What Do Upper and Lower Indices in Tensor Notation Signify?

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Upper and lower indices in tensor notation signify different types of quantities: upper indices represent contravariant vectors, while lower indices denote covariant vectors or covectors. In the context of manifolds, covectors correspond to differential forms that can be integrated without a metric, whereas vectors require a metric for their definition. The distinction is crucial for understanding tensor manipulation in general relativity, where both types of indices can coexist in a single tensor. This relationship allows for a one-to-one correspondence between covectors and vectors when a metric is present. Clarifying this conceptual difference is essential for further comprehension of tensor calculus in physics.
taylrl3
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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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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.
 
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