Physical intrepretation of contra-variant and covariant vectors?

[IxRxPhysicist]

Hey all,
I starting to study QED along with a slew with other materials. (I read in the QED book and when I don't understand a reference I go to Jackson's E&M and work some problems out, it has been beneficial thus far!) Most of the topics are not too far fetched but I am struggling to understand the notation of contra-variant and covariant vectors. I have found a really good pdf (http://www.physics.ohio-state.edu/~mathur/grnotes1.pdf) that has helped out very much but I would still like to know the merits and purposes of using contra-variant and covariant vectors. I haven't started space-time yet!

Thanks,
IR

 

[DEvens]

http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

The difference is related to how the components vary under a change of coordinates. Keep in mind that the components of, say, a vector are projections onto a coordinate basis. If you transform the basis, such as into some set of curvilinear coordinates like spherical polar or cylindrical polar, then you need to transform the coordinates in particular ways to keep the underlying physical thing the same.

The examples in the wiki article should be instructive.
Dan

 

[WannabeNewton]

This has been asked to death on the forum. While looking at it in terms of how they behave under coordinate transformations may be useful in physics, there are geometric objects to which the terms you speak of are associated with and their definitions are far deeper than what the *physically meaningless* notion of coordinates can afford to give. Unfortunately I have no idea how much formal manifold theory or linear algebra you know. Regardless, see here: https://www.physicsforums.com/showthread.php?t=679735&highlight=covariant+vector and here: https://www.physicsforums.com/showthread.php?t=689904

What I wrote up in that thread might be of immediate help as well: https://www.physicsforums.com/showpost.php?p=4374094&postcount=18

 

[IxRxPhysicist]

I have stumbled upon manifolds and manifold theory recently but haven't had the time to plunge down the rabbit hole, trust me I want to but I'm scatter-brained as it is so I want to remain on track. So to reflect my understanding, if I cast a vector into a particular coordinate system but I want to map it to another basis or different coordinate system I must transform it. The manner in which it transforms to retain the original construction dictates whether it is a contra-variant or covariant vector. Yes, no, maybe?

 

[WannabeNewton]

If you are talking about vectors in $\mathbb{R}^{n}$ then yes what you said is fine. If you are talking about more general spaces (e.g. space-time manifolds in general relativity) then you have to be much more careful in how you word things but I don't think you are working with more general spaces at the moment.

 

[Fredrik]

These questions are extremely hard to answer. How would you answer if someone asks for a physical interpretation of e.g. functions or matrices? I don't think there's a meaningful physical interpretation of the terms you mention, other than what's stated explicitly in the "definitions" you have seen in books that use that terminology. I had to put that in quotes, because those "definitions" are usually stated in an incredibly sloppy way. (I see that the pdf you linked to is no exception).

The question about merits and purposes of "covariant vectors" and merits and purposes of "contravariant vectors", is even harder to answer. There's often no merit at all. Maybe you have seen the notation $\eta_{\rho\sigma}\Lambda^\rho{}_\mu\Lambda^\sigma{}_\nu =\eta_{\mu\nu}$. This is just what you get when you apply the definition of matrix multiplication to the matrix equation $\Lambda^T\eta\Lambda=\eta$, and use a specific convention for where to put the row and column indices. A lot of the "tensor" calculations you will see in these books are nothing but matrix multiplication done in a weird way.

So the only meaningful answers I see to the questions you asked are examples of how this terminology can be used. But it would take a lot of work to show you examples, and you're soon about to see lots of them anyway, in the course you're taking.

What I said in the following quote could be useful (if you click the link to get to the next one, and then click the link in that one, and then keep clicking my links for a while).

There's no short answer I'm afraid. You can start with this post, but you will also need to look at the one I linked to near the end of it, and then the three posts that I linked to at the end of that one.

The terms "covariant vector" and "contravariant vector" are the two terms I dislike the most in all of mathematics. It's not just the terms I dislike, but the disgusting "definitions" that they come with. "Any quantity that transforms as..." Great. What's a "quantity"? What does "transform" mean? Hey physicists, if you're going to use the obsolete covariant/contravariant terminology, at least try to define the terms in ways that make sense. (Sorry about the rant. These things have irritated me for a very long time).

 

[IxRxPhysicist]

Not yet at least. Pretty much just rotations, on a tangent, from what I have read in the linked threads it sounds like I should delve into a differential geometry book. I have Kreyszig's book and just reserved O'Neills book from the library.

 

[IxRxPhysicist]

Well you are on a physics forum but if you point me in the right direction I'll update my vocabulary, I don't wish to be ignorant.

 

[WannabeNewton]

The terms "covariant vector" and "contravariant vector" are the two terms I dislike the most in all of mathematics. It's not just the terms I dislike, but the disgusting "definitions" that they come with. "Any quantity that transforms as..." Great. What's a "quantity"? What does "transform" mean? Hey physicists, if you're going to use the obsolete covariant/contravariant terminology, at least try to define the terms in ways that make sense. (Sorry about the rant. These things have irritated me for a very long time).

Heck yeah, you go girl!

 

[WannabeNewton]

Not yet at least. Pretty much just rotations, on a tangent, from what I have read in the linked threads it sounds like I should delve into a differential geometry book. I have Kreyszig's book and just reserved O'Neills book from the library.

Do not use Kreyszig's book. Mother of god please do not use it. His differential geometry book is horribly outdated and relies on coordinates more than we rely on water to live. By O'Neil I assume you mean his elementary differential geometry text. This is a very good text (another one is Do Carmo "Differential Geometry of Curves and Surfaces"). If you want to go somewhat more advanced then most people here would recommend Lee's "Smooth Manifolds".

 

[Fredrik]

So to reflect my understanding, if I cast a vector into a particular coordinate system but I want to map it to another basis or different coordinate system I must transform it. The manner in which it transforms to retain the original construction dictates whether it is a contra-variant or covariant vector. Yes, no, maybe?

Close enough. Note however that if you're just given a 4-tuple $(t,x,y,z)$, there's no way to tell if these are the components of a contravariant vector or a covariant vector. In fact, you could define a contravariant vector V by saying that V is the unique contravariant vector whose component 4-tuple in the current coordinate system is (t,x,y,z), and a covariant vector W by saying that W is the unique covariant vector whose component 4-tuple in the current coordinate system is (t,x,y,z).

This means that it's absurd to call a 4-tuple a contravariant vector or a covariant vector, and yet you will find that the books do this all the time. It's the association of a 4-tuple with each coordinate system that can be called a contravariant vector, a covariant vector, or neither.

 

[IxRxPhysicist]

This explains why it was published by Dover. But yes O'Neil's elementary differential geometry book.

 

[IxRxPhysicist]

I'm getting the impression that this is a TOmato-toMAto ambiguity.

 

[micromass]

Do not use Kreyszig's book. Mother of god please do not use it. His differential geometry book is horribly outdated and relies on coordinates more than we rely on water to live. By O'Neil I assume you mean his elementary differential geometry text. This is a very good text (another one is Do Carmo "Differential Geometry of Curves and Surfaces"). If you want to go somewhat more advanced then most people here would recommend Lee's "Smooth Manifolds".

I second this. Please do not use Kreyszig for differential geometry. O'Neil and Do Carmo are both pretty good. Lee is probably too difficult for now. But do not use Kreyszig unless your life depends on it.

 

[Fredrik]

Well you are on a physics forum but if you point me in the right direction I'll update my vocabulary, I don't wish to be ignorant.

It's all in those posts that I linked to. If you want to get to the point quickly, skip the first one, and the first two paragraphs in the second one. Start reading the second one at "A manifold...".

 

[IxRxPhysicist]

Got it Kreyszig is best used as a paperweight.

 

[IxRxPhysicist]

Any suggestions on a manifold text?

 

[Fredrik]

Differential geometry is the mathematics of smooth manifolds, so just go with the recommendations you got above. I doubt that there's a better book than Lee, but micromass is right that you may find it too difficult, because it assumes that you're already pretty good at point-set topology.

 

[IxRxPhysicist]

Thanks for the feedback everyone, I'm just going to leave this here: