Gravitational field has no curl? What about gas discs around stars, black holes, etc.? So everybody says the gravitational field has no curl, and is not comparable to a liquid swirling around a drain.  Observationally, of course, there are many examples of vector fields (which I think are gravitational fields) which look like they have some curl.  A pair of stars, for example, one being slowly devoured by the other.  The path of the swirling gas seems to trace out a field with a lot of curl in it.  Just as if you were to pour dye into water that is swirling around a drain.
How do you reconcile the observed vector field with the gravitational field which is not supposed to have curl?  Is there some way to work them into a single model?
 A: I think you are confused about vector fields and motion/trajectory.  A vector field is a function such that given any point in space there exists a vector assigned to that point.  For example, the gravitational field is a function of coordinates that tells you the strength and direction of gravitational force at any given point in space.  For a field to have no curl it means that you if you start anywhere in space you cannot follow a path given by the direction of vectors and arrive back at the same place you started.
This is completely different then the motion of the object under such a force.  For example, if the object initially has some velocity perpendicular to the force of gravity then it's trajectory will start to curve.  A trajectory however is not a field.
A: Well, the classical gravitation field has identically zero curl. On the other hand, frame dragging is a general relativistic effect in which angular momentum can affect the local shape of space. But as mcFreid says, that is not generally responsible for the rotational motion of gravitationally bound systems which is adequately explained by simple angular momentum.
A: Is important to remember that curvature does not imply curl. Here is an analogy from fluid mechanics which deals with lack of vorticity (curl) in curved flow. This may seem counter-intuitive at the start, but hopefully you can see the similarities between the velocity field in the flow and the gravitational field.
https://www.youtube.com/watch?list=PL0EC6527BE871ABA3&feature=player_detailpage&v=loCLkcYEWD4#t=268
A: The easiest way I try to think about the force of gravity having no cruel is by example of electric potential in Kirchhoff law, the circuit on a whole must equal zero.
