Toy Universes

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will briefly point out a sixth interpretation of MP of our list of 21, namely;

Mach Principle 6: “As induction of inertial forces by accelerated masses by analogy with electromagnetic induction.”

There is a formal analogy between the equations of electrodynamics, described by Maxwell’s equations, and the weak-field limit and slow motion approximation of Einstein’s General Relativity equations (GR). Such description is referred to as “gravitomagnetism“.

One can show that the angular momentum of a stationary mass-energy current, in the above-mentioned limit of GR, plays a similar role as the magnetic dipole moment of an equally stationary charge current. Therefore, similarly to electromagnetic induction, the orbital plane of a test body is dragged along the sense of rotation of a massive central body, the so-called “frame-dragging” phenomenon.

So the question is: how far such a result, intrinsically manifested in the equations of GR in the weak-field, slow motion limit, could be extended to a stronger principle? That is essentially MP #6.

The interested reader is referred to Ciufolini’s review article on Nature (subscription required), and his article in [Barb95] , page 386. A freely available paper of interest is also his arxiv review.

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will briefly point out a fifth interpretation of MP of our list of 21, namely;

Mach Principle 5: “As the generation of inertial forces in any body accelerated with respect to distant masses”.

You can tell when a body is accelerated by the emergence of the so-called inertial forces acting on the body, which do not exist if the body is at rest or moving at constant velocity(*). The statement concerns the question of whether such inertial forces arise from the motion of the body with respect with distant bodies.

It is well-known that Newton’s bucket experiment shows that inertial forces inducing the concavity of the surface of water in the rotating bucket arise independently of whether the water is rotating with respect to the bucket. In other words, it indicates that inertial forces seem not to be related to motion relative to other bodies.

Mach, however, pointed out that we do not know whether the concavity of the surface of the water would arise if in the experiment the bucket’s walls were increased many orders in width! (A brilliant observation, by the way). In other words, that experiment may rule out the effect of nearby masses, but not necessarily the integrated effect of distant masses in the Universe. This is the point of MP # 5.

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(*) …with respect to what… ? – you would certainly claim to add some reference against which you establish that a body is at rest or moving in constant velocity. According to Newton, the reference is, of course, the Absolute Space. In a Machian view, velocity can only make sense if measured relatively to other bodies, so one could say that the average motion of all bodies in the Universe makes such a reference frame. Today, one would use the Cosmic Microwave Background Radiation (CMBR) “rest frame”, that is, if a body does not move through the average CMBR, then it is “at rest”, and accordingly for a constant velocity movement. In any case, General Covariance extends the notion of inertial frames of reference, so that what is important is that the physical laws have the same mathematical form under arbitrary coordinate transformations. More on that later.

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will discuss a fourth interpretation of MP of our list of 21, namely;

Mach Principle 4: “As a mechanical interaction of masses, in particular through as interactive Machian ‘kinetic energy’”.

That particular interpretation is not only interesting by itself, but also for its apparent historical origin. The above-mentioned ‘Machian kinetic energy’ is explained in the article by John D. Norton on pages 32-33 of [Barb95], from which my brief exposition is here based.

The idea emerges as a Machian hypothesis by its own essentially because it arises from a quote by Einstein in a 1913 paper, where he discussed “the hypothesis of the relativity of inertia” and cites an “ingenious pamphlet of the Viennese mathematician W. Hofmann”.

Essentially, Hofmann (1904) argued that the standard kinetic energy of a body of mass () was unsatisfactory, due to an intrinsic asymmetry: the kinetic energy of with respect to would not be the same as that of with respect to . This consideration would be a necessary outcome of the assumption of the relativity of inertia.

Therefore, in order to have a symmetric law for the kinetic energy, Hofmann proposed that

,

where is a constant and an undetermined function. The idea was: since the mass measured of a given particle should arise as a contribution of all masses in the Universe according to that law, then by an integration of all masses, that formula would have to converge to the usual formula for the kinetic energy.

Further work along that line was carried out by H. Reissner in 1914-15, but he cited Mach for the modified kinetic energy law, and Hofmann was not mentioned.

(Well, I was unsuccessful to find any further online references for W. Hofmann, either a bio or link to his “pamphlet”…)

For historical details, curiosities and references, the reader is referred to Norton’s article.

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will discuss a third interpretation of MP of our list of 21, namely;

Mach Principle 3: “As determination of inertial frames from relative motions of masses”.

A very clear and elegant (albeit extremly general) definition of an inertial reference frame can be found in [Lan76], page 5, namely: an inertial frame is that in “which space is homogeneous and isotropic and time is homogeneous”. That means that all inertial frames move in constant (non-accelerating), rectilinear motion with respect to each other.  This also implies that, in particular, “in such a frame a free body which is at rest at some instant remains always at rest” [Lan76].

That is, however, a definition. One can accept that definition as it is and he/she can use it to operationally describe physical phenomena. However, one could also ask whether there is an underlying reason for such inertial frames to exist to begin with, that is, whether there is a physical explanation for the fact that there are frames in nature in which the laws of mechanics take the simplest form.

Mach’s Principle, despite its many formulations, generally indicates a global origin for such frames, that is, that they are somehow determined by a physical property of the Universe at large. MP #3, in particular, refers to the possibility that the inertial frames arise from the relative motions of masses.

In the context of general relativity, this principle should be translated into the question of whether local inertial frames are determined completely by the energy-momentum tensor of matter. Yet, there remains several questions, as for instance whether the gravitational degrees of freedom should or should not contribute to that formulation. It also seems fundamental to include a constraint condition, that is, the need to impose a Cauchy surface while specifying the energy-momentum tensor field. [See discussions in Barb95 pages 92-96].

In any case, it remains the question of why nature would have “chosen” such constructions. In other words, are there natural explanations for the origin of the reference frames? Or are they just what they are, an operational definition that trivially translates into a simpler description of laws?

In fact, it comes from experience that an inertial frame, as required, e.g., from Newton’s law of inertia, can be defined as “coinciding” with the “fixed stars” (in modern terms, with the reference frame in which the cosmic microwave background radiation appears globally homogeneous and isotropic), within accuracy requirements of a given experiment. The question is: is such a coincidence fortuitous? MP#3 statement says “no”, in the sense that the orientation and movement of a inertial reference frame can only be made meaningful from a complete determination of all masses, namely, their individual values, relative separations, etc.

One could also argue that such a need comes from more philosophically grounded ideas, as those which aims at avoiding altogether the concept of absolute space, to which inertial frames are referred to in Newtonian mechanics, therefore being dependent only on physically measurable entities.

In any case, either in the context of 19th century physics (in which Mach constructed his ideas), or in the modern developments of the 20th century (evidently, from Einstein’s general relativity theory), it is still a matter of controversy what to make of MP #3.

We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will discuss a second interpretation of MP of our list of 21, namely;

Mach Principle 2: As a mere redescription of Newtonian Mechanics.

In fact, it is clear that the main criticism of E. Mach in his book “The Science of Mechanics” is aimed at Newton’s notions of absolute space and time. Specifically, these notions are constitutive parts of Newton’s mechanics, but Mach regards them as superfluous metaphysical ingredients, as for him motion should be described only relatively to other bodies, and not to an absolute space.

Also, absolute time would be seen as unnecessary in the description of physical laws, as one could always substitute it for some other phenomenon able to trace a movement, as for example, the earth’s angle of rotation.

Mach often emphasizes that there is no point for science to use a speculative element which is beyond our experience for describing a physical law. Thus, for instance, Mach wrote:

“When… we say that a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe.”

However, it is not clear from his writings whether he was arguing for a mere redescription of Newtonian Mechanics — be that with without change of its physical content (see Barb95, pages 15-19; 215-218). An example emerge in the excerpt:

“But what would become of the law of inertia if the whole of the heavens began to move and the stars swarmed in confusion? How would we apply it then? How would it have to be expressed then?”

That passage clearly is a simple example that raises the need to reformulate Newton’s law of inertia under a particular extreme situation, namely, a chaotic distribution and movement of the stars. Would they affect the law of inertia? Is a new law necessary? There is still no clear answer to whether Mach had a final position to that question.

Mach’s famous bucket experiment example raises a concrete issue of whether we can eliminate the notion of absolute space, e.g. by redescribing that experiment in terms of a law that does not include a reference to the ‘absolute space’. However, even those who interpret that Mach was in fact arguing for a new law of inertia, not a mere redescription, have not reached a final consensus on what form such a law would take under a final, more precise formulation of Mach’s Principle. Interestingly, as Barbour points out (Barb95, page 218):

“My view is that (…) all the disagreement has arisen because, ironically, Einstein himself never really sorted out the matter.”

I had to give a long break in our tensor course. More on that sometime. Now I have to get back a little to what I had promised in one of my first posts, concerning Mach’s Principle.

So here it is, hopefully, the first of a series of notes about the various (at least, 21) interpretations of Mach’s Principle. These notes will certainly be incomplete, fragmentary, and the exposition will not be as deep as the subject deserves. In fact, I warn the reader that this will be in fact a very short set of statements.

As it is a blog post, the idea is just to collect and state the interpretations. The subject is controversial and many physicists still ponder about the issue. Many do not agree that Einstein’s theory of relativity really solved/implemented Mach’s Principle (completely or at a degree). So here it goes the first one and what it is supposed to address:

Mach Principle 1: As the problem of defining velocity (motion).

This problem can be rephrased as: “If all motion is relative and everything in the universe is in motion, how can one ever set up a determinate theory of motion?” (Barb95, p.7).

Note, first of all, that Einstein’s special relativity theory treats that problem in the context of the class of inertial reference frames (namely, any frame that moves at a constant velocity relative any other frame belongs to this class). It is an operational way to develop the theory, yet it is elegant and powerfull. We will not discuss for the moment if it actually solves the fundamental problem stated in MP 1.

A remark related to Mach Principle 1, by Berkeley, is that our imagination cannot conceive the motion of two globes around a common center in empty space. But if the sky of the fixed (important!) stars is created, a mental representation of the motion can be made (Barb95, p.8) (I find this particularly remarkable).

A related question is whether the Machian Principle 1 presupposes is its statement a finite, in contrast to an infinite universe. Such an establishment seems to be an important issue: do Machian boundary conditions in the latter case arises naturally or must be imposed arbitrarily?

There is in fact the question whether there is, even in principle, any relation between a general Machian supposition (regarding the structure of spacetime needed to define motion — which must, on the other hand, arise itself from these motions) and the closure of space (Barb95, p. 88).

Finally, there is the idea that different types of relative configurations (R) give different notions of an instant, therefore different frameworks for defining motion consistently. For instance, in Newtonian mechanics, R is defined by mass points is Euclidean space. In a Machian field theory, R is defined by field intensities. Machian geometrodynamics can be constructed with R as Riemannian 3-geometries (Barb95, p. 224).

There is a lot to ponder about this first interpretation. If possible, I may explore more in upcoming posts. For the moment, the essential elements to begin an analysis are, I believe, listed in this post. Recall that this is just one interpretation out of at least 21 possible ones.

We start by stating a few facts about Mach’s Principle, without going into details. These will be explored later, in future posts.

Mach’s Principle as an initial guide to General Relativity

The initial inputs that guided Einstein to conceive his General Theory of Relativity (GR) were, basically:

• Mach’s Principle;
• The Principle of Equivalence;
• The Principle of General Covariance.

Although the last two principles of the above list are relatively well understood, the first one is not.

The blurred Mach’s Principle

Mach’s Principle is on people’s mind as basically a general statement on the origin of inertia. This is often understood by many as a problem that has been solved by Einstein’s GR, although Einstein himself disregarded Mach’s Principle later on.

From time to time, you may find obscure papers by obscure people on Mach’s Principle, but also authoritative works written by well-known experts of various areas of physics. This curious phenomenon serves as a testimony to the enthusiasm and reverence to a subtle and deep matter, even if stated so imprecisely.

But what exactly is Mach’s Principle?

The problem is the term “exactly”.  There is no consensus on what Ernst Mach really meant by the principle which carries his name. In fact, the term “Mach’s Principle” was coined by Einstein. But, there is no clear statement of the principle in Mach’s book, “The Science of Mechanics” [Mach10], as cited by Einstein as one of the books which impressed him mostly while constructing his ideas towards GR.

If you want to know all there is to know about Mach’s Principle, you should start by reading a collection of contributions to the Tübingen meeting (1993) entirely dedicated to Mach’s Principle, edited by Barbour and Pfister [Barb95]. This collection is extremely interesting and includes transcriptions of the discussion sections as well.

The various interpretations of Mach’s Principle

Here I merely list the various interpretations identified on that meeting. Hopefully, we will be able to go into each of them (or some of them) in more detail in future posts.

1. As the problem of defining velocity (motion).
2. As mere redescription of Newtonian Mechanics.
3. As determination of inertial frames from relative motions of masses.
4. As mechanical interaction of masses, in particular through an interactive Machian ‘kinetic energy’.
5. As generation of inertial forces in any body accelerated with respect to distant masses.
6. As induction of inertial forces by accelerated masses by analogy with electromagnetic induction.
7. As requirement that the metric tensor be completely determined by matter.
8. The same, but by matter and gravitation (geometrical) degrees of freedom.
9. As prediction of the future given relational initial data (initial-value approach).
10. As cosmic derivation of inertial mass.
11. As generalization of the special principle of relativity to nonuniform motions.
12. As requirement of general covariance of the laws of nature.
13. As need for generally covariant boundary conditions.
14. As need for nonexistence of boundary conditions.
15. There should be no matter-free, singularity-free solutions of GR.
16. As an appeal to the principle of sufficient reason (observable facts must have observable causes).
17. As expectation there will be dragging effects.
18. As explanation of nonrotation of compass of inertia relative to distant masses.
19. As proposition that the universe at large influences local physics (Dicke-type approach).
20. As selection principle of solutions of GR that are intuitively ‘Machian’.
21. As requirement that dynamics should not contain absolute elements.