The discussion clarifies the definition and properties of (k,l) tensors in the context of General Relativity, specifically referencing Robert M. Wald's "General Relativity" and Sean M. Carroll's "Spacetime and Geometry." A (1,0) tensor is definitively identified as a vector, while a (0,1) tensor is a dual vector. The confusion arises from misinterpretation of the tensor's mapping properties, particularly regarding the relationship between dual vectors and vectors in tensor contraction.
PREREQUISITESStudents and professionals in physics, particularly those focusing on General Relativity, tensor calculus, and mathematical physics. This discussion is beneficial for anyone seeking to deepen their understanding of tensor properties and their applications in theoretical frameworks.
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?