Covariant Derivative Wrt Superscript Sign: Explained

In summary, the covariant derivative of a tensor with respect to a superscript has a minus sign in the partial derivative term. However, this depends on the metric and there is no fixed rule for the index position in covariant derivatives.
  • #1
cr7einstein
87
2
Dear all,
I was reading this https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor, and it said that if you take the covariant derivative of a tensor with respect to a superscript, then the partial derivative term has a MINUS sign. How? The Christoffel symbol should have a minus sign, but I don't understand how does the partial derivative get one?

Also, does covariant derivative always have an index opposite to that of the tensor(e.g. a contravariant tensor will be differentiated wrt a covariant tensor, and a covariant tensor wrt to a covariant index)? If so, why? Is there a relation between the two(which the minus sign mentioned above indicates)?
Thanks in advance!
 
Physics news on Phys.org
  • #2
cr7einstein said:
How? The Christoffel symbol should have a minus sign, but I don't understand how does the partial derivative get one?

None of that is correct actually. ##\nabla^{\mu}A^{\nu} = g^{\mu\delta}\nabla_{\delta}A^{\nu} = g^{\mu\delta}\partial_{\delta}A^{\nu} + g^{\mu\delta}\Gamma^{\nu}_{\delta \sigma}A^{\sigma}## and similarly for tensors of arbitrary rank, so it depends entirely on ##g_{\mu\nu}##.

cr7einstein said:
Also, does covariant derivative always have an index opposite to that of the tensor(e.g. a contravariant tensor will be differentiated wrt a covariant tensor, and a covariant tensor wrt to a covariant index)?

No.
 
  • #3
I'm afraid you didn't choose a bright source for reading. So, in general relativity there's no [itex] x_{\mu} [/itex] (and in special relativity shouldn't be either). Next:

[tex] \nabla^{\mu} T_{\alpha} = g^{\mu\beta}\nabla_{\beta}T_{\alpha} [/tex]

is just a shorthand whenever necessary. Because the nonmetricity is 0, then you can play around freely with the position of the index in the covariant derivative. The metric also allows you to play with the index position also for the tensorial objects being differentiated:

[tex] \nabla^{\mu}T_{\alpha}^{~~\gamma} = g_{\alpha\delta} g^{\mu\lambda}\nabla_{\lambda}T^{\delta\gamma} [/tex]

Relativists call this 'idex gymnastics'.
 

What is a covariant derivative with respect to superscript sign?

A covariant derivative with respect to superscript sign is a mathematical concept used in differential geometry and tensor analysis. It is a way to differentiate a vector or tensor field along a given direction or coordinate basis, taking into account the curvature of the underlying space. This helps to account for the changes in the vectors or tensors due to the curvature of the space they are defined on.

How is a covariant derivative with respect to superscript sign calculated?

A covariant derivative with respect to superscript sign is calculated by first defining a connection, which is a way to relate vectors at different points on a manifold. This connection is then used to define a covariant derivative operator, which acts on a vector or tensor field to give the rate of change along a given direction. The superscript sign indicates that the derivative is taken with respect to a specific coordinate basis.

What is the difference between covariant derivative with respect to superscript sign and subscript sign?

The main difference between covariant derivative with respect to superscript sign and subscript sign is the direction of differentiation. A superscript sign indicates differentiation along a given coordinate basis, while a subscript sign indicates differentiation along a given direction. Additionally, the covariant derivative with respect to superscript sign takes into account the curvature of the space, while the covariant derivative with respect to subscript sign does not.

How is a covariant derivative with respect to superscript sign used in physics?

In physics, a covariant derivative with respect to superscript sign is used in many different areas, including general relativity, electromagnetism, and quantum field theory. It is used to describe the behavior of fields in curved spacetime, the interactions of particles with electromagnetic fields, and the behavior of quantum fields in curved spacetime. It is a fundamental tool in modern physics and is essential for understanding the behavior of physical systems in curved spaces.

Are there any applications of covariant derivative with respect to superscript sign outside of physics?

Yes, there are several applications of covariant derivative with respect to superscript sign outside of physics. It is used in computer vision and image processing to analyze and differentiate image features in curved spaces. It is also used in machine learning and data analysis to handle data that is non-linear or has a curved structure. Additionally, it has applications in engineering and calculus, where it is used to solve problems involving curved surfaces and manifolds.

Similar threads

  • Special and General Relativity
Replies
16
Views
2K
Replies
5
Views
1K
  • Special and General Relativity
Replies
4
Views
3K
  • Special and General Relativity
Replies
10
Views
2K
  • Special and General Relativity
Replies
16
Views
2K
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
7
Views
2K
  • Special and General Relativity
Replies
20
Views
5K
  • Advanced Physics Homework Help
Replies
5
Views
2K
Replies
2
Views
3K
Back
Top