# Raising and Lowering Indices and metric tensors

• Physicist97
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.
Physicist97
The metric tensor has the property that it can raise and lower indices, but this is on the assumption that it (the metric) is symmetric. If we were to construct a metric tensor that was non-symmetric, would it still raise and lower indices?

I think raising and lowering follow from the fact that ## T^k_{l+1}V ## ( k covariant, (l+1) contravariant) is canonically isomorphic to ## (V^{*} \times V^{*} \times...\times V^{*}) \times V\times \times...\times V## (l copies of ##V^{*}##, k copies of ##V##). So I think you only need the metric to be positive definite for the canonical isomorphism to hold.

If ##v\in V##, and ##g:V\times V\to\mathbb R## is a metric, we can denote the map ##u\mapsto g(v,u)## from ##V## into ##\mathbb R## by ##g(v,\cdot)##. This is an element of ##V^*##. The map ##v\mapsto g(v,\cdot)## is an isomorphism from ##V## to ##V^*##. Since a metric is symmetric, the map ##v\mapsto g(\cdot,v)## is the same isomorphism from ##V## to ##V^*##. The problem with a non-symmetric "metric" is that these would be two different isomorphisms from ##V## to ##V^*##.

You can make whatever definition of raising/lowering indices you want. But if the definition is not useful, people might not use it. Using a non-symmetric "metric" may not be very useful, but if you can find a use for it, then go ahead and develop the machinery! :D

Well, yes, but my point that the symmetric positive-definite form allows you to produce a natural isomorphism ( as k-linear maps)
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.

## 1. What is the purpose of raising and lowering indices in tensor notation?

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.

## 2. How do you raise and lower indices in tensor notation?

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.

## 3. What is the role of the metric tensor in raising and lowering indices?

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.

## 4. Can you raise and lower multiple indices at once?

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.

## 5. How is the metric tensor related to the concept of distance in a space?

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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