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Ask Slashdot: Math Curriculum To Understand General Relativity?
First time accepted submitter sjwaste writes "Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade."
Save yourself some trouble and get Relativity; The Special and the General Theory by Einstein himself. In his words "The work presumes a standard of education corresponding to that of a university matriculation examination..." however note those words
were written in 1916 and education standards are somewhat lower now. What used to be required for admission are often not
learned during university at all.
I know I have read it several times now and when I finish and sit and think a bit I'll almost 'get it' before retreating from the gates of madness. Think Cthulhu.
But I think it boils down to not only can we not exceed C we can't go slower either. Everything moves at C and the axis of that motion we perceive as time. And everything else we call reality is the contortions required to make that so under all circumstances.
Democrat delenda est
start with this pdf and then slog through the wikipedia articles on GR http://web.mit.edu/edbert/GR/gr1.pdf
Linear Algebra, Differential Equations, Advanced Calculus, Partial Differential Equations, Electromagnetism, Waves, Introduction to Astronomy, Special Relativity, Differential Geometry
Feynman lectures are fun to read
What do you really want to do ? (My guess is that you are not sure.)
If you want to be able to write down and solve Einstein equations for some case, you need vector and tensor algebra, geometry and calculus. Many people who work in GR never do this (for others, it's all they do). If you are interested in some more particular case (black holes or gravitational radiation, say), you need to understand Einstein's equations at some level, plus whatever approximations or simplifications are used in that area (transverse traceless gauge or post-Newtonian approximations, for example). Also, you should get to where you understand Lorentz transforms in your sleep. If you can't do and understand Lorentz transforms, the actual GR math will likely be beyond you.
What I would recommend is to buy Misner, Thorne and Wheeler, and read and follow "track 1." I would allocate 1 year for that.
the one you draw, assuming one the |x> is one glyph means semi-direct product. http://en.wikipedia.org/wiki/Semidirect_product
if you meant |${SOME_NAMES}> it is the bra-ket notation : http://en.wikipedia.org/wiki/Bra_vector
for more help with the notations, wikipedia is your friend @ http://en.wikipedia.org/wiki/List_of_mathematical_symbols
Jehovah be praised, Oracle was not selected
The ''problem'' with General Relativity is that it's differential geometry, so one should understand geometry of manifolds first...
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 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.
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'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.
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?
Gravity, by Hartle. It's the textbook we used in the undergrad GR course, so geared towards those with some math, without being too difficult, abstract, or esoteric. If you know college calculus and vectors, I think it does a good job of explaining any of the other math you need along the way. And if you have any questions, a bit of web searching will fill in any holes.
I 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.
I didn't follow Bra-Ket notation at all until I read up on the history of it. For me, it helped a lot to know Dirac invented it, and that it was needed because it applied to Hilbert spaces, and that Hilbert developed that concept a few years before Dirac got started, and that John von Neumann was the guy who actually named Hilbert's concept "Hilbert Spaces". Why did those things matter?
1. Hilbert was discussing infinities, and he was familiar with Cantor's work (and liked it) so he was using the modern definition of infinities (plural), where there are multiple trans-finites possible. His math was meant to cover all that, and the use of it for QM was a limited case. Some events can be described using a quite limited number of spatial dimensions and the results will be understandable with a little calculus or even trig if you just understand how to take the notation used and put it into actual equations. For example, there's a Hilbert for a three dimensional Euclidean space. Other (particularly in QM) events need many spatial dimensions to describe, sometimes even an infinite number.
2. The Ket part of the notation is about those vectors in a Hilbert space. You could represent that Euclidean space I mentioned with just a Ket notation, for example. Since Hilbert spaces can have either a finite number of dimensions or an infinite number, and can entail complex numbers, the Bra part becomes needed when the Hilbert space has complex numbers involved. The Bra and Ket together are a short way of writing a formula for a complex conjugate, and the whole can be expressed just as a complex number. These can be mathematically manipulated by partial differential equations. Any person with a fair knowledge of Linear Algebra can derive information from them, secure that the treatment is mathematically both complete and rigorous. That seems to be the real point of the notation, it gets results into a form where the rest of the process uses math that's regarded as rock solid.
3. Dirac invented other math for areas where the completeness condition of all Hilbert Spaces didn't apply. He called some of these "rigged Hilbert Spaces" . He proved people could use the Bra-Ket system and similar operations to describe those QM events, but the results won't technically be proven to be correct in an absolute mathematical sense. many working physicists do it anyway.
4. People tend to refer to Feynman for a good source to understand all this and not mention von Neumann as much, but it looks like von N. was historically quite involved in it. Maybe some of what he wrote on QM could clarify Bra-Ket notation better for you than the standard modern textbooks.
Who is John Cabal?
Mod parent up. Feynman Lectures in Physics Volume 3 is a great old man's story giving you a lot of handles for working your way through QED.
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.
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.
I found MTW to be rather schizophrenic when I used it - probably because there were 3 different authors trying to write a single book, and there seemed to be differences in style as you go from one chapter to the next.
The first time I went through the subject I found it difficult to comprehend some of the concepts. It was later that I was taking solid-state physics where we were doing a lot of work in K-space that it became clearer what they meant by MTW.
Understanding how tensors work really does help a lot, but if general relativity is the first exposure to the subject, it might be a little harder. A more common everyday example would be stress and strain tensors that are used to describe how objects are deformed under pressure. Again, my studies of solid state physics helped me here in that I ended up dealing with non-uniform solids.
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.
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).
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?
"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.
but... but... I googled! I swear. Thanks for the link!
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?
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!
Most of the hard parts come down more to notation than actual mathematics. Once you figure out the notation it's a lot less complicated. The guys who wrote books on this sort of subject would invent their own symbol for addition if they dared to do so.
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?
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.
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 "
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
Great question. There are already some answers, but I'll try to give my own try:
The bra-ket things are just a convenient notation for working with vectors and dot product in old boring linear algebra. Standard courses of linear algebra don't usually teach the bra-ket notation, but if you know linear algebra, it's very easy to get used to it. When I first tried to, I still sometimes re-wrote things in the standard notation, especially when things started to get a little confusing, but after a while, you begin to realize that things get more compact and easy to visualize in the bra-ket notation (when you're restricted to just what you need for quantum mechanics).
As for quantum mechanics: if you really want to learn full-blown QM, there's no escape from learning a lot of classical mechanics (including electromagnetism, etc), which takers a long time and maybe is not really what you're interested. But if just want to have a good understanding of exactly how QM is strange, and even learn to do some calculations with it, you might like these series of lectures in youtube given by Leonard Susskind:
Quantum Entanglements: Part 1 (Fall 2006)
The first two lectures are a little slow but, but he starts from the absolute beginning, and tells you exactly what is meant by the bra-kets, and goes from there to teaching about interference and eventually entanglement, which are the two insanely counter-intuitive and strange things about QM.
After that (or instead of that, if you know calculus), you might want to watch this one:
Modern Physics: Quantum Mechanics
For this one you need a little bit of calculus, but you get to play with the Schrodinger equation and see where exactly the Heisenberg's Uncertainty Principle comes from (at least math of it).
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!
#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 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.
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!
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.
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
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.
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
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.
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.
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.