Tensor question

[asmodel]

Is it possible to have a tensor, whereby one or more of its components are themselves vector? This relates to an earlier post.

[studiot]

It can be a little confusing and rather depends what you mean by 'vector'.

 

Tensors are uniquely defined and can generally be written as matrices (though not all matrices represent tensors).

As such they contain row and column 'vectors'.

Tensors do not, however, have 'components' in the vector sense, that represent one quantity (the vector magnitude) that can be resolved in separate directions.

Tensors have 'elements', that may not all be of the same type.

These elements correspond to the elements or entries of the matrix.

So for instance the stress tensor has 3 axial stress and 6 shear stress elements.

These cannot be lumped (like forces which are vectors) into a single magnitude and resolved into particular directions.

 

So in this sense tensors are more complicated objects than vectors.

Scalars are the simplest, they offer one piece of information only eg the mass.

Vectors are next and offer two pieces of information eg magnitude and direction.

In this sense tensors are the generalisation that extends the idea to n pieces of information.

 

On this scheme we would refer to the order of the tensor as (n-1)

So a scalar is a tensor of zero order

A vector is a tensor of first order

and the first tensor we would recognise as such would be a second order tensor, such as the stress tensor, that contains 9 pieces of information.

 

Set against this is the subject of linear algebra which defines Vector Spaces and the members or elements of such spaces as 'vectors' if they obey the axioms of linear algebra.

Tensors obey these linear axioms so can themselves be classed as vectors in this wider sense.

 

There are various algebras or rules for combining scalars, vectors and tensors.

 

Does this help?

[ajb]

Is it possible to have a tensor, whereby one or more of its components are themselves vector? This relates to an earlier post.[/size]

You can have tensor and tensor-like objects that take their values in vector spaces or more generally vector bundles. For example vector-valued differential forms are very important in differential geometry. So with your earlier post we are not sure what you are really trying to do, but maybe the information on vector-valued forms is enough for you to see what you need.

[asmodel]

I understand what a tensor is and what a vector are, and also the relationship they share, and the methods of computing the contra/covariant forms etc.

 

But in a Tensor, even though you can components of a tensor that can be derived from vectors, can a single component of a tensor, itself, be a vector... E.g. a 3-d Tensor, D:

 

 

Where lambda is itself a 3-D vector?

Edited September 3, 2014 by asmodel

[studiot]

 

Tensors do not, however, have 'components' in the vector sense, that represent one quantity (the vector magnitude) that can be resolved in separate directions.

[asmodel]

I understand but what if I wanted to describe a bi-metric?

 

For example a bi-metric is where each point is a space N-D is represented by two metric tensors,instead of one. My idea it to represent a bi-metric tensor, with only 1 tensor. Do you see where I am going now?

Edited September 3, 2014 by asmodel

[studiot]

Individual elements (or parts of one) could be a (coordinate) vector.

 

Unlike vectors, it is not required that all elements represent the same physical quantities, as in my stress tensor example.

 

This means that you must have enough equations to handle different relationships.

[ajb]

I understand but what if I wanted to describe a bi-metric?

What is a bi-metric?

 

The closest I know is gravity theories that have two metrics, but I think this is not what you mean.

 

 

For sure you can have components of a tensor taking values "elsewhere". I gave you an important example of vector valued differential forms. Your D looks like a vector-valued matrix to me, you have to take the zeros to be the zero vectors.

 

It would help if you made the context here very clear.

[asmodel]

Here is Bi-metric theory and yes it relates to gravity.

 

http://en.wikipedia.org/wiki/Bimetric_theory

[ajb]

So, you want to combine the two metrics into one object, or something like that? I don't quite see what your opening question has to to with bi-metric gravity as you have posed it.

[asmodel]

Well a bi-metric is a good example of what I am doing, its not really a bi-metric, but its similar.

 

I want to represent multiple tensors, that map different spaces, into one tensor.

Edited September 4, 2014 by asmodel

[ajb]

I want to represent multiple tensors, that map different spaces, into one tensor.

If you have two initial spaces can't you just take their tensor product and build tensors in that way?

 

This is more or less what you do with vector-valued forms; you can write them as the tensor product over smooth functions on your manifold of sections of a vector bundle and differential forms.

[asmodel]

Thanks, I suppose I can keep them as separate Tensors but I would combine them in some way. I appreciate your advice.

[ajb]

Out of interest, and possible further advice, try to tell us a little more about what you are trying to do and why.

 

I understand if you don't want to say too much as you may be looking to publish this at some time.

[studiot]

 

Out of interest, and possible further advice, try to tell us a little more about what you are trying to do and why.

 

I understand if you don't want to say too much as you may be looking to publish this at some time.

 

 

 

 

In the past I have helped preserve research confidentiality by use of the private messaging system.

 

However I am not a cosmologist, and ajb knows a great deal more about tensor gravity models than I do so I am not the right person to help further here, since the last time I did that with tensors was in fracture mechanics.

Edited September 5, 2014 by studiot

[ajb]

In the past I have help preserve research confidiality by use of the private messaging system.

Of course, I am willing to engage via private messaging. However, if you are based at a university or similar then my official work email may be more appropriate. Anyway, send a private message if you would rather not openly discuss something.