Let us analyze this.
Generalized principle of Mach. Masses in the universe determine the inertial frame in some way, not necessarily like Ernst Mach described it as through the average motion of the masses in the universe.
Newtonian mechanics
Newtonian mechanics satisfies Generalized Mach's principle: an inertial frame is one where noninteracting masses move with a constant velocity vector v.
M ● ---> v
o
/|\ observer
/\
v <--- ● M
But newtonian mechanics does not satisfy the original principle of Ernst Mach. We may well have a two masses moving to opposite directions as in the diagram, and the observer does not feel any centripetal "force".
In this blog (June 20, 2024) we have shown that the Einstein field equations do not have any solution for the two-body problem. In that sense, it is not defined if general relativity satisfies Mach's principle, or any principle.
A "linearized" version of general relativity should give approximately the same prediction as newtonian gravity. Such a model of gravity does not satisfy the original principle of Mach.
The syrup model of gravity: neutron star forces objects on the surface to move with the star – a "linear" principle of Mach
In general relativity, close to a very heavy neutron star, the coordinate speed of light is very slow in the standard Schwarzschild coordinates. It is like syrup: forces everything to move very slowly relative to the neutron star.
photon
• --> v'
● ---> v
M
very heavy
neutron star
A faraway observer sees that even a photon must move approximately along the the movement of the neutron star. Does this mean that some kind of a principle of Mach is true?
Yes, at least in some sense.
The observable universe is quite close to being a black hole or a "very heavy neutron star"
The amount of visible matter and dark matter in the observable universe is something like 10% of the amount required to place us inside the event horizon of a Schwarzschild black hole.
What does this imply about the inertial frame for rotation?
Could it happen that we could see the entire observable universe to rotate around an inertial frame?
Since the "current" radius of the observable universe is ~ 30 billion light years, any rotation of the universe would have a very small angular velocity. Otherwise, faraway galaxies would move faster than light.
Mach's principle fails for electromagnetism
Let us assume that a spherical shell is electrically charged. The electric field inside the shell is zero. If the shell is rotated, there is no electric or magnetic field inside. Mach's principle inside the shell fails for electromagnetism.
Let us then analyze orbits of test charges outside the rotating shell.
Mach's principle for a rotating, heavy shell of matter
Albert Einstein was excited about the possibility that inside a rotating, heavy shell of matter, the "inertial frame" would rotate along the shell.
Does this make sense?
If the shell is not heavy, then we empirically know that the inertial frame does not rotate appreciably. Could it be that the inertial frame rotates a little bit?
A similar question is what is the orbit of a test particle close to a very heavy rotating neutron star.
Conclusions
Do faraway masses determine what is the non-rotating inertial frame? One form of Mach's principle claims that faraway masses determine that frame.
In newtonian mechanics, the motion of faraway masses has no bearing on inertial frames. Mach's principle is not true.
If general relativity is formulated in a way which matches newtonian mechanics for a few small masses in an otherwise empty Minkowki space, then Mach's principle is not true in general relativity.
For a large moving or rotating mass, the syrup model of general relativity says that all objects are forced to move along. But does this mean that the "inertial frame" moves with the large mass? It is not clear. It is hard to define what Mach's principle precisely means in these cases.
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