# Do two bodies spinning about the same axis in the opposite direction repel each other?

I recently read about the GEM equations, which look very much like the Maxwell's equations.

Does this mean the behavior of mass is like the the behavior of the electric charge?

So for example if you spin a ring so you have ring "mass-current" do you get a "gravitomagnet"? If you get two of such spinning rings can you make them repel each other, like electromagnets do when the currect is opposite?

The Gravitoelectromagnetic equations are exactly the same as Maxwell's equations with $\epsilon_0$ replaced by $(-4\,\pi\,G)^{-1}$, so, to the extent that the GEM equations approximate the Einstein field equations, the behavior of mass is very much like that of electric charge. Here are the differences:
1. The minus sign in the "gravitoelectric constant" $(-4\,\pi\,G)^{-1}$ cannot be spirited away by a co-ordinate transformation and this has the physical meaning that, whereas like electrical charges repel, like masses gravitationally attract. So in your spinning wheel example, two wheels spinning in opposite directions about the same axis will have attractive gravitomagnetic force component: further to their static gravitational attraction.
2. The analogue of the Lorentz force has a factor of four in it: $\vec{F} = m\,(\vec{E}+4\,\vec{v}\times\vec{B})$;
3. The relativistic mass $m$ in GEM is not Lorentz invariant, whereas the electric charge is: the latter is a true scalar;