Help Covariant Derivative of Ricci Tensor the hard way.

[nobraner]

I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with

$\nabla_{μ}$R$_{αβ}$=$\frac{∂}{∂x^{μ}}$R$_{αβ}$-2$\Gamma^{α}_{μ\gamma}$R$_{αβ}$

or


$\nabla_{μ}$R$_{αβ}$=$\frac{∂}{∂x^{μ}}$R$_{αβ}$-$\Gamma^{α}_{μ\gamma}$R$_{αβ}$-$\Gamma^{β}_{μ\gamma}$R$_{αβ}$

Any assistance will be much appreciated.

 

[jvicens]

Hello,

I understand your frustration with trying to calculate the covariant derivative of the Ricci Tensor in the same way that Einstein did. It can be a challenging task, but I am here to offer some assistance.

Firstly, I would like to point out that both of the formulas you have presented are actually correct. The difference between them lies in the choice of notation and the specific way in which the covariant derivative is defined. So, depending on which notation and definition you are using, either one of these formulas could be considered the "correct" way to calculate the covariant derivative of the Ricci Tensor.

Now, I would like to offer some tips for calculating the covariant derivative of the Ricci Tensor. Firstly, it is important to remember that the covariant derivative is a tensor operation, meaning that it is invariant under coordinate transformations. So, when calculating the covariant derivative, it is helpful to choose a coordinate system that simplifies the calculations.

Secondly, it is important to remember the definition of the covariant derivative of a tensor. In the case of the Ricci Tensor, it is defined as:

\nabla_{\mu}R_{\alpha\beta} = \frac{\partial R_{\alpha\beta}}{\partial x^{\mu}} - \Gamma^{\lambda}_{\mu\alpha}R_{\lambda\beta} - \Gamma^{\lambda}_{\mu\beta}R_{\alpha\lambda}

This formula may look slightly different from the ones you presented, but it is essentially the same thing. The key is to remember the summation convention, where repeated indices are summed over. In this case, the index \lambda is summed over, so it appears twice in the formula.

Finally, it is important to be careful with index placement and keeping track of the order of operations. The covariant derivative is a complicated operation, and it is easy to make mistakes along the way. I would recommend double-checking your calculations and going through each step carefully.

I hope this helps in your calculations. Don't get discouraged, as it can take some practice to become comfortable with calculating the covariant derivative. Keep at it and don't hesitate to ask for assistance if needed. Good luck!