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?
(k,l) tensors are mathematical objects that are used to describe the geometric properties of space and time in general relativity. They are composed of multiple components that transform in a specific way under coordinate transformations, allowing them to describe physical quantities in a coordinate-independent manner.
(k,l) tensors are directly related to the curvature of space-time in general relativity. They are used to define the metric tensor, which describes the curvature of space-time and is essential for solving the Einstein field equations that govern the behavior of matter and energy in space-time.
The k and l indices in (k,l) tensors represent the number of covariant and contravariant components, respectively. Covariant components transform in the same way as the coordinate system, while contravariant components transform in the opposite way. The number of indices determines the rank of the tensor, with higher rank tensors having more indices.
(k,l) tensors are used extensively in general relativity to describe the geometric properties of space-time and the behavior of matter and energy within it. They are used to define the metric tensor, the curvature tensor, and other important quantities that are necessary for solving the Einstein field equations and making predictions about the behavior of objects in space-time.
(k,l) tensors have numerous applications in real-world scenarios, including predicting the behavior of objects in strong gravitational fields, such as black holes and neutron stars. They are also used in the study of gravitational waves and the expansion of the universe. Additionally, (k,l) tensors are essential in the development of mathematical models and simulations for cosmological and astrophysical phenomena.