There is a giant black hole in the center of our Milky Way galaxy, and like any concentration of mass it exerts gravitational force on the rest of the Galaxy. However, the galactic center is very far away, approximately 30,000 light years,
so it has negligible effects on the solar system or the Earth. There are no special forces from the galactic plane or the galactic center. The only important force that acts on the Earth is the gravitation of the Sun and Moon.
As far as the influence of the galactic plane, there is nothing special about this location. The last time the Earth was in the galactic plane was several million years ago. Claims that we are about to cross the galactic plane are untrue.
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