Analogies between electricity and gravity

This is a brief history of physical analogies that begins at the time of Coulomb. Then it was universally believed that gravity, electricity and magnetism all obeyed Newton's inverse square law. The current era—dominated by quantum mechanics, general relativity, strings and inflationary cosmology—is clearly more complex. A new analogy may be helpful. Evidence is given that, in the non-relativistic limit, electricity is described by a Proca potential ϕ e satisfying the equation ∇ 2 ϕ e − k 0 2 ϕ e = −ρ e / 0 and gravity by a Helmholtz potential ϕ m satisfying ∇ 2 ϕ m + k 0 2 ϕ m = 4π G ρ m . In both equations k 0 is a proposed universal constant, whose value is about 2 π(20/ R 0 ), where R 0 is the distance from the solar system to the centre of the Milky Way. Relativistic treatments are also given.