Mach’s Principle Number 6
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.
Mach’s Principle Number 5
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.
–
(*) …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.
Mach’s Principle Number 4
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 
Therefore, in order to have a symmetric law for the kinetic energy, Hofmann proposed that
where 
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.
Book Review: A First Course in Loop Quantum Gravity
Title: A First Course in Loop Quantum Gravity
Authors: R. Gambini and J. Pullin
Oxford University Press, 183 pages
If you are looking for an introduction to Loop Quantum Gravity that *really* guides you through every *minimum* required piece of knowledge at the basis of this field, including an outline of the mathematics involved, in order to put you in a comfortable position to enter the research literature, then there is (currently) no better place to start.
It has always been a mystery to me why Rovelli’s book, for instance, is generally considered as “introductory”. Although it is a very well written book by one of the experts in the field, and obligatory reading for those seriously interested in studying the matter, readers who had never – or only slightly – been exposed to Loop Quantum Gravity will find it quite advanced at various places. The same occurs with other similar books in this field, considered “introductory”: they may be so at places, but… only to the extent of your background, which seems an amusing way to put it, if not frustrating: “introductory-but-not-exactly-so”.
That does not happen with Gambini and Pullin’s book. It is indeed the only current *real* introductory, (not laymen, but mathematically-based!), self-study textbook on Loop Quantum Gravity. All other current books in the field – Rovelli’s, Kiefer’s, Thiemann’s, etc – are excellent references to be studied *after* Gambini and Pullin’s book. Finally, the basics of Loop Quantum Gravity is available in textbook form!
“A First Course in Loop Quantum Gravity” is in fact aimed at the advanced undergraduate level, and it does a great job to follow that difficult requirement. However, it is unavoidable that even an average reader at that level will have to accept some concepts without further detailed background. Gratefully, the authors are careful enough to state that explicitly, what is going on and why, and to offer a good, minimal list of references to follow at the end of each chapter, so that the reader can attempt studying the details. Various of these references are freely available in the internet as expository or review papers. There are also problems at the end of each chapter, suitable to self-study, as they are of the type: “prove”, “show”, “derive”, “compute”… I believe that some of these problems will be regarded somewhat difficult for an undergraduate student. They are however excellent complements to the main text and should be seriously attempted.
The book is very pleasant to read and very clear. It is also amazingly concise and objective. I really like the way the authors expose every subtlety behind the theory; the “tips” and analogies offered throughout the text are really excellent, and effectively helped me to finally understand a few concepts which were confusing to me until recently.
The reader should, however, be advised that the contents are intrinsically advanced. It is definitely not for the common reader, it is a technical book. It is not an easy field anyway, as Loop Quantum Gravity obviously requires deep knowledge of quantum mechanics and general relativity, as well as several advanced mathematical concepts. But be sure that the authors will teach you what are the minimal required elements of these pre-requisites, so you will at least be aware of them and be able to pursue them further by yourself.
I highly recommend this book for anyone with complete undergraduate courses in physics and mathematics. With my own main background in astrophysics, I can say this book is also a great introduction for astronomers/astrophysicists/cosmologists who never studied Loop Quantum Gravity. A good complement in that regard (applications) would be Bojowald’s books “Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity” and “Quantum Cosmology: A Fundamental Description of the Universe”, apart from the more advanced books already cited above (e.g., the obligatory and excellent books by Rovelli, Kiefer, Thiemann…). Finally, there is of course freely available review material in the arxiv site.
Congratulations to the authors for the great, concise, effective presentation of this challenging field to students and interested researchers coming from other fields.
Mach’s Principle Number 3
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.
Mach’s Principle Number 2
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.”
Mach’s Principle Number 1
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.
Book Review – The Direction of Time
Title: The Direction of Time
Author: Hans Reichenbach
Edited by Maria Reichenbach
270 pages
Published by Dover Publications, Inc. 1956, 1984.
Review by Christine C. Dantas, 2011.
“The Direction of Time” is an important contribution on the nature of time, based on an advanced draft left by H. Reichenbach after his death in 1953. The manuscript was published posthumously as a self-contained book, thanks to the care and attention of his wife, M. Reichenbach, who edited the manuscript for publication.
The book is divided into five chapters, namely: (1) introduction (basically, on the emotive significance of time); (2) the time order of mechanics; (3) the time direction of thermodynamics and microstatistics; (4) the time direction of macrostatistics; and (5) the time of quantum physics.
A final chapter for the book was planned by the author, summarizing a relation between subjective time (experienced by humans) and the objective time (as given by the formal analysis of physical time presented in the book). Unfortunately, the author died before being able to finish that final chapter. The editor was careful, however, to add an appendix outlining possible directions, based on a related paper by the author, published in 1953.
The book is basically a summary of the author’s lifetime contributions to the problem of time, from the point of view of philosophy and physics. In particular, the author is very attentive to clarify as much as possible any philosophical inquiries with logical argumentation and to never loose sight of which mathematical models, based on known physics, can actually bring consistency to the discussion (with the exception of the first chapter, which is purely philosophical throughout).
The book is very clearly written, and any obscurities (there are some, in fact) are mainly due to the difficulty of the subject matter itself, not of his discourse. It is not, for that matter, a popularization book. Some prerequisites are necessary; in particular, with a good understanding of statistical mechanics, the reader will be able to get several insights (starting from chapter 3), based on original ideas and clever constructions, without which he/she would otherwise not be able to fully appreciate the richness and deepness of the arguments involved.
When I opened the book for the first time, I thought that the first chapter, given its title, would be too much subjective for my taste, but on the contrary, it was a nice summary of philosophical schools and inquires on the nature of time. I only found one thing regrettable: his critique of Bergson’s “duration” and “intuition” concepts is based on a too literal interpretation — revealing in fact a (common) misunderstanding. The best here is to read Deleuze’s book “Bergsonism”, which shows that Bergson’s “intuition” concept is in fact a proposal of a method for knowledge, based on three distinct acts: 1 – the positioning and creation of problems (how to denounce false problems); 2 – the discovery of truthful differences of nature; and 3 – the apprehension of real time. Deleuze is usually confusing for me, but his reading of Bergson in particular is quite illuminating and serves here as an opposition to Reichenbach’s brief critique.
The second chapter was quite surprising as well, as I thought that I would learn nothing new about the concept of time from the point of view of classical mechanics. I was wrong. Making time order distinct from time direction was quite revealing. It is a very strong principle to always have in the back of one’s mind. Chapter 3 gradually constructs the hypothesis of branch structure based on the statistical isotropy of the universe as the origin of time direction. This is something to take serious note as well. I have previously seen arguments linking time and the increase of entropy, but Reichenbach’s shows how this notion is not as straightforward as basic statistical mechanics indicates, and further assumptions seem to be necessary. Finally, I have found Chapter 4 the most obscure part of the book, and further thinking and re-reading are necessary.
As the book was written in the 50′s, there is of course some outdated information. The idea of relating entropy to information has evolved significantly from the time the book was written, specially with respect to quantum computation and information. The same concerns the material on the last part of chapter 5, which, for instance, still addresses Dirac’s hole theory interpretation. A treatment of time in modern quantum field theory is, understandably enough, missing. However, the book is, for its most part, still valid and an excellent source for investigation on this never-ending puzzling subject. It certainly does not present a final theory of time, but offers an excellent outline of several lines of thought, reveals some intriguing possibilities and summarizes the foundations of the subject.
In some parts of the book, I have found myself surprised by some subtle conclusion based on gradually complex argumentation which nevertheless was founded on very simple concepts. Overall, I highly recommend the book for those seriously interested in the nature of time, and who are eager to spend good — time — thinking over the matter. Being a Dover publication, the price is quite accessible and it is, of course, a classic for your bookshelf. Five stars (“Amazon scale”).
Tensor symmetries – basic considerations
The transpose of a tensor
In our previous post, you may have wondered why we have chosen to compute
instead of using a different order in the slots of T, say:
The answer comes clearly if you try to see what happens if you do so. Write:
Then
It means that the
So you see, the vector valued function
So the final answer to the above question is simply: there was no reason for chosing one specific order in the slots. But once you have chosen one, you must be consistent throughout on what is the transpose of what.
Symmetries of tensors
Matrices can be classified as symmetric and antisymmetric, and since tensors can be represented by matrices, what can we say about symmetric and antisymmetric properties of tensors?
First, recall that a symmetric matrix is a square matrix such that it is equal to its transpose. And an antysymmetric one is equal to the negative of its transpose.
Now, since:
we see that the above set of components, which form a matrix, will be equal to its transposed set if and only if:
Similarly,
The above equations mean, respectively:
You may want to extend the above concepts to tensors of higher rank. For example, consider a third rank tensor, viewed as a 3-linear functional. If you want to make that tensor a completely symmetric one, you will have to impose the following conditions:
that is, to all permutation of the input vectors. For a completely antisymmetric one:
and I leave as an exercise to you to derive when you use the minus or plus signs above.
As a final note, you may wonder whether it its possible to consider symmetry properties in mixed tensors. Yes, it is. You may just consider symmetry on interchange of the arguments in a given pair of input slots (say, the 1st. and the 2nd.), and antisymmetry on another (say, the 2nd. and the 3rd.), and so on.
Ref.: [Nea10].
Representation theorem for linear functionals in component form
When discussing functionals and their relation to tensors for the first time, in this post, we mentioned the representation theorem for linear functionals (see also our Theorems page at the side bar).
Recall that it states that if
Now we just want to check how this theorem fits with what we have been doing in the last few posts, namely, working with tensors in component form, that is, through indexes, in order to track how tensors behave in terms of their components in some specified basis. Recall that we are working with Euclidian tensors, and already know how to deal with them in orthogonal and non-orthogonal bases. Also recall that we made a change in notation in order to make distinct sub- and superscripts. (Review our previous posts in the Courses page at the sidebar).
What we claim now and show immediately is that the unique vector
Let us prove that quickly. Just take
As an exercise, prove the analogous result for
Let us expand that to bilinear functionals. Let
Again, you can prove, in complete analogous way as in the linear functional case, that such unique vector-valued function is directly computed from:
You can extend that to multilinear functionals in a completely analogous way.
Such computations lead us to infer simply the following cases.
If
If
If
Etc. So you see the appearance of covariant (subscripts), contravariant (superscripts), and mixed (both sub- and superscripts) components, respectively.
As an example to put into context, consider a bilinear functional called the metric tensor:
In a manner consistent with what we have been working until now, we see that the linear vector function that is found by pulling off the
Indeed:
More on the metric tensor later.
Ref.: [Nea10].
