So confused: Tensors and reletivistic cosmology

[Baggio]

I'm on the verge of ripping my hair out

I understand the basics of tensors but I just can't get my head around the need have having contra and covariant vectors.. what is the point??! A vector is a vector right? why have a sub and superscripts why can't they just stick to one or the other? I've read the definitions for both in so many textbooks and they all define them by the way they can be transformed but it still makes no sense to me. Is there a good website or book that explains the use of these and how they relate to cosmology??



thanks

 

[Hurkyl]

Think about this:

Suppose we have a vector that's representing a counter-clockwise rotation of a disk that's spinning in the x-y plane. (So, it's a vector pointing along the positive z-axis)

Now, suppose I reflect all of 3-space through the y-z plane.

What does this reflection do to a vector representing the coordinates of a point in 3-space?

What does this reflection do to the spin of the disk?

What does this reflection do to the vector representing that spin?

 

[Baggio]

ok so the spin is reversed and so the vector points along -z and all plane co-ordinates becomes x->-x, y -> -y

and so..?

 

[Hurkyl]

This reflection takes displacement vectors and reverses their x coordinate -- it does nothing else.

However, the reflection acted differently on this rotation vector!

In general, this reflection will reverse the y and z coordinates of a rotation vector, but leave the x coordinate unchanged!

This exemplifies how there are two different ways the reflection acts on vectors. Your displacement vectors are your covariant vectors, but your rotation vector is an example of a contravariant vector!

(WARNING: I may have covariant and contravariant backwards -- I can never remember which is which... just that they're opposites of each other. )


Actually, I think I fudged something up -- I may have been mixing this up with a closely related topic when I selected this example.


But I hope it demonstrates the point -- there two different kinds of vectors because there are two different ways they can behave under transformations.

 

[HackaB]

Hurkyl: it sounds like u are describing axial and polar vectors.

Baggio: see if ur library has "A Brief on Tensor Analysis" by James Simmonds. In engineering they sometimes call them "reciprocal" vectors. For a given vector basis e^i (contravariant), there is another vector basis (covariant) e_j such that the dot product e^i dot e_j = 1 if i=j and 0 otherwise.

add: oops I got the upper and lower indices reversed. The index is upper on the contravariant components, but lower on the actual contravariant basis vector.

 

[Hurkyl]

Anyways, another way of seperating covariant and contravariant vectors is to think of one type as a column vector and the other type as a row vector. (I forget which one is supposed to be which)

 

[Baggio]

Hurkyl: it sounds like u are describing axial and polar vectors.

Baggio: see if ur library has "A Brief on Tensor Analysis" by James Simmonds. In engineering they sometimes call them "reciprocal" vectors. For a given vector basis e^i (contravariant), there is another vector basis (covariant) e_j such that the dot product e^i dot e_j = 1 if i=j and 0 otherwise.

add: oops I got the upper and lower indices reversed. The index is upper on the contravariant components, but lower on the actual contravariant basis vector.

Yeah they have a copy, I'll take a look at it tommorow

 

[selfAdjoint]

Here is yet another way to think of contravariant and covariant. Your tensors and vectors have components and are multiplied by coefficients. Typically in general relativity the coefficients are real numbers, but in other applications they might be complex numbers or members of some other ring. Now suppose you have an ordinary vector; this we call contravariant. Now consider all the linear functions from the set of contravariant vectors to the real numbers (or the complex ones, or whatever your "coefficient ring" may be). Since it's linear, you can determine the image for any vector by showing what a L.F. does on the basis elements. Your contravariant vector v is c1*e1 + c2*e2 + ... and the image F(v) = c1*F(e1) +c2*F(e2) + ..., and you can add the F(v)'s, multiply them by constants, take inner products of them, and so on, all by using the linearity of F and the properties of the contravariant vectors. Such functions F are called covariant vectors.

Misner Wheeler and Thorn in their big book Gravitation call a covariant vector a machine that accepts a contravariant vector and spits out a number, linearly.