Dual of Newtonian gravitational field

Question

In the static state, the laws of Newtonian gravity and Coulomb force have exactly same formulas, $$F = K \frac{A_1A_2}{r^2}.$$ In the electrical case, moving materials produce a field, say a dual field, named magnetic field. Does in Newtonian gravitational case moving materials produce a dual field? Why?

Answer 1

No. In Newtonian theory, the only field present is the Newtonian potential. You don't have a dual field.

This is no longer true in general relativity. In relativity, the full gravitational field involves ten fields arranged in a $4\times4$ symmetric tensor. This is much more complicated than the situation with electromagnetism, and the Einstein equations (which rule the behavior of the gravitational field) are nonlinear, while the Maxwell equations are linear.

You can, however, linearize the Einstein equations by making the approximation that the gravitational field is sufficiently weak. Within this sort of approximation, you can get a gravitomagnetic field analogous to the magnetic field. In this approximation, gravity obeys a set of equations very similar to the Maxwell equations. The only difference is a sign in the Gauss law, which is due to the fact that gravity is attractive, and which implies that, in this approximate theory, energy is not conserved, proving it inconsistent.

From a deeper point of view, the difference between the two theories in this aspect (complementary fields to the Coulombic field) is due to the fact that electromagnetism is mediated by a spin $1$ particle (the photon), while the gravitational field is mediated by a spin $2$ particle (the yet unseem graviton).