Is there anything different about the gravitation around a non-spinning black hole and a neutron star of the same mass? If we perform a simple experiment by orbiting identical spacecraft around a non-rotating black hole and a non-rotating neutron star of equal mass at a &quot;safe&quot; distance, and ignore differences other than gravitation (e.g. magnetic fields and charge) would there be any differences in their orbits predicted by general relativity?
&quot;Safe&quot; distance means far from the event horizon of the black hole and where tidal forces don't disrupt the spacecraft.
After writing this answer I am now wondering if it's 100% correct.
Birkhoff's theorem is very useful: in general relativity, if you are in vacuum and there is a spherically symmetric gravitational field, then it will be the Schwarzschild solution. This solution only depends on the mass, not on the size of the object. So the neutron star and the black hole will give rise to exactly the same orbits.