Working Rules for Lorentz Covariance

A theory is Lorentz covariant if it is invariant in form under a Lorentz transformation. The rule for forming a Lorentz invariant is to make the upper indices balance the lower indices. If an equation is Lorentz covariant, we must ensure that all un-repeated indices (upper and lower separately) balance on either side of the equation, and that all repeated indices appear once as upper and once as lower indices.

An equation such as $\partial_\mu F^{\mu\nu} = j^\nu_{\textrm{em}}$ is ``Lorentz covariant'', the word ``covariant'' used here means both sides of the equation transform in the same way (i.e. consistently) under a Lorentz transformation. This new meaning of ``covariant'' is actually much better captured by an alternative name for the same thing, which is ``form invariant''.