A simple question about tensor notation

In summary, the difference between upper and lower indices in tensors signifies whether they are associated with "covectors" or "vectors". On a manifold, covectors correspond to "forms" which can be integrated without a metric. In general relativity, there is a one-to-one correspondence between covectors and vectors in the presence of a metric. Another explanation is that upper indices correspond to covectors while lower indices correspond to vectors, as explained in a physical context in the provided link. It is important to understand this concept before delving further into general relativity.
  • #1
taylrl3
61
0
Hi,

I am very new to general relativity and have only just started to learn how to do some very basic manipulation of tensors. I can understand the methods I am using and have some idea of what a tensor is but am not sure what the difference between upper and lower indices signifies. I can identify that one is covariant and another contravariant but what is the difference between the two and what about when a tensor has both indices? I feel I need to clear this conceptual issue up before I can understand things further. Thanks :-)

Taylrl
 
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  • #2
Usually one of them is associated to "covectors" and the other to "vectors". Both are vectors in the linear algebra sense, but on a manifold, the former correspond to "forms" which are things that can be integrated even without a metric. In the presence of a metric, as in general relativity, there is one-to-one correspondence between covectors and vectors. http://www.math.ucla.edu/~tao/preprints/forms.pdf

A perhaps more physical explanation goes something like https://www.physicsforums.com/showpost.php?p=3361102&postcount=14.
 
Last edited:

1. What is tensor notation?

Tensor notation is a mathematical notation used to represent tensors, which are multidimensional arrays of numbers. It is a concise way of writing complex mathematical expressions involving tensors.

2. Why is tensor notation important?

Tensor notation is important because it allows scientists and mathematicians to express and manipulate complex mathematical equations involving tensors in a concise and efficient manner. It also makes it easier to communicate and understand these equations in fields such as physics, engineering, and computer science.

3. How is tensor notation different from matrix notation?

Tensor notation is different from matrix notation in that it can handle multidimensional arrays, whereas matrix notation is limited to two dimensions. Additionally, tensor notation uses different types of indices to represent the dimensions of a tensor, while matrix notation uses row and column indices.

4. Can you give an example of tensor notation?

Yes, an example of tensor notation is the Einstein summation convention, which uses a subscript and superscript index to represent the dimensions of a tensor. For example, Tij represents the ith row and jth column of a tensor T.

5. How can I learn more about tensor notation?

There are many online resources available to learn more about tensor notation, including textbooks, online courses, and tutorials. It is also helpful to have a strong foundation in linear algebra and calculus to understand tensor notation. Practicing with examples and solving problems can also improve your understanding of tensor notation.

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