I will get right to the question; for readers unfamiliar with its genesis, I append a background section below.
I want to know how testable the prediction is that the water in a rotating bucket would experience a centrifugal pseudo force in the closest approximation to empty space we have access to (i.e., as far away from the Solar System as we could reasonably deploy a suitable test station). I envision the simplest possible experimental design, along the lines of the one described in the following post:
[https://physics.stackexchange.com/questions/416011/newtons-bucket-artificial-gravity-absolute-rotation-and-machs-principle][1]
A fancy satellite experiment has been done in near-Earth orbit (Gravity Probe B, launched in 2004 with data collection continuing until 2011). As I understand it, the results were consistent with predictions of standard GR: the so-called Lense–Thirring effect, which dominated behavior of the gyroscopes on the satellite, arose almost entirely from the effects of Earth's rotation on adjacent spacetime.
So, let's get as far away from local masses as possible and see how well standard GR's predictions hold up. Is this idea ridiculous?
Background to the question
In its simplest variant, a Newton's bucket experiment involves spinning a bucket of water in Earth's gravitational field and observing that the surface of the water is not flat due the centrifugal pseudo force. In this case, it is obvious that the bucket is spinning relative to the patch of Earth's surface underneath it.
Einstein attributed his development of general relativity (GR), in part, to the thought experiment of spinning the bucket in "empty space" where there is no obvious way to determine whether or not the bucket is actually spinning. Mach had earlier posed a version of this thought experiment; he invoked distant celestial objects (e.g., stars) as defining a reference frame relative to which the bucket's rotational motion could be judged.
There is a HUGE subsequent literature about whether or not distant celestial objects are relevant to behavior of the water in the bucket. As I understand it, distant celestial objects are not relevant in mainstream variants of GR.
The issue has been addressed dozens of time on the Physics Stack Exchange. In my survey of these previous posts, no one attempts to answer my empirically oriented question in their responses to them. All of them seem to pose the problem as one of prediction rather than observational testing of predictions.
Best Answer
It is clear that your underlying question is the validity of Mach's principle in the context of General Theory of Relativity (GR). It is known that Einstein was quite enamoured with Mach's principle and tried to formulate GR to satisfy it in a more general way.
However, it seems to be the case that GR does not satisfy Mach's principle. This is easy to see in the solution of the Kerr black hole, whereby the angular momentum of the black hole plays a rôle, even as the spacetime asymptotically approaches Minkowski spacetime far away. There is no distant stars in the solution of the ideal Kerr black hole. This means that the solution for the Newton's bucket problem in GR should have bulges to account for centripetal acceleration, even though it would not in any way look like the usual stuff we see on Earth.
A good textbook on GR would emphasise that the solution to a rotating system in GR is not obtained by taking Minkowski spacetime and setting it to rotate.