Background:(you might not be interested so you can skip if you want) I am trying to learn general relativity using the Book Gravitation by Misner, Thorn and Wheeler. The book for the most part seems easy for me to understand but once in a while words i neither heared nor can find the meaning of anywhere are being used. Such a word for example is noncoordinate basis frequently used in exercises in chapter 8. The book is divided in 2 paths for learning 1 for basic things and path 2 for deeper understanding. I am trying path 1 only(lack of time). So now to my questions: If anywhere in the book anything is written about or even defined what a noncoordinate basis: My question is where is it defined or described what it means? If not: What is a noncoordinate basis? Does it have any kind of meaning that distinguishes it from the normal meaning of a basis?(in my own idiot terms a basis of a vector space is a collection of objects from that vector space that when scaled with objects from the field connected to this vector space can reproduce any object of the vector space). If so what distinguishes it?
Have you got to the tangent and cotangent vector spaces of a worldline and their basis vectors ? You could try looking up 'frame field' because frame field basis vectors are examples of non-coordinate bases. A change of basis is equivalent to a coordinate transformation, from global (coordinate) space to the local space carried along the worldline. I'm sorry if this is vague, but you haven't given one context where the phrase is used.
OK, first off, do you know that the basis vectors exist in the tangent space? I'm not quite sure if MTW uses the term "tangent space", I don't recall them using it offhand, but it's helpful. Consider a manifold consisting of the set of points on the surface of the Earth. Then at any point on the manifold, there will be some plane that's tangent to the sphere The set of points on this tangent plane is the tangent space. The tangent space will be spanned by vectors, vectors in this flat plane. On the example of the Earth, you know that a plane is 2 dimensional and needs two sets of vectors. One set could be tangent to great circles that go north-south, the other set of tangent vectors could be tangent to the circles of constant lattitude that go east-west. In a coordinate basis, the tangent vectors are given by [itex]\partial / \partial x[/itex], where x is some coordinate. For our example, [itex]\theta[/itex] and [itex]\phi[/itex] might be our coordinates, so the coordinate basis for these coordinates would be [itex]\partial / \partial \theta[/itex] and [itex] \partial / \partial \phi[/itex]. In our example, the coordinate basis vectors will point north-south, and east-west, but the lengths will not be unity. In particular, in the coordinate basis the east-west vector will vary with the lattitude, i.e [itex] \theta [/itex]. So the basis vectors will be orthogonal, but not normal. If you normalize the coordinate basis vectors, you are in one example of a non-coordinate basis. Another example: in polar coordinates, [itex]\hat r[/itex] and [itex]\hat \theta[/itex] are a non-coordinate basis for the plane - because they're unit vectors, rather than what one might write as dr and d[itex] \theta [/itex], which aren't unit length. The later are actually one forms or co-vectors. dr always has a numeric value. It could be just a number, but if you treat it as something that takes a vector for an input and spits out a number, dr becomes a one-form. MTW uses bold face to distinguish between dr, which is just a number, and dr, which is a one-form.
I was actually looking for a rigorous mathematical definition instead of examples. There are several possibilities of what i could deduct from the examples as a definition for noncoordinate basis. Possibilities: a) a normalized basis(yes of course i know its only tangent vectors that means basis vectors in the tangent space to the manifold). So here the difference between coordinate basis and noncoordinate basis would only be normalization. b) a non-coordinate basis is a basis where each basis vector has the same unit as the other basis vectors. So the only difference to a coordinate vector would be that all its vectors also have to have the same unit(example given 1/m) c) a combination of a) and b) However this of course would also mean for each of those that any noncoordinate basis would also be a coordinate basis while the other direction of reasoning is not necessarily true. I certainly do not have any idea which one of these is true or if any. I would very much appreciate a rigorous definition(best if it used basic mathematical notions so i have a tool of checking what basis is a coordinate basis and which isnt).
A coordinate basis is a basis for which there exist coordinates that produce it. (The vectors of the basis at one point are the derivative of the coordinates functions at this point.) A non-coordinate basis is one for which there don't exist such coordinates.
http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf Eq 8.259 gives an equation for bloby's definition
So it is a basis that where the base vectors do not commutate but instead fulfil those commutation relations. Ok with that the questions make in MTW on this subject make a lot more sense. Also i learned a more commonly used word for this type of coordinates: non-holonomic basis You were all very helpful thank you! (I also saved the script on differential geometry you sent(i like it is short and to the point))
I'd suggest at least checking out Wald's "General Relativity" if you like mathematical rigor. I like MTW's chatty smile, I must admit that a few things I did find easier to grasp with a more mathematical treatment from Wald. There's some synergy in using both.
Wald briefly defines non-coordinate bases in the context of tetrads only in one brief section in the first part of the book. I don't know if he goes into it in more detail in the second part of the book since I haven't read every chapter in the second part (I use it more for reference). MTW is quite rigorous, but just really long because they want to develop everything from a "pictorial" perspective, as well as a mathematical perspective. I haven't really found anything in Wald that wasn't in MTW, but whereas Wald might have a section on some subject, MTW might have 2 chapters on it...lol.
Actually i think i will be staying with MTW since one of my professors suggested it. At my university we dont have any General Relativity courses so i think MTW is a pretty readable book. At least i have much less trouble than with Landau Lifgarbagez that is sometimes nearly unreadable. Some claim Landau Lifgarbagez is rigorous but to me he just seems confused. So i am pretty happy with MTW as a beginner's book on general relativity.
There's a useful comment on terminology like "tetrads" and "nonholonomic" in Winitzki's notes, at the end of section 6.1.1 http://sites.google.com/site/winitzki/index/topics-in-general-relativity
MTW is my personal favorite. But if I want something rigorously defined, I turn to Wald. If I want a short proof, I also turn to Wald. If I want pages and pages of pictures and text, I'll turn to MTW. Usually I prefer the longer exposition, but there's a lot to be said for terseness and rigor.
Try to find a formal definition of a manifold in MTW, for instance. You'll probably find an evocative, pictoral description - and not a formal mathematical one. Wald has a good formal definition, plus some formal discussion of topological spaces as a bonus. Wald uses the time honored scheme of giving the explicit axioms, and proving theorems. MTW is more likely to give you nice pictures or analogies, such as describing tensors as "machines" with "slots", complete with "bongs of a bell" as an output medium. Other things I like about Wald: The nabla notation for derivative operators (MTW uses the semicolon notation) - along with a formal mathematical definition of derivative operators and what their mathematical properties are. A two paragraph proof that the Riemann tensor is a rank (1,3) tensor defined by [itex]\left( \nabla_a \nabla_b - \nabla_b \nabla_a) \right)[/itex] . A discussion of abstract index notation. I think the two books work well together - especially if you want formal definitions. Even if you decide you don't particularly need formal definitions (and I think the OP was looking for some), having access to Wald is a great help in understanding more modern papers with more modern notation. This assumes you have a budget big enough for two GR books....
MTW was the first GR book I bought, and I hate it. They use way too many words without actually explaining anything. I learned much more differential geometry, including non-coordinate bases, from reading sections of Nakahara (not a GR book, but a geometry/topology book). I thought Nakahara's presentation was much more concise and transparent than MTW's. Of course, simply learning differential geometry will not teach you GR, because you need to know physical concepts. I would highly recommend Sean Carroll's book for GR, it is also much clearer than MTW's. And then of course Wald for more advanced things.
You probably don't like Mahler too. Probably the main problem with MTW is that after a month or two it becomes light, so one needs to get another copy or get a regular weight set.