Earth rotates on its axis and revolves around the sun, the sun revolves around the galaxy, the galaxy is also moving. So Earth's net rotation as observed from a fixed inertial frame consists of all these contributions (and is rather complex).
Now a Foucault pendulum on earth is supposed to tell the experimenter whether the earth is rotating or not. See a recent question on this forum Proof that the Earth rotates? Basically for a Foucault's pendulum, the plane of the oscillation of the bob rotates, as the Earth rotates. But, the Foucault's pendulum does not single out one type of rotation, it gets affected by all. Therefore the rotation of plane of oscillation can be used to measure rotation of Earth around its own axis, around the sun, everything.
But the question is, around what is the Foucault pendulum measuring Earth's rotation with respect to? That is, finally what is the Earth rotating around/revolving around? Isn't it intriguing that the effect of the entire Universe on Earth can be measured by a pendulum?
I understand that the effect on the pendulum due to say the revolution of solar system around the galaxy is going to be small, but still, it can be measured say in an experiment going on for 10 years. You can keep on increasing the accuracy by measuring for longer periods.
Best Answer
Your question will eventually lead you to Mach's Principle. It is an old, yet unsolved question, that still remains at the stage of "philosophical idea".
I understand that your question is equivalent to "What would be found if we could measure all effects on the pendulum with infinite accuracy?", what if even the tiniest contributions could be registered? (Please read the note at the end as well, regarding the effect on any pendulum of the proximity of mass, whether that pendulum is in a free-fall orbit or not. The effect of earth's orbital motion is not zero because it affects the speed rate of proper time)
Yes, some components of the acceleration on the pendulum allow to deduce that the pendulum belongs to a rotating frame. That leads to think that the pendulum and the whole Universe may eventually be found to be rotating around some point, but that idea makes no sense (what is that point then, if everything is rotating? Rotation relative to what?). Then Mach's principle comes to the rescue, telling us that inertia effects on your pendulum arise somehow from the influence of all the other objects of the Universe, from here to the most distant ones. But there is no mathematical model for such thing, not even in General Relativity.
The pendulum is blindly affected by the local conditions of space and time, which constantly change in time and from one point to another (although all effect other than those arising from the rotating frame on top of the bulk mass of the Earth are extremely tiny). Those conditions are determined by the arrangement of energy/mass and momentum around. In the newtonian model, by the mass distribution. This is useful because you can idealize a portion of the Universe in a model that allows you to predict some behaviour of the system: for instance the Schwarzschild metrics allow to accurately synchronize the clocks of the GPS satellites in their motion around the Earth, and to accurately model orbits close to the Sun. The homogeneous and isotropic Universe model allows to derive properties of the expansion in the past, etc. But there is no model for an accurate description of how the whole universe is affecting your pendulum.
In other words, the essential origin of inertia is still unknown. What is a Foucault pendulum eventually rotating around? There is no answer to that question. Moreover, it is not yet clear whether the question makes sense or not.
The most close answer to your question may be found in our motion relative to the Background radiation, found by means of the dipole anisotropy of the CBR. This is the closest thing that there is, to an "absolute reference frame" but it makes sense only for us. Other distant observers in out expanding Universe will have a completely different perception.
EDIT:
As correctly stated by Ben Crowell, the orbital motion is a free fall, and therefore its dynamical effects on the pendulum are different from those of being on top of the rotating Earth. However, that free fall happens along places with different values of the gravitational potential (bigger in January, for instance) and therefore the speed rate of pendulums is affected. Thus, your pendulum, as any other clock-alike device, is affected by all the other masses in the Universe.
You might think about placing several synchronized pendulums at different distant points on the surface of the Earth and, by measuring (with infinite accuracy) their speed rate differences, map some properties of the gravitational potential in which you are embedded, deducing for example the direction of a center of mass. This makes an interesting question if you want to start another post.
As for Mach's principle, let me stress that it is merely a philosophical idea, that may or may not some day lead to a real theory. It is neither correct nor incorrect.
There is often a fallacy motivated by the Equivalence Principle, in which people ignore the different speed rate of proper time inside the free-falling elevator. Yes, the man inside the free-falling elevator is unable to distinguish if he is in a gravitational field (but in free fall), or if he is floating in interstellar space, far away from any mass. But in the second case, the man inside the elevator is ageing faster that the one that is in free fall (orbit) around the Sun. This is another kind of twins paradox that is often forgotten.