Covariant derivative of the Christoffel symbol

[redstone]

Homework Statement

Is the covariant derivative of a Christoffel symbol equal to zero? It seems like it would be since it is composed of nothing but metrics, and the covariant derivative of any metric is zero, right?

 

[nicksauce]

This seems like a meaningless question. The covariant derivative acts on tensors, but the Christoffel symbols are not tensors.

 

[redstone]

That's the first thing that came to my mind, but then the covariant derivative can still act on all the terms of the Christoffel symbol, since it is composed of tensors, so it seemed like it might still be a meaningful question. With that in mind, would you still consider it to be meaningless?

 

[nicksauce]

You mean because it is built out of metrics? But it is built out of partial derivatives of metrics, which makes the terms no longer tensors.

 

[paranormal]

Your intuition is correct. The covariant derivative of a Christoffel symbol is indeed equal to zero. This can be seen by considering the definition of the covariant derivative and the properties of the Christoffel symbol.

The covariant derivative of a tensor is defined as the derivative of the tensor along a given direction, while taking into account the effects of the metric tensor. In the case of the Christoffel symbol, it is a tensor that is constructed solely from the metric tensor. This means that the covariant derivative of the Christoffel symbol will only involve derivatives of the metric tensor.

Now, it is a well-known property of the metric tensor that its covariant derivative is equal to zero. This can be seen by considering the metric tensor in a local coordinate system, where it takes the form of a diagonal matrix with entries equal to the squares of the scale factors. In this form, it is clear that the metric tensor is independent of the coordinates, and therefore its covariant derivative is zero.

Since the Christoffel symbol is constructed solely from the metric tensor, its derivatives will also be zero. Therefore, the covariant derivative of the Christoffel symbol is indeed equal to zero.

In summary, the covariant derivative of the Christoffel symbol is equal to zero due to the properties of the metric tensor and the way the Christoffel symbol is constructed from it. This result is important in the study of differential geometry and general relativity, as it allows us to simplify calculations involving the covariant derivative of the metric tensor.