If we imagine two suns of equal mass, and a small object in their combined center of gravity, which is not moving, it will stay there forever.
If the object is displaced a little bit towards one of the suns, not moving, and then released, it will accelerate towards this sun.
I wonder what happens if the object is displaced in the same way, but already given a movement towards the other sun. If we observe the three bodys from a static position relative to the two suns, we first see the body move toward the sun more far away. We may expect the body to slow down, as it receives a stronger gravitational pull from the closer sun. However, if we think of gravity effects traveling with a finite speed, the body would receive less gravitational pull from the sun it left behind and more pull from the sun it moves towards. At certain speed, the effect may compensate the gravitation difference induced by distance difference. The body then would seem to accelerate towards the sun more far away from it, if viewed from a location perpendicular the center of the line between the two suns.
Can we expect to observe this effect?