Help Covariant Derivative of Ricci Tensor the hard way.

In summary, the conversation is about trying to calculate the covariant derivative of the Ricci Tensor in the same way that Einstein did it. The first expression given is incorrect due to an error with the indices, and the second expression is almost correct but also has an issue with indices. The person offering help suggests using the second Bianchi identity, while the person trying to calculate it in the same way as Einstein points out that Einstein didn't have access to symbolic algebra programs. They suggest using the definition of the Ricci and scalar curvature tensors in terms of the metric instead.
  • #1
nobraner
13
0
I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with

[itex]\nabla_{μ}[/itex]R[itex]_{αβ}[/itex]=[itex]\frac{∂}{∂x^{μ}}[/itex]R[itex]_{αβ}[/itex]-2[itex]\Gamma^{α}_{μ\gamma}[/itex]R[itex]_{αβ}[/itex]

or


[itex]\nabla_{μ}[/itex]R[itex]_{αβ}[/itex]=[itex]\frac{∂}{∂x^{μ}}[/itex]R[itex]_{αβ}[/itex]-[itex]\Gamma^{α}_{μ\gamma}[/itex]R[itex]_{αβ}[/itex]-[itex]\Gamma^{β}_{μ\gamma}[/itex]R[itex]_{αβ}[/itex]

Any help will be much appreciated.
 
Physics news on Phys.org
  • #2
Your second expression is almost right, except you have some trouble with indices. If alpha and beta are free indices on the left, you should not use them as summation indices on the right. Use gamma as your summation index, and then alpha and beta should remain free indices.
 
  • #3
Then, how did Einstein get

[itex]\nabla_{μ}[/itex]R[itex]_{αβ}[/itex]=[itex]\frac{1}{2}[/itex]g[itex]_{αβ}[/itex][itex]\nabla_{μ}[/itex]R
 
  • #4
Try the second Bianchi identity, contract it twice and you get the result.
 
  • #5
Thanks for the advice, but anyone can do it that way. I'm trying to do it the way Einstein did it; the hard way. Einstein didn't know about the Bianchi Identities.
 
  • #6
You can put the definition of the Ricci and scalar curvature tensor in term of metric...
 
  • #7
Einstein didn't have symbolic algebra programs, I would guess. But that's what I'd use.
 

FAQ: Help Covariant Derivative of Ricci Tensor the hard way.

What is a covariant derivative?

A covariant derivative is a mathematical operation that measures how a vector field changes as it moves along a given path. It takes into account both the change in the vector itself and the change in the coordinate system along the path. It is commonly used in the study of differential geometry and tensor calculus.

What is the Ricci tensor?

The Ricci tensor is a mathematical object used in differential geometry to describe the curvature of a space. It is a 2nd-order symmetric tensor that is derived from the Riemann curvature tensor. It is used to describe the curvature of a space in a more compact and useful form.

Why is calculating the covariant derivative of the Ricci tensor difficult?

Calculating the covariant derivative of the Ricci tensor can be difficult because it involves a lot of complicated mathematical operations and notation. It also requires a thorough understanding of differential geometry and tensor calculus. Additionally, the process can be time-consuming and prone to errors if not done carefully.

Why is it important to calculate the covariant derivative of the Ricci tensor?

The covariant derivative of the Ricci tensor is important because it allows us to study the curvature of a space in a more detailed and precise manner. It is also used in the study of general relativity and other fields of physics to understand the behavior of matter and energy in curved spacetime.

Is there an easier way to calculate the covariant derivative of the Ricci tensor?

Yes, there are other methods or shortcuts that can be used to calculate the covariant derivative of the Ricci tensor. For example, using index notation and symmetry properties of the Ricci tensor can simplify the calculations. Additionally, computer programs and software can also be used to perform the calculations quickly and accurately.

Similar threads

Replies
4
Views
2K
Replies
10
Views
2K
Replies
10
Views
1K
Replies
11
Views
2K
Replies
7
Views
2K
Replies
11
Views
2K
Back
Top