Covariant and contravariant

In summary, if the notions of covariant and contravariant tensors were not introduced, it would not be possible to express the Einstein equation Guv=8πTuv in its current form. The concept of tensors, whether expressed in index notation or index-free notation, is crucial for the coordinate independence and tensorial nature of the equation. Additionally, the extension of tensors to spinors is also important in understanding the equation and its implications.
  • #1
extrads
16
0
If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?
 
Physics news on Phys.org
  • #2
extrads said:
If the notions of covariant and contravariant tensors were not introduced,what would happen?

I'm not sure what you mean by this. Covariant and contravariant tensors represent distinct kinds of physical things; if you know the metric, you can compute correspondences between them, but they are still distinct concepts. So if you're going to use tensors at all, you need both kinds.
 
  • #3
If the OP is asking whether we could express Guv=8πTuv without tensors, I would have to say that it can be done, but the central property of coordinate independence would still be there ( ie 'tensoriality').
 
  • #4
The notion of contravariant and covariant is always there. It is made explicit in index notation but you can just as well write it in index-free notation as ##G = 8\pi T## but you cannot get rid of the tensorial nature of the classical EFEs. The extension of the concept of a tensor is a spinor: http://en.wikipedia.org/wiki/Spinor
 
  • #5


Covariant and contravariant tensors are important concepts in tensor analysis, which is a mathematical tool used in fields such as physics and engineering. These tensors represent linear transformations between vector spaces and are used to describe physical quantities that can change with respect to different coordinate systems.

If the notions of covariant and contravariant tensors were not introduced, it would be much more difficult to accurately describe and analyze physical phenomena. Without these concepts, the Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy, would not be able to accurately account for the effects of different coordinate systems. In fact, the equations would likely be much more complicated and difficult to solve.

The Einstein field equations, Guv=8πTuv, would also be drastically different without the use of covariant and contravariant tensors. These tensors allow for the equations to be written in a covariant form, meaning that they are independent of the choice of coordinate system. This is important in the study of general relativity, as it allows for the equations to accurately describe the effects of gravity on spacetime in any given coordinate system.

In summary, the introduction of covariant and contravariant tensors has greatly enhanced our understanding and analysis of physical phenomena, particularly in the field of general relativity. Without these concepts, it would be much more difficult to accurately describe and solve complex equations, ultimately hindering our ability to understand the fundamental laws of the universe.
 

1. What is the difference between covariant and contravariant?

Covariant and contravariant refer to the way in which a mathematical quantity changes when the coordinate systems used to measure it are transformed. In general, covariant quantities change in the same direction as the coordinate system changes, while contravariant quantities change in the opposite direction.

2. How do covariant and contravariant vectors relate to each other?

Covariant and contravariant vectors are related through a mathematical operation called the metric tensor. The metric tensor relates the components of a covariant vector to the components of a contravariant vector, allowing for the transformation between the two types of vectors.

3. What is the significance of covariant and contravariant in tensor calculus?

Covariant and contravariant are essential concepts in tensor calculus, as they allow for the proper transformation of tensors between different coordinate systems. Without taking into account the covariant and contravariant nature of tensors, calculations and equations in tensor calculus may result in incorrect results.

4. How are covariant and contravariant quantities represented in tensor notation?

In tensor notation, covariant quantities are represented with lower indices, while contravariant quantities are represented with upper indices. This notation helps to distinguish between the two types of quantities and is used in various mathematical operations involving tensors.

5. Can you give an example of a covariant and contravariant quantity?

A common example of a covariant quantity is length, as it changes in the same direction as the coordinate system changes. An example of a contravariant quantity is velocity, as it changes in the opposite direction to the coordinate system changes. This can be seen when comparing the velocity of an object from an observer on the ground and an observer on a moving train.

Similar threads

  • Special and General Relativity
Replies
10
Views
1K
  • Special and General Relativity
Replies
3
Views
801
  • Special and General Relativity
2
Replies
36
Views
2K
  • Special and General Relativity
Replies
1
Views
1K
Replies
4
Views
1K
  • Special and General Relativity
Replies
8
Views
1K
  • Special and General Relativity
Replies
4
Views
3K
  • Special and General Relativity
Replies
19
Views
2K
  • Special and General Relativity
Replies
2
Views
846
  • Advanced Physics Homework Help
Replies
5
Views
2K
Back
Top