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
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