Are 10 dimensions related to the tensor of 3d

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Discussion Overview

The discussion revolves around the relationship between the number of components in tensors defined in three-dimensional spaces and the concept of ten dimensions as seen in string theory. Participants explore the mathematical properties of tensors, particularly in the context of dimensionality and symmetry, while also touching upon the implications for string theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the tensor in three dimensions has ten elements and questions if this is related to the ten dimensions in some forms of string theory.
  • Another participant clarifies that the tensor of order ##k## in three-dimensional vector spaces has ##3^k## components, indicating that none will have ten components unless symmetries reduce the number of unique components.
  • A different viewpoint mentions the possibility of considering nine spatial dimensions plus one time dimension, leading to a total of ten, but questions the relevance of this to string theory.
  • One participant proposes that the metric tensor in 3+1 dimensions could be what is meant, noting that string theory's dimensionality has a different origin.
  • Another reiterates that the metric tensor of General Relativity has 16 components, which reduces to 10 independent components due to its symmetry.

Areas of Agreement / Disagreement

Participants express disagreement regarding the relationship between the number of tensor components in three dimensions and the ten dimensions in string theory. There is no consensus on whether these concepts are related.

Contextual Notes

Participants highlight the importance of symmetries in determining the number of unique components in tensors, and there are unresolved questions about the definitions and implications of dimensionality in relation to string theory.

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I think I've read the the tensor in three dimensions has 10 elements in its matrix(?). Is this related to the 10 dimensions in some forms of string theory?
 
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There is more than one tensor over three-dimensional vector spaces. The tensor of order ##k## has ##3^k## components. None of the tensors will have ten components, as three does not divide ten. However, if there are symmetries in the type of tensor you are considering, that will reduce the number of unique components, as some will be the same as others, and you may get ten that way.
This has nothing to do with string theory though, which seems to be about vector spaces with many more dimensions, eg eleven.
 
None of the tensors will have ten components, as three does not divide ten.
I was thinking 9 plus 1 (zero).

eg eleven
I was thinking 10 spatial plus 1 (time).

But if they aren't related they aren't. Thanks
 
Maybe you mean the metric tensor in 3+1 dimensions? The number of spacetime dimensions in string theory has a different origin :)
 
andrewkirk said:
There is more than one tensor over three-dimensional vector spaces. The tensor of order ##k## has ##3^k## components. None of the tensors will have ten components, as three does not divide ten. However, if there are symmetries in the type of tensor you are considering, that will reduce the number of unique components, as some will be the same as others, and you may get ten that way.
This has nothing to do with string theory though, which seems to be about vector spaces with many more dimensions, eg eleven.

This is the case with the metric tensor of GR. It has 16 components but being symmetric it reduces to 10 independent components.
 

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