Are 10 dimensions related to the tensor of 3d

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SUMMARY

The discussion clarifies that a tensor in three dimensions does not inherently have ten components, as three does not divide ten. The tensor of order k has 3^k components, and while symmetries can reduce the number of unique components, they do not lead to ten components in standard tensors. The conversation also distinguishes between tensor components and string theory dimensions, noting that string theory often involves higher dimensions, such as eleven. The metric tensor in General Relativity (GR) is highlighted as having 16 components, which reduces to 10 independent components due to its symmetry.

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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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