Setting up a field theory (contra/covariant)

[Pengwuino]

I'm trying to go through the Reissner-Nordstrom solution to the EFE's and since I'm trying to do this correctly, I find myself running into trouble about how to define everything.

I set my coordinates up as $$x^a = x^a(r,\theta,\phi,ct)$$

Now, I need to use the fact that $$\nabla^b F_{ab} = 0$$ as I am looking for solutions in source-free space. However, the construction of the field strength tensor gets me. I'm attempting to follow Stephani's convention and he has his field strength tensor as

$$F^{ab} = \[ \left( {\begin{array}{*{20}c} 0 & {B_z } & { - B_y } & {E_x } \\ { - B_z } & 0 & {B_x } & {E_y } \\ {B_y } & { - B_x } & 0 & {E_z } \\ { - E_x } & { - E_y } & { - E_z } & 0 \\ \end{array}} \right) \]$$

Now, the question I have is how would one know that this is how I should setup my field strength tensor? More to the point, why is it $$F^{ab}$$ and not $$F_{ab}$$? My instinct tells me that you should simply define all of your tensors to start with either contravariantly or covariantly (ie. $$x^a, u^a, g^{ab}, F^{ab}, T^{ab}$$ etc.) and you work your geometry in with the covariant guys. Is this the right path to take?

Also, to what extent are you allowed to do things like $$A^{ab}B_{cabd} = A_{ab}B_c^{\;ab}_d$$? It seems like in general you shouldn't be able to do that because, say in B, your a,b indices could be things in partial derivatives and your metric would have to act on them. On the other hand, I see things like $$F^{ab}F_{ab}$$ (the Maxwell field tensors) and know that it should equal $$F_{ab}F^{ab}$$. Is it the fact that it's a scalar that makes them equal? Is it the anti-symmetry of $$F^{ab}$$?

 

[sgd37]

you can raise and lower indices using the metric so it doesn't really matter whether you define everything covariantly or contravariantly i.e.

with metric $$g_{ab}$$ and inverse metric $$g^{ab}$$

$$F^{ab} = g^{ac} g^{bd} F_{cd}$$

dummy indices you can lower and raise at your convenience in GR as long as they're in alternating positions

 

[grey_earl]

Your a,b indices cannot be partial derivatives, since then the object in question wouldn't be a tensor under general coordinate transformations. They must be covariant derivatives, and since the covariant derivative of the metric is zero, you can raise and lower them with the metric without problems. Since $$F^{ab}$$ is antisymmetric in a and b, the Christoffel symbols actually cancel and the covariant derivatives reduce to partial, but that is a special case.

 

[Pengwuino]

What about something like the Christoffel symbols? They're constructed using partials. So I would think $$\Gamma_a^{\;b}_{\;c}A^a$$ is not necessarily the same as $$\Gamma^{ab}_{\;\;c}A_a$$.

 

[sgd37]

Christoffel symbols aren't tensors so you can't raise and lower their indices with the metric