Understand (k,l) Tensors in Gen. Relativity

In summary, (k,l) tensors are mathematical objects used in general relativity to represent physical quantities such as mass, energy, and momentum. They have a certain number of indices, k and l, which determine their rank and indicate how they transform under coordinate transformations. They are used to describe the curvature of spacetime, calculate quantities such as geodesics, and are fundamental in understanding the behavior of matter and energy in curved spacetime. One example of a (k,l) tensor is the stress-energy tensor, which is used in Einstein's field equations to relate the curvature of spacetime to the distribution of matter and energy.
  • #1
GeoffFB
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TL;DR 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.
 
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  • #2
GeoffFB said:
In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors
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:
a (1,0) tensor is a vector and a (0,1) tensor is a dual vector.
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.
 
  • #3
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.
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.
 
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  • #4
GeoffFB said:
In both Wald and Carroll
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.
 
  • #5
Sorry!
General Relativity by Robert M. Wald, Page 20.
Spacetime and Geometry by Sean M. Carroll, Page 21.
 
  • #6
GeoffFB said:
General Relativity by Robert M. Wald, Page 20.
What on this page led you to believe what you said in the OP?
 
  • #7
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.
Thanks. Now I understand.
 
  • #8
PeterDonis said:
What on this page led you to believe what you said in the OP?
I was confused by thinking that type (0, 1) tensor meant (k, l) tensor, not understanding about the mapping to R.
 

Related to Understand (k,l) Tensors in Gen. Relativity

1. What are (k,l) tensors in general relativity?

(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.

2. How are (k,l) tensors related to the curvature of space-time?

(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.

3. What do the k and l indices represent in (k,l) tensors?

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.

4. How are (k,l) tensors used in general relativity?

(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.

5. What are some real-world applications of (k,l) tensors in general relativity?

(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.

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