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Maxwell's electromagnetic field tensor, $F^{ab}$, is completely antisymmetric rank-2 covariant tensor. The Lorentz transformation can be written as $\Lambda^a_b$, a rank-2 mixed tensor, a.k.a. a transformation matrix. It is not sufficient, however, to use matrix multiplication of the Lorentz matrix with an adjusted Maxwell tensor $F^{ab}g_{bc} = F^a_c$ to find the components of the boosted Maxwell Tensor.

Why is this? In Euclidean space, shouldn't co- and contravariance be interchangeable? Is it because of the nature of the Lorentz matrix, or because of the nature of the Maxwell tensor?

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The misconception is that everything has to transform by a single matrix multiplication.

The field tensor is a rank two tensor, so it takes two (co)vectors as arguments. The components of the tensor are found by passing in basis vectors, e.g. $$F^{\mu\nu} = F(e^\mu, e^\nu).$$ Under a Lorentz transformation, both of these input vectors change, so the components change with two factors of the transformation matrix $\Lambda$. In particular, if the coefficients $F^{\mu\nu}$ are put into a matrix $\bar{F}$, the transformation is $$\bar{F} \to \Lambda^T \bar{F} \Lambda.$$ This has nothing to do with the 'nature of the Lorentz matrix' or 'nature of the Maxwell tensor', this is just the generic transformation law for rank 2 tensors.

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