The precise correspondence between inertial reference frames and what used to be called “the fixed stars” cannot be a coincidence, and yet Mach’s Principal has struggled to find a place in modern cosmology. This essay discusses Mach’s Principle and develops a version of it that does not involve action at a distance.
A Brief History
Reference frames are needed to describe motion and dynamics properly and the concepts associated with them have a long history in physics. See for example (Barbour and Pfister, 1995).
In the first half of 17th century Sir Isaac Newton described an experiment involving a bucket of water suspended by a long rope. The surface of the water is flat. The bucket is set spinning and the water eventually spins with it. Now the surface of the water is higher at the edges and depressed in the middle. But the water is more or less at rest with respect to the bucket. So it is not the motion of the water with respect to the bucket that matters, but rather the motion with respect to something else. Newton assumed that there must be some sort of absolute reference frame.
Three centuries later Ernst Mach disagreed that there was an absolute reference frame as such. Mach argued that a body only has motion relative to other bodies. He also noted that inertial affects only arose if the body was accelerating or rotating with respect to the “fixed stars” in the heavens above. His line of reasoning led to speculation that inertia itself was a purely relative phenomenon and that the stars “up there” had something to do with the manifestation of inertia “down here”.
Quote from Robert Dicke (R. Dicke, 1959): “It is interesting that only two ideas concerning the nature of space have dominated our thinking since the time of Descartes. According to one of these pictures, space is an absolute physical structure with properties of its own. This picture can be traced from Descartes’ vortices through the absolute space of Newton, to the ether theories of the 19th century.
The contrary view – that the geometrical and inertial properties of space are meaningless for an empty space, that the physical properties of space have their origin in the matter contained therein, and that the only meaningful motion of a particle is motion relative to other matter in the universe has never found its complete expression in a physical theory. This picture is also old and can be traced from the writings of Bishop Berkeley to those of Ernst Mach.
These ideas have found a limited expression in general relativity, but it must be admitted that, although in general relativity spatial geometries are affected by mass distributions, the geometry is not uniquely specified by the distribution. It has not yet been possible to specify boundary conditions on the field equations of general relativity which would bring the theory into accord with Mach’s principle.
The Foucault Pendulum
Foucault pendulums are popular displays in science museums and universities. In essence, the pendulum is a weight at the end of a long wire attached to a point above in such as way that the weight can swing in any direction. If such a pendulum is set in motion, it does not just go back and forth. The plane of such motion can be seen to slowly turn around, i.e. it precesses.
Such pendulums are named after the French physicist Léon Foucault (1819-1868). The first public exhibition of a Foucault pendulum took place in February 1851 in the Paris Observatory. A few weeks later, Foucault made his most famous pendulum when he suspended a 28-kg brass-coated lead bob from the dome of the Panthéon in Paris using a 67m long wire and set it swinging. The plane of the pendulum’s swing rotated clockwise approximately 11.3° per hour, making a full circle in approximately 31.8 hours.
Imagine such a pendulum suspended in a sheltered enclosure at either the North Pole or the South Pole. It will be observed that the plane in which it swings will slowly turn around through 360 degrees every 23 hours 56 minutes and 4.09 seconds. If the observers go outside and look up then (weather permitting) they can notice that the plane of the pendulum is remaining exactly aligned to the stars above. While clearly moving under the influence of gravity from the Earth, the pendulum is choosing not to revolve along with the Earth but instead to remain fixed relative to the heavens above. The correspondence is so exact it cannot be a coincidence.
The period of the 360 degree rotation relative to the icy ground is not exactly 24 hours because the Earth is also in orbit around the Sun. On places other than the poles the pendulum tries to maintain alignment with the stars but its point of suspension is being carried around due to the daily rotation of the Earth.
There can be little doubt that Ernst Mach (1838-1916) was familiar with Foucault pendulums.
A Thought Experiment – The Ball and Ring
Thought experiments are a powerful, cheap and convenient tool. They can take us wherever our minds can wander. They can enlarge our thinking and discipline it at the same time. And they are easy to discuss.
Imagine a massive ball surrounded by a rubber ring in an otherwise empty universe. Now set the ring revolving with respect to the ball. Which is spinning – the ring or the ball? There are several possible answers:
- The ring
- The ball
- Both of them
- We can’t say unless we surround the system with some other matter or field
- We can’t say unless we put the system into the context of an external universe.
The answer is not obvious. If the ball is spinning then it might display some inertial effects. For example, it might become oblate, or develop a bulge around its equator. Or the water in a bucket placed at one of its poles might develop a concave surface. If such effects are present then that could be taken as evidence that the ball was spinning. Conversely, if the ring displays some stretching then that would be evidence that it is the ring that is spinning.
Can we assume that if the ring and the ball are not displaying any inertial effects then this is proof that they are not spinning? No we cannot. We have no evidence or proof that inertial effects would be present or not in an otherwise empty universe.
Principle of Consistency
One of the good things about our Universe is that Nature is not random or capricious. Physical laws do not change without reason and physical laws seem to hold just as well “here” as “over there”. And if they are true one day we can expect them to be true the next (unlike economics for instance). And if two systems are exactly identical then (apart from quantum fluctuations) they tend to remain identical. This suggests a principle that will be referred to herein as the Principle of Consistency. It can be stated as:
- Identical physical systems remain identical unless there is a reason for at least one them to change; and
- The laws of physics affect identical systems in identical circumstances in identical ways.
In an otherwise empty universe, a system with the ring spinning around the ball is identical to a system with the ball spinning inside the ring. Under the Principle of Consistency there should be no differences.
Outcome 3 has superficial merit. It has a bet each way on whether it is the ring or the ball that is spinning.
Suppose that the ring and the ball both show some stretching distortions. This would incline an observer to deduce that both are spinning. But now imagine the ring becoming smaller and smaller until it disappears. Does this leave the ball still displaying some signs of spinning? No, because that could lead to violations of the Principle of Consistency – a ball in an otherwise empty universe could show signs of spinning or not for no apparent reason.
Now let us introduce a third element. A blob of matter at a large distance away from the ball and ring. Position an observer on the ring. If the blob is observed to travel round and round then the observer would conclude that the ring itself is spinning. The alternative is to assume that the blob is travelling around and around in some enormous orbit for no apparent reason and at an incredible speed.
If anyone tries to argue that it could be the blob that is moving and not the ring, thus creating an illusion of rotation for our observer, just think of this. Imagine a second ball and ring system just above the first pair, and imagine that the second ring is rotating in the opposite direction to the first. Now the two observers can be absolutely sure it their rings that are spinning. A far away blob cannot create opposite illusions of rotation at the same time.
Okay, delete the second ball and ring and put observers on both the ring and ball that we started with. And keep the same far away blob.
If observers on the ball observe that the blob seems to be going round and round they can safely conclude that it is their ball that is spinning. If observers on the ring observe that the blob seems to be going round and round they can safely conclude that their ring is spinning. The observers can then hold a meeting and it is safe to assume they can decide who is spinning and who isn’t, and if both are spinning then who is spinning the fastest and in what direction.
It is strange, is it not, that a small blob of matter a long way away can be so critical to resolving the situation? And if you do not think it is strange, then conceptually reduce the size of the blob by umpteen orders of magnitude and move it a million times further away. Keep doing this until you do find it strange.
Outcome 5 is much more familiar. It is our own Universe. We can expect observers to be able to agree who is spinning and who isn’t. They can do so by observing not just a small blob of matter but an infinity of stars and galaxies overhead. And if that is not good enough they can get out a bucket of water. Or close their eyes and check for dizziness.
But to prefer outcome 5 seems to admit that the Universe is somehow a key factor for determining which objects are spinning and which are not, and that the Universe has something to do with the manifestations of behaviors that we call inertial effects.
This idea was given a name by Albert Einstein. He called it Mach’s Principle.
But this raises a new issue. How can the Universe ‘out there’ affect the ring and ball ‘down here’? It looks like action at a distance. But so does gravity. Perhaps the two are related? After all the gravitational mass of a piece of matter is exactly the same as its inertial mass.
Einstein went on to develop a mathematical description of gravity using curved (i.e. non-Euclidian) four dimensional spacetime and a set of ten differential field equations. Einstein never claimed that he had replaced gravity by curved spacetime. Nor did he disclaim such a view. Einstein simple said that curved spacetime was useful. However, many others came to think of curved spacetime as being more than a reference frame and a set of equation. They came to think of it as a physical entity in its own right. It is usually taught as such today, often using the misleading analogy of a distorted rubber sheet.
But wait, astute readers will notice that not all of the Options have yet been considered. Option 4 has been overlooked. What if the ball and ring are immersed in some sort of invisible field and it is this field that determines which bodies are at rest, which are moving in a constant straight line, which are accelerating in a constant straight line (boosting or braking), which are curving, spinning, rotating or orbiting in a steady way and which are curving, spinning, rotating or orbiting in a changing way.
Mach’s Principle and Einstein
Ernst Mach argued that somehow the entire Universe was responsible for giving mass the property we call its inertia, and for giving rise to inertial effects whenever masses spin around or move in curves. (By entire Universe he meant all the stars in our Milky Way galaxy because it was another forty years or so until it was discovered our galaxy is only one galaxy amongst countless others).
But how can distant matter affect what is going on close by? The only thing that seems to work over such large distances is gravity. Mach wondered if perhaps there was a connection. Perhaps gravitational mass and inertial mass had some sort of common origin, and this has something to do with Universe at large.
Einstein was happy to credit Mach as being an influence on his own thinking. Einstein came up with the idea that some inertial effects looked exactly like gravitational effects on a small local scale and incorporated this into his theory of General Relativity. General Relativity became used for models of the entre cosmos and Einstein always wanted to incorporate Mach’s Principle into such models somehow e.g. as a constraint on the multiple solutions afforded by the complexity of the equations. But by the time Einstein died in 1955 he held the opinion that neither he nor anyone else had satisfactorily incorporated Mach’s Principle into General Relativistic models of the Universe.
Modern cosmologists are divided on the question of whether General Relativity offers a satisfactory explanation for inertia in accordance with Mach’s Principle.
Mach’s Principle Reimagined
Over the last century some things have changed a lot in physics and some haven’t. Quantum mechanics and atomic physics have progressed mightily at the small scale. Observational astronomy has progressed wonderfully well at the large scale. But attempts to bring General Relativity and quantum mechanics together have struggled. And now new astronomical evidence from the James Webb Space Telescope is bringing into question critical aspects of the standard model developed over many decades by hundred of theorists, cosmologists and astrophysicist to ‘explain’ how the Universe came to be what we can currently observe. Even the basic dynamics of the motion of stars in spiral galaxies is a big problem, and over 90% of the Universe seems to be missing in action.
It may pay to go back and check the evidence, assumptions, ideas and interpretations of evidence that have led up to the current standard paradigm. What might have been missed? What might have been misinterpreted? What might bet wrong? What issues did we ignore and neglect? What alternatives did we ignore?
There are valid questions to consider in relation to Mach’s Principle. Now that we know the Milky Way is one galaxy amongst countless others, what is meant by the “fixed stars”? Would a Foucault pendulum at the South Pole align to the Milky Way, or to the visible Universe, or the cosmic microwave background, or to something else?
The attempt to find a connection with gravity seems problematic. Why should the plane of a Foucault pendulum’s swing maintain an orientation to stars whose gravitational pull is almost infinitesimally weak compared to the gravity field of the Earth? And how might a connection work anyway?
The idea of action-at-a-distance seems to be outdated. The short and long range nuclear forces are described by an exchange of ephemeral smaller entities. General Relativity models gravity by distorting the nearby spacetime. The idea that the Universe as a whole determines local inertial effects through some sort of action at a distance is left with no friends at all.
However, just because there is a close correspondence between alignment with the heavens and the absence of so-called fictitious forces such as centrifugal forces and coriolis forces it doesn’t mean that there has to be a direct connection. Perhaps the Universe is aligned to something and an absence of the fictitious forces is also related to that same thing.
Suppose the whole Universe is embedded in some sort of universal background field. Galaxies could relate to such a field millions of light years away and a Foucault pendulum could relate to it right here on Earth. There would be no action at a distance. All physics would be local.
And suppose such a field raises no resistance to matter moving at a constant rate in a straight line but does resists changes of speed, changes of spin and changes of direction unless some external force is applied. It would give matter its inertial properties and also underlie Newton’s laws of motion.
And suppose such surrounding field affects or distorts whatever is meant by distances and time in ways described by Einstein’s field equations. It would literally become “the fabric of spacetime”. It would also become the medium in which light travels.
It would be a very useful field indeed! It is a pity then that the existence of such a field has been ruled out by experimental evidence. Or has it?
This is a question that will be examined in later essays (Van de Vusse, 2024).
References
J. Barbour, Julian; and H. Pfister, (eds.) 1995, Mach’s principle: from Newton’s bucket to quantum gravity. Boston: Birkhäuser. pp. 188–207. ISBN 3-7643-3823-7. (Einstein studies, vol. 6)
Robert Dicke, 1959, Review of Modern Physics, 29. p363 ff
Van de Vusse, Sjoerd B.A., 2024, Some ideas and experiments for issues affecting modern physics, https://hereticalphysics.com.au
Author contact: SBAvan@utas.edu.au
Author’s location: Hobart, Australia
