Does space itself fall into a black hole? Long time ago I heard someone say that it is space itself that falls into a black hole. Yesterday I saw a little animation that suggested the same (although I´m not sure, because the person who put the animation in the answer didn´t reply to my question). So I asked myself the question again. If that´s the case then this would also be true for the Earth. I´ve always learned that space is stationary around a massive object, and that the curvature of space-time causes objects to move to the mass. And not the picture that an object falls to a big piece of mass, because it´s riding along with the falling space. This suggests a completely different picture how gravity works. What´s playing here? 
 A: You sound as though you may have heard of Gullstrand Painlevé co-ordinates, which are a particular system of co-ordinates for labelling spacetime defined by the Schwarzschild metric around a nonspinning, noncharged black hole. 
The analogy is often made of a "spacetime river" with this depiction; if you stand still with respect to the co-ordinates you are in fact accelerating towards the hole, rather like being stationary in a strong rip or stream in the sea, watching the land go by as you are borne along by the current. To some degree the analogy is a pretty one as it evokes a fairly accurate feel for the motion of someone in freefall as seen by an observer at infinity or hovering just above the event horizon.
But it is important always to bear in mind that co-ordinates are simply the labels we give to the underlying spacetime manifold. So the physical reality is, rather obviously, exactly the same whether parameterized by GP co-ordinates or by Schwarzschild's originals. Spacetime is no more "flowing" in GP co-ordinates than it is for any other. And it's very hard to see a flowing analogy with other co-ordinates for the same, physical metric.
So in these lay pictures, you need to look at the underlying mathematics / physics and decide for yourself whether a particular analogy is helpful for your way of thinking. Flowing rivers do help many people to picture the situation, but always be clear that what you imagine is only an analogy and therefore cannot be relied upon to make all physical predictions.
By the way, these co-ordinates are almost certainly what Brian Cox has in mind when he describes passing through the event horizon by analogy with a river current.
