Is space covariant or contravariant?

Click For Summary

Discussion Overview

The discussion revolves around the classification of space as covariant or contravariant, particularly in the context of tensor components and the choice of basis. Participants explore the implications of these classifications and the interpretations of their professors regarding the nature of space in relation to tensors.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant asserts that space itself is neither covariant nor contravariant, but rather that the classification applies to the components of tensors based on the chosen basis.
  • Another participant suggests that coordinates are contravariant, implying that this may be what the professors were referring to.
  • A different viewpoint indicates that tensors can be represented in mixed coordinates, which complicates the classification of their components as covariant or contravariant.
  • One participant encourages consulting the professors directly for clarification, noting that the distinction between space and basis vectors may have been overlooked.
  • Another participant references an external source that supports their understanding, indicating that there are differing opinions on the topic.
  • Concerns are expressed about potentially missing key concepts, prompting a request for further assistance from the forum.
  • A later reply mentions that the meanings of covariant and contravariant may have changed historically, suggesting inconsistency in usage.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether space is covariant or contravariant. Multiple competing views remain, with some emphasizing the role of basis choice and others suggesting a definitive classification of space itself.

Contextual Notes

Participants highlight the dependence on definitions and the potential for historical changes in terminology, which may affect the understanding of covariant and contravariant classifications.

RudiTrudi
Messages
4
Reaction score
0
I often meet the question whether the (physical) space is 'covariant' or 'contravariant'.

I once replied to that question with: Space is space. The COMPONENTS of a tensor are covariant/contravariant if the basis is CHOSEN TO BE contravariant/covariant. As far as I know tensors the 'covariance' or 'contravariance' of space (or tensor) itself is not even the question.

The thing is that the professors were not satisfied with the answer. They said the space is contravariant.

So, I am confused. As far as I know tensors, or better saying tensor components, the 'covariance' or 'contravariance' depends on the choice of basis, i.e. contravariant or covariant basis. In other words, tensor cannot be covariant or contravariant, it can only be represented in covariant or contravariant BASIS and therefore having contravaiant or covariant components.


Does anyone have any ideas about that?
 
Physics news on Phys.org
Coordinates are contravariant. I suspect that is all that the professors meant to ask.
 
the coordinates can be covariant or contravariant if the basis is chosen to be contravariant or covariant. The are also cases when tensor is represented in mixed coordinates, i.e. as a linear combination of base vectors, some from tangent space and some from its dual space. So I don't understand why can an answer 'contravariant' be 'correct'.
 
Last edited:
I think you should ask the professors. I am really bad at reading your professor's minds.

However, if you look at your explanation you will note that justifying your answer required use of basis dual vectors from the dual space whereas the professor's question was about space. If he was intending to emphasize the distinction then you missed his point.
 
Last edited:
thanks for the effort. It is obvious that reading someone’s mind is difficult if not impossible. But still, I found the answer on imechanica forum. Prof. Sia Nemat-Nasser clearly confirms my understanding of the topic. Here is the link to the post http://imechanica.org/node/4356. I recommend reading it. And once again thanks for your responses. I appreciate it.
 
Last edited by a moderator:
But still I am afraid I am missing something. That is why I am asking for some help on physics forum instead of (solely) on mathematical one.
 
You need to speak to your professor. He obviously wants you to learn something that you have not learned. Looking up websites that you think somehow agree with you may make you feel better but is not going to help you learn a concept you are struggling with.
 
Keep in mind that, some time ago, the meaning of covariant and contravariant was reversed. I'm not sure if the usage is consistent these days.
 

Similar threads

Graduate Contravariance, Covariance, stress and position vector
  • · Replies 24 ·
Replies
24
Views
3K
Undergrad Covariant and contravariant tensors
  • · Replies 8 ·
Replies
8
Views
9K
Undergrad Physical Difference Between Co- and Contravariant Vectors
  • · Replies 6 ·
Replies
6
Views
2K
Showing that the gradient of a scalar field is a covariant vector
  • · Replies 5 ·
Replies
5
Views
4K
High School A covariant vs contravariant vector?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Why do we use covariant formulation in classical electrodynamics?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Is a vector itself contra/covariant or just its components?
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Purpose of Tensors, Indices in Tensor Calculus Explained
  • · Replies 10 ·
Replies
10
Views
4K
Undergrad GR: Can Units Tell You if Quantity is Covariant or Contravariant?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Differential forms as a basis for covariant antisym. tensors
  • · Replies 1 ·
Replies
1
Views
2K