Abstract Index Notation and Tensors

Date: 07/15/2002 at 18:11:54
From: Thomas Mainiero
Subject: Abstract Index Notation and Tensors

Hello, 

I am currently studying the mathematical side of General Relativity. 
The problem is that I do not entirely understand abstract index 
notation for tensors. I comprehend what they are, what they are used 
for, tensor ranks, and representation of tensors by matrices. I know 
the notation cannot be typed out in ASCII, so just use ^ for 
superscript and _ for subscript. I tried searching for an introduction 
to the notation, but the very few sites I have found are not that 
clearly typed. Also can you provide an example of how tensors are used 
in coordinate transformations?  

Thanks in advance!

Date: 07/18/2002 at 09:08:58
From: Doctor Nitrogen
Subject: Re: Abstract Index Notation and Tensors

Hi, Thomas:

Here is an example of the index notation:

If there is no index, that is, if the tensor has a form, say, like
                          
  v,

then it is a tensor of rank zero, meaning no index, and v is a scalar. 
If it looks like

  v_i, or v^i, i = 1, 2, ...n,

then it is a tensor of rank one because it has one index. If it has a 
form like

  v^i_j, 

or covariant in j, contravariant in i, it is a tensor of rank two, and 
so on. Summation for the indices is implied, so there is no summation 
symbol. 

This is called Einstein summation. The rank is equal to the number of 
indices. What the indices sum to is determined by the space you are 
in. For Euclidean 3D space it would be just i = 1, 2, 3, j = 1, 2, 3. 
Since you say you are studying General Relativity, you must know that 
for the Riemannian Geometry used in it, the indices are summed from
1 to 4. 

Covariant tensors transform thusly:

  A_ij = (&x_i/&x'_p)(&x_j/&x'_q)A'_pq. 

(Here "&" denotes the partial derivative symbol.)

Again, that transformation is for a covariant transformation. There is 
also a similar one for a contravariant transformation.

The modern terminology for tensor calculus  has become pretty 
abstract, using terms like "charts," "atlases," "manifolds," 
"diffeomorphisms," "p-forms," etc., but the basic idea of tensors is 
that one can quickly transform from one coordinate system (chart) to 
another in a few simple steps. 

Suppose you know what the metric tensor g_ij is for a General 
Relativity problem, maybe spherical coordinates. Then using tensor 
methods, you can compute the square of the arc length using the 
coordinates involved.

  ds^2 = g_mn(dx^mdx^n)

There are also things called Christoffel symbols, which help you 
compute the "covariant derivative" and the "intrinsic derivative" of a 
tensor.

Be innovative when you try to find information on tensors and their 
indices online. If you enter as keywords 

  tensors AND indices, 

or 

 (tensors and their indices)

you will find links that include both those expressions and you will 
probably find a tutorial somewhere on the topic.

There are math/physics professors who might scoff at their use, but 
McGraw Hill publishes excellent books in a book series on a variety of 
pure/applied math topics. Their Schaum Outline Series has an excellent 
book called _Vector and Tensor Analysis_.

I hope this was helpful.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/18/2002 at 16:24:52
From: Thomas Mainiero
Subject: Thank you (Abstract Index Notation and Tensors)

Thank you a lot, it was really helpful and clear! You also answered a 
question about how to find the squared distance between two points 
with the metric tensor that I was about to ask. You have been really 
helpful to me.

I have two other questions. Is the matrix that represents a tensor 
always a square matrix (cube matrix, hypercube matrix, etc.) except in 
the case of vectors? Also, what is it meant by the "contraction" of a 
tensor?

Thomas

Date: 07/21/2002 at 19:43:21
From: Doctor Nitrogen
Subject: Re: Abstract Index Notation and Tensors

Hello again, Thomas:

To my knowledge, the matrix representations for tensors generally 
involve square matrices. For example, mechanical engineers use special 
tensors called the stress and strain tensors, which are represented by 
3X3 square matrices, and physicists in General Relativity use a 
tensor, the metric tensor g^ik or g_ik, which is represented by a 4X4 
matrix.

The following illustrates the "contraction" of the indices in a 
tensor. Let:

   R^i_jkl

be a tensor of rank four, contravariant index i, covariant indices 
j, k, and l. You can lower the number of indices if, say, i = j:

   R^i_ikl.

If you multiply a tensor  A^i_jklm  by g^jk, you can also get 
contraction of the indices:

   g^jk(A^i_jklm) = A^i_lm.

Here, j and k were contracted. 

I hope this answers your questions. 

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/26/2002 at 22:28:37
From: Thomas Mainiero
Subject: Thank you (Abstract Index Notation and Tensors)

Thank you once again. You have been a great help in answering my 
questions. I also recently purchased that Schaum Outline book on 
Vector and Tensor analysis and it is a great resource; thank you for 
recommending it.

Thomas