Let g be the spacetime metric, and A be the electromagnetic 4-potential.
1. Suppose you could write the metric as g_{μν} = A_μ A_ν, i.e. the symmetric product of A with itself.
2. Conclude that the Einstein-Hilbert action is just the electromagnetic action plus a correction term A_μ ∇^μ ∇_ν A^ν = g(A, grad(div A)).
3. Assume that J^μ = ∇^μ ∇_ν A^ν = grad(div A).
4. The correction term from step 2 then becomes the usual electromagnetic coupling A_μ J^μ. As a consequence, the Einstein-Hilbert action for g is just the usual full (non-vacuum) action of electrodynamics on a background curved by g = Sym(A⊗A).
5. Consider the vacuum Einstein equations and, thus, a Ricci-flat spacetime. Show that this is equivalent to ∇² A^μ = J^μ which are the inhomogeneous Maxwell equations in Lorentz gauge. The fact that we're in a vacuum spacetime but still considering electromagnetism seems odd but I guess their idea is that if electromagnetism is a purely geometric property of spacetime, then the electromagnetic action (including any potential electric current) shouldn't appear on the right-hand side of Einstein's equations in the first place – because the Einstein equations are Maxwell's equations.
6. Identify the 4-current J^μ with terms involving the electromagnetic field tensor and the metric's Weyl curvature. (Meaning, once again, that J^μ can be non-trivial even though we're considering a vacuum/Ricci-flat spacetime.)
7. Identify the remaining (homogeneous) Maxwell equations with the first Bianchi identity for the Riemann tensor.
8. Impose the continuity equation ∇_μ J^μ = 0, i.e. assume conservation of charge.
9. Conclude from 3) and 8) that div(A) fulfills a homogeneous wave equation.
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Comments and observations:
- In step 5, I don't see how Maxwell's equations (18) are supposed to follow from equation (17). But it's late, maybe I'm just being blind.
- As other comments have already pointed out, step 1 seems unreasonable because the metric will no longer be of Lorentzian type (with determinant -1) but instead will be positive-semi-definite. (To see this, diagonalize the metric at a given point => g = (A_μ)² (dx^μ)².) In particular, the metric might even be degenerate(!) It seems section 2.1 in their paper is supposed to address the signature issue but from my POV it's insufficient.
- The paper basically claims that gravity is just the theory of a vector 4-potential. That doesn't seem right, given that much effort was spent in the past 100 years to find such a theory. AFAIK it's pretty much ruled out these days.
- Given step 5 and the fact that the EM field no longer seems to contribute to the field equations, I have even more doubts this theory could ever turn out to be true. There are lots of solutions to the Einstein-Maxwell equations and I'm sure some of them have been confirmed experimentally by now. (I'm thinking of black hole jets etc.)
- For instance, IIRC there's a paper showing that from Einstein-Maxwell's equations it follows that photons move along null geodesics (which in the beginning of GR was merely an axiom of the theory). I wonder what would happen to this result. Hypothetically, photons might no longer move along geodesics in this new gravito-electromagnetic theory but the theory might still reproduce gravitational lensing. I don't think that's very likely, though.
- More generally, I think their theory is even difficult to reconcile with classic electrodynamics in the first place. In the absence of strong gravitational and quantum effects, we know that Maxwell's equations describe ED very well. However, the equations ∇² A^μ = J^μ above no longer are the classic (linear!¹) vacuum Maxwell equations we know – they are now highly non-linear since the covariant derivative ∇ now also involves the vector potential A. To reobtain classic ED in flat space one would basically need to ensure that in every-day situations A is "constant enough" not to produce any significant curvature through g = Sym(A⊗A) but still dynamic enough to reproduce the classic wavey nature of light. This doesn't seem likely. Plugging any known (experimentally proven) solution to the Maxwell equations into the equations here should invalidate the theory.
¹) in the absence of charges, i.e. J^μ = 0