How to explain centripetal force in terms or relativity At the end of a video of dropping a ball and feathers in a vacuum,  Brian Cox explains that the Ball and Feathers, as understood in terms of General Relativity, aren't falling. (apologies I can only post one link)
There was lots of discussion on a Reddit thread and consensus seemed to be that objects, while travelling in straight lines, warp space time which cause them to move together over time. There was a nice practical demonstration in this video.
My question is how does spinning an object counteract the effect of gravity? If I am on the surface of an object[edit: sorry by object here I meant something like a planet!]  and (this is the tough bit) all frames of reference are equally valid, why would it make a difference if the object is spinning or not?
I'll confess to being very confused.
 A: The basic idea of general relativity is that a freely moving object follows a path through spacetime called a geodesic. By freely moving I mean the object experiences no force i.e. if you were that object you would be weightless just as if you were floating in space.
In flat spacetime geodesics are straight lines i.e. a freely moving object moves in a straight line at constant speed. This is just Newton's first law. However general relativity tells us that spacetime is curved due to mass, and in a curved spacetime geodesics are not straight lines. For example the astronauts floating around in the International Space Station are weightless because they are moving along a geodesic. However the mass of the Earth curves the geodesic so it goes in a circle round the Earth.
I'd guess (I haven't seen the video) that this is what Brian Greene is getting at. When we drop the ball and feathers we see them accelerate downwards. However if you were sitting on the ball or amidst the feathers you'd consider yourself to be weightless and not accelerating at all, just like the astronauts in the ISS do.
However if you whirl a stone on a string around your head, or drive a car round a roundabout, or any other form of rotational motion you are not following a geodesic. You know you're not following a geodesic because you can feel a force - the centrifugal force. This force is quite distinct from the gravitational force calculated using general relativity. It does not not have the same origin and it does not have the same effects. For example gravity causes time dilation, e.g. time runs slower near black holes, but the centripetal/centrifugal force does not cause time dilation.
