4) It gives the underived equations for many of the effects of special relativity. Once you get that, you can move on the to derivation of the equations, and then eventually General Relativity. -- Mitch
Hi, Try looking for Giancoli's Physics Textbook. It explains is and makes it quite easy to understand
When I was an undergraduate engineering student, I learned relativity from my university's physics department as part of a lower-division series of classes. A typical series looks like this:
Now, as for the math classes, you would usually take many previous math classes (or concurrently) as part of the physics prerequisites. These classes would include three in calculus, linear algebra, differential equations, and vector analysis. I believe this is fairly typical for U.S. college engineering programs.
before you take anything, read "sphereland" to help open your mind.
repeat as necessary until you "get it"
then take vector calculus, field theory, and tensor analysis
(and of ourse, any pre-requisites)
you should now be well eqipped to understand both the
concepts and undrerlying math.
cheers
My understanding is that, while related, general relativity requires tensor analysis (aka vector calculus). Special relativity can be thought of as a 'correction' to Newton's laws of motion. General relativity is more kin to 'altering the topology of the universe' (lack of a better phrase).
prerequisites:
calc I and II
Math for special relativity:
-linear algebra (possibly modern algebra)
good pdf:
http://www.math.rochester.edu/people/faculty/chaessig/students/Adams(S10).pdf
Math for general relativity:
-vector/tensor calculus (class after calc III)
-(optional) complex analysis (adding the point at infinity gives you a rough idea of how topologies can be manipulated/changed. The business of finding poles and using the location of poles in integral domains might help to form some intuition, I'm not sure.)
As pointed out elsewhere, go straight to the source, as well. You'll want to study more than just Einstein's papers, possibly.
PS: I don't reply to ACs.
http://www.youtube.com/playlist?list=PL6C8BDEEBA6BDC78D
Leonard Susskind has a series of free lectures on GR on youtube. They're quite excellent, and they don't assume much beyond basic multivariate calculus (partial derivatives)
Id just recommend reading a "dumbed-down" book first that covers the basic outlines. 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. Otherwise, read up on differential and integral calculus, multivariable calculus, linear algebra, differential equations, electromagnetism, and introductory astronomy. You don't need much more advanced than that to understand the basics. I doubt you will be proving theorems and such. You can get some Schaums Outlines books on some of those topics that would guide you through the process.
That brings me to an interesting point, / . is just "the ramblings of socially-inept, technology-literate news-mongers".
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
Read this pdf online, chapter by chapter, and do the exercises. It should take weeks:
http://virtualmathmuseum.org/Surface/a/bk/curves_surfaces_palais.pdf
If you understand the pdf well, you can probably then take on a graduate level general relativity text directly. If not, you should refresh your trigonometry and calculus first, I suppose.
If you haven't already, you could start with "einstein's Universe" by Nigel Calder. It's a great introduction to relativity without the heavy math. Then learn the math as needed to explore specific parts in depth.
First and foremost, you need a full introductory calculus course and calculus based physics course. There are many options- as far as books are concerned, I'm a fan of Stein's Calculus book and Halliday and Resnick's Physics book but you should also check out MIT's OpenCourseWare as well as Carnegie Mellon's open learning initiative. Once comfortable with the basics, you should get a book (or find a website) on linear algebra, a book on tensors, and a higher level geometry book that includes non-euclidian geometry. At this point, you could move on to a book on general relativity but you might consider getting a book on electromagnetism- it will give you a background on special relativity (David Griffiths book is great). As for general relativity, Robert Wald's text is a good intro, as is Sean Carroll's or James Hartle's. Carroll also has lectures online about gravitational waves (http://elmer.tapir.caltech.edu/ph237/CourseOutlineA.html). You might want to also check out Kip Thorne's Applications of Classical Physics (http://www.pma.caltech.edu/Courses/ph136/yr2008/ - not just GR but many interesting physics subjects). If you get this far and are still interested, look at Misner/Thorne/Wheeler's Gravitation (don't start with this book).
You might also consider getting the Schaum's outlines for the subjects in addition to/ instead of textbooks- many good examples and explanations.
http://www.lightandmatter.com/lm/
This is a decent online textbook that covers basics physics concepts.
I have a degree in theoretical physics, from the UK's top science university, and in my final year I did a course on General Relativity, for which I scored 70% (i.e. a 1st). I then went on to do a PhD in maths (or math for the non-Brits).
Despite the above, I don't fully understand the maths of general relativity. It is really, *really* hard! Likewise for advanced particle physics and quantum mechanics. I get the principles (I think), but unless you're an Einstein type genius, the maths is essentially about learning the rules and applying them. It is not intuitive, and unless you're prepared to write down the equations and work through them for each situation you come across, the maths is going to remain completely opaque.
That said, I still enjoy reading about these subjects on Slashdot and elsewhere. I think it's much more a question of finding good explanations of what the maths means than feeling obliged to work through it yourself.
If you're really keen, I suggest starting with special relativity. The maths is much simpler, but it still requires working through to make sense of the more complex relativistic situations, e.g. questions of simultaneity and so on. If you can manage that and are still keen, come back to general relativity at that point!
https://alephnull.uk/
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.
I would recommend getting Schaum's Outline on Tensor Calculus and working through it. Calculus is a must... and linear algebra is useful. However, don't let the matrix math infest your brain too deeply, because that will make learning tensors harder.
Make sure you understand special relativity before you start... knowing a couple of different way to derive the Lorentz Transformation is a must. Learning how to derive Poisson's field equation is also important. It's also helpful to read about Mach's Principle.
Misner, Thorne, and Wheeler is a great book on General Relativity. I recommend working through it slowly.
It's a lot of work to learn GR on your own, but it you do so, you can gain a much deeper understanding that most people get from a class. The theory is deeply beautiful and profound.
Nobel Prize Gerard 't Hooft has already done that for you: HOW to BECOME a GOOD THEORETICAL PHYSICIST.
Gerard 't Hooft, who won the nobel price in physics by his theory of the holographic priniciple,
has written a nice list of subjects to master to become a theoretical physicist.
http://www.staff.science.uu.nl/~hooft101/theorist.html
I'd take Calc 1,2,3 Then linear algebra, diff eq, partial diff eq. Then a tensor calc class and you should be ready.
Just read "black holes and time warps" by Kip Thorn.
"Many of us, myself included, don't have the background to understand them."
OK, step one: As a college grad you should have learned the difference between an object and a reflexive pronoun. So, let's make that "Many of us, me included, don't have the background to understand them." Or, perhaps "Many of us, including myself, don't have the background to understand them."
On to things mathematical. Don't try to gain a general understanding of differential geometry. It'll be years before you get where you want to be.
Step 2: Learn the basics of tensor notation in a Euclidean setting. Learn wha a metic is and how to measure infinitesimal distances using the local metic. This is trivial in a Euclidean space.
Step 3. Learn how to do this on a sphere. It's one of the easier nontrivial cases. Practice calculating geodesics and Christoffel symbols on the sphere.
Step 4. Learn how to put the GR mass-energy density into the metric tensor. Do this for a simple nontrivial case of a single massive object in an otherwise flat 2+1 dimensional space-time.
Without committing massive amounts of time to become a genuine expert, this is probably about as well as you will come to understand GR. It will certainly put you in a position to appreciate the popular articles on the subject.
Now, many interesting things in the world of physics are not actually GR, but relativistic QM for which you'll want to instead just study the special theory and probably the Dirac formulation of QM.
Can I ask the same question for particle physics -- specifically non-abelian gauge theories. I'd like to be able to under stand the Higgs mechanism and supersymmetry properly and how the particles emerge from the symmetries of the fields.
My pure maths background is quite strong, but I stopped doing applied somewhere in my second undergraduate year and have forgotten most of the more advanced bits of it. So I have a hazy memory of curvilinear coordinates, and an even hazier one of Hamiltonians and Lagrangians. I can still more or less remember my SR course. On the positive side, I understand Lie groups and Lie algebras and their representation theory pretty well.
The internet is a terrible, TERRIBLE, source for a proper scientific education free from bias.
Right. Because of humans. Luckily humans don't make books or any other sources of information. They just dwell on the internet, and there's absolutely no useful information there! That's why you can accept everything you hear or read as long as it didn't come from the internet.
Filthy, filthy copyrapists!
Leonard Susskind's Modern Physics lectures on the Stanford University's channel on youtube are excellent.
http://www.youtube.com/watch?v=hbmf0bB38h0
First off, you don't state how much knowledge of maths and physics you _actually_ have beforehand, This makes answering the question an awful lot harder -- a 'college course in calculus' could be evaluating simple derivatives, or it could be some nasty vector calc and differential equations. In the order that they come into my head, you need to understand _intimately_ vector calculus (leading to Einstein notation -- play with it and become comfortable with it!), methods of solving partial differential equations, multivariate calculus, and how to properly play with differentials (i.e. proofs that start with statements like "df(x, y) = \partial f / \partial x dx + \partial f / \partial y dy"). You'll also need to properly understand matrix algebra, and ideally what tensors are (hint: generalisations of matricies that follow certain properties). You should be able to prove vector identities in Einstein notation, and be quite comfortable manipulating 'hardcore maths'. Honestly, just go away and play with maths until you understand it fully, you understand where it comes from, and you can use it without thinking about it at all. After that, try and become familiar with special relativity. This will be hard. Feynman explains everything very well in his lectures, but he doesn't list any problems: the best way to learn physics is to derive a true statement (like the lorentz contractions) and go away and shove it in all sorts of different situations (i.e. answer problems with it). The book by French & Taylor is commonly well-received; there are many different textbooks. Find a good set of problems, and answer them. Then, when you understand modern Special Relativity, get a large GR book -- there are many; Gravitation, or "General Relativity for Physicists" is a good one -- and read it. _Think_ about it, and answer the problems at the end of every chapter. If your book doesn't have questions at the end of each chapter, go away, and get one that does. Make sure you do them, and if you don't get something, find out why. If you can't find out why, ask someone who can. Finally, a taught undergraduate level course in GR would be a fantastic introduction after a well-defined amount of knowledge has been acquired. The lecture notes from the course at my home institution can be found here.
My UID is prime. Is yours?
To work with GR mathematically, you have to understand differential geometry. You have to be able to work with tensors, like the metric tensor.
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.
*That* was useful...
At each step I have digested the basics of that subject before moving on. And later reiterated and expanded my knowledge on them on a need to know basis. It has taken seven years and still counting.
Real analysis -> Euclidean vector analysis -> linear algebra -> functional analysis -> manifolds and differential forms -> tensors -> Riemannian geometry
I found a copy of Feynman's book (including a CD audio copy) "6 Not So Easy Pieces" on quantum mechanics and related topics, the companion to "6 Easy Pieces" on general physics, about 10 years ago. It is remarkably easy for someone with basic college math and science to understand - once you whack your head against the wall a few times! :-) Anyway, here is a link to the Amazon page for the book: http://www.amazon.com/Six-Not-So-Easy-Pieces-Relativity-Space-Time/dp/0465025269/ref=sr_1_17?s=books&ie=UTF8&qid=1314560980&sr=1-17
Sometimes, real fast is almost as good as real-time.
One can learn quite a bit about General Relativity without breaking out any math. In fact, GR is taught in introductory physics using introductory level math to great effect. It is a sexy topic and draws a broad audience. And like with many of Physics' big theories, a student can learn a tremendous amount about the cause and effect of forces and behaviors without necessarily learning the math behind them. It is very much like a philosophy.
Fundamental to the theory of GR is that time is constrained by the speed of light. That in itself is not obvious and has implications that trickle down to (among other things) solid-state physics. Understanding how those implications are manifested in the real world is fascinating. You can read layman books that give you a pretty broad understanding of GR (and other big concepts).
If on the other hand, your intent is to get in the business of postulating and predicting outcomes, then you do need to understand the math behind the concepts. But beware that the math may not bring you closer to understanding the concepts. Additionally, it is a rare topic in physics that can't be explained to the layman in words they understand. The best test of a student's understanding of physics is to have them explain to another (non-science) student the principles of the theory. Without that, it's all to easy for a student to learn the math without really grasping the reasoning behind it.
It might not go up to relativity, but should get you most of the way there.
www.khanacademy.org
GR has some hairy tensor equations that have never been fully solved. You are correct is saying the principle of relativity goes back the Galileo in its most basic terms, just requiring algebra then. SR is not that much harder.
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.
Try out the Youtube lectured of Stanford university.
Leonard Susskind has made a bunch of lectures starting easy & building up to the maximum my head can handle without exploding;
http://www.youtube.com/watch?v=25haxRuZQUk&feature=list_related&playnext=1&list=SPA2FDCCBC7956448F is the course where my head said pop in lecture nr. 9 or so..;)
Thes are full-length classes (2 hours each, 12 vids in total in this course alone; & there are about 10 of them so 10*2 * 10 = 200 hours+ of vids explaining in a very good way (even i can understand the buildup going on there).
I've been doing this for the last year or 2; & it is hard; but very interesting.
(& it starts off 'easy' & builds on previous lectures).
Really; this is probably the best vids i've seen about our universe, string theory, relativity, & whatnot..
Why the guy has not yet gotten a Nobel Prize is beyond me; but probably will be given in the next 4-5 years or so :)
Good luck; & remember to take a brake after each vid; & watch them again after a week or so just so you start to grasp what you dont know in this world :)
http://www.noob.us/humor/the-office-dwight-faces-nerd-torture-of-the-highest-form/
I did my PhD in GR,
you need:
- calculus in several variables
- linear algebra
- some topology
- ideally something on partial differential equations
- differential geometry
- classical mechanics
then, take a deep breath... and go read General Relativity by Wald.
The book isn't long at all. None of the underlying concepts are difficult. However if reading the book a few times is enough for a person to "get" relativity, it would be much more widely understood.
Reading the book and "thinking" that you grok relativity is a much easier task.
I know plenty of people that think they have it down pat. However there are quite a few time dilation scenarios that will cause a paradox if you don't have the model dead right. The frames of reference are a bitch.
Welcome to slashdot where innocent questions are met with derision and insults.
It was also a lazy question, one that a simple Google search for "general relativity" could have answered. I agree with the parent poster that if he can't be bothered to dig a little on his own, he's never going to take the time to study it anyways.
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.
General relativity is only one small part of physics, and focusing on it wouldn't help you understand a lot of the physics articles that go through here. I would suggest a more balanced approach -- with your background you should be able to work through Griffith's E&M and Quantum books which many undergraduate physics majors use. All the purists out there may scoff at them, but let's face it, your not actually going to work through Zee's "QFT in a nutshell" or many of the other books suggested above on your own. With a bit more of a background in the field, you would be in a better place to evaluate what you wanted to study next.
During the 80s I wrote an interactive three-dimensional special relativity simulator. It was a wire frame simulation and ran under DOS. I recently tried it on a Windows XP machine and it still works. (It did not work when I tried on a Mac under Parallels/XP, so it appears that one needs an actual Windows machine, not a virtual machine.) When I first ran it during the 80s I simulated a famous scene from the first 3D relativistic simulation done at MIT during the 50s and I got the same results: lamp posts that curve inward as one travels down an avenue. It was a sublime moment.
I found that when I ran the simulator I was able to grasp many of the classic special relativity paradoxes, such as the "pole in the tent" paradox. When one sees what happens it becomes "oh yeah, I see". For example, it turns out that Lorentz contraction is really a time effect: the time at the leading edge of an object is different than at the trailing edge, so you perceive the leading edge at an earlier point in time than the trailing edge, and so the object effectively contracts in your reference frame. The simulator has options to include/exclude the effects of (1) the travel time of light (causes apparent rotation, known as "Terrell rotation"), (2) time dilation, (3) perspective, etc. It also attaches clocks at various points of the moving object, and you can orient the object anywhere in space in any direction.
I will post the simulator on my personal website late tonight for anyone who is interested. The url is http://cliffberg.com/
As for General Relativity, one needs to know tensor calculus. I was going to build a simulator but it was a large undertaking and I never got around to it.
Differential geometry
Partial differential equations
Special relativity
Each of these topics has their own list of required material. I couldn't take a GR course until the fourth year of my B.Sc in Physics, and it was still a challenging course.
First of all, you need to decide, first and foremost, if you are a mathematician or a physicist (or an engineer). I hate to make this stereotype but i have to start this post from somewhere (i know usually there is tons of overlap). If you are an aspiring mathematician, you would want to learn "general relativity from a mathematical point of view": let the math guide your quest, and the physics follows. If you are an aspiring physicist, you want to basically "understand general relativity from a physical point of view": let your own physical intuition guide you, and the math follows. That is the first step; figuring out who you are and how you "view the world".
*****Usually mathematicians are good at solving the relativity equations that arise, while physicists have "developed" the equation that needs to be solved. A subtle yet extremely important difference. Figure out if you like to solve equations or create equations. *****
Secondly, you can begin to research authors who are usually physicists or mathematicians (i.e. they think like you), and find the correct books. Don't be fooled: usually an author is at his or her heart, either a mathematician first (like Witten) or a physicist first (like Feynmann, Einstein, etc.). Judging from your post, you want the math first, so dont buy physics textbooks, buy math texts which have physics problems, so you can understand the physics from within a mathematical framework.
You wont get much out of a book written by Feynman if all you care about are the mathematical principles, because guys like Feynmann and Einstein let the "physics do the talking" instead of the math, so usually there is more text than equations.
Thirdly, you need to activly start solving problems. Problems problems problems. And I do mean problems. Did I mention problems?
*****Do not read a paragraph or chapter from a book, then convince yourself you understand it. You need to _test_ your understanding. Learning is an interactive process. *****
You need to solve as many problems as you can find (this is probably the hardest part about independent learning, finding good questions for your skill level). Unfortunately the best problems are usually from expensive textbooks. Drill yourself stupid. Start with easier ones, the proceed to the harder ones. This is also critical that you invest some time in Step 1, because if you dont know by this step, you will work on problems that aren't especially suited to developing your skill set, and you will give up frustrated.
You need problems that are explained in a way that relates to you.
Now that your doing problems, over time, as you solve them and get good at it, your "toolbox" will generalize and your intuition will sharpen, and you can begin to solve harder and harder problems. Eventually you will get to understanding the framework behind relativity.
I used to teach this stuff, so here's the prerequisite stack I would recommend:
* On the math side: You'll need to be comfortable with math through all 3 semesters of calculus, linear algebra and differential equations. You don't need to be a grand master of PDE's, but knowing at least the basics is pretty essential.
* On the physics side: Electromagnetism is actually critical -- a lot of the stuff built up in GR is built up on top of E&M technology. I recommend D. J. Griffiths' _Introduction to Electrodynamics_ -- it's the best undergrad book, and if you actually read through it and do the exercises it's the best way to really learn how to do integrals and so on in the real world, too. You also need to know special relativity well; by an interesting coincidence, chapter 10 of Griffiths is one of the best texts on that, too.
In theory, that should be enough for you to start on GR. I don't have any favorite GR books to recommend -- Schutz (which lots of other people have recommended) is probably the lesser of several evils. I would avoid Misner, Thorne & Wheeler (aka "the telephone book") because, while it gives some very nice explanations, it also picks conventions and notations which are completely different from what everyone else in physics uses, for no particular reason.
Anyway, the key thing you should remember when doing any of these: don't just read the books, do the exercises. The books are all written in a way that you'll never really get what they're saying until you do them.
No, humans inhabit both spaces. But, give people anonymity and suddenly they feel free to say and do anything (including say what ever they think you want to hear or will make them sound smart).
I had to post this anonymously for all the ironical reasons.
If you have no prior knowledge of differential geometry, then Misner, Wheeler, Thorne 'Gravitation' offers a smooth learning experience where mathematics and physics go hand-in-hand. However, the mathematics is only presented as needed and you will not really understand the mathematics unless you consider a good book on differential geometry and topology such as M. Nakahara: "Geometry, Topology and Physics" as a follow-up read. A very clear presentation can also be found in Steven Weinberg's 'Cosmology', but it is less intuitive as a first read compared to 'Gravitation' unless you happen to be a particle physicist.
People who recommend Einstein's books on the topic probably haven't read those nor any other book on the topic; they lack a good clear presentation style and do not benefit from the research that has been done since Einstein wrote them. Einstein was a genius in his own right, but he certainly was not a great teacher nor writer. Understanding does not mean to be 'wow-ed' by the fame of the author, but to be able to reason on the topic. Thorne's popular science book 'Black Holes and Time Warps' is perhaps the best book to read aside and to keep you motivated. Good luck!
This is a standard graduate textbook on the subject. I recommend getting it if you're serious. Even if you can't get through it at first, at least you'll have some idea what you're actually missing. After I became comfortable with the material in here, I felt like I could think for myself.
The mathematics of GR isn't too bad, but it likely has a few more moving parts than you've seen before. It'll probably be confusing at first, but with some persistence you can work it out.
I highly recommend making friends with someone who at-least-sort-of-understands this stuff and is willing to talk about it with you.
So I've both taken GR as an undergrad/grad student, and now taught it to both. My undergrad was in math, grad school physics. To understand modern GR (singularity theorems, black holes, cosmology, lensing effects etc) from a math background the subjects that really help are:
1) Special Relativity. This is an easier intro that really comes out of the end of electrodynamics courses (ie, why there's that pesky 'c' in Maxwell's equations that doesn't seem Gallilean invariant). There are outstanding lecture notes available from, say, oxford university on both SR and GR - see www.maths.ox.ac.uk and go to lecture notes for undergraduates and dig around a bit.
2) Differential Geometry. I started out with 2D shapes in 3D spaces (Geometry of surfaces) which actually taught me all I need to know about how the idea of a metric is formed etc. Then I moved on to general differential geometry (book: Differential Maniforlds by Hitchin: http://people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html) . If you can wrap your head around Riemannian geometry, moving over to the Lorentzian case isn't too hard.
Anything you can get your hands on to do with tensors will help a LOT, as all modern interpretations are based on the abstract index notation which is written in tensors.
For learning GR itself, the standard book is Wald's General Relativity. Carrol's book is pretty good too, but Wald seems to be the one that just about everyone I know cuts their teeth on.
I found GR a hell of a leap from everything I'd understood so far, so I took a long, long time reading through notes again and again until I understood the ideas behind things like connections, covariant derivatives, tensors, Christoffel symbols etc. Don't expect to learn it quickly or easily like most concepts in statistics, but rather be prepared for it to take a long time. As you probably know by now, maths is a participation sport, so really flex those muscles by working through any examples/problems you can get your hands on - that was really what made concepts sink in for me.
Let me know if I've assumed too much background (to get to these you need prerequisites like topology, analysis, euclidean geometry etc). But I'm assuming that you want to understand the modern mathematical background of curved space-times rather than just the general philosophy (if so, as someone else suggested Einstein's original book on the special and general theories is a delight to read).
should about cover it.
I reposted your question to Physics Stack Exchange so you can get input from an additional group of people, several of whom have actually studied GR. (Disclaimer: it's not my website, but I'm a frequent contributor) Of course, most of the prerequisites I would think of have already been mentioned here (Newtonian mechanics, electromagnetism, special relativity, linear algebra, multivariable calculus, differential equations, differential geometry), but on PSE you won't have to filter out a bunch of irrelevant comments ;-)
For what it's worth, the main "thrust" of GR is encapsulated in two equations, which you can find here among other places: the geodesic equation and the Einstein field equations. You can use those to guide your progress: once you know enough to understand what they mean, you've successfully learned the basics of GR.
GR is suitable as a 4th year or graduate course in physics. The undergrad is a bit sketchy but manageable. So really whatever the math requirements at your school are for 3rd year or 3.5 years of an undergrad in physics and you'll be there. As with most problems in physics there's a few different ways to formulate them, so your instructor may choose the one most appropriate given the available prereqs (and depending on how much time they have they might teach a lot of the math you need in the class).
Typically you'll want PDE's, Linear algebra and and hopefully in there you'll get some tensor analysis, but really, all courses depend on what the instructor chooses to teach of the overall topic, and how your school wants to organize the material so you can't really get handed a list of course names and hope to have a lot of success with only that.
It really does depend a lot on how your school formulates its programme. When I went to school our physics and maths were separate courses, taught by separate departments, but had I been 3 or 4 years earlier it was all one big blob of "mathematics for physics" + the various physics courses.
Unless you're already a BSc in math or physics your best bet is just progress along the path to take it as a regular course, and if not the easiest bet is to just look up the prereqs on a particular schools GR course and go with those.
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
Has anybody mentioned "Einstein's Legacy" by Julian Schwinger?
I studied for a 4 year undergraduate maths degree and tailored my module choices towards general relativity and advanced quantum mechanics. After taking all possible calculus, linear algebra, differential equations (Ordinary and partial) and vector and tensor calculus I began a module on special relativity and reading around general relativity concepts.
I was very comfortable with special relativity and felt well prepared for general relativity but even with an excellent lecturer and very supportive class mates it was very complicated. It was extremely interesting but not something I would like to do on my own, not to deter you but it will take a long time to get yourself to a level if maths knowledge you need to attack the concepts thoroughly.
"What is happening on Mars right now?"
If you know that this question is meaningless and why, then you are ready to study general relativity.
Otherwise take a course in Special Relativity or read and study "Spacetime Physics" by E F Taylor and J A Wheeler. Wheeler once told me that he believed that every figure should have as much information as 10 pages of text, and some figures in "Spacetime Physics" come near his goal.
IMHO most scientists who can perform the algebra and solve problems in Special Relativity do not really understand the implications of their answers.
You should just buy a time machine, go back, and ask Einstein himself to explain it to you :)
From my preface: "The purpose of this little book is to provide a clear and careful account of general relativity with a minimum of mathematics. The book has fewer prerequisites than other texts, and less mathematics is developed. The prerequisites are single variable calculus, a few basic facts about partial derivatives and line integrals, and a little matrix algebra. The algebra of tensors plays only a minor role." Available at: faculty.luther.edu/~macdonal
bra-ket notation is very well written down "dumbed down" for chemists instead of mathematicians and physicists :-) in Szabo and Ostlund's "Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory"
But you need to have a background in high-school level linear algebra first, i.e. you need to know how to work with a simple n-dimensional orthogonal basis set of vectors, otherwise Szabo and Ostlund is probably too difficult.
I
To be, or not to be: isn't that quite logical, Slashdot Beta?
Read any examples of airborn contemplations during wagon/sled rides gone awry with Calvin and Hobbes :)
The book by Albert E. is a good choice. Another approach is books by Martin Gardner, starting with Relativity Simply Explained and The Relativity Explosion (which I read years ago in high school as " Relativity for the Million ", the older title before it was revised). Your local library should be able to get them for you by inter-library exchange if they don't have copies. Or you might even be able to find good info right on the web. Start with Special Relativity, which is easy enough to understand even without a college education. From there General Relativity is almost trivial (it is just a matter of understanding that there is no point of reference that is "Special").
Of course, there is plenty of other math that goes well beyond simple relativity. Quantum mechanics will make General Relativity look like Sesame Street. But that's a different issue.
I'm an American. I love this country and the freedoms that we used to have.
and realize there is no spoon.
Yeah I know, what a horrible opening, but it really applies.
Think about the utter simplicity and beauty of the equation of E=MC^2.
Read a "Brief History of Time" cover to cover about 10 times but don't try to dig into what he is saying, take it on face value, because he is explaining it, you just have let it sink in.
What will really bake your noodle is when you realize that everything has infinite energy.
Hey KID! Yeah you, get the fuck off my lawn!
I suspect that to understand general relativity you also need a text on tensors, e.g. Schaum's outline of tensor calculus. Probably many physics textbooks have enough about tensors as well but I wouldn't know; :-(
It was all a little beyond me; a friend once tried to explain to me the metric tensor but I couldn't get it in my thick head
Steps to take: if that wikipedia article is gobbledygook, go read Schaum first (you probably don't need to understand the whole book but you need the tensor notation at least). If you can't read Schaum, brush up on you linear algebra first.
To be, or not to be: isn't that quite logical, Slashdot Beta?
most physicists only have a cursory understanding of GR, at best. It is typically only taught in mathematics programs, and requires a very strong background in calculus, differential equations, tensor analysis, and manifold theory. not to mention a solid grasp on classical mechanics, electromagnetic theory and special relativity. Having taken a course on it, I would recommend you not even attempt such an undertaking, unless you are working under a professional mathematician or physicist. the understanding which you will achieve here through self-study will be on the level of misinformation, at best. honestly, it is a subject for professionals. anyone who claims to have a non-professional understanding of it is deeply confused about the state of their knowledge.
I wouldn't trust anyone who doesn't admit to being at least somewhat confused about the state of their own knowledge. But maybe you're right, if you can't master a subject then it's your responsibility to the esteemed status of credentialed experts to kill off any curiosity you might have on advanced subjects. As a lowly member of the masses I just wait until the experts tell me what I should think instead of exercising my individual curiosities.
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.
That's a little unfair, I think. I'm asking for help in becoming an "armchair physicist" if you will, not a PhD. I want to get through the material and learn it to better my own understanding, not to master it and get into research. It's not as though I didn't google the subject, but if you went with that approach for a complex subject that you didn't already understand, you'd realize it's hard to know which path provides any sort of focus for what you want to understand. For instance, if you have a small company that needs to raise cash and want to know the details of American Securities Law, you might google it and get a little overwhelmed with the overlap of the states' Blue Sky laws, Exchange Act, etc. I, on the other hand, could pare the list down to what you might need to research to understand how to issue stock for your small, closely held corporation. It's a good intermediate step to have an expert filter your reading list, after all.
It's easy to call me lazy if you know the material and I don't, because you can look at what's out there and sort the material into "important," "good to know" and "discard" lists.
The real question is, did you really want me to actually put what I've already read and have queued up to read in the submission? I suppose I could have done that, but I've gotten some succinct responses already that suggest a totally different path than I've already started down. Maybe I wouldn't have received such good information otherwise.
In any case, I do appreciate the folks that have responded constructively. I'm lazy but I'm not that bad. Jeez.
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.
Differential Geometry will give you the mathematical foundation for expressing non-flat spaces. From there, GR is "just" the Einstein Field Equations and the implications thereof. And compared to, say, quantum mechanics, there's very few solvable exact solutions to make case studies out of (black holes and possible evolutions of the universe, really).
Springer has an OK book on Differential Geometry, and then you want to move on to Gravitation, by Misner, Thorne, and Wheeler.
You're going to need tensor calculus. Probably the best way to get a curriculum is to look at whether your school offers this, then look at the prerequisites for the class and work your way down. It will require a minimum of several semesters of calc (these would have been calc 1, 2 and 3 at my school), a theory or proofs course, probably abstract algebra/real analysis, linear algebra, differential equations (if it's offered as a separate course from calc 2 & 3), and a solid grounding in vectors.
No, because of annonimity and ass hattery. Slashdot may have been written by humans, but bears no resemblance to good solid scientific texts.
Well I did Google the subject, and there were lots of pages describing the topic at a high level, including the mathematics and pointers to deeper treatments. Considering that you were on a ten-year plan, I don't think you needed the specific help you as claimed in your analogy.
Reading up on general sources and diving deeper as you saw fit doesn't require a post to Slashdot.
To fully appreciate special and general relativity, you should really take the normal courseload of physics and calc that work up to it.
Because, in the beginning you learn algebra and then you learn physics with it using standard equations like d=rt.
Then, you take your first or second calc class and take something like mechanics or dynamics and realize everything you learned was lie. Everything was a special case and physics is truly based on calculus.
Then, you take your third and fourth calc (vector calc and differential equations) classes and take general relativity. Then you find out once again everything you learned in mechanics was a special case and really a bunch of lies.
One of the best thought puzzles, and one that still sticks in my head to this day, is one that Feynman (I believe) used to illustrate how reference frames change things. He basically used an example of light bouncing between two mirrors on a moving train. For an observer on the train, the light is simply going up and down. To an observer on the platform, the light is bouncing in a path like a "wwww" shape. Since Michelson-Morley had proved the speed of light is constant the only explanation for how the basic distance=rate*time equation could hold true is if each observer experiences time in a different manner. Each sees a different "distance" and the rate, the speed of light, is constant for both. So the only other variable that can change is time.
----- obSig
No. Googling is equivalent to asking "what are the available approaches?" not "what is the best approach?"
There's a big difference between finding out what's available and getting advice on how to use 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 "
Its just a theory.
This book chapter, by Kip Thorne and others, plus a heavy does of vector calculus, will get you there: http://www.pma.caltech.edu/Courses/ph136/yr2004/0424.1.K.pdf
Google summarizes the best approaches by ranking the search results. It's easy from there to browse the top results and pick what suits you.
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.
It is no wonder how the math department always fielded the best baseball team. ...crickets...
In theory, there is no difference between practice and theory; in practice there is.
Anyways for when I took GR you needed to have an understanding of tensor analysis. This was covered in our 6th course (yes sixth we took more calculus than the math majors). So you'd want an understanding of basic calculus (derivatives, limits, integrals). Then move on to differential equations and vector calculus (particularly line integrals, continuity equations (Green's theorem and its physical consequences). Then off to tensor analysis which is really just the vector calculus equivalent to differential equations. Then you can happily do classical GR.
That said as other people mentioned a lot of things are just concepts and there are several lower level introduction to the concepts and consequences of GR. Also if you don't care to know how to derive things in GR then just skipping to the final formulas in a lot of texts will help. Ie what is the time dialation between a guy this far from a star and one that far, etc. Ultimately for simple geometries at least you end up with just algerbraic formulas that you can plug values into, if you have more than 2 things in your model universe then the problems aren't solvable by math (seriously it is that complicated) anyways and you are back to the first principles and simulations.
What you want is a course in Continuum Mechanics. The progression could go as follows:
High School - > single variable calculus - > linear algebra -> multi-variable calculus -> differential equations (ordinary and partial) -> various physics courses (this is important to put the math in context) -> Continuum Mechanics (Mechanics of a continuous media)
See the wikipedia article on this. Continuum mechanics will teach you all about tensors (or about 3/4 of what the wikipedia article talks about).
You can top it off with some Statistical Mechanics and Quantum mechanics (these are about non-continuous media, such as atoms) if you like. You would get most of this with an applied math degree I'd presume. To get just the math down, you could probably get there in a year or so of intense study if you've got some time and some wits.
Best of luck!
Tamran
No, humans inhabit both spaces. But, give people anonymity and suddenly they feel free to say and do anything (including say what ever they think you want to hear or will make them sound smart).
There's not much you could do even if they weren't anonymous. Not only that, but not everyone is anonymous on the internet. There are, as far as I know, plenty of reputable sources. Just as there can be incorrect information in books, the same can be said of the internet. You just have to know where to look for the good information (and double checking the information would help, I think).
Filthy, filthy copyrapists!
As I said in my other comment, the same could be said of just about anything. And not everyone is anonymous on the internet, anyway (and even if everyone wasn't, there wouldn't be a whole lot you could do if they said something you didn't like). Again, you just have to know where to look. Verify the information, too.
Filthy, filthy copyrapists!
Never have I read such a diatribe of complete BS before - the sort of crap you'd expect from a bunch of recent academic graduates who are now discovering that they actually need to make a living doing real work. The key to understanding GR is simple:
- Watch all four Jaws movies in reverse order.
- Setup four Othello boards and play yourself in the different matches stretched out over a period of no less than four years.
- Never get less than eight hours of sleep
- Never gamble with a man named after a city or state.
- Never data a chic with a dagger tattoo
Follow those instructions and you'll realize that the math needed to understand GR is as much of a waste of time as thinking about GR.
#1. The universe has no edge, no center, i.e., that no matter where in the universe you are, it stretches out in all directions as if you were at the center.
That's it. That's all insight you need to understand the theories. Everything else follows from it.
From #1 follows:
#2. The position of any object in the universe can only be defined in terms of other objects in the universe. For example, the position of the earth is generally defined relative to the position of the sun. "Absolute" positions (i.e., not defined in terms of other objects) do not exist.
#3. Since the position of objects can only be defined in terms of positions of other objects, this automatically also holds true for velocity. The speed of an object can only be defined in terms of speeds of other objects. For example, the speed of the sun in our solar system is (close to) zero (by definition), but generally non-zero relative to other stars. Any non-accelerating object may equally well be viewed as being stationary. There are no "absolute" velocities in the universe, since measuring an absolute velocity would require a stationary object holding a fixed absolute position in space, but we said absolute positions do not exist (#2).
#4. The speed of light traveling through space is constant.
Now imagine a non-relativistic universe. Then, #4 would contradict #3 (and therefore #2 and #1). Since if the speed of light is constant, an observer standing on some rock in space could measure its absolute velocity in the universe by measuring up how fast photons pass it by. If the observer finds that the speed of photons coming from some direction is 99% of c, then the observer would rightfully conclude that his rock was moving at 1% of c in that same direction.
Einstein understood that "position is relative" and "speed of light is constant" were both true. But that means that it must be impossible for an observer to measure his speed relative to the speed of light:
Imagine an observer in a spaceship who wants to establish its absolute speed in the universe. He switches off all engines and measures the speed of light in all directions and finds it to be exactly c. Not knowing the universe is relativistic, he concludes he is exactly stationary. Next, he speeds to 10% of c in some direction, switches off his engines and again measures the speed of c. To his surprise, he again finds the speed of light is c in all directions!
No matter how fast the observer moves (relative to its original speed), he always measures the speed of light to be c in all directions. The observer always sees photons pass him at a speed of c. Even when travelling at 99.9999% of the speed of light relative to a photon source, he still sees these photons passing him by at the speed of c.
The observer establishes the velocity of a photon by is measuring how much time it took the photon to travel from A to B. If the speed of c is constant, and at the same time the observer always measures c regardless of his own velocity, this must mean that clocks and dimensions of his spaceship must vary.
For instance, when moving away from a planet at 99% of c, photons coming from that planet are still being measured to have a speed of c. The time a photon coming from the planet takes to travel some fixed distance is constant regardless of the speed of the spaceship relative to the planet. This means that clocks on board of the spaceship must be moving slower than clocks on that planet, and such that the time the photon takes to travel a fixed distance, is fixed and c for the observer.
See also http://en.wikipedia.org/wiki/Consequences_of_special_relativity
My karma ran over your dogma
Misner, Thorene, and Wheeler's Gravitation is an excellent book. It explains the ideas behind the mathematics, shows you what the mathematics does, and how it expresses the physics. It's visual, as a lot of math really is once you figure out what the symbols mean. I spent a happy summer vacation reading it while sunbathing many decades ago.
And it uses the theory of differential forms where appropriate. Often where antisymettric tensors show up, the geometrical intuition is differential forms.
That leads to a question; Which part of taking the time to write up snotty replies that aren't in any way helpful was required?
Ask yourself the same question, and maybe you'll find the answer.
You could have left off the first paragraph and provided an informative response. ... 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.
Welcome to humanity. Such behavior wasn't invented here on slashdot. It's the universal response of "experts" to questions from non-experts who are trying to learn something.
Something I learned long ago was to discount such put-downs, and pay attention to whether the arrogant jerk happened to impart useful information while insulting me. If they did, I thank them, and look for their name in future discussions. I they only insulted me and didn't provide any information, I file them in the "ID10T" bin, and try to avoid their comments in the future.
Actually, in some arenas, you see the opposite problem: People sometimes give a "Don't worry your little head about it" answer, and fail to give any useful information while being oh-so-friendly to the n00b. I ran across this a year or so back, when I tried to learn something about drupal. All the forums I found were full of excruciatingly friendly people - who never answered my questions. I eventually gave up and stopped bothering them with my dumb questions. Then I implemented the sites that I was working on, in less time than I'd wasted in trying to figure out whether drupal could help.
I also use Macs a fair amount and I'm typing this on a Macbook Pro. The Mac forums are full of people who are the friendly-but-unhelpful type. You get very familiar with the mantra "It just works", and come to understand that while something may "work" in some fashion, it may not be doing what you're trying to get it to do due to your misunderstanding of what it was designed to do. And the experts are often oh-so-friendly but unable to explain how to achieve the result you're trying for (perhaps by using a different tool that was designed to do what you want), so they just say "It Just Works" in a friendly, reassuring, and very condescending way.
But the insult-without-answer jerk is a lot more common. /. is certainly infested with this sort of person. And the two kinds of non-answering people cover most of the human species, in great part because most people won't answer "I don't know", which is usually the correct answer.
But in all too many cases, the best you can find is the insult-while-answering sort of person. In that case, the best approach is to use their information without becoming one of them.
Those who do study history are doomed to stand helplessly by while everyone else repeats it.
I really liked Stephen Hawking's, "A Brief History of Time" for an accessible description of Special and General Relativity.
I googled the subject many times over the years and found some good stuff, yet some things that have been recommended here I had not stumbled upon. So I found this slashdot story quite helpful.
Google's ranking system is a poor substitute for the judgment of people who actually studied the subject.
On the other hand I would have never asked for help here at /. due to the high anti-social quota.
No reason to invite unwarranted abuse.
This is only true if one holds it as a given that 'popularity' (in this case, popularity being defined by the sooper-secret search ranking algorithm, but popularity nonetheless) implies 'quality.'
It doesn't take long to look around the world around us and doubt the truth of this relation.
Yes, the best judge of the validity of a scientific approach is whether it's named to match popular search queries and how well websites that talk about it do self promotion.
Totally, I've found exactly the same thing on several occasions. It kills the atmosphere in such places for me. It's worse than useless and gives the impression people are posting from some kind of fanboyism that they don't believe anyone else should be having problems with the things they love so much.
The fact that offensive replies can still fly, stupid questions get stupid answers (though I think this question about relativity personally is a really interesting one) and the AC system is one of the things that keeps me coming back to Slashdot. If it ever got suport nice and trolling died off I probably wouldn't read the comments anymore.
My homepage for years was an obscure PC gaming support/discussion forum I used to post on back in the early noughties until it died a slow death - almost no moderation and a constant stream of airheads asking stupid questions answered in the FAQ or stuff like "We played Team XXX and lost, we tink they r cheaters". All the regular posters were trolls posting just to wind up the idiots and fanboys in tow for the entertainment. I don't know how many imbecilic teenagers we discouraged from online gaming or asking questions in forums, but I do hope that we made a difference.
I don't proclaim to know everything and I am primarily self-taught. I found this combination of books (some of them mentioned by others) immensely useful:
1. "Spacetime Physics" by Taylor and Wheeler (yes, the same Wheeler of the "MTW" trio). It is a gentle introduction to special relativity using nothing beyond first year calculus and a qualitative introduction to the concepts of general relativity. This book is very well geared for self study because it contains the solutions to all the problems (at least it did in the earlier edition I used).
2. "The Electromagnetic Field" by Albert Shadowitz. This is an older book (Dover Publications), which, IMHO, does not receive the attention it deserves. I think it stands heads and shoulders above Griffith's book, the darling of many college professors today. It will expose you to subtle concepts of electromagnetism from different reference frames and will introduce you to the concept of stress tensor and other coördinate transformations. It will be a nice complement to Shutz's book.
3. "A First Course in General Relativity" by Bernard Shutz (now in its second edition). It is mentioned elsewhere in this post.
4. "Gravitation" by Misner, Thorp, and Wheeler (referred to as "MTW"). Nothing more to say here as it has been mentioned in several other posts.
If you have not been exposed to calculus of variations, I would recommend reading chapter 19 of the second volume of "The Feynmann Lectures in Physics."
I would suggest instead "Relativity Simply Explained", by Martin Gardner. Even my best friend, a Lit. teacher with little patience for Algebra or Math. Analysis, understood special and general relativity after reading it.
Something that I always noted in explanations about relativity is that they never tell you which problems it solves, or why it's even necessary to come up with such a crazy theory. That book explained those to me in layman's terms.
I rarely respond to comments. Also, don't ask for clarifications: a brain and Google are faster, believe me!
Well I just spent this weekend trying to find some neat physics to pep up my interest in amateur radio.
I am also angling to pep up my resume so I can wiggle into a job where there is a particle accelerator.
Here is an introduction to quantum physics with an emphasis on modern gadgets that use quantum phenomena.
http://www.colorado.edu/physics/2000/index.pl?Type=TOC
Here is a pretty reasonable home quantum physics project.
http://www.instructables.com/id/Homemade-Quantum-Laser-Micrometer-Nestors-Microm/?ALLSTEPS
An introduction to the Planck Constant and emission spectra.
http://www.radio-astronomy.org/educ/tutor2.htm
As I master the math, I plan to write my own tutorial and computation scripts using this tool.
http://sagemath.org/
Don't check your skepticism at the door. Science is all about skepticism. There is quite a bit of pressure to conform to consensus science, particularly when it comes to relativity. Here is what I have found:
Special Relativity = TRUE. You don't have to be a true believer to believe in the truth of SR. The ideas are understandable by human minds and are mostly testable and mostly well tested. With SR the ideas came first and the math came second. There are certainly aspects of SR that are non-intuitive, but they still maintain a certain base of plausibility. And of course the math is well proven.
General Relativity = Partially TRUE.
AFAICT, the math has been experimentally proven to a large degree. The equations can make more accurate predictions than Newton's simple equations. As such the field equations are incredibly useful tools and should be seen as just that: tools.
The field equations are a kind of mathematical model of reality which works and thus reflects the nature of reality, but it is no more a direct representation of that reality than a map is of a region of the earth. You wouldn't use a map to perform earth science experiments on regardless of how accurate that map may be. Instead you would use the earth directly.
My advice is to learn the tensor calculus field equations so that you can use them to make useful predictions, but be wary of the analogies that will be trotted out to try to give you a pseudo-understanding of how the mathematics relates to the real world.
Those 'implications' require further experiment to prove and verify. In some cases this may not be possible as some of the alleged implications of the mathematics are non-falsifiable and unprovable and can only be taken on faith. OTOH, some aspects of the general theory have experimental evidence. A scientist will accept the ideas which have sufficient experimental evidence and withhold judgment on the ideas that don't.
I think the difficulty of the mathematics makes people less skeptical about the theory itself than they would otherwise be. And Einstein's reputation makes it difficult to doubt any of his theories. So people tend to just accept the analogies in lieu of a genuine understanding of the ideas because they cannot understand the mathematics itself and the ideas are usually so non-intuitive that they seem impossible. Also the analogies seem kind of interesting and cool, which makes it more tempting to believe in the pseudo-understanding that they provide.
Quite an experience to live in fear, isn't it? That's what it is to be a slave.
You don't even have the ability to understand the question you ponder.
"Computers are a lot like Air Conditioners" "They both work great until you start opening Windows"
I must say that this thread is Slashdot at its best. Knowledgeable people, whose knowledge was earned from years of study, freely share their knowledge with the rest of us. I am humbled by their knowledge, but more importantly, I am stunned by the generosity of these posters.
one website will teach you everything you need to know about math or anything really...
http://www.khanacademy.org/
Time dilation works largely because as we approach C our mass also increases.
You mass is something called a "Lorentz invariant" - IT DOES NOT INCREASE and in fact is constrained by relatively to be constant in ALL inertial frames. The gamma factor in relativistic momentum comes from the velocity NOT the mass. Try using a gamma factor for a mass increase with Newton's second law and you will get it spectacularly wrong!
Google summarizes the best approaches by ranking the search results. It's easy from there to browse the top results and pick what suits you.
Yes, of course, and the best way to resolve which conflicting school of thought in an area of theoretical physics is correct is to have a google war between relevant search terms. The one that has the most hits is obviously the correct one. Google's ranking system is obviously the answer to all of life's problems.
Umm, no.
Umm, no!
At 0.5% of c, your mass will have increased by 0%, and your time will have dilated by ~1.0000125 as viewed by a stationary observer.
At 50% of c, your mass will have increased by 0%, and your time will have dilated by ~1.155 as viewed by a stationary observer.
At 95% of c, your mass will have increased by 0%, and your time will have dilated by ~3.2 as viewed by a stationary observer.
Key points: mass is invariant and does not change, and you do not notice any change to the passage of time only a "stationary" observer notices that time apparently passes more slowly for you.
First time comment , so please bear with me:) I have been searching for the same answers for a while now. The best study plan i have come across is from Gerard 't Hooft here, HOW to BECOME a GOOD THEORETICAL PHYSICIST (http://www.staff.science.uu.nl/~hooft101/theorist.html) He outlines the logical order to be followed with links to study material .
Was it indeed d'Inverno in the Introduction chapter where he writes having learned tensor calculus and the basics of general relativity by himself in high school? Not an impossible task, one only really needs a strong calculus background and after that you're pretty much set. For the simpler approaches to GR it's just about index manipulation and a couple of big, but simple, ideas, really. Having mastered calculus, the concept of manipulating indices shouldn't sound too unnatural. Just pick up a book used in basic GR classes and off you go. If you prefer free stuff, my favourite is http://www.physics.mcgill.ca/~maloney/514/. Susskind's GR lectures are also quite decent, although hurried.
That's not really true. Dirac went looking to remove the square from E=mc^2
Actually he went looking for a way to factorize E^2=m^2c^4+p^2c^2. Using E=mc^2 you have already taken the root which means you have assumed a stationary, matter particle.
Maths is the language of physics. You cannot get far without it but, as a language, it is a tool to describe the physics so you need to understand the physics too because it is easy to write down a maths expression which is non-physical and/or has non-physical solutions. Dirac's case is an excellent example. Dirac was NOT the first person to come up with negative energy solutions - the Klein-Gordon equation was already known and was considered a problem because of the negative energy solutions. Dirac's genius was that he came up with a valid, physical interpretation for negative energy solutions i.e. why a negative energy solution was a valid physical one.
So you need an understanding BOTH of the physics AND the maths to describe it if you really want to get to grips with a particular topic.
To elaborate, he did it from the Klein Gordon equation
No he did not. The Klein-Gordon equation is for scalar particles - the wave function you get is a scalar one i.e. it has a single value at each point in space. With the Dirac equation the wavefunction gives a 4-component spinor at each point in space. These are fundamentally different. Dirac started with the Einstein energy-mass-momentum relationship and attempted to factorize it.
Well, according to relativity the world must contract to zero time
Not quite. Relativity is concerned with observers so here there are two: the person watching the photon and the photon itself. To the person watching the photon time is infinitely dilated for the photon so, as far as they are concerned no time passes for the photon between emission and absorption. However for the photon space is infinitely contracted in the direction of motion so it sees that there is no distance to travel to the object which absorbs it so it too thinks that no time will pass before it is absorbed.
A bit over 25 years ago, I and ten other students derived E=mc(2) in a semester as frosh undergrads in Morley's old lab.
I had nothing more than pre-calc before the course. The threater major in the class, didn't have that much. We did have a great TA who made electronics for us (showing us how-- and who was more dedicated to us learning, than his own GPA), some darn good lasers, far better mirrors than M&M had access to, and of course, the fact that someone had done it before and could guide us though it nudging the way, but not giving away the secrets -- making us find them ourselves.
Hard work-- 20 hours a week, at least, not that hours mattered-- but everyone did it themselves. Half of us are now physics profs (not I!), but I don't thing any of us didn't come out, without being profoundly aware of what science was and what we could achieve.
Then again-- maybe the key, was being in Morley's lab.
The famous british physics expert Arthur Eddington was once asked if there are really only three people in the entire world, who truly understand general relativity. After a long science he replied: "I was wondering who the third one might be!"
Therefore GR is probably not really about match curriculum, but a weird mindset. After all, Einstein got a fail grade for math at the university.
Learning relativity is really simple, even Marilyn Monroe could do it:
http://en.wikipedia.org/wiki/Insignificance_%28film%29
Soo... You're an abrasive cunt?
Seems that the two or three people who are offended at the prospect of this guy asking for advice for his self education could just wander away and leave those interested in the conversation to have it. Though I suppose you cannot bear to rob the public of your ...insights? Seriously just give it up you angry little creature.
Remember that at some point in time even educated people at a hard time doing multiplications, divisions..
While we have not changed our math system since Einstein, trying to understand something "already done" is much easier than inventing it, so this is a very different task..
That said, I agree that tensors are quite difficult to understand and it isn't helped by the fact that most books aren't very good (not progressive enough so that you can learn without too much difficulty).
The end goal is an interactive textbook, so far it's just (somewhat buggy) simulations.
You're a month or so early for it to be useful (bugs galore, limited browser compatability (chrome and ff>4) and it's my first piece of programming over 50 lines or so), but feel free to keep an eye on it. Here's the preview:
http://schroedingers-hat.github.com/jsphys/jsphys.html
If anyone else feels like popping in and taking a look/helping out you're most welcome. Even a critique/pointing out of mistakes at this stage is most appreciated.
On the subject of GR:
You'll need a heavy helping of calculus, including vector calculus. Decent linear algebra. Geometry and some understanding of tensors. There are a lot of books around that start from about that level.
Also there's some stanford lectures on youtube that may be useful:
http://www.youtube.com/watch?v=hbmf0bB38h0
it's differential geometry. particualarly, at the time it was dscovered "riemenian geometry" and/or "tensor calculus". basically, the math is all about how one measures distance and angle on a space that isn't flat. one defiles the space as a matrix of partial differential equations - the change in each dimensiion with respect to the change in each other dimension. so one has to start with "partial differential equations". so partial diff eg, then differntial geometry. particularly riemenian geometry and/or tensor calculus.
If you are curious about the mathematical tools necessary to deploy GR effectively, The Road to Reality is your book. It was written by Roger Penrose, one of the foremost mathematicians of our time.
Get a copy of the Phone Book (Misner, Thorne & Wheeler, _Gravitation_) and go at it.
Wheeler introduces the mathematics necessary to understand it, better than the mathematicians. Particularly differential geometry.
Salman Khan (http://www.khanacademy.org/) has created some great videos for learning math.
I have to disagree with a lot of the posts out here on this subject so far. Yes, general relativity is about the physics, but as a physics grad student, I've had some of my greatest frustrations just trying to "learn the math through the physics". I think that it's very helpful (at least it has been to me) to learn the math on it's own from mathematics text and then, once the math is understood, pick up a physics text so you can focus solely on the physics ideas. Again, this is all my own humble opinion, but it is what has worked for me, and many of my friends in the field have related similar stories.
To answer your question more specifically, here is what I would recommend:
On the math side:
1) Review single variable calculus
2) Multivariable calculus
3) Linear Algebra (check out Axler's "Linear Algebra Done Right" - this book is amazing)
4) Differential Equations
5) Differential Geomotery
General relativity is all differential geometery, so understanding this is what you're shooting for. However, just knowing the math isn't enough; you'll need to get up to speed on physics as well. So, assuming you've had an introduction to physics somewhere (high school or undergrad):
1) Intermediate mechanics (The book by Taylor is brilliant)
2) Electricity & Magnetism (Griffiths is the way to go here, no question)
3) Special Relativity
4) General Relativity
I know you know technically need a course in E&M to understand general relativity, but a lot of Einstein's work on special relativity was motivated/inspired by ideas from E&M (and a lot of his work on general relativity was inspired by his own work on special relativity...).
Finally, on last word of warning: relativity is something you'll have to approach multiple times before you fully understand it. First, try to understand special relativity on a very simple like (Feynman has a very simple exposition on this). This doesn't take any math beyond algebra. Look at the equations for the Lorentz transformations and do some problems on time dilation and such. Next, try to understand special relativity from a more advanced point of view using Minkowski space and all the fancy linear algebra and calculus that comes with it. Then try to understand general relativity as a generalization of this, where space-time is curved by the matter in it. It's a very incremental process. If you get an undergrad degree in physics, you would probably see relativity, in some level of increasing complexity/subtlety, at least three times on your way from Newtonian mechanics to general relativity.
Though let me stress again: I think it's worth it to learn the math for the sake of the math which will free you to really focus on the physics when you go to tackle the actual ideas within. I've seen so many people get disheartened as they struggle to understand both at the same time and end up strangling themselves on the twisted mess that you get when you try to do both at once.
http://www.amazon.com/Manga-Guide-Relativity-Hideo-Nitta/dp/1593272723/ref=sr_1_1?ie=UTF8&qid=1314635810&sr=8-1
I think my professor in modern physics class said it best on the first day:
"Anyone who claims to be able explain and understand modern physics, is lying to you"
This coming from a theoretical physicist PhD really set the bar high. Also not to blow my own horn, but I got the highest grade in the class on every single test in that class and in the intro to quantum mechanics one and still feel like I don't quite understand modern physics completely.
I understand you're asking for a list of courses, and what I've seen above seems to be a good treatment. I have several texts on General Relativity, as that is my area of research specialty.
One of the most accessible is _Introducing Einstein's Relativity_ by Ray D'Inverno. His goal was to make it extremely accessible while still teaching you the math you need to know. I strongly encourage you to check it out. Ray did a good job of letting you get to the meat as soon as possible.
Also Bernard Schutz's _A First Course In General Relativity_ is an excellent study.
OK, I can assist. I decided back in May 2010 to study applied mathematics on my own until I could approach quantum mechanics with reasonable probability of success. I've done that. But, I took a detour through general relativity first.
By way of background? Before starting out, I'd already taken courses and worked very hard at understanding differential calculus, integral calculus, multivariate calculus, differential equations, and calculus based statistics.
However, to get brain cells back in shape required books (must use texts to write in; a U-tube video isn’t good enough) -- and sad to say none of them is inexpensive. However, hunting by the ISBN of the text, I managed to control costs compared to even Amazon retail. Old books (like Dover reprints) just do not work for me. Nor, Schaum’s outlines. All of them are outmoded too, including the “new” one on tensor calculus!
++++++To begin -- despite their age these volumes are widely used -- and still mainstream college texts (aka expensive):
A. Applied mathematics for physics.
***** 1. Shankar, R. Basic training in mathematics. (a fitness program for science students -- the subtitle!) Plenum Press 1985.
The must have starting point! Will take you from where you are to where you need to be. Designed to get new Yalies in physics equipped with the minimum standard applied math for tackling quantum mechanics and special relativity. Only 350 packed pages.
Useful, no nonsense material on vector calculus, matrix algebra, generalized vector calculus, functions of a complex variable, ordinary differential equations, fundamentals of wave theory. But, look elsewhere for fourier analysis. More important, the approach to tensor calculus is outdated.For tensor calculus, however, see Schutz below.
**** 2. Boas, Mary. Mathematical methods in the physical sciences. Wiley 1980.
Covers material in Shankar and more. It ought to at over 800 pages. I use this book as a reference and a backup volume to Shankar. For tensor calculus, Schutz is a must-use.
**** 3. Nahin, Paul. Dr. Euler’s fabulous formula. 2006.
An inexpensive volume! Nahin’s many works in applied mathematics are popular in tone -- but only for the mathematically educated who enjoy other aspects of culture, such as history. This volume contains the best introduction to the vital topic of Fourier transforms that I’ve seen.
B. Classical physics
**** 1. Taylor, John. Classical mechanics. University science books. 2005.
Wish I’d had this text when doing physics. Well-written and carefully illustrated. It reflects the impact of quantum mechanics and relativity by directing fundamental classical physics toward those goals.
*** 2. Griffiths, David. Introduction to electrodynamics. 3rd ed. Wiley. 1999.
Useful compendium of classical electrodynamics leading to and going slightly beyond establishing the four familiar equations of Maxwell, garbed in Victorian vector analysis. A chapter on relativity in electrodynamics provides a useful transition to Einstein.
C. Tensor calculus and general relativity
1. ***** Schutz, Bernard. A first course in general relativity. 2nd ed. Cambridge U Press. 2009
Just 200 pages in Chapters 1 through 8 devoted to special relativity and the mathematical apparatus of tensor calculus. Applications of general relativity take up the remaining 170 pages, Chapters 9 through 12. Schutz is an expert in gravitational radiation theory and the on-going work to detect gravitational waves. Recent developments in study of black holes and cosmology put the general theory to work.
By using a modern approach to tensor calculus -- in particular, employing a construct, the “1-form”, Schutz does away with so-called “contravariant” tensors. In this he follows MTW (Gravitation) -- the benefits in clarity and logical development of general relativity are obvious.
2. **** Misner, Thorne, and Wheeler (MTW). Gravitation. WH Freeman. 1973.
My Math curriculum included:
- Introductory Calculus (Freshman Level) - Differential and Integral Calculus of Algebraic and Transcendental functions of one and several variables
- Intermediate Calculus (Sophomore level) - sequences and series
- Ordinary Differential Equations (Sophomore level)
- Matrix Theory (Sophomore level)
- Partial Differential Equations (Junior level)
- Linear Algebra (Junior level)
- Advanced Calculus (Senior level) - fields and interval arguments to establish fundamental principles
- Mathematical Statistics (Senior level) - moment-generating functions and distributions
- Intermediate Statistical Methods (Master's level) - advanced correlation and regression, auto-regressive, integrated moving average models
- other math junk not likey to promote a grasp of relativity
My Physics curriculum included (only Sophomore level):
- Mechanics
- Electricity and Magnetism
- Heat, Light, and Sound
My Chemistry curriculum included (lots of physical chemistry principles):
- General Inorganic (Freshman level)
- Organic (Sophomore level)
- more chem junk not likely to promote a grasp of relativity
That's a collection of prerequisites (the chem might be optional, altho' it helped me). That was just enough get me a basic, general sense of how relativity matters and operates in our observable spacetime continuum (circa the early 20th century). I don't have the capacity directly to read Einstein or pretty much anything purely technical, i.e., beyond the depth of a Scientific American article, since.
In short, it's a pretty long row to hoe. I'd recommend taking up a musical instrument instead.
You'll want to search for "introduction to tensor calculus for general relativity." It's course notes from a course at MIT. Most of general relativity is an extension of the concepts found in linear algebra and calculus/diffeq. Like others have said here, it's important to understand the physics behind this and to put some thought into the thought experiments before trying to tackle math. The concepts are more important for the avocational reader, IMO.
Some old memories come up from jmorris42's post recommending Relativity; The Special and the General Theory. I read that when I was in junior-high, did a book-report on it (I wish I had the book report to read now), and phoned the university to ask some anonymous physics professor questions about it. I haven't looked at it since, so I can't really judge how accessible it was.
I would say that Steven Weinberg's "Gravitation and Cosmology" was the most accessible book that I studied at university.
A book that tried to be accessible, but was all over the map was Misner, Thorne, and Wheeler's "Gravitation". If you just go through and pick and choose sections, it's probably good too.
Here's others's opinions at physics forums
You'll have to decide what you mean by "understanding" the theory. There are many different levels of understanding and only you can decide what you are comfortable with, and what level of understanding meets your needs.
That's special relativity, not general. General is a whole different ballgame.
If you don't understand any of my sayings, come to me in private and I shall take you in my German mouth.
As Hawking was told, for every equation you put into the book, you halve the readership. You miss the point. The OP adds an entire edifice of tools and disciples on top of the questions (most of which were added post-hoc); you don't need all of that, you can derive most of what is needed on the spot and it's probably better for you.
By Jean-Pierre Petit, a french physicist (translated in english): http://www.savoir-sans-frontieres.com/JPP/telechargeables/free_downloads.htm#english
This gives a really nice introduction to many of the concepts involved with relativity.
All basic relativity can be explained with simple high school algebra but if you want a higher understanding of these concepts you need to learn about tensors.
First, you might start to enhance your understanding of advanced calculus.
At some early point along the road, get yourself a copy of
The ABC of Relativity, by Bertrand Russell, first ed. in 1925.
(Reading this book will just take the better part of a rainy day, breaks included. Enjoy it.)
Later on, read the Master's own writing:
Relativity. The Special and General Theory, by Albert Einstein, first ed. in 1920
http://www.bartleby.com/173/
Meanwhile, don't forget to continue your calculus efforts. ;-)
Remember, Einstein had a very pragmatic approach towards mathematics, he just used it.
To understand GR, you won't necessarily have to become more of a mathematician than Einstein wanted to be.
As a nuclear physics grad student, I have some comments on this.
sjwaste, this is an excellent idea. I like what you're trying to do, and I'd love to see more people get a basic understanding on physics. Really I would.
However, your choice of specific ideas to pursue is nearly the worst way to accomplish your underlying goal that I can come up with. I suppose you could've asked about string theory, and that would be worse. The fact is, most normal physicists with PhDs who study physics for 50 years do not understand general relativity. It isn't a required part of our curriculum in grad school, and it only influences a tiny, tiny sub-sub field of physics research or physics theory. It is an interesting, extremely specialized, and extremely impractical quirk of the world. Lots of physicists and engineers use special relativity, which is the dumber cousin of general relativity. Special relativity is all around you. Special relativity is, generally, good enough. As in, it is good enough to run the vast majority of particle accelerator experiments and all the high end theoretical calculations that accompany those experiments.
What I'm trying to say is, if you want a better basic grasp of the mathematics and science underlying neat Slashdot articles, pick something else as the vehicle for your educational efforts. If you want to understand a very specialized bit of information that most physicists don't really grasp, and which won't improve your general science-article-consumption, then pursue general relativity. It's a neat subject - but lacks any strong ties to any science article you will read for the next several decades.
The best mathematics subject that you could study in order to understand physics articles is statistics. Statistics is how you tell a good experiment from a crackpot experiment from a marginal experiment. Statistics can tell you when to be skeptical and when to be impressed and when someone is probably fudging their error bars.
Most good physics articles shouldn't require any strong math background at all. Physics is tied strongly to math, but physical concepts are not. We need math to do our simulations, to extract meaningful quantities from detectors, to figure out whether a study is feasible. We don't really need math to explain what we're doing to the general public. Think of it like architecture - the architect needs a good understanding of several mathematical formulas and physical properties to design a stable building. You don't need to know anything about the tensile strength of steel to appreciate a neat-looking building or to figure out that the tallest skyscrapers are marvels of modern science.
What do I do? I slam two bits of the building blocks of the universe together, and I see what comes out. It's very much like trying to figure out how a car works, but the car's hood has been welded shut. So, I slam that car into a wall, or into another car, and I study the bits that fly out to see what I can piece together about how the car works. More specifically, I'm trying to figure out where one of the elements comes from. We know elements are made in stars from hydrogen that was produced in the big bang. Cue They Might Be Giants music, "Why Does the Sun Shine," if you want the basic idea and a catchy tune. We have a pretty good idea of where most elements are made (turns out that there are different types of stars which make different types of elements). There are one or two elements that we haven't figured out yet, though. I try to figure out how these elements are made by smashing them to bits - because un-making an element has lots of similarities to making that element. What's the point of this paragraph? No math needed to understand the general gist. There's some physics missing that explains a lot of details I left out. It takes a lot of math to actually make the experiment happen. But the result shouldn't require a math degree to explain to a Slashdot audience.
If you don't want to totally neglect the human side, don't miss this: Einstein's Dreams by Alan Lightman
I actually took general relativity these are the requirements:
Calc 1, 2, 3
Foundations of Math
Linear Algebra
Ordinary Differential Equations ( ours also covered Partial differential equations, despite the name)
Foundations of Algebra I & II
Foundations of Calc I & II
Set Theory
Topology
Ring Algebra ( covered some what in Foundations of Algebra, not 100% )
Differential Geometry ( we actually used the same book for Differential Geometry and General Relativity)
Now, I, myself begged the prof to let me take the class as it was only offered once every 3 years or so. I didn't have the Topology, Set theory, Ring Algebra, or foundations of Algebra requirements. He let me in, but I had to struggle to get a C. If you are doing it on your own, read up. I kept reminding myself, It took Einstein a decade to learn the math to formulate his theory, struggling in a single year to learn it all isn't that bad.
Well.. maybe. Or Maybe not. But Definitely not sort of.
meaning explained .. Groosh
Relativity A to B
Journey into spacetime Wheeler