If you start with general relativity, and assume small perturbations around a nearly flat metric, it is possible to obtain linearized equations of gravity that look a lot like Maxwell's equations, also called GEM equations. One of them reads:
$$\nabla \times \mathbf{B}_\text{g} = 4 \left( -\frac{4 \pi G}{c^2} \mathbf{J}_\text{g} + \frac{1}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t} \right)$$
Where $\mathbf{E}_g$ is the (static) gravitational field, $\mathbf{B}_g$ is a gravitational analogon to the magnetic field that encodes some general-relativistic effects. You see that it is coupled to the mass current density $\mathbf{J}_g$, which has the effect that rotating or moving masses have a distinct (attractive or repulsive) gravitational effect. For example, two massive rings rotating in the same direction along a common axis will experience a small additional repulsion (or a slightly reduced net gravitational attraction). These effects are typically very small, but have been observed, e.g. by Gravity Probe B.
Now compare with the corresponding electromagnetic equation:
$$ \nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$$
The factor in front of the electric current $\mathbf{J}$ is the magnetic permeability $\mu_0 = 1/\epsilon_0 c^2$. In a medium, it becomes $\mu_r \mu_0$ instead. This relative permeability $\mu_r$ is material dependent and source of a lot of interesting phenomena and technical applications, including transformators.
You can simularly identify the gravitomagnetic permeability $\mu_0^g = 4\pi G/c^2$. (Don't know if I should put the minus sign in the constant or not.)
Now I'm wondering if there is a corresponding relative $\mu_r^g$. Are there materials that show a different GM permeability? Maybe event time-dependent, or manipulable through external fields?
From a microscopic persepective, it seems this would be somehow related to the spin or angular momentum of particles in the material. The effect is probably tiny, but if one would be able to vary it on very short time scales, it could maybe produce measurable effects.
If found a mention of this idea in a 1963 paper (there's a PDF on the net if you Google the title):
- Robert L. Forward: Guidelines to Antigravity - American Journal of Physics, March 1963, Volume 31, Issue 3, pp. 166
Unfortunately, more recent publications on this topic seem to be of dubious veracity (going in the direction of Podkletnov et al.).
