Electromagnetic Tensor: Vector Space Explained

[Gavroy]

Hey,

does anoyone of you know to which vector space the electromagnetic tensor belongs to?

thank you for your ideas...

 

[Naty1]

maybe Minkowski space??

http://en.wikipedia.org/wiki/Electromagnetic_tensor#Definition

 

[dextercioby]

Assuming the vacuum case, then the vector space of 2-forms on flat 4D space-time ?

 

[Gavroy]

okay, i don't think that it is the minkowski space.

but the vector space of 2-forms sounds good.

i saw somewhere the notation: $$TM* \otimes TM*$$

sorry, this does not work: TM* (tensorproduct) TM*

what does this T stand for, does anyone know? this space, could have something to do with the 2-forms, but i am not really sure.

 

[fzero]

$$T_x M^*$$ is called the cotangent space of $$M$$ at the point $$x$$. It is the vector space of 1-forms. Higher forms are in the vector space made from tensor products of the cotangent space, so 2-forms are in $$T_x M^*\otimes T_x M^*$$.

When we write $$T M^*$$ we mean something different, but related. This is the cotangent bundle, which is the total space of the manifold $$M$$ together with the cotangent space at every point.

 

[Gavroy]

ah okay...thank you all

 

[Phrak]

An electromagnetic field tensor, or Faraday tensor, F = F_uvdx^vdx^v is an element of the space of two forms over the field of reals, or a type [0,2] antisymmetric tensors. This is a subspace of all type [0,2] tensors, so any Faraday tensor with lower indeces is also a member of the space of type [0,2] tensors.

Sometimes the Faraday tensor is given with upper indeces. It is still antisymmetric but a member of the antisymmetric tensors over the field of reals, but with upper indeces, so is called a type [2,0] tensor.

Or it could be presented in mixed form, type [1,1]. A vector space doesn't need or involve a manifold in it's set of axioms but can, however, be identified with the tangent space of a point on a manifold, which fzero has discussed.