Is Mach's Principle the Key to Understanding Rotatory Motion?

[Alfredo Tifi]

TL;DR Summary

I am not sure to fully understand Einstein's reasoning in the second paragraph (reasons for the extension of the relativity to non-inertial motion) of his 1916 book on General Relativity.

I refer to the second paragraph of 1916's book, "Die Grundlage der allgemeinen Relativitätstheorie", translated here.

First issue
There are two distant stellar bodies, with unchanging shapes: S₁ (spherical) and S₂ (ellipsoidal), made of the same amount and kind of matter. Their centres of mass are still for an observer placed in between, who observes both the fluid bodies as rotating.
The conclusions of Einstein's, in searching for a physical cause for the different shape of S₂, are the following (fully agreeing with Mach's principle):
"The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine especially the forms of

and

must be of such a kind, that the mechanical behaviour of

and

must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration."

Here there could be a second possible explanation, IMHO: in spite of the relative character of inertial motion, rotation is absolute. So, S₂ is rotating in an absolute sense, while S₁ is not-rotating (in an absolute sense) because it is not an ellipsoid (if it were rotating, it would have become ellipsoidal). More, the observer sees S₁ as rotating because itself is rotating (in an absolute sense), as could perceive from some internal effect.

Second issue
In search of a suggestion to extend the principle of relativity, Einstein considers an inertial system K and uniformly accelerated system K'. His definition of an inertial system is quite satisfactory, and should be used in any manual of physics: "Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses moves uniformly in a line."
Now, inasmuch as all the distant probe masses would have a centre of mass, those CMs can be considered as in-built reference systems O, O', O", which are inertial (that move uniformly in a line). Einstein excludes that the probe mass is itself uniformly accelerated, and that means that acceleration is absolute and could be recognized by some motion effect of O. We have to exclude linear uniform acceleration because K' could appear as not being inertial only because Einstein's mass built-in with O is the only really linearly and uniformly accelerate. Moreover, to find a distant probe mass m' and reference system O' which would move "uniformly in a line" respect to any portion of K' is impossible whatever the acceleration of O'. Thus is not possible for circularly accelerated systems to find an inertial observer moving uniformly in a line. That means that circular motion is absolute. This is confirmed by the second part, in the words of Einstein:
"Let

be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to

any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction of acceleration is independent of its material composition and its physical conditions."
In other words, the motion of the probe mass will appear as accelerated if referred to any non-linearly accelerated system. That is enough to demonstrate that circular acceleration is absolute.

 

[Ibix]

I'd be wary of using the word "absolute", because it doesn't seem to have an unambiguous technical meaning. But it is certainly the case that whether or not an object is undergoing proper acceleration is a locally detectable invariant fact - i.e. a person sealed in a box can detect proper acceleration from within the box, and all observers will agree what that person's accelerometers will read.

 

[Dale]

I agree with @Ibix. When I hear someone else use the word “absolute” I tend to interpret that as meaning “invariant”, but I prefer to use the word “invariant” myself.

Sometimes people do not mean “absolute” = “invariant” but rather “absolute” = “with respect to an absolute/aether frame”

 

[Alfredo Tifi]

Thank you Ibix and Dale. Absolute circularly accelerated motion here is specular to relative uniform motion (in a qualitative Galileian sense of having or not having a velocity, that is just a matter of reference systems, not of invariance).
If the reference system O' associated with the probe mass m' is linearly accelerated with the same acceleration, the motion of the centre of mass of K' will appear as still. Nevertheless, both the mass m' and the cosmic body K' will perceive the physical effects of being accelerated "from within".
If the circular motion is "absolute" in this sense, if it has physical effects that can't be reproduced as caused by the point of view of a different point of view of a second observer, contrary to the accelerated rocket whose effect can be obtained with a locally uniform gravitational field (that can't be completely uniform in any cosmologic object), I don't understand the main topic of Einstein need to generalize laws of non-inertial motion to gravity. In particular, it is not true that "for we can "create" a gravitational field by a simple variation of the co-ordinate system". This is possible only locally for a planet. Gravitational effects are not linear for a whole planet or a stellar body, but centripetal. Whereas the effects of a rotatory uniformly accelerated motion are centrifugal, and they are increasing with distance from the centre, while the gravitational effects are decreasing.

 

[Dale]

Absolute circularly accelerated motion here is specular to relative uniform motion (in a qualitative Galileian sense of having or not having a velocity, that is just a matter of reference systems, not of invariance)

Then if I am understanding your comment rotation would not be absolute. It is invariant, but not absolute in this sense.

In particular, it is not true that "for we can "create" a gravitational field by a simple variation of the co-ordinate system"

Unfortunately, in GR this is a bit of a vague statement. There are several quantities which could be identified with the term “gravitational field”. You could be referring to the Riemann curvature tensor, the metric, or the Christoffel symbols.

Of those, the first two cannot be changed by changing your coordinate system , they are tensors. However the Christoffel symbols can be created by changing the coordinate system. So if you are talking about inducing gravity by changing coordinates then you are talking about the Christoffel symbols.

Fictitious forces, like the centrifugal force and the Coriolis force show up in the Christoffel symbols as does the gravitational force on the surface of the earth.

 

[Ibix]

This is possible only locally for a planet.

Yes. The formal statement turns out to be that you can make second derivatives of the metric vanish at any chosen point, but not everywhere in general.

 

[PeterDonis]

If the circular motion is "absolute" in this sense

It is invariant (as others, I prefer that word to "absolute") in the sense that the distribution of proper acceleration will be different in the two bodies S1 and S2. In S1 it will be spherically symmetric; in S2 it will not.

I think what Einstein was asking was why the distributions of proper accelerations should be different; and the Mach's Principle answer he gave was that the "background" spacetime geometry in their vicinity--the geometry that would be present if S1 and S2 were not there--is determined by the distribution of matter and energy in the rest of the universe. Cuifolini and Wheeler's Gravitation and Inertia is a book-length exposition of a similar viewpoint.