I know that the Foucault pendulum rotation in relation to Earth is a proof that the object is inertial in relation to the distant stars. But what makes them more important than the Earth? Are they an absolute and universal inertial frame? How can we prove that? Please elaborate.
General Relativity – What Inertial Frame Explains the Foucault Pendulum?
classical-mechanicsgeneral-relativityinertial-framesuniverse
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Your argument is actually more or less right, but some of the details are wrong.
First you have to realize that Newtonian mechanics and general relativity have different definitions of an inertial frame. According to Newtonian mechanics, the coffee cup sitting on my desk right now defines a (very nearly) inertial frame, but a falling rock is extremely noninertial, because the rock has an acceleration of 9.8 m/s2. According to GR, free-fall is the preferred inertial state, so the rock is considered to define an inertial frame, but the coffee cup has a proper acceleration of 9.8 m/s2.
The Newtonian definition is actually impossible to define 100% rigorously, but traditionally the "fixed stars" have been taken as a pretty good standard for Newtonian frames. Any frame in which the stars have a very small acceleration is considered a very good inertial frame.
So if you have the Newtonian definition in mind, then your argument only goes wrong at the end, where you refer to a "very fast speed." What's relevant is the stars' acceleration, not their speed. If a rocket ship is gliding through our solar system at 1,000,000 m/s, then it's an inertial frame. It doesn't matter that the stars have a velocity of -1,000,000 m/s in its frame; what matters is that they have a=0.
According to the Newtonian definition, a frame of reference fixed to a point on the earth's surface is not an inertial frame. You can tell this because in that frame, the stars have large centripetal accelerations. However, the earth-fixed frame comes very close to being inertial, because you can find other frames that are inertial and that differ from it only by a very small acceleration. Therefore experiments on the earth's surface need to be pretty sensitive in order to detect any noninertial effects. The classic example of such an experiment is the Foucault pendulum.
In GR, a frame of reference fixed to a point on the earth's surface is not an inertial frame, and it doesn't even come close to being one. It differs from a valid (free-falling) inertial frame by a huge amount -- an acceleration of 9.8 m/s2. Even an extremely crude experiment can determine this. For instance, I can tell because I feel pressure from my chair on the seat of my pants. A secondary issue is that the earth's frame is rotating, and GR does consider rotating frames to be noninertial as well. (There was a lot of historical confusion on this point, including some early mistakes by Einstein, who thought GR would embody Mach's principle better than it actually did.)
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.
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Actually the path of the Foucault Pendulum is not "fixed" (even approximately!) to the "fixed" stars. Unless the pendulum is installed at one of the Earth's poles (as someone has done), then the point of suspension is in constant rotation with the Earth itself. $\therefore$ the pendulum is really not in an intertial frame.
A very good discussion of the forces (real and fictitious) on the pendulum can be found at this UNSW site. The vector that points from the suspension point toward the Earth is in constant acceleration and has a precession period that varies according to latitude.
This animation from the Wikipedia article on the Foucault pendulum may help show how the plane of the pendulum is rotating.