Tensor Rank vs Type: Explained

[ddesai]

Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2 tensor" and also "the metric is a covariant 2-tensor or type 2 tensor" I assume the two concepts, that of "type" and "rank" are unrelated, but I want another perspective.

Also, in GR mostly we deal with tensor fields as well as tensors. At different points the rank (as in matrix rank) may be different. Is this true?

 

[WannabeNewton]

When books say "the metric tensor is a rank 2 tensor" they really mean it's a tensor field of type (0,2). In this context rank doesn't mean the dimension of the image of a linear operator acting on a vector space.

 

[jcsd]

Rank = n+m. The tensors of a tensor field will always be of the same type. For example you wouldn't have field that was vector-valued at some points and scalar-valued at others.

 

[ddesai]

@jcsd. That's clear. But you use the term "Rank" in a different way than for example, this paper: http://www.its.caltech.edu/~matilde/WeitzMa10Abstract.pdf. If we assume that rank is defined as it is in this paper, then can you still say it doesn't change as you move from point to point?

@Newton. So then folks mix the terminology.

 

[jcsd]

@jcsd. That's clear. But you use the term "Rank" in a different way than for example, this paper: http://www.its.caltech.edu/~matilde/WeitzMa10Abstract.pdf. If we assume that rank is defined as it is in this paper, then can you still say it doesn't change as you move from point to point?

@Newton. So then folks mix the terminology.

I completely missed the analogy with matrix rank. I suppose the metric must always be rank 4 for this meaning of rank as it has a non-zero determinant. I also would guess that the rank of a tensor field could change from point to point, for example in any tensor field that was zero at some point, but wasn't a zero tensor field.