The Earth's magnetic field varies with time over geological time scales (we have phenomena such as pole inversion). Can this affect the acceleration of gravity at Earth's surface? I would think that since the magnetic field has an energy density, such energy density affect the gravitational effects of the magnet which generates it (the liquid metal currents inside the Earth). However, even if this is correct, I would expect the effect to be extremely small. So, is there such an effect, and if yes, what's the order of magnitude? 1/10^6 $g$? 1/10^9 $g$? Etc.