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Physicist97

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In summary, the conversation discusses the properties of the metric tensor and whether it can still raise and lower indices if it is non-symmetric. It is mentioned that the metric tensor must be positive definite for this to hold, and that the symmetric nature of the metric allows for a natural isomorphism between tensors. However, it is also noted that a non-symmetric metric may not be as useful, but this does not prevent one from developing the necessary machinery for it.

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Physicist97

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WWGD

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Fredrik

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between the initial tensor and the contracted one. This is what allows you to contract indices: you are substituting a tensor by an

isomorphic copy of it.

Raising and lowering indices allows for a more compact and efficient representation of tensor equations. It also allows for easier manipulation and comparison of tensors.

To raise an index, you multiply the tensor by the metric tensor and sum over the repeated index. To lower an index, you multiply the tensor by the inverse metric tensor and sum over the repeated index.

The metric tensor defines the inner product between vectors and allows for the transformation of tensors between different coordinate systems. It is also used to raise and lower indices in tensor notation.

Yes, you can raise or lower multiple indices at once by multiplying by the corresponding number of metric tensors and summing over all repeated indices.

The metric tensor is used to calculate distances in a space by defining the relationship between different coordinate systems. It is also used to determine the shortest distance between two points in a curved space.

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