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if the universe is constantly expanding then the distance between black holes and neutron stars must be increasing. Then
- What makes them to come under the influence of each other's gravitational field and start rotating, or are they rotating around each other from birth?
- Why is the rotary motion spiraling one instead of just rotating around each other in a stable fashion like planets around sun?
The expansion is on a scale larger than galaxies, so within a galaxy, two black holes or neutron stars might pass close to each other from time to time as they move under the influence of the galaxy's gravity. If a third object gets involved (eg a star) then they could end up being captured into orbit . Much more likely, however, is both components of a binary star (which are common) evolve inbto black holes or neutron stars so they are orbiting one another "from birth".
This is a difference between general relativity and Newtonian gravity. In GR no orbit is ultimately stable -- energy is slowly lost as gravitational waves and the objects spiral in. If the gravitational fields are not very large, however, this effect is much too small to detect, and overwhelmed by other effects, so it seems like planets around the sun are orbiting stably. Only when you have objects with a very strong gravitational field in a very close orbit is it noticeable. Even then we're only detecting a few such events per year within a billion light years of Earth, so it's not common.