Tensors with both covariant and contravariant components

[noahcharris]

Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index. And yet there must be a mixture of sorts because of all the upper/lowercase indices in GR. Any illumination would be much appreciated, thanks!

 

[Orodruin]

Are you familiar with and understand how a type (1,1) tensor is a linear map from the tangent vector space to itself?

 

[noahcharris]

Are you familiar with and understand how a type (1,1) tensor is a linear map from the tangent vector space to itself?

I was not aware of that. Could you explain further or perhaps point me towards the right resources? I don't understand what it means for a tensor to have a tangent vector space. -_-

 

[Orodruin]

The tensor does not have a tangent vector space, the underlying manifold does. Tangent vectors are what you would also call vectors or contravariant vectors. A linear map from the tangent vector space at a point of the manifold to itself is a type (1,1) tensor - it must be for the vectors to have the correct transformation properties.

 

[noahcharris]

The tensor does not have a tangent vector space, the underlying manifold does. Tangent vectors are what you would also call vectors or contravariant vectors. A linear map from the tangent vector space at a point of the manifold to itself is a type (1,1) tensor - it must be for the vectors to have the correct transformation properties.

What does it mean for a tensor to have an underlying manifold? And I would think that a linear mapping from a point to a point in vector space would simply be the kronecker delta, but maybe that is a type (1,1) among others?

 

[Orodruin]

If you are studying GR you should be familiar with the concept of a manifold and vectors on the manifold, that is what we are dealing with is it not?

Yes, the Kronecker delta is a prime example of a type (1,1) tensor - it has one covariant and one contravariant index, but far from the only one.

 

[Fredrik]

Tensors are multilinear maps $T:V^*\times\cdots\times V^*\times V\times\cdots\times V\to\mathbb R$, where V is a vector space and V* its dual space. In GR, and in differential geometry in general, V is the tangent space $T_pM$ of a smooth manifold M, at a point p in M. A tensor field is a map that takes each point p (in some subset of M) to a tensor at p.

 

[FactChecker]

I think that if you had a tensor where some indices were for distances or volumes (dx) and other indices were for rates or densities ( ∂y/∂x ), then the tensor would be a mixture of contravariant (dx) and covariant ( ∂y/∂x ). I am not well versed on this subject, so I don't have any real-world examples.