If the rubber sheet model of general relativity is accurate enough, then a weightless perfectly rigid object repels masses.
We assume that the rigid object was cast in outer space, under the Minkowski geometry. When we bring it close to a mass, where the geometry is Schwarzschild, the object refuses to obey the Schwarzschild geometry. There will be negative and positive pressure within the object. The pressure straightens up the geometry within the object.
It is like bringing a long straight steel bar embedded into a rubber sheet close to a metal ball whose weight has pushed the membrane down. It is obvious that the metal ball will roll farther when the steel bar comes closer. The deformation energy of the sheet is less when the ball is farther away.
In Newtonian gravity, this kind of a repulsion does not exist.
If we cast a perfectly rigid object under a Schwarzschild geometry, then it will refuse to adapt to the Minkowski geometry if moved away. Does it distort the geometry also in the neigborhood of the object or just inside the object? There is energy in the deformation of the rubber sheet. Does that energy cause gravitation?
The rubber model says that the object does change deformation also outside the object. It is like embedding a bent steel bar into the rubber sheet. If there are several bent steel bars, there will be various attractive and repulsive forces between parts of the bars.
We find that the rubber model predicts a rich spectrum of phenomena, while Newtonian gravity is always attractive.
In theory, we could make "anti-gravity" vehicles which would be able to float in the gravitational field of Earth.
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