How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Question

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.

Answer 1

The repeated indices are Einstein summation. So that means

$$F^{\mu \nu}F_{\mu \nu} = \sum_{\mu=0}^3 \sum_{\nu=0}^3 F^{\mu \nu}F_{\mu \nu}$$

$$=F^{00}F_{00}+F^{01}F_{01}+F^{10}F_{10}+F^{11}F_{11}+F^{02}F_{02} + ...$$

There are 16 terms in total, one for every combination of $\mu, \nu =0,1,2,3$.

Answer 2

The upper indices are related with the lower ones through the metric. If you know for instance $F^{\mu\nu}$, you can specify automatically $F_{a\beta}$ as $$F_{a\beta}=\eta_{\alpha\mu}\eta_{\beta\nu}F^{\mu\nu}$$ and vice versa, given that the space time is flat, which I assume it is...

The summation between repeated indices is implied.

Answer 3

The upper indices are contravariant and the lower are covariant. Covariant components change as the basis and the contravariant inversely to basis. If in the equation there are repeated contravariant and covariance component, a sum over those components is assumed. There is a good book called "a student guide to vectors and tensors" by Daniel Fleisch. The subject is well explained.

Answer 4

To start, there exists two types of indices, dummy indices and free indices. In your expression we have $$F^{\mu\nu} F_{\mu\nu}.$$ Notice that our indices are $\mu,\nu=0,1,2,3$. Becuase of the fact that we have $\mu,\nu$ both appearing in the contravariant and corvariant index slots we say that these are being summed over and that they are so called dummy indices. A dummy index can be renamed to any greek letter because they are being summed over. This notion is called the Einstein Summation Convention. The Einstein Summation Convention says that for any index that appears exactly once in the covariant index and once in the contravariant index, there exists an implied summation over the range of that index. So we just have this way of hiding the summation sign, that is $$F^{\mu\nu}F_{\mu\nu}=\sum_{\mu=0}^3\sum_{\nu=0}^3F^{\mu\nu}F_{\mu\nu}=F^{00}F_{00}+F^{01}F_{01}+F^{02}F_{02}+...$$ Notice that the expression itself stays the same but the summation symbol vanishes, that is the Einstein Summation Convention. Again the indices could have been called anything because they are just summation/dummy indices. If you had an arbitrary tensor expression $$T_{\rho\alpha}S^{\alpha\sigma}$$ then we would only sum over the $\alpha$ index because its the dummy index and then that leaves $\rho,\sigma$ as our free indices. See this link on Einstein Summation Convention.