What is the solution to Newton's bucket problem?
Answers
The general consensus is that Newton's laws are defined such that they hold true for the inertial reference frame, which could be described as a constant displacement in euclidean space. This seems to work for the frame of the fixed stars relative to earth. (by experiment) Absolute space is a mathematical constraint invented to allow Newton's laws to work properly. It's almost circular reasoning if it weren't for the accuracy of the predictions of his laws of motion in most any situation if taken from this 'frame'.
I would say that it's entirely possible to hold Newton's law's of motion as good within a rotating frame if you introduce new force fields. For example the centrifugal and coriolis forces. We can't on the other hand have something like an elliptical frame, unless velocity is defined as sum of displacements from two points. Incidentally, there is one other frame for which we can keep the laws of motion. an object accelerating at a constant rate will be able to assume it's frame if it simply introduces a new force which acts constantly from some arbitrary direction. One frame might explain forces more generally than another, but no rest frame is safe from question. Even the one Newton described breaks down for objects travelling at a (high) velocity due to special relativity.