(Apologies for the long post. I can’t point off the top of my head to a textbook reference, but [1] may be enough for a start.)

This is not an unsolved problem. The key is to understand how inertial motion and inertial forces are explained in Newtonian mechanics vs. general relativity.

Let’s take two friends, Charlie and Isidor. Charlie is sipping coffee while reading a book sitting on a chair in his home, while Isidor is reading the same book while floating freely on the International Space Station. At first glance, it may seem that Charlie is experiencing inertial motion (with velocity 0, since he is at rest) and Isidor is in accelerated motion around the earth at a very high speed, since he is moving in a circular orbit. (Recall that in Newtonian mechanics, circular motion is always accelerated motion since the velocity vector varies with time, even though only in its direction.)

However, this is backwards. In fact, Charlie feels the constant force of the chair pushing on his bum and lower back, which is why he keeps changing his position when it gets uncomfortable, while Isidor is weightless, experiencing no forces acting on his body whatsoever. Also, absent any collision or explicit application of rocket thrust, the ISS and Isidor inside will maintain their current trajectory forever, while maintaining Charlie’s ‘at-rest’ position requires the constant application of force on his body, which is why when the chair eventually breaks (Charlie is a heavy dude), Charlie moves quickly towards the ground, spilling his coffee.

This observation is at the core of general relativity. Einstein remarked that, if we surround Charlie with a tent, he cannot possibly tell whether he is sitting on the beach or whether his chair is actually on the floor of a rocket accelerating constantly at 9.81 m/s^2 towards Alpha Centauri. At the same time, if the ISS had no windows, there is no experiment Isidor can conduct that would tell him whether the ship is orbiting the earth or is hurtling unaccelerated in a straight line in deep space. IOW, Charlie is in fact the one experiencing accelerated motion, while Isidor is in inertial motion.

So what, then, is inertial motion? In general relativity, a body is in inertial motion if its trajectory in spacetime follows a local spacetime geodesic. Earth’s gravity distorts space time around it to a shape in which the trajectory of the ISS at that altitude and velocity is actually a straight line according to the local metric imposed by the Earth’s gravitational field. Conversely, if we plot Charlie’s at-rest position on the beach through the same spacetime according to the same metric, it corresponds to a trajectory that deviates from the geodesic right up until the moment when the chair breaks, when for a very brief duration, his motion will actually follow the local geodesic.

What does this have to do with the water in the bucket? Once the water molecules rotate in sync with the bucket, their trajectories all deviate from the local geodesic lines. We can see that by imagining the bucket wall instantly disappearing; the water molecules would cease following the previously enforced circular trajectories, and they would start traveling in inertial motion away from the bucket, freely falling towards the ground (i.e., inertial motion in the Earth’s gravitational field). The sum of the forces required to maintain the water molecules on circular trajectories happen to lead to a curved water surface. The water molecules at rest in the non-rotating bucket are still in non-inertial motion similar to Charlie’s, but the forces acting on them to maintain those particular trajectories simply lead to a flat surface.

The point is that every time we want a body to deviate from its free, inertial trajectory in its local spacetime, we must apply a force. The water curvature is a direct result of all the forces which are necessary to steer the water molecules away from the local spacetime geodesics.

[1] https://physics.stackexchange.com/a/3988