# Definition of gauge freedom in electromagnetism and general relativity

The freedom we have in choosing the vector potential $$\vec{A}$$ in E&M is referred to as the gauge freedom, whereas in general relativity (GR), we refer to the freedom to choose any coordinate system as the gauge freedom.

We do have such freedom of choosing coordinate systems (related by a Lorentz boost) in E&M. Why then this freedom is not called the gauge freedom (taking inspiration from GR)? In fact, this freedom is not given any special name/mention. Or is it?

General relativity is special (pardon the pun!) in the sense that (a) the metric is the primary field of the theory, and (b) the coordinate freedom is inherently wrapped as part that field itself. Not just the coordinates used to label values of the field, but actually the field. In particular, when you do a 3+1 decomposition, the $$00$$ and $$0i$$ components are specified by a lapse function and a shift vector. You can show that those, in turn, are in 1-to-1 correspondence with your freedom to choose coordinates.
Since there is already another answer, this is just meant to provide a different perspective on the matter. For clarity, it might help to think about Kaluza-Klein theory, which brings the two kinds of gauge symmetry into the same picture, so that we can appreciate the difference. We start with an arbitrary five-dimensional metric $$G_{MN}$$, then postulate that one of the spatial dimensions is periodic, and the resultant circle has small enough radius that the world appears four dimensional. An observer in this apparently four-dimensional world (with metric $$g_{\mu\nu}=G_{MN}, M,N=0,1,2,3$$) will see the action of regular four dimensional gravity, and also U(1) electromagnetism. On the other hand, an observer who knows that the world is five dimensional will see no electromagnetism, just 5d general relativity. In other words, the 4d observer interprets the dynamics of the extra metric degrees of freedom ($$G_{4M}, M=0,1,2,3$$), in the language of a traditional gauge theory of electromagnetism. We now have a setting where we can ask the question, how do the 4d and the 5d observer see gauge freedom?