How to find the mixed tensor, contravariant tensor and tensor trace of $F$?

Question

I have a question in particle physics that ask me to find the mixed tensor, contravariant tensor and tensor trace of $F$:

Our professor didn't teach us that much about the math of tensor, which makes it very difficult in doing this question.

I have calculated $A_\mu$, $A_\mu A^\mu$, $\partial_\mu A^\mu$, $\partial_\nu(\partial_\mu A^\mu)$, $F_{\mu\nu}$. However, I have trouble dealing with the mixed tensor:

$F_\mu{}^\nu = g^{\rho\nu} F_{\mu\rho}$, which we can get $g^{\rho\nu}$ by $g_{\mu\rho} g^{\rho\nu} = \delta_\mu{}^\nu$.

However, I am confused in how to find $g^{\rho\nu}$ by $g_{\mu\rho} g^{\rho\nu} = \delta_\mu{}^\nu$, since it introduced a new variable of $\rho$. How do we deal with $\rho$?

Is there any kind of rules in dealing with tensor manipulation?

Answer 1

The inverse metric tensor $g^{\mu\nu}$ (or with any other choice of index naming) is the inverse of $g_{\mu\nu}$, in the matrix sense. Since it's usually $4\times 4$ or larger, I would shug it into Mathematica or Wolfram Alpha to get $g^{\mu\nu}$.

If $g_{\mu\nu}$ is diagonal, then $g^{\mu\nu}$ is element-wise the reciprocal of each element of $g_{\mu\nu}$. Once you have it, you can find your "mixed" tensor via,

$$F^\nu_\mu = g^{\rho\nu}F_{\mu\rho} = \sum_{\rho = 0}^3 g^{\rho\nu} F_{\mu\rho}.$$

Answer 2

I read your question as : how to find $g^{\rho\nu}$ while I have the metric tensor $g_{\mu\nu}$

Actually if your metric tensor is $g_{\mu\nu}=diag(1 \ -1 \ -1 \ -1)$ (minkowskian flat spacetime) than it's easy to find contravariant of $g$ (indices doesn’t matter). In flat spacetime geometry, metric tensor is same all way around. So you can choose $g_{\mu\nu}=g_{\rho\nu}$. Another identity you have to remember that is whenever you convert covariant to contravariant, you just have to inverse them $(g_{\mu\nu})^{-1}=g^{\mu\nu}$.

Take a look at raising and lowering indices