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We are going to prove that locally the inertial frames and gauge states ofthe electromagnetic field are equivalent. This proof will be valid forEinstein-Maxwell theories in four-dimensional Lorentzian spacetimes. Use willbe made of theorems proved in a previous manuscript. These theorems state thatlocally the group of electromagnetic gauge transformations is isomorphic to thelocal Lorentz transformations of a special set of tetrad vectors. The tetradthat locally and covariantly diagonalizes any non-null electromagneticstress-energy tensor. Two isomorphisms, one for each plane defined locally bytwo separate sets of two vectors each. In particular, we are going to use theplane defined by the timelike and one spacelike vector, plane or blade one.These results will be extended to any tetrad that results in a local Lorentztransformation of the special tetrad that locally and covariantly diagonalizesthe stress-energy tensor.

The equivalence between local inertial frames and electromagnetic gauge in Einstein-Maxwell theories