Why is A_\mu a vector field on spacetime? In the standard treatment A is the (pullback to the base of the) connection 1-form of a connection on a principal U(1)-bundle on spacetime. Technically it's valued in the Lie algebra of U(1), but as that can be identified with i times the real numbers, we can ignore that here. Does the product A_\mu A_\nu happen to transform tensorially? Because as the parent pointed out, the transformation rule for A involves an extra term, so it's not obvious.
You're right – maybe I should have been more precise with my wording.
Let me address curt15's (and your) question in two ways:
####### The physicist's argument #######
In classic electrodynamics it doesn't make much sense to talk about a U(1) gauge symmetry (whether global or local) – because there's no electron field exhibiting such a symmetry in the first place. Gauge symmetries really only become important when talking about quantum mechanics, Dirac's equation and, more generally, coupling particle fields to force-carrying boson fields in a QFT. There it turns out that imposing local gauge invariance would guarantee you a way to accomplish the coupling while preserving the conservation of the Noether current associated with the global gauge symmetry.
But in classical electrodynamics, none of that is needed and the vector potential is simply a vector field on Minkowski space. In fact, the value of the vector field and the choice of gauge are not even important because there's no classic analogue of the Aharonov-Bohm effect.
Now pretty much the same applies to Einstein-Maxwell theory: There is no electron field and no U(1) gauge symmetry. It's all classical. Indeed, in my experience relativists just view the vector potential as a vector field on spacetime and ignore the QFT-inspired classical gauge theory stuff (principal bundles, associated vector bundles, etc.) altogether. You might disagree with this approach but given that no one has yet managed to write down any 4-dimensional, coupled quantum field theory in a mathematically coherent fashion (let alone on a curved background), I wouldn't say there's too much pressure to incorporate results from QED into GR.
Back to the paper: The authors basically start from the same perspective, i.e. the foundations of Special Relativity and classic electrodynamics where A was simply viewed as a vector field. They then propose an alternative to General Relativity and, in fact, electrodynamics(!) Like true physicists, they first worry about the functorial nature of their objects (what do they do & how do they relate to one another), not about the category the objects live in. If that means redefining[2] what A is in a mathematical sense, so be it. At the end of the day, experiments are what matters.
####### The mathematician's argument #######
Could it be that you and curt15 are thinking of the transformation behavior of A under local gauge transformations[0], not coordinate transformations? Because, unless I'm mistaken[1], the transformation of A under coordinate transformations is the same as for a vector fields, see definition 5.4.1 on p. 270 of [3]. The relevant portion reads:
Let p: P -> M be a principal G-bundle on the manifold M and A a connection 1-form on P, let s: U -> P be a local gauge of the principal bundle on an open subset U of M. Then we define the *local connection 1-form* (or *local gauge field*) A_s on U by
A_s := A ○ Ds = s* A
If we have a manifold chart on U and {∂_μ} (μ=1,…,n) are the local basis vector fields on U, we set
A_μ := A_s(∂_μ)
Now since the basis vector fields ∂_μ transform in the usual way and A_s is linear, the components A_μ should transform just like those of a section of the cotangent bundle T*M. Am I missing something?
I'm not too familiar with the physics side of things, so maybe I'm misunderstanding the notion of gauge symmetry, but even in the absence of matter I think there is a U(1) gauge symmetry present. The homogeneous Maxwell's equations are expressed purely in terms of the curvature F of the connection, if I remember correctly. So you might add a flat U(1)-connection to A without changing the equations of motion.
As for the math: the key point is that the metric is globally defined on the spacetime manifold M. I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U. They depend, in particular, on the choice of local gauge s:U \to P. So the question about covariance (or globalization or coordinate-independence or whatever you want to call it) of both sides of the equation g_{\mu\nu}=A_\mu A\nu very much involves the question of local gauge transformations.
Either the principal bundle is assumed to be trivial, in which case there exists a global gauge and the connection A can be identified with a global 1-form on M (connections form an affine space modeled on such 1-forms; a gauge effectively converts this affine space to a vector space by choosing an origin), or we need to check that the right-hand side A_\mu A_\nu is indeed a symmetric two-tensor on M. The latter is not clear to me.
I admit I haven't looked at the paper carefully, and physicists typically don't approach things in such an explicitly mathematical way. So perhaps there's some (physical?) justification for why that equation typechecks. I don't quite see it though.
> but even in the absence of matter I think there is a U(1) gauge symmetry present
I don't think there is. The local U(1) gauge symmetry really comes from the (complex-valued) Dirac field and its coupling to the photon field (i.e. A). In classic electrodynamics you can add the 4-gradient of any function to the 4-potential A without changing the equations of motion, so the space of valid gauge transformations is infinite-dimensional. (Which is not that interesting – given that you can't measure the potential A –, so all those degrees of freedom are non-physical.)
> I agree that the A_\mu transform as the coefficients of a differential form (A is a connection after all), but the notation elides the fact that the A_\mu are defined only on U.
That's a very good point indeed! Though I think the authors simply interpreted A as a regular vector field on the manifold, meaning that it is defined everywhere. But following your train of thought for a moment: Could you solve this issue through a partition-of-unity argument? I.e. cover the manifold with neighborhoods where you have local gauges and then construct the metric locally as a finite sum of those A_{\mu,s} (s being the gauge).
The issue of a metric constructed this way not necessarily being a Lorentz metric (let alone non-degenerate) of course nonwithstanding. Then again, the authors didn't worry too much about this, either… :)
