Is the Contravariant Derivative Defined by Contracting with the Metric?

[Pacopag]

Hi everyone;
I'm new to both PF and GR, so please bear with me if I'm not being very clear, or using standard syntax and such. Here is my question.

Given a vector v^a, the covariant derivative is defined as v^a _;b = v^a _,b + v^c GAMMA^a _bc.
(here I'm using ^ before upper indices and _ before lower indices).
The object I'm interested in now is v^a;b (where the whole thing a;b is upper). Is this called the contravariant derivative? Is there a similar definition in terms of the Christoffel symbol? or can we only obtain it from contracting v^a _;b with the metric? I can't seem to find the definition of this object in any books, and when I try to do my calculation via contraction with the metric, I'm getting the wrong answer.

Thanks.

 

[pervect]

You might want to look at the Latex capabilities of PF. Quote this post for examples.

The short answer is that you should already know how to or be able to find out how to raise and lower indices with tensors. Then

$$v^{a;b} = g^{bc} v^a{}_{;c}$$

 

[Pacopag]

Ok. Thanks pervect.

 

[haushofer]

The term "covariant" is used in the sense of how the components of a tensor transform. But covariance also means that the laws of physics can be written down tensorially, and so have a certain behaviour under coordinate transformations.

The covariant derivative is the demand that we use a derivative in our physical formulation that transforms tensorially. The partial derivative doesn't, so we need an extra term. This is the connection, but to specify it you have to demand certain properties on it. The same happens in the standard model: you impose a local symmetry, and demand that the derivative of your fields transforms exactly as the field itself under the symmetry transformation.