Simple identity for antisymmetric tensor

In summary, the identity \nabla_\mu \nabla_\nu F^{\mu\nu}=0 holds for all antisymmetric tensors F^{\mu\nu}, as long as the connection is torsion-free. This is because the symmetric part of ∇µ∇ρ zeroes out anything antisymmetric in µ and ρ. However, for a general affine connection, the identity may not hold and is dependent on the definition of the Ricci tensor and presence of torsion.
  • #1
paweld
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Is it true that for all antisymmetric tensors [tex]F^{\mu\nu} [/tex]
the following identity is true:
[tex]\nabla_\mu \nabla_\nu F^{\mu\nu}=0 [/tex]
(I've checked it but I'm not absolutely sure).
 
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  • #2
hi paweld! :smile:
paweld said:
Is it true that for all antisymmetric tensors [tex]F^{\mu\nu} [/tex]
the following identity is true:
[tex]\nabla_\mu \nabla_\nu F^{\mu\nu}=0 [/tex]
(I've checked it but I'm not absolutely sure).

yup, because ∇µρ is symmetric in µ and ρ, so it zeroes anything antisymmetric in µ and ρ :wink:
 
  • #3
That depends on how you define [tex]\nabla_\mu[/tex]. For a general affine connection you get, more or less, [tex]\pm R_{\mu\nu}F^{\mu\nu}[/tex] (plus or minus depending on which convention is being used in the definition of the Ricci tensor). When there is no torsion, Ricci tensor is symmetric and you get zero. But not so for a general connection.
 
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  • #4
Thanks, I always assume that connection is torsion-free.
 
  • #5
BTW:

∇µ∇ρ is not symmetric in µ and ρ. Its antisymmetric part is related to the curvature tensor.

d493bbad067a502909d1ae33781994cc.png


The above holds for u,v commuting vector fields like [tex]\partial_\mu,\, \partial_\nu [/tex]
 
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1. What is a simple identity for antisymmetric tensors?

A simple identity for antisymmetric tensors is a mathematical expression that relates the components of an antisymmetric tensor to each other. It is often used in the study of vector calculus and differential geometry.

2. How is the simple identity for antisymmetric tensors derived?

The simple identity for antisymmetric tensors is derived from the properties of antisymmetric tensors, which include skew-symmetry and the fact that the components of the tensor are equal to the negative of their transposes.

3. What is the significance of the simple identity for antisymmetric tensors?

The simple identity for antisymmetric tensors is important in the study of vector calculus and differential geometry because it allows for the simplification of calculations involving these types of tensors. It also helps to identify relationships between different components of the tensor.

4. Can the simple identity for antisymmetric tensors be generalized to higher dimensions?

Yes, the simple identity for antisymmetric tensors can be generalized to higher dimensions. In three dimensions, the identity takes the form of the cross product, while in four dimensions it is related to the wedge product.

5. Are there any practical applications of the simple identity for antisymmetric tensors?

Yes, the simple identity for antisymmetric tensors has practical applications in various fields such as physics, engineering, and computer graphics. It is used in the study of electromagnetic fields, fluid mechanics, and rotations in 3D space, among others.

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