Covariant derivate problem (christoffel symbols)

In summary, the conversation discusses the calculation of a specific equation involving the covariant derivative and its components. The conversation also references several sources for additional information and clarifies the notation used.
  • #1
Fisica
12
1

Homework Statement



I need to calculate [tex] \square A_\mu + R_{\mu \nu} A^\nu [/tex] if [tex] \square = \nabla_\alpha \nabla^\alpha [/tex], and is the covariant derivate

SEE THIS PDF arXiv:0807.2528v1 i want to get the equation (5) from (3)



Homework Equations



[tex] A^{i}_{{;}{\alpha}} = \frac{{\partial}{A^{i}}}{{\partial}{x^{\alpha}}} + \Gamma^{i}_{{j}{\alpha}} A^{j} [/tex]

Lool arXiv:hep-th/0504052v2 for the conections and christoffel symbols o ricci tensor components

The Attempt at a Solution



for example The 00 component for Ruv is

[tex] R_{\mu \nu} = -3(\dot{H}+H^2) [/tex]

see the links for more compoents
 
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  • #2
Do you see why only one equation remains?
How ##\square ## is expressed with low index nabla's only?
Do you know the expression for ##A_{{\alpha}{;}{\beta}}## ?

(I use latin indices for space components and greek indices for space-time components)

note: if you past an url it becomes clickable, for inline latex wrap it with ##.

http://arxiv.org/pdf/0807.2528v1

eq (8)
http://arxiv.org/pdf/hep-th/0504052v2
 
Last edited:

What is a covariant derivative?

A covariant derivative is a mathematical tool used in differential geometry to describe how a vector field changes as we move along a curved manifold. It takes into account the curvature of the space and ensures that the resulting vector is parallel transported along the manifold.

What are Christoffel symbols?

Christoffel symbols are a set of coefficients used in the covariant derivative to account for the curvature of the manifold. They represent the components of the connection, which describes how vectors change as we move along the manifold.

Why is the covariant derivative important?

The covariant derivative is important because it allows us to define a notion of parallel transport on curved manifolds. This is essential in many areas of physics, such as general relativity, where the curvature of space-time must be taken into account.

How do you calculate the covariant derivative?

The covariant derivative is calculated using the Christoffel symbols and the partial derivatives of the vector field. It involves taking the dot product of the vector field with the connection and then subtracting the terms involving the Christoffel symbols. The resulting vector is the covariant derivative of the original vector field.

What is the difference between a covariant derivative and a partial derivative?

The covariant derivative takes into account the curvature of the manifold, while the partial derivative does not. This means that the covariant derivative is a more general and powerful tool, especially when dealing with curved spaces. In flat spaces, the covariant derivative reduces to the partial derivative.

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