The discussion revolves around the understanding of (k,l) tensors in the context of General Relativity, specifically addressing definitions and interpretations found in the works of Wald and Carroll. Participants explore the nature of tensors, dual vectors, and the implications of their definitions.
Participants exhibit disagreement regarding the interpretation of tensor definitions and their implications. There is no consensus reached on the initial confusion, but some clarification occurs through discussion.
Some participants express uncertainty about specific definitions and their interpretations, indicating a need for precise quotes from the referenced texts to resolve misunderstandings.
No, it doesn't. A type (k, 1) tensor can be contracted with k dual vectors and l vectors to produce a scalar. That does not mean the k dual vectors and l vectors are "part of" the tensor.GeoffFB said:In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors
Yes, because a (1, 0) tensor can be contracted with 1 dual vector to produce a scalar, so it's a vector; and a (0, 1) tensor can be contracted with 1 vector to produce a scalar, so it's a dual vector.GeoffFB said:a (1,0) tensor is a vector and a (0,1) tensor is a dual vector.
A tensor, as they define it, is a multilinear map, it has k dual vectors and l vectors as input. So a (0,1) tensor will be a linear map that has a vector as an input i.e. it will be a dual vector.GeoffFB said:Summary:: Type (k,l) tensors, dual vectors and vectors.
In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors, yet a (1,0) tensor is a vector and a (0,1) tensor is a dual vector. I must be missing something simple. Please explain.
You are evidently misinterpreting statements from these sources, but unless you give specific quotes and where they are from, it's impossible to tell exactly what you are misinterpreting.GeoffFB said:In both Wald and Carroll
What on this page led you to believe what you said in the OP?GeoffFB said:General Relativity by Robert M. Wald, Page 20.
Thanks. Now I understand.PeterDonis said:No, it doesn't. A type (k, 1) tensor can be contracted with k dual vectors and l vectors to produce a scalar. That does not mean the k dual vectors and l vectors are "part of" the tensor.Yes, because a (1, 0) tensor can be contracted with 1 dual vector to produce a scalar, so it's a vector; and a (0, 1) tensor can be contracted with 1 vector to produce a scalar, so it's a dual vector.
I was confused by thinking that type (0, 1) tensor meant (k, l) tensor, not understanding about the mapping to R.PeterDonis said:What on this page led you to believe what you said in the OP?