What Do Upper and Lower Indices in Tensor Notation Signify?

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SUMMARY

The discussion focuses on the significance of upper and lower indices in tensor notation, particularly in the context of general relativity. Upper indices represent contravariant components, corresponding to vectors, while lower indices denote covariant components, associated with covectors or forms. Understanding this distinction is crucial for manipulating tensors effectively, especially when both types of indices are present in a single tensor. The conversation highlights the importance of these concepts in the study of manifolds and their applications in physics.

PREREQUISITES
  • Basic understanding of tensors and their manipulation
  • Familiarity with the concepts of covariant and contravariant vectors
  • Knowledge of manifolds in differential geometry
  • Introduction to general relativity principles
NEXT STEPS
  • Study the differences between covariant and contravariant tensors in detail
  • Learn about the role of metrics in general relativity and their impact on tensor notation
  • Explore the concept of forms and their integration on manifolds
  • Review examples of tensors with both upper and lower indices in physical applications
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Students and researchers in physics, particularly those studying general relativity, as well as mathematicians interested in differential geometry and tensor analysis.

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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