Why are contravariant and covariant vectors important in general relativity?

In summary, the conversation discusses the concepts of Contravariant and Covariant vectors and their relationship to scalar quantities. The speakers also touch on the use of these vectors in General Relativity and express a desire for a deeper understanding of their purpose and significance.
  • #1
superbat
12
0
1) I read different texts on Contravariant , Covariant vectors.
2) Contravariant - they say is like vector . Covariant is like gradient
From what I see they have those vector spaces because it eventually helps get scalar out of it if we multiply contravariant by covariant

Also Contravariant like chaneg in displacement and covariant is like change in function (may be the curvature of space here)

Is that right?

i understand contra/co are like 2 independent vector spaces and they act on each other to produce kronecker deltas but i fail to see why GR uses it so heavily and any physical meaning other than what i mentioned above

Thank You
 
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  • #3
Thanks man
I have read a lot about contra covariant but most texts articles I read fail to provide motivation behind the same.
I was looking for that
I will read article you shared anyways
 

1. What is the difference between contravariant and covariant in general relativity (GR)?

Contravariant and covariant are two different mathematical concepts used in general relativity to describe the behavior of tensors. Contravariant tensors are defined as objects that transform in an opposite way to the coordinate system, while covariant tensors transform in the same way as the coordinate system. In simpler terms, contravariant tensors change their components when the coordinate system changes, while covariant tensors keep the same values. This difference is essential in the formulation of the equations of general relativity.

2. How are contravariant and covariant tensors related in GR?

In general relativity, there is a relationship between contravariant and covariant tensors known as the metric tensor. The metric tensor is a mathematical object that relates the contravariant and covariant components of a tensor. It is used to raise and lower indices in tensors, allowing for the transformation between contravariant and covariant tensors. This relationship is crucial in the formulation of the Einstein field equations.

3. What is the significance of contravariant and covariant tensors in GR?

Contravariant and covariant tensors play a crucial role in general relativity because they allow for the proper formulation of the theory. They are used to describe the curvature of spacetime, which is the fundamental concept in general relativity. The use of these tensors in the Einstein field equations allows for the prediction and understanding of the behavior of matter and energy in the presence of gravity.

4. How are contravariant and covariant tensors used to describe the curvature of spacetime?

In general relativity, the curvature of spacetime is described by the Riemann curvature tensor, which is a combination of contravariant and covariant tensors. The contravariant and covariant components of this tensor represent the curvature of spacetime in different directions. This tensor is then used to calculate the Einstein tensor, which is a key component in the Einstein field equations and describes the relationship between the curvature of spacetime and the distribution of matter and energy.

5. Are there any practical applications of understanding contravariant and covariant in GR?

Yes, there are several practical applications of understanding contravariant and covariant tensors in general relativity. For example, these concepts are essential in the development of gravitational wave detectors, which use the distortion of spacetime to detect gravitational waves. Additionally, understanding these concepts is crucial in the study of black holes, as they are objects with extreme curvature of spacetime. Furthermore, the use of contravariant and covariant tensors has also been applied in cosmology, where they are used to study the large-scale structure of the universe.

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