Thought experiment in Mach's principle - Can a void universe be considered with special relativity?

Question

Mach's principle is based on a thought experiment in which an astronaut floats in the middle of a space devoid of all matter and all landmarks. No star, no source of energy is present, whatever the distance considered. The question then arises as to whether the astronaut has a means of determining whether he is rotating on himself or not, despite the absence of a reference point.

If Mach's principle is false, i.e. if inertial forces exist even in the absence of any matter or energy, then the astronaut could know this, by feeling inertial forces, like the centrifugal force pushing his arms outward. This idea clashes with common sense, insofar as it is difficult to conceive of a movement, in this case a rotation, without any point of reference. This would imply the notion of a space and an absolute frame of reference, which is called into question by the principle of general relativity.

I would like to know if by making the universe void like this thought experiment, shouldn't we consider the space-time as Minkowski's space time, i.e treat the Universe as entirely euclidean from a metric point of view.

But in this case, if we add suddenly an observer which could spin on it own with arms outward, we add matter in the same time, so we should consider the general relativity and no more special relativity, is it right ?

I need to grasp the subtilities of this thought experiment.

Answer 1

If reference frame is not an inertial one, this can be detected "from inside" as laws of physics are not invariant ones in non-inertial reference frames.

In case system is rotating, it is experiencing bunch of pseudo-forces like centrifugal force or Coriolis force which in principle can be detected by the observer within frame, as a movement with "no apparent reason".

In this case, for example take a spring in the equilibrium state, tie both ends together with a rope, (so that spring couldn't expand), then attach considerable mass at one of it's ends and attach another end to an astronaut belt (better to astronaut COM, but this can be hardly done due to hazard reasons). Then suddenly cut the rope in the middle, if astronaut is rotating,- spring mass should be pointing outwards from the pivot point and spring should expand due to centrifugal force acting according to Hooke law $F_s = kx$. Greater angular velocity invokes greater centrifugal force and consequently - greater shift in spring.

Similarly the fact that planet rotates, can be drawn from the "insiders" by observing own Coliolis effect,- object trajectory shift from a line with no direct forces acting. As for example it is seen by wind direction change on Mars Hesperia Planum area (credit NASA) :

Answer 2

Any complex object (like an astronaut) consisting of many parts provides a way of detecting absolute rotation or acceleration by comparing the movements of the parts. For a rotating astronaut, the atoms on the axis of rotation are not moving, atoms a little ways away from the axis are moving slowly, and atoms further away are moving faster. The relative motions of all these atoms can (at least in principle) be detected.

Reducing the object introduced into the void to the simplest possible (a featureless point particle) would appear to make detecting motion impossible, at least in a classical framework. But the real world is quantum, not classical, and in quantum mechanics there is no such thing as a featureless point particle: there are only fields, and particles are excitations of fields. The typical pop-science view of this is that there are "virtual" particles popping into and out of existence all around the "real" particle. That's a misleading picture in many ways, but it does illustrate that the fields provide a reference that would allow us to detect acceleration even for a "point" particle. An accelerating particle would experience radiation from the Unruh effect which could in principle be detected. So Mach's principle seems to be false in our actual (quantum) universe.

Could there be a universe which is purely classical and not quantum, and in which these questions arise? Frankly we just don't know at this stage -- but if you start changing the laws of physics, all bets are off, and even special relativity need not hold any more.

Answer 3

It helps to consider the situation in four dimensional space, where the fourth dimension is not time but additional coordinate $x^{0}$. In such space astronaut moves with velocity $c$. The coordinate time $t$ is there an affine parameter. An infinitesimal length of the curve element is defined as $dl\equiv c~dt$.

If astronaut's trajectory is a straight line: $$x^{0}(t)=c~t,~~~x^{i}(t)=0,~~~~~~i=1,2,3$$ then he feels no acceleration. If he rotates his trajectory becomes a spiral about $x^{0}$ direction: $$x^{0}(t)=\sqrt{c^2-\omega^2 R^2}~t,~~~x^{1}(t)=R~\cos(\omega t),~~~ x^{2}(t)=R~\sin(\omega t),~~~x^{3}(t)=0. $$ In this case he feels radial acceleration $a_{r}=R~\omega^2$ and, correspondingly, the inertial (centrifugal) force which is proportional to the distance $R$ from axis of rotation.

The Mach’s principle is false. Inertial forces are consequence of non-inertial trajectories (not straight lines) in the four-dimensional spacetime. In General Relativity this should be understood locally.

An excellent and comprehensive discussion of Mach's principle in context of General Relativity can be found in Briane Greene's book “The Fabric of Cosmos” in Chapter 2 and 3. By the way, this book should be on everyone's shelf who is interested in physics. For a quick glimpse into it see:

https://rcsstewa.com/wp-content/uploads/2020/12/The-Fabric-of-the-Cosmos-Space-Time-and-the-Texture-of-Reality-by-Brian-Greene-z-lib.org_.pdf

Answer 4

I believe that an empty spacetime would not be flat because a positive cosmological constant is a property of the vacuum. The rate of cosmic expansion is accelerating because the expansive force of the cosmological constant increases in proportion to the amount of vacuum in the universe; whereas the attractive force of gravity does not grow: it would grow in proportion to the amount of mass -energy in the universe, which is constant. I believe the best classical model of the cosmological constant is that it is (up to a constant factor) equal to the curvature of the vacuum. The differential geometry of general relativity works fine if you use a space of constant curvature in place of a flat Minkowski space in the fiber bundle over spacetime (Petti R.J. 1977 GRG Journal, Vol 8, Issue 11, pp 887–903).