Subject: The differentiable manifold concept Dear Professor Trautman, Thank you *very much* for your kind email from Sat, 14 Apr 2007 10:02:19 +0200 and for the pp123.pdf file attached (the scanned three pages from [Ref. 1], which I requested in my email from Sun, 8 Apr 2007 21:54:21 +0300). As I mentioned in my preceding email, I am definitely sure that "the You clearly demonstrated the need for new physical theories based on new assumptions about the structure of space and time, "but up to now, no such attempt has met with much success" [ibid., p. 102]. Let me try to pose two questions. Q1: Can we suggest a new degree of freedom of the 4-dimensional http://www.god-does-not-play-dice.net/Straumann.html#dark If we succeed, Élie Cartan's 'La Géométrie des espaces de Riemann' As an example of the need for new assumptions about the structure of space and time, consider the common belief in 'countable infinite' http://www.god-does-not-play-dice.net/download.html#lamp and with the lessons from the Hole Argument, http://www.god-does-not-play-dice.net/Straumann.html#crux As Henri Poincaré predicted, "point set topology is a disease from which the human race will soon recover", http://www-history.mcs.st-andrews.ac.uk/Quotations/Poincare.html The sooner, the better. So, if we succeed with the first task, the next question follows: Q2: Can we modify the Einstein-Cartan Theory [Ref. 2] by "inserting" the new degree of freedom in the Christoffel connection [Ref. 3, Eq. 1]? Given the totally unclear outcome from the first task, I can only offer some raw ideas about the "torsion" degree of freedom, http://www.god-does-not-play-dice.net/Xiao.html#IMHO http://www.god-does-not-play-dice.net/Xiao.html#Cooperstock If we don't leave for India, how can we discover America? Perhaps this is at least a well-posed question :-) With best regards, Dimi Chakalov
[Ref. 1] A. Trautman, Foundations and current problems of general relativity, in Lectures on general relativity, ed. by Andrzej Trautman, F.A.E. Pirani, and Hermann Bondi, Englewood Cliffs: Prentice-Hall, 1965, Sec. 5.1, pp. 101-103 p. 103: "From now on we shall always assume that space-time can be represented by a 4-dimensional differentiable manifold. This is why the differentiable manifold concept was defined with care and discussed in detail in the preceding chapter. Any changes in this assumption would result in a very profound revolution in physics".
"It is possible that the Einstein-Cartan theory will prove to be a [Ref. 3] José G. Pereira, In Search of the Spacetime Torsion, Talk presented at the Rencontres de Moriond on Gravitational Waves and Experimental Gravity (La Thuile, Val d'Aosta, Italy, March 11-18, 2007), on Thursday, 15 March 2007; transparencies at
Note: Hermann Weyl says: "We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context." Let me comment on the last sentence, in reverse order. 1. The deeper context: The 'grin of the cat without the cat' (cf. Alice) facilitates the negotiation (cf. the ontology of relational reality) with 'everything else in the universe', which "takes place" in the putative global mode of spacetime. The putative 'local mode of spacetime' is a perfect continuum of already-negotiated (or linearized) facts cast in the absolute past of the universe. Stated differently, the main hypothesis here is that the relational ontology is the fundamental principle by which every "point" from a differentiable manifold is being identified dynamically (cf. dynamical determinism) by its transient and covariant physical content. In the classical limit of 'relational ontology' (classical mechanics and STR), the "negotiation" in the global mode of spacetime is spanned over the "immediate neighborhood" of the point, hence the effect of the Holon is vanishing small, the spacetime is "flat", and neither non-local nor quasi-local effects are present. In this highly contrived case, all "points" from the differentiable manifold are uniquely fixed/locked by their physical content, hence cannot be 'moved around' by what people call 'active diffeomorphism'. Consequently, by extending the "neighborhood" of the point and hence "extending" its Holon in the global mode, we introduce both "non-local" and quasi-local effects (cf. Lįszló Szabados) on the individuation of the points by their common Holon, as performed with the rules of relational ontology. The end result is again a perfect continuum of already-negotiated, linearized facts cast in the absolute past of the universe: the local mode of spacetime. The latter is a re-created (cf. Antonio Machado) continuum of points living on already linearized and "always flat" spacetime, with [lambda] tending asymptotically toward zero (otherwise you may lose your night sleep, like Ed Witten). Also, the local mode of spacetime is a perfect continuum, because the Holon in the global mode of spacetime stays always in the absolute potential future of the whole universe as ONE, hence it is nonexistent in the absolute past of the universe -- there is nothing "in between" the points constituting 'the grin of the cat without the cat' (Alice), in the same way that there is no water in between two adjacent molecules of water. Hence the local mode of spacetime is "quantized" from the outset, and the law of continuity, as defined in the standard calculus texts of the 1800's, is fully obeyed: "the consecutive points of the same line succeed each other without any interval". Notice that the global mode "interval" is kept in the global mode of spacetime, and can indeed be eliminated in the local mode with the Aristotelian Connection.
Hence we have (i) a perfect continuum, (ii) locality, and (iii) global retarded causality (no CTCs nor CPP) in the local mode of spacetime: the EPR-like correlation of future potential events is being "inserted" in the gaps of the global mode of spacetime, thus making the local mode of spacetime an already-correlated "back bone" of the whole physical world, at all length scales. The 'global mode of spacetime' is a new degree of freedom in GR: the "vertical shift" in the "GW lake" here is omnipresent in the local mode of spacetime, as explained here. It is also completely hidden in STR, by the virtue of 'interactions on null intervals': light exists only as an already completed interactions on null intervals (Kevin Brown), hence the "proper time" of the flight "over" this null interval is zero to all 'passengers inside the train'; more here. Thus, a relativistic observer confined inside the local mode of spacetime will "see" only one single instant dt of some scalar quantity called "time". We may experience dt one-at-a-time, along the "vertical line" of the global mode (the "explicit (but unmeasureable) time", W.G. Unruh), but in the local mode the "duration" of the elementary "horizontal" step -- the elementary timelike displacement -- is infinitesimal. Perfect continuum, ladies and gentlemen! Its "quantization" is introduced in the global mode of spacetime -- ]between[ the "points" from the local mode -- with 'the universe as ONE' (the atom of Lucretius) and the Aristotelian First Cause; hence the name proposed to this connection is 'The Aristotelian Connection'. You can't find it in the caricature below.
The idea about some spacetime foam is 'not even wrong' !
NB: There is no other way to introduce a proper continuum. Its ontology requires a reference object -- global mode of spacetime -- such that it will (i) define discernible "points" on the continuum, and (ii) create them dynamically: Panta rei conditio sine qua non est. Mathematicians will probably hate these statements and will ignore them, only they can't do better: see Paul Ehrlich's "the wonderful elementary limit" due to the Aristotelian Connection. As to theoretical physicists, they very seldom admit the generic pathologies in present-day GR: see Matt Visser, The quantum physics of chronology protection. In: Gibbons GW, Shellard EPS and Rankin SJ, The Future of Theoretical Physics and Cosmology, Cambridge: Cambridge University Press, 2003; gr-qc/0204022, p. 3: "What general relativity does not do is to provide any natural way of 2. We want to understand the idea of the proof: Place the elephant's trunk in the Holon (global mode of spacetime), and let it choose one state of all sub-systems for their next explication as 'facts' in the local mode of spacetime. Such 'one state' will be unique, since 'the chooser' is the whole universe as ONE: God is flexible. Notice that we can address the non-linear quasi-local dynamics of both gravitational and quantum systems, thus providing a unification of GR and QM from the outset as well. The key idea is a new kind of determinism, dubbed 'dynamical determinism', which exploits the inherent flexibility of quantum and gravitational systems. 3. We want first an overview of the aim and of the road: Start with the "boundary" of spacetime, delivered by the Aristotelian Unmoved Mover and First Cause, and solve the puzzle of matter-geometry "talk" with a third entity, resembling the elephant's trunk. See another elephant story here. Notice also the affine connection puzzle: "The affine structure is a further primitive (not definable from mere differential structure) structure" (Graham Nerlich). We simply postulate that the Hausdorff topological space is "connected", but cannot derive this connection from any physical stuff, because it isn't there yet. We are still working with pure math, yet we tacitly introduce by hand the fundamental connection originating from the Aristotelian First Cause: the Beginning is that which does not have anything necessarily before it, but does have something necessarily following from it [Poetics VII 1450b27-29]. The implications from this story are that we cannot define any 'elementary step' on the differentiable manifold unless we have defined it 'as a whole', which means fixing the "boundary" of spacetime. It's a package. In plain words, the Cosmological Principle reads:
This is the bootstrap ontology of Geoffrey Chew, applied to the whole universe. Obviously, the self-determination of the whole universe will be a bona fide self-action: the universe as ONE entity will act on itself. Such self-action, performed by the whole universe as Aristotelian First Cause, will inevitably look "dark" to all sub-systems, simply because its "origin" cannot be traced back to any sub-system. Isn't that simple? Why is this so difficult to understand, I wonder. Even my teenage daughter knows how to calculate the circumference of a circle, so all we need is to reveal the reference fluid [Ref. 1], which identifies uniquely the "points" from the circle, and then think of such 'continuum of points' as matter fields coupled to gravity, and finally explain the dynamics of GR: from one "point" to the "nearest" one. Basically, all we need is to find the reference fluid in GR -- provided it is there. If it cannot be found in GR in principle, new math may be needed, but nobody seems to be interested. More on Sunday,
21 September 2008.
Subject: AoC ========== Subject: Re: AoC Note: The rich topology hiding in Einstein gravity comes from a "bare" topological manifold that is not (yet) endowed even with metric; it is just a three-dimensional topological manifold, denoted with Q . Marco Spaans explains (arXiv:0705.3902v2, p. 4): "A priori, there is nothing physical about Q. It is merely a means to pick out all the different three-space regions in a four-dimensional space. Still, the logic that leads to Q holds for any physical property of the three-space regions. Given that there is no metric at this stage, one cannot assess the sizes of different regions, i.e. one is dealing with socks because there are none of the physical properties yet that define Einstein gravity." And finally (ibid., p. 9): "Although Einstein-Cartan theory is beyond the scope of this derivation, it is obvious that Q possesses a topological analog because the three-tori are not simply connected." But to avoid the Axiom of Choice (AoC) and hence construct a genuine background-free theory such as Einstein GR, "one needs a distinguishing quality," says Marco Spaans. Here's the full excerpt (ibid., p. 3): "For self-consistency of the procedure one can then best ask the following question: Is it possible, for a connected set of three-space regions (in a four-dimensional space-time), to choose a region from that set through a choice process that is defined solely in terms of the three-space regions in the space-time itself? If such a construction can be found then it is self-contained and does not require the AoC. Which brings us to the fundamental article by A. Trautman [Ref. 1] and the reference fluid in GR, provided by the Aristotelian connection. The latter can be unraveled (not without difficulties) in the following quote (ibid., p. 3, emphasis added): "Because the region is infinitesimal, one can take it to be a simply connected region, i.e. one assumes that the region has no prior properties other than continuity. The assumption of continuity, although plausible, is a necessary one." The very presence of continuity -- from one 'selected' object x to the 'distinguishable' object y -- is the manifestation of Aristotelian connection. It is the fundamental structure enabling quasi-local interactions in GR. It is a genuine fundamental structure that cannot be derived from the physical stuff we introduce later in GR. Hence it looks "dark". You can only notice that x and y are somehow "self-connected" by pure math called "the assumption of continuity". There is no intermediate object "between" x and y. The latter are "self-acting" like Baron Munchausen, because are defined in a background-free manner, without the Axiom of Choice (AoC) but "solely in terms of the three-space regions in the space-time itself." If you don't accept miracles in Einstein GR, try the Aristotelian connection. I shall wait to see if Marco Spaans can suggest a new topological structure that is 'rich enough' to remove some of the generic pathologies of classical spacetime manifold, such as the geodesic incompleteness. Highly unlikely, I'm afraid. Notice that M. Spaans considers pairs of three-space regions [x, y], which yields two (one for x and one for y) three-tori and three "handles" (arXiv:0705.3902v2, p. 4). If I was in his shoes, I would start with a triplet of three-space regions [x, y, z], and would search for some "handles" that can provide for a time-orientability of y with respect to its neighbors x and z . I mean, we should lay out a topological structure that would enable the 3-D space to acquire its "time", otherwise the very assumption of continuity (see above) wouldn't be justified. Notice that neither the topology nor time-orientability can be derived from GR, so we may explore this 'freedom of choice' and try to fix many problems before they appear in GR equations from textbooks. In general, if you wish to "think of points for simplicity", like M. Spaans (see above), you are already outside the applicable limits of GR, since it doesn't hold for "points". Hence you again need the Aristotelian connection to make sense of the "points" in Einstein's GR. Last but not least, I wish to stress that I have great respect for the work by Prof. M. Spaans. Anyone can drop suggestions, but he does the hard work, and I sincerely wish him full success.
==================== Subject: GW parapsychology? ========== Subject: Re: GW parapsychology?
Joshua Goldberg will immediately notice the main problem from the "linearized approximation" of Einstein's GR: it is like building an antennae that can only detect very weak TV signal, because the "linearized approximation" itself cannot cope with any strong TV signal whatsoever. Surely we can use approximations, such as the Schrödinger equation, which does not take into account any effects from the quantum vacuum. Only LIGO Scientific Collaboration (490 distinguished scholars) does not have any theory combining strong gravitational effects of both GWs and spacetime curvature, from which one can derive the theory of detecting GWs in the case of vanishing small "dimensionless amplitude". Their expectations for "increased sensitivity" with the Enhanced LIGO and Advanced LIGO make them look like some harum-scarum kids who claim that can measure the room temperature in their house, but their home-made "theory of thermometers" cannot be applied to the surface of the Sun even with a Gedankenexperiment, which in turn obliterates their "theory of thermometers" (LIGO 'n LISA) and the initial "theory of temperature" (the quadrupole approximation). It is not surprising that LSC failed to measure any "temperature" at all, since they in fact measure the dipole mode, and no "increasing of sensitivity" can help. Most importantly, in a universe dominated by Dynamic Dark Energy (DDE), the mass-energy (cf. Eq. 10 in B. Schutz' article in Encycopedia of Astronomy and Astrophysics here) cannot be conserved: there is indeed energy radiated due to the "dipole" and "monopole" effects in general relativity, only it is totally "dark". It is most likely pertaining to the energy associated with the elementary timelike displacement, which is not an observable in GR (see Slide 29 in Bolen.zip). How do we detect "the time changing quadrupolar distribution of mass and energy" (cf. Scott A. Hughes below), given the fact that 'time changing distribution of mass and energy densities in GR' is totally unknown? Let's briefly recall the basic basics of gravitational energy: we must take into account "nonlocal interactions between the T_uv's at different points" (R. Penrose). "At least two ideal observers are needed to detect gravitation, but only one is enough to detect an electromagnetic field. In this sense gauge fields are local, and gravitation is not" (J. Pereira et al.). Thus, we need at least two observers/test particles, such that they could detect the gravitational energy only by being EPR-like correlated, like a shoal of fish. Locally, they must not "feel" any gravitational energy contributions (H. Weyl). To quote from R. Penrose's book "The Road to Reality": p. 467, emphasis added: "(T)he gravitational wave energy has to be measured in some other way that is not locally attributable to an energy ‘density’. Gravitational energy is a genuinely non-local quantity." p. 458: "The contributions of gravity to energy-momentum conservation should somehow enter non-locally as corrections to the calculation of total energy-momentum. (...) From this perspective, gravitational contributions to energy-momentum, in a sense, ‘slip in through the cracks’ that separate the local equation [XXX] = 0 from an integral conservation law of total energy momentum." More here. Therefore, LIGO's arms are totally incapable of detecting GWs from the outset. This conclusion shouldn't be surprising, since you don't need non-local interactions in the "linearized approximation" of GR -- it can only "detect" very week GW strain. Again, it is like building an antennae that can only detect very weak TV signal, because the "theory" itself cannot cope with any strong TV signal whatsoever. p. 317: "But if the geometry is strongly distorted, the distinction between And again from p. 317 (emphasis added): "To arrive at a conserved energy that can be exchanged between the detector and the wave, we have to treat the wave and detector together. This is not so easy in general relativity, because it is not easy to define the wave separately from the rest of the geometry. (...) Energy is only conserved in situations where external forces are independent of time. For weak waves, it is possible to define their energy with reference to the "background" or undisturbed geometry, which is there before the wave arrives and after it passes." Which is why V. Faraoni (see below) tried very hard to dismiss the objections by Steven Weinberg. But if you introduce some fictitious "background" or undisturbed geometry, which "is there before the wave arrives and after it passes" (B. Schutz), you can play with GR as much as you wish, as long as you manage to separate the undisturbed "time parameter" of the "undisturbed geometry" from the other time parameter pertaining to the "waveforms" that "carry detailed information about the source" (cf. Kip Thorne's slide 5, from Caltech's Physics 237-2002). It sounds like a stupid joke ą la Baron Munchausen, only costs billions. The same type of error is made by those who claim that the so-called Dynamic Dark Energy (DDE) produces the cosmological time, and at the same time evolves in that same time. True, every measurement is relational by its nature, so we need a new referential background, but LIGO Scientific Collaboration had made an incredible error of "producing" it from the very stuff (the metric field) they were supposed to measure. If we are serious about the dynamics of GR and the wave of the spacetime metric (Caldwell & Kamionkowski), I believe we need to clarify the status of "dipole radiation", since the common, and convenient, belief that it is nonexistent (see above) is not valid anymore. And here we enter a terra incognita, because of the unknown nature of the much-needed gravitational shielding: "any manipulation of matter acting as a source of gravitational field will introduce an additional stress-energy tensor as a source of gravitational field" (J. Stachel). Thus, it seems that the only option we may have is to manipulate matter in the global mode of spacetime, before it becomes localized in the local mode of spacetime as "positive matter" (P. Joshi).
See the train metaphor here, and recall that we eliminated such preferred origin/Big Bang and preferred reference frame by "multiplying" The Beginning across an imaginary instantaneous "cut" of 3-D space, in which all observers would have measured/observed the same value of the cosmological time: 13.7 billion years, called 'modern universe' (cf. above). Again, the picture above is a highly misleading, non-relativistic presentation of the Cosmological Principle: the instantaneous "cut" called 'modern universe' is just a mental imagery of the global mode of spacetime, corresponding to an imaginary 'now-at-a-distance' reference frame, in which the "distance" can be stretched to infinity (like a transcendental tachyon that is absolutely everywhere in "no time"), to exhaust the whole 3-D space. If we think of the "balloon center" in the expanding balloon metaphor (courtesy of Ned Wright), we can easily visualize such instantaneous "cut" of the expanding balloon, corresponding to the current surface of the balloon, because we can compare it to a "cut" that was smaller in the cosmological past with a "cut" that will be larger in the cosmological future, as depicted in the picture above; all these 'past' and 'future' would have absolute values, since we'd have an absolute "center" of the balloon (a.k.a. big bang) and an absolute empty space "outside" the balloon, waiting patiently for the balloon to expand into. We would also be able to settle the old dispute about who walks upside down, Matt Visser in New Zealand or the author of these lines in Europe (the way I see it, Matt will win, because he is so good in math!). Mother Nature is smarter, however, because in the real, relativistic case the very "surface of the balloon" -- the 3-D space itself -- is composed of infinitely many "centers" of the balloon: (i) the "center" has been multiplied and spanned absolutely everywhere and evenly inside the 3-D "balloon surface" (cf. again the Cosmological Principle above), and (ii) the "boundaries" of spacetime are being dynamically fixed by 'the universe as ONE' -- the Aristotelian First Cause. Not surprisingly, then, some "remnants" from 'the universe as ONE' will show up in its relativistic presentation (e.g., the omnipresent cosmic microwave background (CMB) radiation and the smooth DDE), and even become observable (the cosmic equator). Once we switch to this relativistic presentation (which is, I believe, the only possible alternative to the misleading non-relativistic picture above), the global mode of spacetime "shows up" in the gaps "between" the infinitely many "centers" of the balloon placed on the balloon "surface", only we cannot directly observe this "dark shift" due to the "speed" of light. Thus, all "dynamic dark energy" effects, all acausal, inflation-like interactions across the expanding "rubber band" (see Fig. 24.3 from B. Schutz' book here), all quasi-local contributions of gravity to energy-momentum conservation (see R. Penrose's book above, p. 458), and the very nature of gravity -- "the whole universe must know about everything instantaneously" (see M. Zucker below) -- are vivid evidence for the "dark", or rather holistic effects of gravity, produced in the global mode of time. Briefly, in the relativistic case both 'the center of the balloon' and 'the empty space before and after the current balloon surface' are converted into 'global mode of spacetime': see again the white area in Fig. 3.1 here. This is a concise, but incomplete, outlook of the structure of spacetime, suggested at this web site. See also the GW lake metaphor here, and on the hypothetical GWs here and here. I am sure Josh Goldberg can elaborate extensively on these issues. Back in 1955, he published a landmark article on gravitational radiation (Phys. Rev. 99 (1955) 1873-1883). From 1956 to 1963, he was responsible for US Air Force support of research in GR, based at the Aeronautical Research Lab at Wright-Patterson Air Force Base in Ohio, and managed to support the 1957 Chapel Hill conference "Conference on the Role of Gravitation in Physics", organized by Bryce De Witt in January 1957, with US Air Force money. At that conference, Sir Hermann Bondi had stressed the following: "The analogy between electromagnetism and gravitational waves has often been made, but doesn’t go very far, holding only to the very questionable extent to which the equations are similar. The cardinal feature of electromagnetic radiation is that when radiation is produced the radiator loses an amount of energy which is independent of the location of the Thus, when EM radiation is produced, it is like a letter sent to all recipients which can "absorb" it; the simplest example with the emission of light from the Sun can be read here. The whole "postal service" with emission and absorption is perfectly local, obeys STR, and energy conservation laws.
If we trust the inflationary scenario, the conversion of gravitational energy into physical energy is precisely the puzzling "localization" of normal-plus-dark gravitational energy, which we don't understand at all, because it had somehow evaded STR during the inflation, and also because in GR there is no tensorial presentation of it. All we can say is about what this "localization" is not: read Mike Zucker here. We cannot accommodate any acausal process ą la inflation nor action-at-a-distance in GR, yet "somehow the whole universe must know about everything instantaneously", as Mike Zucker put it. Therefore, it shouldn't be surprising that GR explicitly forbids any 'simple localization of energy' similar to the case of EM radiation, as is well known since 1917, thanks to Tullio Levi-Civita. Ninety years later, we've only coined a very peculiar term for the gravitational energy: Quasi-Local Energy (QLE), as explained by Bjoern Schmekel [Ref. 1]. Since the localization of GW energy is not a 'simple localization of energy at a point', we also call it "non-localizable" [Ref. 2]. Again, what we cannot use to describe it? NB: We can't use any finite domain of 3-D space either, so the only option left is to define a "boundary" at infinity [Ref. 1], which is nothing but the main issue raised here and elaborated on here. We simply do not have any other choice -- "His thoughts" can only be produced by the Aristotelian Connection, as the quasi-localized conversion of "His thoughts" constitute the putative local mode of spacetime. If Kip Thorne or any of his LSC colleagues had made a professional effort to explain the puzzle demonstrated by Tullio Levi-Civita in 1917, they might have had a chance to gain respect and admiration on behalf of GR community. Instead, "Kip Thorne had no difficulty in 1981 in finding a taker for a wager that gravitational waves would be detected by the end of the last century. The wager was made with the astronomer Jeremiah Ostriker, one of the better-known critics of the large detectors then being proposed. Thorne was one of the chief movers behind the largest of the new detector projects, the half-billion-dollar Laser Interferometer Gravitational Wave Observatory, or LIGO. He lost the bet, of course." (Daniel Kennefick, Traveling at the Speed of Thought, Princeton University Press, Princeton, 2007, p. 1.) Twenty-six years later, Kip Thorne has not yet encountered any difficulties in spending taxpayers' money for his juvenile dream. He just keeps quiet. And so does Josh Goldberg, regrettably. But because Josh Goldberg was involved with these problems for more than fifty years, and also because I consider him a real professional, it is my hope that he will not stay quiet forever. If some day Josh Goldberg decides to teach LSC a lesson in contemporary (after 1960s) gravitational physics, he will do it professionally, and they might eventually read it and start thinking.
p. 2: "Because of the problems associated with defining a local energy density it may be easier to make sense of the energy enclosed by a boundary. For regions of finite extend we expect non-zero values because in general a coordinate transformation can make the connection coefficients vanish at only one point. Therefore, it seems the only sensible way to define energy is by defining energy itself and not energy density.
p. 10: "At the Bern conference Rosen, returning to the cylindrical wave solution of his 1937 paper with Einstein, adduced evidence backing up Scheidegger’s position by proposing the possibility that gravitational waves did not transport energy (Rosen 1955). It is a peculiar characteristic of general relativity that the energy contained in the gravitational field, and thus the energy in gravitational radiation, is not described in a coordinate invariant way. This energy is considered to be real enough, and can be converted into other forms of energy which can be expressed invariantly, but the principle of equivalence prevents one from doing this for field energy in gravity. The reason is that any observer in a gravitational field is always entitled to imagine himself in a locally Lorentz (that is zero gravity) freely falling frame of reference which, locally, contains no field energy. Of course, one is not free to transform away the entire field energy of a planet but one can always choose co-ordinates on an infinitesimally small portion of its surface so as to eliminate the field energy in that region. Thus it is said that gravitational field energy is non-localizable."
====================
===================== Date: Tue, 9 Oct 2007 23:20:04 +0300 ===================== Subject: Re: Netiquette
Unlike in STR, where the metric acts as a background structure given a priori, in Einstein GR the metric is treated as a field which not only affects, but also is affected by, the other fields present. This non-linear, bi-directional "talk" (J. Wheeler) continues at every instant from the non-tensorial "time" [tau] (cf. C. Rovelli and B. Bolen), as depicted with the lake metaphor here. Due to the active diffeomorphism freedom, the geometrical "points" cannot be identified by any fixed material content of 'objective reality out there', as explained by J. Stachel and Butterfield & Isham. Now, if LIGO detects some "wave" of the metric field, such "wave" could only "propagate" on a set of geometrical points that are already-fixed by their physical content, which of course contradicts GR: Angelo Loinger, GW's towards fundamental principles of GR, arXiv:0709.0490v1 [physics.gen-ph] To clarify this issue, I will draw 1-D wave pattern snapshots of the GW lake below, at two instants of time, tm and tn , depicting the periodic wobbling of the metric field:
According to the "linearized approximation" of GR (cf. Scott A. Hughes, [Ref. 1]), your wristwatch and LIGO's arms can read these instants of time: "A passing gravitational wave would change the distance between the weights, first in one arm, then in the other arm, which is arranged at a right angle to the first" (reference here). Which in turn requires that the GW energy should be localized during a finite time interval during which the spacetime "points" x possess fixed content obtained from the right-hand side of Einstein equation (the material stuff of the GW lake), which is, of course, pure fantasy: read again Angelo Loinger. Notice that we can ponder on such "wave pattern" only by taking a bird eye's view on the whole spacetime, that is, by taking the stand of some meta-observer placed in an absolute reference frame. The bold reality of Einstein GR is entirely different: your wristwatch and LIGO's arms are immersed into the GW lake, such that there is no fixed background nor meta-observer "outside" the GW lake. Steven Weinberg was right: “all lengths are stretched at the same rate by the gravitational wave”. You can't enjoy a physical GW wavelength paired with a dimensionless, yet "slowly evolving", GW amplitude. The whole mess of "GW astronomy" begins with the "time parameter" pertaining to the "waveforms" that "carry detailed information about the source" (cf. Kip Thorne's slide 5, from Caltech's Physics 237-2002), which is a ghost that shows up only in the "linearized approximation" of GR: see NB above. Moreover, if GWs had genuine physical "phase", there should be a way to cancel it by Gedankenexperiment, similar to the real experiment with canceling the phase of EM waves. Here's my Research Proposal to LIGO Scientific Collaboration (currently 490 physicists) regarding the GW phase. First, some history. As Richard M. Jones reminded, in late summer of 1991, the House Science Subcommittee passed a bill prohibiting LIGO construction funding, but on 27 September 1991 "conference action on the NSF bill was completed, and LIGO had the full $23.5 million the Bush Administration had requested." I was in the United States in 1991, but cannot recall any major discovery in the late summer of that year, which could have changed drastically the course of action set by the House Science Subcommittee, prohibiting LIGO construction funding. Such 'major discovery' can be explained with an old joke from Martin Gardner: two people whose ship sank were left on an island for many years, and one day they found on the beach a big Coca-Cola bottle (they knew only the small ones). Then one of them said: "Holy cow, we got shrunken!" Suppose they were instead "stretched 'n squeezed" by some local GW passing locally over their island.
What could possibly constitute the referential background with undisturbed metric that defined the dimensions of the small/undistorted Coca-Cola bottle? And how would the two guys keep track on it, so that they can at the same time realize that were indeed "stretched 'n squeezed"? It really requires a major discovery to explain such schizophrenic observation. With a delay of sixteen years, during which at least $500 million taxpayers' money have been "spent", may I offer to LIGO Scientific Collaboration a simple task regarding the GW phase and amplitude, which, if completed successfully, would clear their reputation. It is just a Gedankenexperiment, so all they need is blank notebooks and sharp pencils. Please explain the phase and amplitude of Gravitational Waves. Please don't miss the details explained above and the main objection explained here. In a nutshell, the measuring device will be "stretched and squeezed" by zero distortion: there is no undisturbed background. Why? Because we cannot "split" the metric field into two parts: undisturbed background metric vs. "disturbed" metric (cf. A. Buonanno in [Ref. 1]). In GR, you can't produce a referential background from the very stuff you're supposed to measure, and then measure "it" with respect to itself. Can't have your cake and eat it. In the context of the story from Martin Gardner above, our island is 'the whole spacetime', hence the alleged effect depicted with the drawing from Kip Thorne
is exactly zero. Read Angelo Loinger below. Again, there was no major discovery in the late summer of 1991, which would have recovered the dimensionality of GW "amplitude", h . Should anyone has doubts about the statements above, there is a simple way to refute them and vindicate the scope of LSC research: write down the dynamics of strong GWs in the case when "the geometry is strongly distorted" (cf. B. Schutz above), and then choose a parameter in GR -- your choice -- that could enable you to recover -- reversibly -- the linearized approximation of GR valid for negligible distortion of geometry, and then go back to the initial case when "the geometry is strongly distorted". If you succeed, you will demonstrate the correctness of the linearized approximation, hence clear the reputation of LSC as people doing science (not parapsychology). Needless to say, the rest of the world will learn about such discovery from CNN Breaking News, and we might get a clue how to build classical spacetime 'from scratch', that is, from nonlocal Diff(M)-invariant observables (cf. Steve Carlip and Chris Isham). Notice how LSC member Valerio Faraoni tried to obscure both the crucial issue explained above and Steven Weinberg's objection, by arguing that "the gravitational wave “treats in a different way” the wavelength of light and the length of the interferometer’s arm". But LSC don't have any detected GW in the first place, to prove that it indeed “treats in a different way” the wavelength of light and the length of the interferometer’s arm. They can only offer speculations about some "linearized approximation", and five consecutive failures to detect GWs. As Steven Weinberg wrote to L. Grishchuk (email from 25 February 2003): "I agree that much of what one reads in the literature is absurd. Often it is a result of bad writing, rather than bad physics. I often find that people who say silly things actually do correct calculations, but are careless in what they say about them." The "silly things" in question are the statements of many LSC members regarding the phase and amplitude of GWs. Let's hope this time they will act professionally, and not be careless in what they say about them. To begin with, let us recall some basic prerequisites from Kip Thorne (see his slide 4 below):
By comparison, recall the cancellation of the EM phase with two Polaroid filters, as explained by B. Schutz (reference here): "You can prove that light is a transverse wave by using Polaroid, the semi-transparent material that is used in some sunglasses. If you take two pieces of Polaroid and place them over one another, then if they are oriented correctly they will pass about half the light through that falls on them. But if you rotate one piece by 90o, then the two pieces together will completely block all the light (propagating along the Z axis - D.C.)."
To perform the Gedankenexperiment with GW's phase, I believe LSC will need some object that can be mapped onto itself by 180o rotation ("force pattern invariant under 180o rotation", see Kip Thorne's slide 4 above), in 3-D space and by using Cartesian coordinates. And also keep in mind that "each polarization has its own gravitational-wave field", as Kip Thorne stated in slide 5 from his course Caltech's Physics 237-2002, so you'll have to fit those two independent "gravitational-wave fields" in the same 3-D space as well, and finally ensure that all this happens "with reference to the "background" or undisturbed geometry, which is there before the wave arrives and after it passes", as Bernard Schutz eloquently explained. Once you complete this task, not only will you pass Caltech's Physics 237-2002, but also discover the "direction" of GW propagation along the Z axis, and of course recover the dimensionality of GW "amplitude" projected on x/y axes (in meters or bananas, whichever comes first). That's all.
Notice also that the "direction" of GW propagation can only be determined relationally, with respect to some other direction in 3-D space, in which the same GW does not propagate. For EM waves in Minkowski spacetime, this task is trivial, because we can determine the direction of light propagation with respect to the undisturbed spacetime "grid", and define the volume of 3-D space ahead, in which the photons have not yet arrived, as well as some alternative direction in 3-D space, in which the same EM waves do not propagate. In the case of GWs, however, such exercise does not make sense at all, since it requires a non-relativistic presentation of 'the whole spacetime' -- see the non-relativistic cosmological picture above. Read also my email to Kip Thorne from Sun, 16 May 2004 02:02:03 +0300 here. Moreover, if GWs propagate along the "direction" of the global expansion of 3-D space as well, there will be no "direction" left in 3-D space in which they would not propagate, and subsequently there will be no unique direction singled out by GWs, on which LIGO or LISA would stick to measure the GW strain. They just can't detect an omnipresent stuff like GWs and DDE. These crucial conceptual problems of "GW astronomy" should have been discussed in February 2003, at the American Association for the Advancement of Science's 2003 Annual Meeting (17 February 2003, Denver, Colorado; reference here). What happened instead was that Kip Thorne and his LSC colleagues got additional US$150 million to "discover" the "desired sensitivity" of LIGO, since all their failures to detect some effect from the dimensionless GW amplitude have been interpreted as useful hints for obtaining the "desired level of LIGO sensitivity". The latter was certainly not clear even to Kip Thorne, since in 1981 he bet that GWs will be detected "by the end of the last century" (Daniel Kennefick, Traveling at the Speed of Thought, PU Press, Princeton, 2007, p. 1). The insane quest for detecting GWs continues. Surely we can use a linearized approximation of GR for task-specific purposes, for example, to fix Global Positioning System (GPS) coordinates in Minkowski space (C. Rovelli, arXiv:gr-qc/0110003v2), but detecting GWs is an entirely different challenge. For example, can you detect your "local coordinates" in the reference frame of the equator of the universe? If you can, you might discover the local "push" from the omnipresent Dynamic Dark Energy, and perhaps the "waves" of the spacetime metric. Go ahead, only use your own savings. Again, the answer to the key question 'with respect to what?' cannot be 'locally, and with respect to itself' (see above). Only in a non-relativistic presentation of GW radiation one could "envisage" an unphysical, gauge-dependent "global reference frame" (cf. Butterfield & Isham), as depicted in the (very misleading!) picture above ("Seeing back into the cosmos", cf. above). In order to detect the perturbations of the quasi-local gravitational energy densities caused by the impact from the quasi-local GW energy, we need a unique 'referential background' that can only be provided by 'the whole spacetime', which in turn requires brand new kind of quasi-local GW detectors, resembling the human brain (read a historical remark from 1984 here). As to the "linearized approximation" of GR, it produces artifacts totally incompatible with the full, non-linear GR. If the Schrödinger equation (see above) were the same kind of fake "approximation", it would have predicted effects that contradict QFT. Another comparison with quantum theory goes as follows: There are quantum effects that are quite week too (e.g., Josephson effect), but nobody would treat them classically. Most importantly, nobody would search for some "weak" quantum effects with some classical mechanics approximation, given the indisputable fact that such "weak" quantum effects cannot exist in quantum theory in principle. Now, replace 'quantum theory' with 'full non-linear GR', and 'classical mechanics approximation' with 'linearized approximation', and you will get the full coverage of "GW astronomy". Daniel Kennefick believes that Kip Thorne and his group have not "wantonly spent tax-payer's money is pursuit of a dream", but I haven't read any effort to clarify the status of the dipole radiation in "GW astronomy", despite the facts that the dark energy problem has been established since 1998 (see J.A.S. Lima). Which reminds me of a somehow cruel experiment my son did with our cat two years ago: he boiled her milk in the microwave, and poured it in her cup. Poor thing, she was running around her milk cup but couldn't touch it. But at least she showed genuine interest and dedication. Not so with Kip Thorne and his LSC collaborators, perhaps because they get their "cat food" from us anyway, since we all pay for their totally irresponsible dream. Again, if we interpret the spacetime "points" x (see my clumsy drawing above) as 'EPR-like correlated dice on the table' (local mode of spacetime), then there is indeed a genuine GW, but it cannot in principle be detected with LIGO and the like: read Roger Penrose on the quasi-local gravitational energy above. Hence one day we may have to convert LIGO and the other interferometer-based "GW detectors" to wine cellars, as suggested previously, but LISA will remain a totally unusable piece of junk. Do not tell me you knew nothing about it, Professor Goldberg! To finish this discussion, let me comment on two excerpts from Flanagan & Hughes, The basics of gravitational wave theory, New J. Phys. 7 (2005) 204, gr-qc/0501041 v3: p. 9: "We begin by defining the decomposition of the metric perturbation h_ab, in any gauge, into a number of irreducible pieces. Assuming that h_ab --> 0 as r --> [inf], we define the quantities (...) together with the constraints (...) and boundary conditions (...) as r --> [inf]." And on p. 12, Eq. 2.70: "Although the variables [X1], [X2], [X3], and hTT_ij have the advantage of being gauge invariant, they have the disadvantage of being non-local. Computation of these variables at a point requires knowledge of the metric perturbation h_ab everywhere. (...) Thus, at least certain combinations of the gauge invariant variables are locally observable." NB: I hope Prof. J. Goldberg will (i) clarify the crucial limitations from not knowing the metric perturbation h_ab everywhere, and (ii) disentangle the alleged gauge invariant variables that were "locally observable" from those which aren't simply because they can't be "locally observable" in the first place (cf. Larry H. Ford and Steve Carlip), hence eliminate the poetry in the seemingly innocent expression "certain combinations". This poetry costs billions. And an excerpt from LISA International Science web site (last modified 2006-11-29 14:05): "gravitational waves - disturbances of the fabric of space travelling through the cosmos like ripples on a pond (notice the poetry - D.C.). This poetry will also cost billions. Notice that the main reference in Flanagan & Hughes' article is ref. [51], which is an article by Richard A. Isaacson (Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio) from 1968. Notice also that their Ph.D. Advisor, Kip Thorne, also relies on "crucial" articles from 1960's (the mythical "gravitons" and the "invariance angle" that determined the L-shape of LIGO's arms). Again, there was no breakthrough in the summer of 1991, which would have changed the opinion of the House Science Subcommittee, prohibiting LIGO construction funding. Instead, I guess Kip Thorne and his colleagues and friends have convinced some influential people to play poker with taxpayers' money. Approximately fifty physicists have received the PhD at Caltech under Kip Thorne's personal mentorship; look at the list here, and will see that No. 32 is Eanna Flanagan and No. 37 is Scott A. Hughes. There are many more people on that list: just see Nos. 5 (Clifford Martin Will), 6 (Richard H. Price), 7 (Bernard Frederick Schutz, Jr.), 11 (Saul Arno Teukolsky), 25 (Lee Samuel Finn), and 35 (Daniel Kennefick). So far all these people are keeping quiet, included Josh Goldberg, who was responsible for US Air Force support of research in GR, based at the Aeronautical Research Lab at Wright-Patterson Air Force Base in Ohio, where all this mess started to evolve, up to this day. It will be highly embarrassing to LSC scholars in USA, UK, Germany, Italy, Australia, Japan, Canada, India and Spain if it turns out that some outsider has been repeatedly showing their errors, while they were ignoring Hermann Weyl and Angelo Loinger and wasting money earned with hard labor by their fellow citizens, until -- and finally -- fail miserably again, this time with their "Advanced LIGO". Speak up. Raise your voice. It's time to get real. Can't have your cake and eat it. D. Chakalov [Ref. 1] Chris L. Fryer, Daniel E. Holz, Scott A. Hughes, and Michael S. Warren, Stellar collapse and gravitational waves, See also: Alessandra Buonanno, Gravitational waves, arXiv:0709.4682v1 [gr-qc], 50 pages, 13 figures; to appear in the Proceedings of Les Houches Summer School, Particle Physics and Cosmology: The Fabric of Spacetime, Les Houches, France, 31 Jul - 25 Aug 2006
"Yang-Mills transformations occur at a fixed spacetime point whereas the diffeomorphism group moves points around. Invariance under such an active group of transformations robs the individual points in M of any fundamental ontological significance. (...) In the present context, the natural objects that are manifestly Diff(M)-invariant are spacetime integrals like, for example, "Thus 'observables' of this type are intrinsically non-local. "These implications of Diff(M)-invariance pose no real difficulty in the classical theory since once the field equations have been solved the Lorentzian metric on M can be used to give meaning to concepts like 'causality' and 'spacelike separated', even if these notions are not invariant under the action of Diff(M). However, the situation in the quantum theory is very different. For example, whether or not a hypersurface is spacelike depends on the spacetime metric g . But in any quantum theory of gravity there will presumably be some sense in which g is subject to quantum fluctuations. Thus causal relationships, and in particular the notion of 'spacelike', appear to depend on the quantum state."
See also: Alan Rendall, Approximation methods for gravitational radiation, See also: M. Kunze and A.D. Rendall, Simplified models of electromagnetic and gravitational radiation damping, Classical Quantum Gravity 18 (2001) 3573-3587; http://arxiv.org/abs/gr-qc/0105045
See also Alan D. Rendall's web site,
Subject: The correct answer to the wrong question
Actually, the “3+1” approach requires the knowledge of the data on a whole space-like hyper-surface which is not factual; similarly the “1+3” is not factual because it requires the knowledge of the data on a whole world line, i.e. also in the “future.” The reason why the splitting of spacetime does not and cannot produce any factual presentation of the physical reality can be explained by zooming on the nature of continuum (see above). When people ponder on GR and claim that "the simplest differential identities of the theory, namely the Bianchi identities, implied the existence of conservation laws", and then envisage "match between geometrical (Einstein tensor) and physical (energy-momentum tensor) quantities" (C. G. Böhmer, arXiv:0710.0752v1 [gr-qc]), they ignore a hidden, directly unobservable (recall the quark confinement and my prediction about LHC dated January 9, 2003) entity called here 'global mode of spacetime'. The impact of the latter on the quasi-localized constituents of the local mode of spacetime is "dark" in the sense that it comes from the Holon of the universe (cf. the Cosmological Principle above), hence it cannot be traced back to its origin. Only it is a bit too much: up to 96% from the stuff in the universe is considered "dark". Going back to the forest metaphor, each tree lives in the local mode of spacetime, and gets EPR-like corrections/contributions to its instantaneous state from 'the forest' (the Holon in the global mode of spacetime), due to which the whole forest exhibits wave-like pattern (read a story about a centipede here, and a note on the "dark" brain dynamics here). However, it is manifestly pointless to try to detect such quantum-gravitational "wave" with local interactions: the "amplitude" of the wave can only be dimensionless, as we know from QM textbooks. Besides, in order to eventually understand the quasi-localized gravitational "energy", we need to define the 'whole spacetime', from each arbitrary "point" up to "infinity", in such way that the "dynamic dark energy" will be poured into it. And because "any non-constancy in [lambda] would have to be accompanied by a compensating non-conservation of the mass-energy of the matter" (R. Penrose), the twice-contracted Bianchi identities can only be relevant to the current GR, as "a low energy effective field theory description of something else" [Ref. 1]. One question immediately arises: Can we unravel the 'entry points' of the "dark" impact from the Holon? Of course we can. They are made of "unparticles" [Ref. 2]. And that is not a poetry.
[Ref. 1] Assaf Shomer, A pedagogical explanation for the non-renormalizability of gravity, arXiv:0709.3555v1 [hep-th]. [Ref. 2] Howard Georgi, Unparticle Physics, arXiv:hep-ph/0703260v3.
Subject: The "boundary points" of asymptotically flat spacetime, Sec. 2.3 Note: Jörg Frauendiener explained above the "boundary points" of asymptotically flat spacetime as follows: he can attach boundary points to all null-geodesics, although the general recipe for making a geodesic is far from being understood; see the geodesic hypothesis in Alan Rendall's review above. More importantly, he stated, "these points together form a three-dimensional manifold that is smoothly embedded into a larger extended space-time." That may sound good, but is not good enough, for reasons explained by G.F.R. Ellis. Let me try to elaborate on the Conformal Infinity hypothesis. Imagine those "boundary points" as absolute zero "temperature": it has a finite value, but cannot be reached by any physical system. Further on the analogy breaks down, because all "points" on the numerical axis that go down to the absolute zero "temperature" are indistinguishable: if you happen to live in an asymptotically flat spacetime, any point from it will be as "close" to the "boundary points" as any other one. Check out the Cosmological Principle above. I believe the idea due to Aristotle -- the First Cause -- can be explored for understanding the "boundary points": if you travel with the "speed" of light, your proper time will be frozen, and you will enter the global mode of spacetime in which the whole universe is ONE. It seems to me that this is the most natural way to fix a 'numerically finite but physically unattainable boundary' at which the whole universe is ONE. It cannot be physically reached from the teleological local mode of spacetime, hence it seems as being placed "at infinity". But what is 'global mode of spacetime'? If you live in an asymptotically flat spacetime, it will be both "at infinity" and "inside the singularity", hence you will never reach it. See again the white area in Fig. 3.1 here. As Alan Rendall put it, "the study of these matters is still in a state of flux."
================= Subject: What precisely do we mean by a singularity anyway? Note: Take, for example, Steve Giddings' lecture "Observables in Quantum Gravity" (The Quantum Nature of Spacetime Singularities, KITP, January 8-26, 2007), Slide 3:
"Moreover, locality is only recovered in an approximation, and is in general spoiled by both quantum and gravitational effects. Thus locality is both relative and approximate." On September 13, 2006, I suggested to Giddings, Marolf, and Hartle to elaborate on the implications from their ‘pseudo-local’ observables for "GR astronomy", but I am seriously doubtful they have the guts to do it. To show that locality can be both relative and exact, I will try to elaborate on the expression "the metric is treated as a field which not only affects, but also is affected by, the other fields present" (see above). Imagine the metric field as a thermometer measuring the temperature in your room (isolated system with boundaries). Imagine also that the thermometer is very large, and the size of the room is so small that the thermometer affects the room temperature by measuring it. The dynamics will be non-linear, but to mimic the non-linear dynamics of GR we need to postulate that the thermometer and the air molecules ("the other fields present") engage in a fully "democratic" bi-directional "talk" (J. Wheeler). It is intuitively clear that such non-linear dynamics cannot be modeled with the linear time parameter from classical physics (see Pujol & Pérez). Not surprisingly, it tuned out that such non-linear "time" is not an observable in GR (see C. Rovelli), and because "energy is only conserved in situations where external forces are independent of time" (see B. Schutz above), we cannot have the luxury of 'simple localization of energy in GR', as is well known since 1917, thanks to Tullio Levi-Civita. My proposal from December 1999 for resolving this problem can be read here. Notice that the main idea used to define the "boundaries" of the universe -- the only truly isolated system -- is from Aristotle. Apart from some insults (some polite, some not), nobody has agreed to comment on my proposal. I cannot even post a manuscript at ArXiv.org server anymore, because nobody agrees to endorse it. The only paper I managed to submit was immediately deleted by the moderators of physics.GR: Don Marolf or Matt Visser. This is a hideous communist censorship. Period.
Note: Another excellent physicist who can certainly shed light on the origin of 'asymptotically flat spacetime' is Brett Bolen [Ref. 1], but I doubt he would be anxious to do it. In a universe dominated by DDE of [quantum vacuum], it doesn't make sense to talk about the constraint equations on the energy momentum tensor
But in a universe without any "dark energy" of [whatever] (see a textbook example here), it doesn't make sense to talk about the constraint equations on the energy momentum tensor either. And since we currently don't have any choice but to impose such constraints, we end up with a 'block universe' in which all GWs are dead frozen (cf. George F. R. Ellis).
Contrary to the common belief, the time read by your wristwatch is not an observable in GR, as stressed by Carlo Rovelli; we have to choose the lapse and the shift by hand, since they cannot qualify as 'observables' -- see Slide 29 from Brett Bolen's talk [Ref. 1]. Thus, there is no guarantee that LIGO's arms would play the role of GR clock that could "read" the dynamics of GWs. Don't bet your money on the artifacts from the so-called linearized approximation of GR. What can be done, then? Here's a money-wise proposal: Give LIGO Scientific Collaboration (discreetly, of course) 490 blank notebooks and sharp pencils, and leave them alone to sort out their recipe for 'asymptotically flat spacetime'. If they fail, we will convert those long, air-conditioned tunnels of LIGO to wine cellars, and won't waste more money for LISA. The buck stops with them. If they succeed, we'll get a full-fledged quantum gravity, possibly with quantum-gravitational "empty" waves whose dimensionless amplitudes won't be directly observable. As to those interested in the mundane affairs of classical GR, consider the simple question below (emphasis added), from John Coleman [Ref. 2]: if you cannot measure the direction of the tangent to the geodesic at each and every point from your walk, how do you know that you've been walking on a "curved" path? With respect to what?
If x is a point from a tangent line, the "geodesic" line in Minkowski spacetime will overlap completely with its tangent line, while in "curved" spacetime they can "overlap" only at the point x -- one-at-a-time. Thus, if you wish to talk about spacetime "curvature", you tacitly imply some meta-observer in the global mode of spacetime, who can observe two successive points/events from your geodesic simultaneously, then calculate the angle between their tangents, and say, -- 'Oh yeah, it is damn curved!' All I can suggest is that the point x is created/explicated as an QM "eigenvalue"; more here. If you can sort out this puzzle, I suppose you'll understand the lapse and the shift in ADM formalism (see again Slide 29 from Brett Bolen [Ref. 1], in Bolen.zip), as well as the mechanism by which each of the "leaves" of the foliation of spacetime are conflated/welded together, to produce the elementary step Et --> Et+dt . My wristwatch shouldn't be able to read this elementary step, yet it obviously does it. But how? The only answer I can suggest is that my wristwatch is somehow rooted on the reference fluid in GR, which nobody has found in GR, simply because it isn't there. As I acknowledged above, I've been struggling to understand GR since 1972. Hilbert and Einstein were also having problems, since they couldn't find the reference fluid either. Read Andrzej Trautman above.
D.C.
[Ref. 2] A. John Coleman, Whitehead’s Trilogy and the Curvature of Spacetime, arXiv:0704.2223v1 [physics.hist-ph], p. 7: "Einstein bases GTR on a vicious circle rather like "which came first, the chicken or the egg?". The confusion can be illustrated in different ways. For example:
======================
There is indeed a "center" of the universe, as explained above, but this "center" is nothing but the source [X] of the so-called Dynamic Dark Energy (DDE) in the global mode of spacetime. Once we define the "boundaries" of spacetime with the hypothetical two modes of spacetime, the reference fluid & Aristotelian First Cause act as a 'numerically finite but physically unattainable' cutoff which cannot be reached 'from within' the local mode of spacetime, in no circumstances: see the explanation here, and again the reformulation of the Cosmological Principle above. Thus, the crucial notions of 'relative size of the universe' and the relative and total densities of matter, in "energy" per "unit volume" (cf. WMAP Cosmology 101 in [Ref. 1]), acquire new meanings in the context of the Cosmological Principle above. I believe this is the only logical possibility for eliminating all misleading remnants from the primitive view of The Beginning as a conventional explosion originating from a central, absolute, and privileged point [Ref. 1] in the local mode of spacetime. The global mode of spacetime is crucially important for clarification of such naļve statements as 'the energy density of the quantum vacuum is infinite', because the alleged cutoff at the Planck scale (cf. S. Carroll, Slide 19, in [Ref. 2]) is utterly misleading. Although the Planck length and 'the upper bound on the volume of 3-D space', Vmax, can be calculated, they can never be reached 'from within' the local mode of spacetime, since they act as the ultimate "boundaries" of space which cannot be reached by any physical object. The numerical value of the Planck length is known; to calculate the other "cutoff", the upper bound on the volume of 3-D space, Vmax, calculate the future "size" of the universe, in which the relative vacuum density approaches asymptotically 100%. Hint from Ned Wright: "10 Gyr in the future the vacuum density will be 96% of the total density" [Ref. 2]. If, for some strange reason, you can't calculate Vmax, ask Ned Wright or Sean Carroll for help. I suppose there will be physical bans on reaching Vmax, analogous to the bans on reaching the absolute zero "temperature". Notice that Vmax is the crux of George F R Ellis' 'finite infinity' proposal: the cutoff Vmax is placed effectively at "null infinity", yet is very different from Geroch-Kronheimer-Penrose "ideal points", not to mention the old-fashioned conformal recipe for deriving "boundaries" of spacetime (cf., e.g., Steven G. Harris). Again, please do not bypass the text here and here. Notice also the conjecture below that immediately "after" The Beginning the "dark energy" of the Holon state of the universe (global mode of spacetime) increases at each moment 'now' from the cosmological time arrow. Think of the Holon as the brain of the universe (not "mind") which increases its memory about possible outcomes 'if A then possibly B' in each and every step of its non-unitary evolution along the cosmological time arrow. If it starts with some finite, albeit very small, value 'right after The Beginning', it will again define the Vmax of the universe at this stage. As an illustrative example, look at the circle below, and think of it as a 'finite, albeit very small, value of DDE right after The Beginning', and you'll see that it always contains an infinite (actual infinity) "number" of points (elementary steps of the cosmological time arrow), if counted from the local mode of spacetime. Hence the ratio of [the finite value of the circumference of the circle] to [the number of points in its local mode of spacetime] is always a "constant" that cannot taka any precise value, because it is "infinite". It is this ratio that defines the ever-increasing Vmax of the universe: both the potentialities, kept in the Holon, and their explications in the local mode of the universe change/evolve it time by getting "richer". A meta-observer placed in the global mode of spacetime 'right after The Beginning' may witness the evolution of the scale factor of the whole universe, and "observe" that the current volume of 3-D space is "very small" (e.g., "only a few kilometres across"), but a physical observer locked inside the local mode of spacetime will again "see from inside" an unlimited 3-D space, because she/he can never reach the "boundaries" of the 3-D space, as determined by its current "cosmological horizon" Vmax , to compare her/his "scale factor" to that of a meta-observer. This is because the "number" of geometrical points, explicated from [phi], is always 'the same' in the local mode of spacetime, regardless of the current "size" of 3-D space, as viewed by a meta-observer. Thus, from the perspective of a meta-observer, the Holon "brain" of the universe, which keeps its "dark energy", will indeed increase along the cosmological time arrow, starting with a small but finite value 'right after The Beginning', and after 10 Gyr in the future the vacuum density may very well be "96% of the total density", as Ned Wright speculated [Ref. 2]. The upper bound on the "dimensions" of the universe, Vmax , should be related to some finite but "immensely huge" diameter of the universe, which in turn corresponds to 'vacuum density approaching asymptotically 100% of the total density'. But all the local observers locked inside the local mode of spacetime will not be influenced by the evolution of Vmax , since there is no way for them to find out that their tables and chairs have been "stretched" by the "expansion of the spacetime metric". Only some privileged observer endowed with the faculty of an 'extended now' could keep track of the cosmological time arrow, and eventually observe "online" the "waves" of the spacetime metric. Such privileged observer is explicitly excluded with the Hamiltonian formulation of GR (cf., e.g., C. Kiefer, gr-qc/9308025, pp. 5-9), because he would have to 'act on itself', and since the poor guy can't perform such miracles, his proper time is "frozen" (the Hamilton constraint problem), and cannot observe the quantum reality either. As to "GW astronomy", it is 'not even wrong' to think of GWs propagating like photons (say, from "the center of the Galaxy" towards the Earth -- cf. Marie-Anne Bizouard et al., gr-qc/0701026 v1), because GWs pertain to 'the whole spacetime', and the "center" of their origin [X] is 'spread over' the whole spacetime as well. Surely the "waves" of the metric should start to unfold from particular object located inside 3-D space (much like "from a central point", [Ref. 1]), but they also "cover" the whole spacetime en bloc, just like the expansion of spacetime metric after The Beginning [Ref. 1], hence GWs cannot be detected with any 'local theory of GW detectors', as explained here. Put it differently, if you can take the stand of some meta-observer (cf. above), you can certainly notice that every fish from a shoal of fish follows a modified trajectory (called in textbooks 'geodesic line'; cf. L. Landau and E. Lifschitz, The Classical Theory of Fields, Pergamon Press, Oxford, 1973, Sec. 87) due to its quasi-local "holomovement", but because with the 'local theory of GW detectors' you can't have access to such meta-observer, you and LSC will "detect" nothing but the dipole mode. Now, since nobody agrees even to comment on these proposals, may I suggest to the reader (Adam) to follow the steps below and find out whether he has any other choice. Look at the spacetime "curvature" above, and recall that GR itself tells you nothing about the topology of 3-D space, hence you can't get even a glimpse of its "total energy". Where would you "insert" the source [X] of DDE, and how would you modify the lapse and the shift (Brett Bolen) to obtain an asymptotically flat spacetime, such that its metric can be stretched by the remnant from the "inflation", denoted with DDE [Ref. 2]? Before you begin, please consult Pankaj S. Joshi, and keep in mind that you need a brand new degree of freedom to make the "dark energy of [X]" itself dynamical: unless you're a member of that joyful LSC, you can't say that DDE creates time, and at the same time evolves in that same time. See the dualistic conception of time here. Moreover, to solve the so-called coincidence problem [Ref. 2] -- why are the densities of matter and vacuum of the same order precisely today? -- without introducing CRAP (Completely Ridiculous Anthropic Principle, Martin Gardner) or some post hoc postulated scalar fields (cf. T. Padmanabhan), you may need to endow DDE (vacuum energy density) with the ability to be both 'extremely small' or effectively zero (cf. Ed Witten) in the local mode of spacetime (the energy needed for the elementary timelike displacement), and infinitely large in the global mode of spacetime, as the source of 'the ultimate free lunch'. Namely, the vacuum energy density could start with an infinitesimal or 'vanishing small' value in the global mode of spacetime right "after" The Beginning, and then increase its value in the global mode of spacetime, as 'the ultimate free lunch'. No need to worry that "10 Gyr in the future the vacuum density will be 96% of the total density" [Ref. 2], since it may be infinite in the global mode of spacetime, yet its localized, or rather actualized, fraction will be always 'extremely small' -- just as small as needed to show up in the local mode of spacetime as the elementary tick-of-time-and-shift-in-space. This is the ultimate benefit from working with 'potential reality'. Try to live without it, and you may lose your sound night sleep, like Ed Witten. Anyway. Since I mentioned Max Plank above, it is worth to recall a story about a chat he has had with one of his sons in 1904 (I suppose it was Erwin Planck, but I may be wrong). At that time Max Plank had made all efforts to refute his own hypothesis, and told his son that he believes that something very important could evolve from his 'elementary quantum of action'. But he had to wait until the advent of Quantum Mechanics to see the final confirmation of his "ugly" hypothesis. It goes without saying that my speculations are no match for the genius of Geheimrat Max Plank. The only similarity is that I am also doing all efforts to rebut my hypothesis, and will deeply appreciate the professional help (not insults) from the readers of these lines. Unlike people who promote their ideas, I literally fight with mine, each and every day. I also had a chat with one of my kids, after which I offered the following 'explanation for pedestrians' of what I call potential reality: It may not be easy for a teenager to grasp the complexity and richness of the Aristotelian connection operating at the final frontier of the physical world, which we perceive as "geometry"; Alice had also problems to understand it. Notice a very important issue above: the center of the balloon is spanned everywhere and evenly on the balloon surface, in no time. From the perspective of a material 3+1-D world, the phrase 'in no time' means a transcendental tachyon from a 1+3-D world (cf. below, reference here), which is an extreme case in a mirror tachyonic world. Its counterpart in a mirror 3+1-D world would be an equally extreme case of an object which is at absolute rest, in all reference frames. Notice also that these two extreme cases are indistinguishable, because if an object can be absolutely everywhere in no time, there is no place left to move into, so it will be at absolute rest as well.
In modern jargon, the quasi-local gravitational energy cannot be positive definite, although there is no "obvious conflict" with the positivity of the classical energy (cf. Geoff Hayward, gr-qc/9403039 v1, p. 3, and ref [2] therein). How does Mother Nature eliminate all "obvious conflicts" with the positivity of the classical energy? Does She convert the "negative mass" into a free DDE lunch? How does the "negative mass" fit into this picture? Perhaps Fred Cooperstock can shed light on this crucial DDE issue. The math may look simple, but it only suggests that some sort of 'cutoff' must exist. If we look at it "from the train", such "global time outside the train" corresponds to a kind of "global space" which covers the whole universe as ONE -- its spatial dimensions would seem to be both extremely small or "infinitesimal" and extremely large, as denoted with Vmax above. But if you take a walk "outside the train", you will simply enter the "timeless" mode of the human self, as manifested in the "frozen" time in canonical quantum gravity. The story goes back to the 1967 Wheeler-DeWitt equation. Any time people talk about the 'current size of the universe', they refer to the current Vmax, which implies a meta-observer in the global mode of spacetime (outside the train), who can monitor the genuine dynamics of spacetime and can witness the dynamics of "spacetime curvature", by measuring the dynamics of the angle between any two tangents (see above). There is no trace in GR from the "proper time" of his 'global watch', of course. More here, here, here, and here. Is the Aristotelian connection difficult? Maybe, but do we have a choice? Let's compare it to the established speculations about the origin of time and space. Andrei Linde, a well-known Russian philosopher and student of David A. Kirzhnits, has just posted (2 May 2007) a review article, entitled: "Inflationary Cosmology" (arXiv:0705.0164v1 [hep-th]). We read in the abstract that A. Linde offers "a general review of the history of inflationary cosmology and of its present status". Just a quote:
If you, my dear reader, don't accept such parapsychology (see another example here), notice that the "total duration" of "inflation" (if any), as measured in the local mode of spacetime, could be infinite, tending asymptotically toward The Beginning; some simple math can be accessed from here. There is no direct access or 'short circuit' between the two modes of spacetime: nothing in the teleological local mode of spacetime can reach the final layer of Aristotelian connection. Try it. A Chinese proverb says: "When the student is ready, the Teacher will appear." By the way, the "teacher" might look just like you.
[Ref. 1] Philip Gibbs (1997), Where is the centre of the universe? "In a conventional explosion material expands out from a central point. See also: WMAP Cosmology 101: What is the Universe Made Of?
"The top (red) curve shows a universe in which a large fraction of the matter is in a form dubbed "dark energy" which is causing the expansion of the universe to speed up (accelerate). There is growing evidence that our universe is following the red curve."
"The animation above shows the piston moving in the cylinder filled with a "vacuum" containing quantum fluctuations, while the region outside the cylinder has "nothing" with zero density and pressure. Of course the politically correct terms are "false vacuum" in the cylinder and "true vacuum" outside, but the physics is the same. "If, on the other hand, the vacuum energy density is zero, then it is always 0% of the total density and the current epoch is not special." See also:
Ed Witten on M theory, supersymmetry and appreciating calculus Q: How can the cosmological constant be so close to zero but not zero? More from The Ambrose Swasey Professor of Physics and Chairman of the Physics Department of Case Western Reserve University Lawrence Krauss.
No stellar black holes, no 'intermediate-mass' black holes, and no supermassive black holes exist in Nature. By the same token, there is no "massive object" inside the dark galaxy VIRGOHI 21. Much like the quarks, the Holon cannot be directly observed. Surely its global imprint on the observable universe can be detected, but why use those ugly Bushisms? Take another example of those fictional "black holes", known as 'rotating black holes' or 'Kerr black holes'. If you read Encyclopędia Britannica, you will be struck by the following statement regarding the author of these "rotating black holes", Roy P. Kerr: "New Zealander mathematician who solved (1963) Einstein's field equations of general relativity to describe rotating black holes, thus providing a major contribution to the field of astrophysics." Now, read what Roy P. Kerr [Ref. 1] says on his "black holes" (emphasis added), and notice that they are "still a mystery after more than four decades". Why? Because in present-day GR the Holon and its effects are totally unrecognized, hence you'll reach "negative mass" or geodesic incompleteness. While if you interpret the Holon state of the whole universe as Aristotelian potentia, in line with PR interpretation of QM, you will never actually hit any "negative mass" or "geodesic incompleteness". The effects from the Holon will be very real indeed, but they will always be cast in the local mode of spacetime as some "dark" effects of the Holon. These effects are "dark" in the sense that the Holon itself cannot be reached from the local mode of spacetime, just as you cannot observe 'the quantum state per se' as Aristotelian potentia in the local mode of spacetime. Briefly, if you believe that GR should describe only 'objective reality out there', you will inevitably hit insurmountable problems. For example, you will have to make a 'short circuit' between positive and negative mass, as in the case of Kerr's problem [Ref. 1]. Hence the only choice you have is to examine closely the fundamental object in GR, which is outside the applicable limits of GR: the so-called "point". To get back on academic track, please read the question posed by Sean Carroll above: "We know that virtual particles couple to photons (e.g., Lamb shift); why not to gravity?" Because gravity couples everything that is 'not-gravity'. Hence the latter becomes 'self-coupled' and 'self-acting', just like the human brain. Unlike all type I matter fields constituting 'non-gravity', the gravitational field itself has no proper energy-momentum density. Thus, the energy-momentum for all 'non-gravity' systems -- self-coupled via gravity -- is inherently quasi-local. This fact can be understood as a consequence of Einstein’s equivalence principle: read Chiang-Mei Chen et al., arXiv:0705.1080v1 [gr-qc], p 1. The "dark energy" comes from the Holon, and not from the gravitational field itself. Sean Carroll, however, belongs to the club of 'GW scientific communism'.
pp. 24-25: "However, it should be remembered that this radius is purely a coordinate radius, and that there is no way that the final stage of such a collapse is that all the mass is located at the singularity.
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I hope this answers the question above.
============== Subject: Re: "... it is absolutely unclear how to exclude such singularities from the theory ..."
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Note: Last year, I made a suggestion, which was quickly rejected by Serguei Krasnikov as "poetry". The facts that need explanation are presented here; the update on my suggestion can be read at the links from my recent email to S. Krasnikov above. Very briefly, my guess from Thu, 16 Mar 2006 14:27:41 +0200 about UFO dynamics was about [quote] "a modified geodesic that is "partly" in the 3-D space, and "partly" in the global mode of spacetime" [unquote]. First, notice that UFOs evade all known physics of 'inertial mass': their dynamics resembles that of our thoughts. As to the idea about 'modified geodesic', it can be explained as follows. Suppose you have to drive your car along a distance of 100 m, and have divided the path into 100 "steps" of 1 m each. According to the principle of locality, if you start from step 0, you can reach step 2 only after you've passed step 1, etc. An UFO does the same, only the guys there can divide the same distance intro, say, 10 steps, each of which will be again of 1 m to them. Thus, the UFO will also start from step 0, and will reach step 2 only after it had passed step 1, but to us it will pass not 2 but 20 meters. Perhaps all we need is to modify the metric of spacetime and eliminate (reversibly) the quality of matter called 'inertia'. Looks like some people had done it. That was my guess. I don't mind if someone calls it "poetry". In my opinion, all those "singularities" examined by S. Krasnikov in his latest arXiv:gr-qc/0611047 v2 are indeed Russian poetry. I talk about facts that nobody can explain. He talks fiction that hasn't happened in the past 13.7 billion years: even one case of "geodesic incompleteness" and timelike naked singularity would have created an enormous pathological domain in 3-D space, much worse than a cancer tumor. What matters to us are facts, right? I suppose the reader would wish to ask, 'but where's da beef?'. Read this web site. I've been working on it for over ten years now, and I think it provides a definite proof of the effect of the Holon: read my email below. I am, indeed, just a psychologist, and my math is limited to PDE only. I have never studied tensor calculus, differential geometry, or topology, and never will.
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Subject: Beyond Partial Differential Equations, math7999-01pdf.pdf, p. 2
Note: The non-local initial conditions, mentioned by Horst R. Beyer above, are inevitable in the case of non-linear time, which is in turn inevitable for Einstein's GR: no classical "energy conservation" laws [Ref. 3] are possible due to the non-linear time. If we don't resolve the latter, we may never get rid of the "dark energy" and the generic pathologies of the current differentiable manifold [Ref. 4]; more from Matt Visser and Celine Cattoen here. Notice that the "isolated system" here is the local mode of spacetime (positive mass universe with 3-D space), wrapped by, and "isolated" from, its global mode, -- thanks to the Aristotelian First Cause. Thus, the two modes of spacetime are rooted on the 'ideal monad' and evolve jointly along the cosmological time arrow. Thanks to [phi], their evolution is also manifestly non-unitary. In quantum cosmology, 'unitary evolution' simply doesn't make sense: read John A. Wheeler. All this is quite complicated, so let's ask a simple question about the "local gravitational energy-momentum", which is, as we all know, "searching for the right answer to the wrong question" (C. Misner, K. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco, 1973, p. 467). In the local mode of spacetime, you can easily make it "quasi-local" [Ref. 5]. Why? See a hint for producing quasi-local "eigenvalues" here. It is not a 'simple localisation', as stressed by Tullio Levi-Civita in 1917, because we can play with "simple localisation" only if we can totally ignore General Relativity and fix an "isolated system", to teach kids the meaning of time-and-energy [Ref. 3]. If we wish to teach kids General Relativity, like Bob Wald does, we should first fix a brand new kind of "boundary" with the Aristotelian connection. As of today, we do not know the dynamics of GR. "It seems to me that the concept of probability is terribly mishandled these days. Probability surely has as its substance a statement as to whether something is or is not the case -- an uncertain statement, to be sure. But nevertheless it has meaning only if one is indeed convinced that the something in question quite definitely is or is not the case. A probabilistic assertion presupposes the full reality of its subject." It goes without saying that the Aristotelian potential reality has a totally different nature. Henry Margenau, for example, called it Onta, while Karl Popper called it propensity. Strangely enough, it was John Wheeler who provided a beautiful example of Margenau's Onta, yet he refused to comment on it. More on the issue of 'ontological potentiality' from Christian de Ronde, arXiv:0705.3850v1 [quant-ph], Sec. 3.4 and p. 16, and ref. [44] therein. Also in 1989, the experiments by Fleischmann and Pons were totally rejected, and tons of sleazy insults were poured over them. Now the case is different, since the "undisputable evidence of their nuclear origin" has been confirmed [Ref. 6]. This gives you a glimpse of the attitude of the established scientific community to brand new ideas -- they deeply hate anything that would make them rethink their textbooks. If you ask tough questions that are on the table since 1917, chances are you will also get only a dark silence and insults (some polite, some not). But if you ask nice questions, they will be more than happy to answer. Here's an example: How long would it take an average cow to fill the Grand Canyon with milk? The first job would be to divert the Colorado river, of course. Then the established scientific community will quickly provide the tantalizing answer: read NewScientist from 5 May 2007, p. 93. D.C.
============== Subject: Time in Quantum Theory Note: I put strong emphasis (bold text) on the expression "unitarily evolving global quantum state" above, because if you follow the linked words, you will see what 'global quantum state' means. It cannot evolve "unitarily", because we can use probabilities only and exclusively only if we are dealing with facts, while the 'global quantum state' is Margenau's Onta (also Aristotelian 'potentia' -- see above). Thus, when you observe, say, an electron with particular spin, you see a fact that has been already cast in your past light cone, before you've looked at it. The probability, then, for observing the electron there is unity, while the probability for observing that same electron 'anywhere else in the universe' is zero. The 'global quantum state', however, has not been "collapsed", because it does not evolve, and has never evolved "unitarily". If you disagree, try to trace back the history of the 'global quantum state' on your past light cone, starting from its "collapsed" (or whatever you call it) state. More on this Gedankenexperiment here. This should be a fully legitimate exercise in relativistic QM, because you've surely read in textbooks of non-relativistic QM that "the background Newtonian time appears explicitly in the time-dependent Schroedinger equation" (reference here), and now you are assured by H. D. Zeh that "in non-relativistic quantum mechanics, the time parameter t that appears in the Schrödinger wave function ... is identified with Newton's absolute time." So, if H. D. Zeh wishes to talk on time in quantum theory, the first off task will be to make a relativistic QM out of "Newton's absolute time", so that we can trace back the history of the 'global quantum state' and its cat states, as you might have guessed. If he can't solve this first off task, many statements in his essay "Time in Quantum Theory" may be wrong, starting with his claim about "the time parameter t that appears in the Schrödinger wave function." H. D. Zeh claims (p. 4) that "in the theory of relativity, proper times assume the role of Newton's absolute time for all local systems, that is, for those approximately (notice the poetry - D.C.) following world lines in spacetime. Quantum states are generically nonlocal (they do not define or consist of local subsystem states)." What, then, is the time in quantum theory? You need relativistic QM, but the "time parameter" in it will be entirely different. Try it with your own brain here. Dead matter makes quantum jumps; the living-and-quantum matter is smarter. Please read Erwin Schrödinger above, and recall "that sharp time" in his 1935 paper: "... the special treatment of time forms a serious hindrance to adapting Q.M. to the relativity principle, is something that in recent years I have brought up again and again, unfortunately without being able to make the shadow of a useful counterproposal." The main obstacles are in the following: To sum up, if we wish to speak about 'time in quantum theory', we should never abuse QM by imposing 'the time of facts' from Newtonian mechanics or STR. Here's why. 2. Back in 1935, Erwin Schrödinger wrote (Die gegenwärtige Situation in der Quantenmechanik, Sec. 8, Theory of Measurement, Part One): "The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has. But then what does it mean?" The rejection of classical realism in (1) is inevitable. If we wish to talk about 'time in quantum theory', we must seek a new kind of time that would match the nature of quantum objects. There is, of course, a much broader manifestation of reality, which can be called 'potential reality'. Hence the realism can be restored, in both quantum and gravitational realms. In other words, the question of whether the quantum state represents reality or our knowledge of reality (e.g., Harrigan & Spekkens) is seriously befuddled: the quantum state is 'potential reality' which is both reality and "knowledge" of the whole universe about this reality, kept in the "brain" of the universe -- the Holon. No mental concepts (such as 'knowledge' or 'imagination') are admissible in the ontology of quantum reality. Do not mix apples with oranges. First, it is very instructive to recall what these quantum objects are not, by referring to Kochen-Specker Theorem and Conway-Kochen Theorem. A simple example is the famous tripod of Ernst Specker. As Karl Svozil eloquently explained, the "legs (corresponding to elementary propositions) appear differently colored, depending on the particular tripod they are in!" Thus, the phrase "an incomplete Kochen-Specker coloring" (Helena Granström, quant-ph/0612103 v2) has no physical meaning whatsoever. And if you subscribe to the quantum mysticism -- "the quantum state is not a physical object, it is a representation of our state of knowledge, or belief" (Itamar Pitowsky, quant-ph/0510095 v1, pp. 26-28) -- you will wind up in a schizophrenic state of, say, 68% "knowledge" of the quantum state, and 32% of "[what da hell is that uncolored KS sphere?]". Therefore, the legs themselves, as identified by their color, do not exist as 'objective reality out there' but are 'context dependent', after KS Theorem (see also John's jackets metaphor here). Recall also that an entangled quantum system does not have any "individual parts" that can be identified by their "individual properties" (cf. Yanhua Shih; Ghirardi & Marinatto). See also the problem of quantum state "identification" in Svozil & Tkadlec: "we cannot solve the type of trivalent decision problems as discussed above by a single query", and in Karl Svozil's quant-ph/0206076 v6, p. 4: "this ambiguity (known since 1935 - D.C.) gets worse as the number of particles increases."
Again, if we wish to talk about 'time in quantum theory', the first off task is to reveal the new kind of time pertinent to the unique nature of quantum objects as Aristotelian potentia (see above). Then of course we have to reconcile it with the kind of time pertinent to 'objective reality out there', from (1). The latter (called local mode of time) bears two key features: (i) a macroscopic object possesses two or more macroscopically distinct states, and (ii) it is possible in principle to determine which of these states the system is in, without any effect on the state itself or on the subsequent system dynamics (the principle of noninvasive measurability). The quantum objects exist as Aristotelian potentia (see above), which we call for brevity 'potential reality. Being a reality 'out there', it conforms to the principle of noninvasive measurability, but in this case the principle applies to an UNspeakable holistic object, which keeps 'the context' (after KS Theorem), hence cannot be explicated entirely by a denumerable set of 'localized projections'. For example, the Schrödinger cat per se will never undergo any "collapse" by casting one of its possible states (either |alive cat> or |dead cat>) in the local mode of time. This key feature of 'potential reality' offers a possibility, at least in principle, to develop a proper Relativistic QM: the Schrödinger cat per se is just as 'real' as a macroscopic object, only it evolves non-unitarily. It is also "spread" in the global mode of time, which, if viewed from the perspective of the local time, would "cover" a domain of space literally "in no (local) time". In other words, different observers, in their inertial frames from the domain of space "covered" by the UNspeakable quantum state, will have identical interpretations of the same potential reality explicated in the local mode of time -- they all will observe correlated events from the same EPR-like entangled system, which will occur simultaneously in their common global mode of time. However, they wouldn't be able to verify the simultaneous explications of the entangled state in their local mode of time, hence Eberhard's Theorem wouldn't be violated: the causal relations 'the quantum state per se --> its local explication' do not run in the local mode of time. Locally, all explicated states will be correlated like a shoal of fish swinging along a coral reef, but each and every "fish" will proceed to its next EPR-like correlated state smoothly, without any "quantum jumps", and by fully obeying the principles of locality and relativistic causality. Locally, no "fish" can observe the holistic quantum state per se, which correlates, literally "in no time", the whole shoal of fish/domain of space. Having established such correlation in the global mode of time, all "fish" will inevitably exhibit wave-like behavior, but the origin of their "quantum wave" can never be found in the local mode of time, as we know since the inception of QM. NB: Put it differently, the "quantum jump" and its alleged "uncertainly", which we can tackle in the local mode of time only with the Born rule (the 'shut up and calculate' interpretation of QM), are artifacts from the measuring device. If the latter could access the global mode of time, it would "measure" the Holon state of the global quantum system as well (compare it with the "extended" moment 'now'), hence would not detect any jerky movements, but a perfectly smooth holomovement, much like an EPR-like correlated shoal of fish.
To explain this proposition (not endorsed by Chris Isham), imagine four dice that are EPR-like correlated in such way that the sum of the readings from all dice, explicated in the local mode of time, must be confined in the closed interval [12 - 18]. Let's say we start with the four dice on the table (=quantum phase space) in configuration {5, 4, 6, 2}. They are "shaken" and EPR-like correlated in the global mode of time. Also, each dice has equal "rights" (Popperian propensity) to negotiate its next state with the rest of the dice, in line with the rules of 'relational reality' -- no "background" nor "classical limit" are needed. The only requirement, again, is that the sum of their readings has to be in the interval [12 - 18]. Suppose they have negotiated their next states as 'facts' to be {3, 5, 5, 1}. Now, remove the "dark gaps" of negotiation in the global mode of time, and build a trajectory of "points" exclusively from the already-correlated states (=an operational definition of 'local mode of time'). Each and every such point is an already-correlated joint state of the "shoal" of dice: STR (cf. Kevin Brown) does not allow you to witness 'online' the negotiation of the four dice in the global mode of time. Also, because they behave as a whole (cf. the forest metaphor here), a "quantum wave" pattern will be created, but again you got to have access to the global mode of time to "see" such wave. If instead you use some dumb unanimated measuring device, it will inevitably "collapse" the wave (von Neumann's Process I), and then you'll be deeply puzzled by the apparent "instantaneous" correlation of all dice. Hence all you could suggest would be a recipe for calculating the "probability" for observing the next correlated state of the dice by yet another "collapse", as well as some poetic expressions like 'peaceful coexistence of QM and STR'. Needless to say, the real quantum dice are not "uncertain" but flexible, just like your arm. We can also think of Heisenberg position-momentum relation as a genuine flexibility of the quantum world 'out there': if a quantum object has been pressed to choose from a narrower spectrum of is potential states of position, it will be compensated with a wider spectrum of potential states of momenta, and the end result will be, again, a perfectly continuous (albeit hidden to unanimated macroscopic measuring devices) quantum trajectory of allowed states, each of which will obey the Heisenberg flexibility principle. In other words, instead of noncommutative geometry, we can now use the global mode of spacetime. Nothing can shrink the spectrum of potential states of 'here-and-now' (position & time) to a singular, pre-determined future state (the quantum object would then follow a classical trajectory like a Frisbee), because its "complementary" spectrum of potential states of 'energy & momentum' would have to be infinite. Neither infinite nor singular potential spectrums are allowed due to the Heisenberg flexibility principle, hence the quantum objects are flexible and "smart" in building their unique quantum trajectories. Dead matter makes quantum jumps; the living-and-quantum matter is smarter. This is the motto of Potential Reality (PR) interpretation of Quantum Mechanics. It is also the Proper Relativistic (PR) interpretation of Quantum Mechanics. Briefly, PR2 interpretation of QM. The crux of PR2 interpretation of QM is that we can think of an individual quantum object in a dual way: on the one hand, it is 'potential reality out there' which keeps the elementary tick of time of 'the universe as ONE', hence it "carries" the moment 'now' along the universal/master cosmological arrow. On the other hand, it can be explicated only and exclusively only by its localized "projections" in the local mode of time, which fully obey the principle of relativity: The Beginning is really hidden in the local mode of time, by being multiplied as an infinite (actual infinity) localized "centers" of the same universe. These localized states of the type {5, 4, 6, 2}, {3, 5, 5, 1}, etc., "happen" in a domain of space governed by 'the shoal of dice', hence the question 'in what moment now did the "collapse" happen?' is meaningless, as required in STR. Why? Because the "absolute motion" of the global quantum object (viewed as potential reality correlating the whole space domain en bloc and evolving along the global time) is unobservable in the local mode of time. It does exist, but only in the "dark gaps" of the Aristotelian Connection. If you compare a quantum trajectory in such brand new 'quantum phase space', build with the rule [12 - 18] (see above), to a classical trajectory of a Frisbee in its phase space, the former would look like doing "quantum jumps", but there will be no jumps whatsoever in such quantum phase space: what might look like a "quantum jump" to a Frisbee will be a perfectly smooth transition from one quantum point to the "nearest" one, that is, from {5, 4, 6, 2} to {3, 5, 5, 1}, etc. Just like in GR, we don't have a fixed grid of "points" in the (yet to be discovered) quantum phase space, because the continuum of quantum "points" is being re-actualized (or re-created, if you prefer) from the global quantum object or 'potential reality'. It is a genuine continuum, but the dynamics on it will be very different from that of macroscopic matter, as I tried to explain with the Gedankenexperiment with four entangled dice (see also the speculations on UFO dynamics above). Briefly, we can avoid all those seemingly "non-local" interactions and "instantaneous" or "faster-than-light" correlations in the local mode of time, because they are effects of the holistic Aristotelian potentia living in the global mode of time. We are not dealing with facts but with a different kind of reality, which is outside the applicability of probabilistic calculus, as stressed by Erwin Schrödinger on November 18, 1950 (see above). NB: Perhaps the clearest way to understand the PR2 interpretation of QM is to compare it with Reichenbach's Principle of the Common Cause (reference and discussion here), and with Henry Margenau's Latency Interpretation of QM: the tendencies of 'latent observables' to take on “I propose a shift of attention. The contrast, or at any rate the difference, is now between (…) possessed and latent observables. Possessed are those, like mass and charge of an electron, whose values are “intrinsic”, do not vary except in a continuous manner, as for examples the mass does with changing velocity. The others are quantized, have eigenvalues, are subject to the uncertainty principle, manifest themselves as clearly present only upon measurement. I believe that they are “not always there”, that they take on values when an act of measurement (...) forces them out of indiscriminacy or latency." In other words, the act of localization in the local mode of spacetime not only brings into physical existence the value of the latent observable in question, but also the latent observable itself. Prior to localization, we do not have some stuff in brackets, |XXX> , such as |alive cat> and |dead cat> , in the local mode of spacetime. Prior to localization, the 'cat per se' exists in the global mode of spacetime as Margenau's latent observable and Aristotelian potentia. In the example with the four dice, the 'dice per se' is always in the global mode of spacetime, while its localized 'projections' are always cats in the local mode of spacetime -- one-at-a-time. If we compare it to the standard QM jargon: in the local mode of spacetime, the quantum state 'the dice per se' is always in the "appropriate" eigenstate which ensures that the value [number of dots] of the observable [dots] matches the requirement [12, 18]. Again, if compare it to the standard QM and postulate eigenvalue-eigenstate link (the quantum system has a 'property' iff the quantum state [psi] is an eigenstate of the property’s operator), then the PR2 interpretation of QM says that the system has always a 'potential property' in the global mode of spacetime, which is ready to be actualized upon localization/observation by taking a correlated (if needed) eigenvalue-of-the-eigenstate: one at a time. There are no eigenstates with possible eigenvalues wandering in the local mode of spacetime of the quantum world, firstly, and secondly -- the flexibility of the quantum system, exhibited in the dynamically created quantum phase space in the local mode of spacetime of the quantum world, cannot be displayed in the world of fixed facts ('objective reality out there'), hence the latter imposes an abrupt and indeterminist "quantum jump" -- an inevitable artifact from the (inanimate) measuring device operating at the length scale of the world of tables and chairs. Briefly, if we use the current eigenvalue-eigenstate link, there is no way to make a relativistic QM, for reasons explained by Schrödinger above. Prior to localization/observation, the quantum system does not exist in some superposed or entangled state in the local mode of spacetime. The late Jeeva Anandan also stressed that the "simplest, though dramatic, statement of the measurement problem in quantum theory is that quantum theory does not explain the occurrence of events. So, quantum theory does not explain the first thing we observe about the world around us." I think we should change our QM textbooks (not STR), by underscoring the possibility for many artifacts from the (unanimated) measuring devices, and stop wasting time and money for "quantum computing". There could have been a chance to develop some quantum computer iff the quantum system as Aristotelian potentia were confined entirely in the Hilbert space. No way. The Hilbert space is custom-built to satisfy the requirement that "the probabilities for an exhaustive set of mutually exclusive (or classical - D.C.) alternatives should add up to one. In quantum mechanics, this is generally ensured by using an instant of time to specify such (classical - D.C.) alternatives" (A. Ashtekar). Thus, the Hilbert space is pathologically dependent on some external agent, and therefore has to be replaced by a dynamical entity -- the quantum phase space in the local mode of spacetime. It goes without saying that the dynamics of such brand new quantum phase space cannot be linear and unitary, because it is being determined by/in the global mode of spacetime. One last word about Bell's inequality. James Franson mentioned that, as a graduate student at Caltech, he had taken Richard Feynman’s class on quantum mechanics, and "one of the students asked Feynman if he would explain Bell’s inequality. Feynman’s reply was “There is nothing to it – I will explain it all later”. But he never did." There is really nothing special in Bell's inequality, because it is based on counterfactual statements. The whole story can be elucidated in the PR2 interpretation of QM, without any counterfactual headaches. We simply continue from the point at which Abner Shimony admitted (Bell's Theorem, Aug 22, 2004, Stanford Encyclopedia of Philosophy): "... the domain governed by Relativistic locality is the domain of actuality, while potentialities have careers in space-time (if that word is appropriate) which modify and even violate the restrictions that space-time structure imposes upon actual events. The peculiar kind of causality exhibited when measurements at stations with space-like separation are correlated is a symptom of the slipperiness of the space-time behavior of potentialities. This is the point of view tentatively espoused by the present writer, but admittedly without full understanding." For example, Jacques Mallah wrote, regarding a Stern-Gerlach device (SGD), the following (emphasis added): "Consider an ideal measurement of the Z-component of the spin of an incoming spin-½ particle by a SGD. This will be described for the cases in which the incoming particle is already in an eigenstate of the Z-component of spin with the following notation: ... " But the "incoming particle" is already in a unique and pre-correlated "eigenstate" (if you prefer QM jargon), as explained with the four dice above. Apart from counterfactuals (cf. K. Svozil, arXiv:0711.1473v2 [quant-ph], Sec. 1), there is nothing to Bell's inequality. Suppose you keep a white ball in a box, and anytime you open the box, you see it there. If you are interested in the color of the ball, it will certainly display it for you -- instantaneously, and with unit probability. This is a non-contextual (in the sense of KS Theorem) measurement, because the ball exists 'out there', therefore such case of observation would totally contradict the spirit of QM, after Schrödinger. Now, suppose the color of the ball represents a quantum observable, such that its "state" could be either white or black. Also, suppose the ball is EPR-like entangled with the same kind of a ball, under the condition that if one of the balls show up as 'white', the other will instantaneously, and with unit probability, show up as 'black'. So, once you observe one of the balls as white (or black), you can be dead certain that the other ball has already displayed its state as black (or white). Then you put all this on experimental test, confirm it, and what do you do next? You visit your Ministry of Defense and apply for research grant for quantum computing, or else the "bad guys" will make it first, and then might break all your vital secrets. And of course you get all the taxpayers' money you've asked for. Simple, no? Wrong. You need to employ the quantum entanglement, in order to manipulate it locally. But to get this task done, the quantum entanglement would have to actually fix the color of the other, unobserved by you, ball: at the very instant at which you fix your ball as white, the quantum entanglement will have to "obey your command" and instantaneously convert the other, unobserved by you, ball into black, and vice versa. And because this is actually one single event, which pertains to the entangled system as 'one entity' that shows up instantaneously, and with unit probability (an entangled "system" does not possess any "individual parts"), you will gain the power to manipulate it locally, at the length scale of tables and chairs, and will also map the intrinsic time of the quantum system to the one read by your wristwatch. If you manage to achieve this miracle (e.g., D.L. Khokhlov, A scheme of supraluminal telegraph, arXiv:0801.0528v1), you will literally employ the quantum entanglement. And of course ruin KS Theorem and the whole Quantum Mechanics, because the state/color of the unobserved ball will exist 'out there', disregarding the experimental context -- just as in the first example above. You will also ruin the whole STR, by employing faster-than-light actions, and demolish Eberhard's theorem as well. Everything stated in the above paragraph can be derived from the second (and totally neglected by "quantum computing" experts) part from the famous sentence in Schrödinger's Die gegenwärtige Situation in der Quantenmechanik (see above): "... then measuring it does not mean ascertaining the value that it has." If you wish to apply for research grant for quantum computing (cf. Saul Youssef), be honest with your Ministry of Defense, to deserve the money from your fellow citizens: solve the puzzle of QM & STR first, by deriving the classical limit of QM from STR. A simple Gedankenexperiment is waiting for you here. As John Bell himself acknowledged, "So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics -- the real problems of quantum mechanics -- they must be talking about Lorentz invariance." And this is precisely the scope of PR2 interpretation of QM. The phrase 'immediately prior to the observation, the particle had already been in such and such eigenstate ...' (e.g., Gillespie, 1973) is sheer poetry, which, on top of everything, leads us inevitably to the preferred basis problem. If we stick to the classical determinism from the Fifth Solvay Conference, we abuse Quantum Mechanics, because "a probabilistic assertion presupposes the full reality of its subject", as stressed by Erwin Schrödinger. Due to the "ambiguity" of the quantum state, neither the absolute Newtonian time nor the "time label" in STR can accommodate the state of the quantum system before its observation: try again here a Gedankenexperiment with the classical limit of QM, derived from STR. 3. Finally, if we choose to talk about 'the classical limit' of the quantum world by instructing 'h --> 0', we are sweeping the garbage under the rug. The challenge of describing 'the classical limit' cannot be resolved with any "decoherence", as H. D. Zeh tacitly suggested in his essay. We need to explain the smooth, reversible, bi-directional transition from the quantum world to the world of 'objective reality out there', and back to the quantum world of Aristotelian potentia. Thus, if we wish to talk about 'time in quantum theory', we need to bridge QM with STR by a proper 'Relativistic QM', as explained by Erwin Schrödinger in 1931, in Specielle Relativitätstheorie und Quantenmechanik (no, we can't find the proper Relativistic QM in P. Dirac's textbook). Seventy-six years later, I couldn't find a shadow of a useful proposal in H. D. Zeh's "Time in Quantum Theory" either. I doubt he would reply to my critical comments (as he never did in the past six years). I haven't heard from Claus Kiefer either, despite the fact that these problems are rooted on his ideas on quantum gravity (see C. Kiefer, gr-qc/9308025, pp. 5-9, and ref. [15] in H. D. Zeh's essay above). Instead of suggesting some "anthropic principle" (cf. Steven Weinberg) to tackle the "fine tuning" of constants and the "smart" behavior of DDE (the coincidence problem), consider the oldest proposition by Leibnitz and Pauli & Jung here, and think of the requirements for life as Reichenbach's Principle of the Common Cause placed in the potential future of 'universe's eye' (global mode of spacetime), and you will get a pre-correlated initial and boundary conditions suitable for life, just as in the example with the four dice above. This is the 'chooser' that can handle googolplexes of the Landscape, and we will inevitably wind up in 'the only possible universe' correlated in its Holon state by one quantum-gravitational wave. Besser ein Laus im Kraut als gar kein Fleisch, mein lieber Dr. Zeh.
Note: 31 seconds after I sent the email above, it was rejected (Thu, 05 Jul 2007 18:00:53 +0200). Which means that Hans-Dieter Zeh got serious about QM very quickly indeed! D.C.
============== Subject: "That's what I hope." Note: See a typical 'statement of belief' in [Ref. 1]. I will again highlight the poetry in red, for reasons explained by Erwin Schrödinger in November 1950. To my knowledge, only Alex Kryukov has addressed 'the only mystery in QM' from the perspective of Henry Margeuan's Latency Interpretation: see statements (C) and (S) on pp. 5-6 [Ref. 2]. It will be wonderful if he can suggest the quantum dynamics within the geometric formulation of QM, which is the only viable approach IMHO. The math, however, is staggering. I sincerely wish Alex Kryukov best of luck. D.C. [Ref. 1] F. Alexander Bais and J. Doyne Farmer, The Physics of Information, arXiv:0708.2837v1 [physics.class-ph] "We can arbitrarily designate one quantum state as "spin up", represented by the symbol |1>, and the other "spin down", represented by the symbol |0>. "Unitary time evolution means that the length of the state vector remains invariant, which is necessary to preserve the total probability for the system to be in any of its possible states. p. 44: "... one can argue that historically the field of quantum computation emerged from thinking carefully about the measurement problem [...]."
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Note: See the discussion of the arrow of time on p. 274 in Claus Kiefer's "Quantum Gravity", 1st edition, publication date: 20 May 2004. The second edition (22 February 2007) is supposed to "contain some pedagogical extensions". Some pedagogical instructions about 'isolated gravitational system' maybe? That would be very intriguing, but I doubt that C. Kiefer would say anything on such crucial issue. My impression is that C. Kiefer is a bit reluctant on providing "pedagogical extensions", as compared to his colleague Carlo Rovelli, who also published a monograph with the same title, "Quantum Gravity", but has generously offered his pedagogical insights as follows (p. 21): "As far as we remain within classical general relativity, a given gravitational field has the structure of a pseudo-Riemannian manifold. Therefore, the dynamics of the theory has no preferred time variable, but we nevertheless have a notion of spacetime for each given solution. But in quantum theory there are no classical field configurations, like there are no trajectories of a particle. Thus, in quantum gravity the notion of spacetime disappears in the same manner in which the notion of trajectory disappears in the quantum theory of a particle". Thus, if we recover the notion of trajectory in the quantum theory of a single particle, we have a chance to say something meaningful on the arrow of time, and correct some of the pedagogical statements made by Claus Kiefer in Ch. 10 from the first edition of his "Quantum Gravity". Also, Carlo Rovelli claims that classical determinism will be lost, "because equal initial data could evolve in physically distinguishable ways respecting the equations of motion. Therefore classical determinism forces us to interpret the invariance under Diff(M) as a gauge invariance: we must assume that diffeomorphic configurations are physically indistinguishable" (C. Rovelli, The century of the incomplete revolution: Searching for general relativistic quantum field theory, arXiv:hep-th/9910131v1, p. 3). Once we have dynamical determinism, why would anyone care about 'classical determinism'? Why doing quantum gravity with our knowledge from 1927? Surely "equal initial data could evolve in physically distinguishable ways respecting the equations of motion" in the global mode of spacetime, creating the spectrum of potential states in the Holon, with which we can recover the notion of trajectory in the quantum theory of a single particle. Again, everything goes back to Ch. 10 from Claus Kiefer's "Quantum Gravity", as in its first edition from 20 May 2004. The last time I heard from Claus Kiefer was four years ago, just to tell me that he can't open the CD ROM I sent him by surface mail, because all PCs at the University of Cologne run on Unix. If Claus Kiefer hasn't found any Windows-based PC in the past four years, and has never read any of my email messages sent to him in the past five years, chances are that he hasn't learned anything from this web site, and probably never will. If so, I suppose he will ignore my email above, and will continue to work on the third edition of his "Quantum Gravity". The reason why the energy associated with the elementary timelike displacement is not an observable in GR (see Slide 29 in Bolen.zip) is that it comes from the Aristotelian connection, which makes it "dark". The conservation of total energy-momentum does not hold in the case of DDE or 'evolving [lambda]', so we need quantum gravity, as Claus Kiefer rightly noticed three years ago, on p. 274 from the first edition of his "Quantum Gravity". Hence the first off tasks are to work out new conservation laws for DDE (see above), and define an 'isolated gravitational system': the local mode of spacetime, which is being "wrapped" by, and isolated from, the global mode of spacetime. There is no other choice. Sorry. But if you work under Unix, like Claus Kiefer, you wouldn't know. On the positive side, you might qualify for being invited to discuss the nature of gravity, scuba diving, and jumbo shrimps at Virgin Islands.
D. Chakalov
Subject: Netiquette ==================== Subject: arXiv:0707.2593v1 [quant-ph] and Nature, 448, 23, July 2007 ==================== Subject: 13 Frequently Asked Questions, arXiv:0705.2222v1 [gr-qc] Note: The hardest thing of all is to find a black cat in a dark room, especially if there is no cat, says Confucius. Abby Ashtekar does not have a sound physical theory in the first place, which is why he and his younger colleagues can never solve the Hamiltonian constraint problem in Loop Quantum Gravity (LQG). An example: nobody would support a proposal for Perpetuum Mobile, simply because such machine would ruin the whole thermodynamics, which is absurd, hence we reject such proposals from the outset, even if some sophisticated math is used to describe such absurd projects. Similarly, if A. Ashtekar and his younger colleagues could solve their Hamiltonian constraint problem, they would immediately offer, by the same token, a solution to the problem of time in Wheeler-DeWitt canonical quantum gravity, which is absurd, hence we should reject such proposals from the outset, regardless of the sophisticated math used to describe them. There is nothing fundamentally different in LQG, compared to the Wheeler-DeWitt geometrodynamics. On the contrary, A. Ashtekar himself acknowledged the same "grave injustice to space-time covariance that underlies general relativity" (reference here; see also Alan Rendall). To the best of my knowledge, nobody from LQG community has posed the following questions: 'Suppose we resolve the Hamiltonian constraint problem. How would this "solution" look like? Wouldn't it "describe" the self-acting action of the universe (which may be the reason why its Hamiltonian dynamics is dead frozen, as we know from Wheeler-DeWitt geometrodynamics)? Can we, at least in principle, get our job done ensuing from 'splitting the spacetime', or is it like building a Perpetuum Mobile?' Nobody from LQG community asks such questions. They do blind guessing only. There are two kinds of blind guessing, however. One is when the black cat is in the dark room, as opposed to the cat being on the street. If Einstein was doing blind guessing only, as depicted in the caricature below, he would at least have a chance to come up with his famous equation, E = mc2 .
However, in the case of LQG the "black cat" has been on the street from the outset, and no blind guessing can invite it into the "dark room". Details about the mess of LQG porridge can be read in Claus Kiefer et al., arXiv:0705.1688v1 [gr-qc] and in Hermann Nicolai et al., hep-th/0501114 v4, Sec. 5.2. Since I am not able even to post a paper at ArXiv.org, I will be very brief, and will also assume that you've read the current web page here and the text regarding the "dark energy" of the reference fluid here. Regarding the issue of gravitational energy, notice how A. Ashtekar has 'swept the garbage under the rug' in his Lectures on Non-Perturbative Canonical Gravity, as explained (politely, of course) by his Polish colleagues here. Once you do that, you chase the black cat out, and nothing can bring it back in the dark room. The reference fluid of GR cannot be found in GR, because it is "dark". The reference fluid (cf. A. Trautman) is "dark" for two main reasons. Firstly, its "dark energy" is spanned "over" the whole 3-D hypersurface and acts on it en bloc, which of course contradicts STR, as we know from inflation theory. Secondly, the reference fluid is "dark" because it comes from the Aristotelian connection: no physical stuff in the local mode of spacetime can physically reach the Aristotelian connection in the global mode of spacetime, not even with Kurt Gödel's theorem. Surely the reference fluid and its Aristotelian connection must exist, otherwise we wouldn't have any 'isolated systems' and finite volumes of 3-D space and durations of time in the local mode of spacetime. On the other hand, the reference fluid and its Aristotelian connection cannot be reached by any physical stuff whatsoever, because they exist as ONE, as noticed by Lucretius some 2060 year ago. Their dynamics is quite different from the alleged "dynamics of GR" in the current GR textbooks. Namely, in the local mode of spacetime, [lambda] interpreted as vacuum energy density will be always a "constant" that can be considered 'tending asymptotically toward zero', while in the global mode of spacetime the value of [lambda] will start from 'infinitesimal', right after The Beginning, and will reach its potential value corresponding to the current, and ever increasing, spacetime "horizon" at Vmax . Hence we can have our cake and eat it! In my view, there is no other possibility for reconciling the two requirements from [lambda] being the vacuum energy density: it has to be both "zero" (local mode of spacetime) and evolve freely (global mode of spacetime) along the cosmological time arrow, being the ultimate free lunch. We need new conservation laws for 'isolated gravitational system with DDE', as argued above. More from John Baez here, and from Carl Hoefer here. I could be wrong, of course, but at least I don't chase the black cat out of the dark room, like A. Ashtekar and his colleagues who kindly contributed twelve utterly genteel questions in arXiv:0705.2222v1 [gr-qc]. The most troubling fact is that many young physicists have forgotten that GR is not a complete theory, as Einstein himself acknowledged. A typical example is the latest efforts by A. Corichi and J.A. Zapata to "explain" the basic ideas of GR (arXiv:0705.2440v1 [gr-qc]) in the following manner: "In other words, does the concept of a point on space even make sense? In the context of classical general relativity we know the answer: given that the theory is invariant under diffeomorphisms, the concept of a point as an abstract entity dissolves. Instead what makes sense is the point not as an abstract mathematical object but as the location where matter fields and gravity have some (quasi-local - D.C.) particular property (for instance the point where two word-lines intersect, or the point where light is emitted by a source, etc.). Nevertheless, any point on the manifold is as good as any other point given that all fields are smooth objects and are thus ‘well defined’ on any point of space." But the matter fields, self-coupled via gravity, cannot made themselves "well defined" on any point of space, unless the reference fluid has been already introduced. The latter was expelled from GR from the outset, however. We need a brand new "background" that can act without being 'acted upon': the Aristotelian First Cause. We can't bind the points with their fleeting quasi-local physical content only. Einstein's GR is incomplete because it provides only the necessary conditions for describing gravity, while the sufficient conditions are delivered by the Aristotelian connection. It's a bundle. Ignore it at your peril. To make this quandary as clear as possible, I will refer to the two currently available options for interpreting the Einstein equation, which were discussed previously in the context of "GW astronomy". I will now label them with 'GR psychokinesis' and 'GR scientific communism'. I believe these two options are wrong, which is why a third possibility has been outlined, based on what I called 'the Aristotelian connection'. Hence the dilemma is to choose either one of the first two options, or the third one. Regarding the puzzle of vacuum energy, Sean Carroll posed above a very illuminating question: "We know that virtual particles couple to photons (e.g., Lamb shift); why not to gravity?" This is a clear cut case of 'GR psychokinesis', in the sense that it presupposes some direct interaction of geometry and matter during their bi-directional talk: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John Wheeler). Its only merit is that it complies with the incomplete theory of gravity as 'metric field'. It has the severe shortcoming of leading inevitably to the task of "searching for the right answer to the wrong question" (see above): just like the human mind, the gravitational field itself has no proper energy-momentum density. Instead of talking about 'matter coupled to gravitational field' and invoke some tacit 'GR psychokinesis', I suggest the notion of 'matter fields self-coupled via gravity', much like all neurons in the human brain are self-coupled by their common "context", via the brain's Holon.
Thus, matter interacts with matter only, but their interaction is quasi-local due to gravity. As to the second option, dubbed ''GR scientific communism', read Stanislav Babak and Leonid Grishchuk here. There are literally dozens of alternative theories of gravity, in which the gravity is modeled with some physical field. Many people found it difficult to think about 'the grin of the cat without the cat', and have also tried to present gravity as some quantum field, which in turn has led Stanley Deser to muse "why, unlike all other fields, does the gravitational "metric" variable not have zero vacuum?" It is a blind alley from which nobody has come back with a clear answer to the nature of gravity. Most importantly, in such scenarios the task of "quantization" of gravity is self-contradictory: read Angelo Loinger here. The prerequisites of the third option are presented here. It is not yet explored, because the basic notion of 'potential reality' requires new math. It may be a tough mathematical challenge to unravel [phi], but at least the basic ideas are very simple. The choice is yours. To get you started, read Steve Carlip's notes "Conceptual problems in quantum gravity" at http://www.physics.ucdavis.edu/Text/Carlip.html#problems S. Carlip: "As a probabilistic theory, quantum mechanics gives time a special role: we would like to say, for example, that an electron has a total probability of one of being somewhere in the Universe at a given time. But if spacetime is quantized, we don't know what "at a given time" means." 1. If you wish to talk about QM in the context of quantum gravity, do not abuse QM by imposing 'the sharp time' from STR. More here. 2. If you claim that an electron has a total probability of one of being somewhere in the Universe at a given time, then you must specify 'the rest of the Universe' in which the probability of the same electron to be there is zero at the same given time. Hence you need to treat 'the Universe' as an "isolated system" with "boundaries": see above. 3. "But if spacetime is quantized, we don't know what "at a given time" means", says S. Carlip. But the spacetime is automatically "quantized" from the outset, by its two modes, local and global. Join the club of Aristotelian connection! As I acknowledged above, I've been struggling to understand Einstein's GR since 1972. The first time I spoke on these issues was twenty years ago, on February 5, 1987. It was during an informal meeting of theoretical physicists at the Institute for Nuclear Research and Nuclear Energy at BG Academy of Sciences, bearing the loose title 'Philosophical and interdisciplinary problems of physics'. I addressed a group of particle physicists and relativists (with PhDs mainly from Dubna, USSR), from 11:00 to 11:40, but the only feedback at 11:40 AM, right after I finished, was this: "Oh, it's time for lunch now! Thanks, Dimi, it was very interesting." Eleven days later, on 16 March 1987, I was summoned at the Office of the Director, and was informed that the Bulgarian Academy of Sciences doesn't have money for my salary, so the next day I wind up on the street. Now the situation is different, in the sense that I got old, don't work for anybody, and don't depend on any institution whatsoever. However, apart from some insults, the attitude of the established theoretical physics community has not changed. Since I have nothing to lose, I decided to be very frank with them. And every time they produce bullshit, I will tell them that their stuff is indeed bullshit. For example, 'GW parapsychology'. They don't care and won't respond anyway. To quote Max Planck: "An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out and that the growing generation is familiarized with the idea from the beginning: another instance of the fact that the future lies with youth."
================== Subject: "Fundamental discreteness at Planck scale in LQG" is an empty statement. Note: There is a whole bundle of issues in my email to T. Thiemann above, which I will try to disentangle and explain as clear as I can. Will be very brief though -- please follow the links. Einstein GR is manifestly blind and deaf to a sustained, ongoing non-conservation of energy-momentum due to the dynamic "dark" energy of [X], whereby [X] is most likely, and naturally, the quantum vacuum: "any non-constancy in [lambda] would have to be accompanied by a compensating non-conservation of the mass-energy of the matter" (R. Penrose, The Road to Reality, Jonathan Cape, London, 2004, p. 777.) Once we postulate, after Dirac and ADM, some globally hyperbolic spacetime that can be time-orientable due to some "global time coordinate" and convenient metric, we then impose the "obvious" constraint of the twice-contracted Bianchi identities, which would guarantee the conservation of total energy-momentum, provided the cosmological "constant" [lambda] were a constant. Only it isn't. Thus, we impose a "filter" that prevents us from seeing any dynamic "dark" energy (DDE) whatsoever: it is excluded from the outset -- it cannot be "gauge independent" in the first place -- hence we call it "dark". Consequently, we cannot talk about the dynamics of an object with normal-plus-dark energy, as mentioned in the email above. As an example, T. Thiemann tried to tackle the Problem of Time in GR with some ad hoc postulated "phantoms" and "k-essence", only to reach what he called a "devastating conclusion" (astro-ph/0607380 v1): "Either the mathematical formalism, which has been tested experimentally so excellently in other gauge theories such as QED, is inappropriate or we are missing some new physics." So, let's go back to the first days of GR and try to pinpoint the origin of this whole mess of normal-plus-dark energy of matter fields self-coupled via gravity. If we can clean up the "can of worms" (J. Baez) in the "normal" energy conservation, perhaps we would get a glimpse at the true dynamics of GR, without imposing any "filters" that make the latter blind and deaf to 96 per cent from the stuff in the universe. As is well known, proper energy conservation laws in GR are impossible: see, for example, Kenneth Dalton and Anatol Logunov; general outline in Bjoern S. Schmekel, Quasi-local definitions of energy in general relativity, arXiv:0708.4388v1 [gr-qc], p. 2. The "normal" gravitational energy cannot be "gauge independent", so we cannot claim that our wristwatch is reading a chain of some well-defined Dirac observables constituting the cosmological time arrow driven by DDE. Just by looking at your wristwatch, you prove the Dirac observables wrong by reductio ad absurdum. Unless of course we can actually observe non-tensorial entities as well (the proper time [tau] along spacetime trajectories), in which case GR will have to be substantially updated, and then we may have to forget about those Dirac observables and the whole "canonical" approach to GR. Even A. Ashtekar acknowledged that the splitting of spacetime into space and time is "doing grave injustice to space-time covariance that underlies general relativity". What he and his younger colleagues failed to realize is that the 3-D space in present-day GR is in fact a dead frozen background, so if we "instruct" certain diff-invariant "data" to "evolve" on it, they will inevitably hit the Cauchy problem of the "block universe" [Ref. 4]. The Cauchy problem is an inherent problem of the "canonical" approach to GR, since the latter inevitably produces a "block universe". Forget it. Drop it. We must utilize and harness the non-tensorial quantities in GR. But how could such "dark", non-tensorial entities be "smuggled" into an updated version of GR? To answer this question, let's try a careful examination of the relational ontology of today's GR, as understood by Thomas Thiemann and Bianca Dittrich (cf. the so-called "weak Dirac observables" here). There is a very clear paper by their colleagues Hans Westman and Sebastiano Sonego [Ref. 2], which was posted this morning (August 15, 2007). NB: Notice that the space of "point-coincidences" can only be defined locally, that is, "over" an infinitesimal point [Ref. 2, p. 3]. This is a crucial feature of present-day GR, which has been made exceptionally clear by Hermann Weyl and Lįszló Szabados. It is the cornerstone of the problem of the dynamics of today's GR: see Matt Visser, gr-qc/0204022, p. 3, above. We need to explore and harness the absence of 'classical determinism' in GR (see above) with 'dynamical determinism'. Back in 1990, Hermann Bondi stated: "In relativity a non-localizable form of energy is inadmissible, because any form of energy contributes to gravitation and so its location can in principle be found." I believe it all depends on the mechanism for 'localization': if we make it 'quasi-local' (local mode of spacetime), as outlined above, then all non-tensorial and non-localizable quantities can and will co-exist peacefully (global mode of spacetime) with the actualized, localized ones: see a hint here. In the context of Henry Margenau’s Latency Interpretation of QM, 'possessed observables' in GR are those which comply with the rules of active diffeomorphism ('possessed observables in GR' are invariant under change of coordinate "time", therefore they are "constant" in this "time" by such gauge invariance) and keep 'the sameness of objects' (Kurt Lewin), while 'latent observables' are non-tensorial; in the local mode of spacetime, the two are always present and intermingled over extended domains. To sum up, "even if we start with genuine tensorial variables, then certain important physical quantities turn out to be non-tensorial" (Laszlo Szabados). The alternative viewpoint shared by many researchers (B. Dittrich, T. Thiemann, H. Westman, S. Sonego, C. Rovelli, to name but a few) is the following: if we can fix a space of "point-coincidences" relationally [Ref. 2], then we may have some suitable approximation to the genuine dynamics of GR, in the sense that we can 'let the data evolve' from such instant. No way. Relationally, we can only fix a dead frozen instant, a kinematical snapshot of the spacetime: see the Buridan donkey paradox here. If we wish to unravel the true dynamics of GR, we need an explicitly non-local constraint (global mode of spacetime) on the whole spacetime, up to its "boundaries". Otherwise no donkey can make any next step, as dictated by its relational ontology. We can 'let the data evolve' from some instant only if we have a fixed background spacetime, as in STR. In GR, there are no generic coordinates nor paths; paths are made by walking. But in present-day GR we cannot "walk", because there are no Perennials which we can 'hold onto': read Karel Kuchar. In canonical gravity, we have only 'laws of an instant'. Surely the proper way to define such instant is by adopting the relational ontology [Ref. 2], but it is the latter that will freeze the dynamics of GR. Carlo Rovelli could only flout at Karel Kuchar by talking about 'evolving constants' and 'partial observables', but has so far produced nothing but poetry. If he or anyone else had solved the Cauchy problem for the Einstein equations, we would have heard about it from CNN Breaking News. As to Bianca Dittrich, she admitted [cf. p. 2 in ref. [8] in Ref. 2]: "For gravity coupled to matter, in some cases gauge invariant functions describing matter are known but in general no phase space functions which describe the gravitational degrees of freedom (with the exception of the ADM charges). Yet there are infinitely many gauge invariant degrees of freedom." Very important observation. But let's suppose, just for the sake of her argument, that some Dirac observables (phase space functions that would be invariant under gauge transformations) might exist. Here's Bianca Dittrich's recipe (ibid.): "Let us assume that this gauge degree of freedom corresponds to reparametrizations of an (unphysical) time parameter." It is not "unphysical" but "dark". See again the Buridan donkey paradox. Not surprisingly perhaps, her colleague Thomas Thiemann did not allow me to speak at GR17, but decided to bury my work in a poster session. Same did Mike Cruise one year later, on a different occasion: GW parapsychology. Surely Gravitational Waves exist, but their proper detector will have to be build in line with the true dynamics of GR. And this is already a very serious issue, which will not be discussed here, nor at any poster session. Suffice it to say that, unlike Angelo Loinger, I claim that GWs exist, and the quasi-local detector of their quasi-local propagation requires (i) a brand new kind of quasi-local detector, and (ii) a brand new kind of "shielding", as stressed by John Stachel. Just follow the links.
[Ref. 2] Hans Westman, Sebastiano Sonego, Events and observables in generally invariant spacetime theories, arXiv:0708.1825v1 [gr-qc]
The lack of a rigorous mathematical proof (read: fishing in murky waters) is evident in the confession of B. Dittrich and T. Thiemann that "the explicit construction of Dirac observables in general relativity is very difficult since it requires the solution of the dynamics of the theory" [Ref. 1, p. 2]. Which means that 'the proof of the pudding' for some "Dirac observables" is by trying them for solving the dynamics of GR: the Cauchy problem for the Einstein equations (references here and here). C. Rovelli tried very hard to explain his “partial observables” [Ref. 3], but in my opinion has produced nothing but poetry (highlighted in red). Six years ago, C. Rovelli stated: "But success, I think, can only be granted by scrupulous intellectual honesty." Only he failed to mention the opinion of Claus Kiefer and Karel Kuchar regarding the mythical "evolving constants" and "partial observables". He never replied to my email enquiries -- no matter how polite -- sent since 26 November 1999 either. If his success with Loop Quantum Gravity (LQG) "can only be granted by scrupulous intellectual honesty", then I'm a bit skeptical. No need to reply by email: just unravel the dynamics of GR by solving the Cauchy problem for Einstein equations, hence prove that you understand the alleged Dirac observables in GR, the "evolving constants" and "partial observables", and the emergence of time and space in quantum gravity [Ref. 4]. We all will hear about it from CNN Breaking News. D. Chakalov
[Ref. 3] Carlo Rovelli, Comment on “Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?” by B. Dittrich and T. Thiemann, arXiv:0708.2481v1 [gr-qc], August 20, 2007 p. 2: “partial observables”: "These are physical quantities that can be measured, but are not necessarily predictable [8]. (An example of a quantity considered “measurable but not predictable” is the usual time variable t). "What is relevant for the present discussion is the following. The geometrical operators that have discrete spectra in LQG can be considered partial observables. If one follows interpretation II, the conclusion that a physical measurement of these quantities yields quantized values is immediate, because physical quantization depends on the spectra of kinematical operators in K in this interpretation. If, instead, one follows interpretation I, as Dittrich and Thiemann, then the possibility is open in principle that the spectrum of a quantity f be discrete but a corresponding p. 4: "As clarified in [12], the reason we must assume that only the gauge-invariant quantities can be measured and predicted is that the equations of motion do not determine the evolution of the non-gauge-invariant ones. The only quantities whose evolution is well determined are the Dirac observables. This fact is taken into account within interpretation II, where transition amplitudes describe gauge-invariant correlations, and all predictions are indeed gauge-invariant. (Try then the Cauchy problem for the Einstein equations - D.C.) (Only this "independent variable", according to C. Rovelli, "does not correspond to anything directly observable". Read also his teacher, Chris Isham, here - D.C.) p. 5: iii) Gauge fixing. "According to the first, the coordinates xµ are irrelevant mathematical labels that can be changed at will. The only quantities that have a physical interpretation are those that are independent on the choice of these coordinates. This is the interpretation which is is most commonly considered in quantum gravity. "This is the analog of the fact that the Maxwell potential Aµ describes a physically observable quantity, if we work in a formalism in which we have entirely fixed the gauge." (Try then to use this "analogy" for solving the Cauchy problem for the Einstein equations - D.C.)
See also: Edward Anderson, Emergent semiclassical time in quantum gravity, Class. Quantum Grav. 24 (2007) 2935-2977 See also: Brett McInnes, Inheriting The Arrow of Time, arXiv:0705.4141v2 [hep-th] "This is no mere detail: the observed properties of the cosmic background radiation means that “there is nowhere else to store” the low initial entropy. Thus, again, it is not enough for a theory of the Arrow to generate “low” entropy initial conditions. The theory must explicitly give rise to low geometric entropy [whatever the latter’s precise definition
Note: Do you happen to know exactly how energy-momentum flows from matter to gravitational fields and back? With "covariant derivatives", said Gerard 't Hooft in March 2004. It is tantamount to saying that the planets in the Solar system are orbiting the Sun by solving differential equations. If Gerard 't Hooft would read my email above, he'd probably say that the Rindler space has nothing to do with the cosmic equator, and will repeat his bold claim that my web site contains "too much obvious nonsense" (Nobel Prize laureates need not elaborate). We can show the moon with a finger, but the finger is not the moon. To look at the moon means to look "over" the finger. Those who say that there is no direct link from the Rindler space to the cosmic equator are right, but they see only the finger. If you unravel the bigger picture behind the Rindler space and the cosmic equator, perhaps you will "see" the Aristotelian connection which binds the points in 3-D space with 'the universe as ONE', as I tried to explain to my teenage daughter above. There exists an absolute reference frame of 'the universe as ONE', only it is in the global mode of spacetime, being "attached" to all points in the local mode of the train/balloon surface. Subsequently, there are various "dark", or rather holistic effects from 'the universe as ONE' (see above), which enter the local mode of spacetime via its geometry: 'the grin of the cat without the cat'. Once Gerard 't Hooft gets his article "200 wrong theories for the cosmological constant" published, I will quickly elaborate by providing for the 201st reference. So far he has only promised the following (quant-ph/0604008 v2, footnote on p. 4): "Special and general relativistic transformations are left for future studies." No rush, take your time (with all "covariant derivatives", of course). If my daughter decides to study physics, I hope she will never get hit by a Nobel Prize. Looks like some people just can't recover.
I will assume that the "binding forces" (Bernhard Riemann) act upon the local mode of spacetime "from outside" -- the so-called global mode of spacetime -- and will argue that the minimal or "infinitesimal" displacement in time and space, which is also believed to represent a 'geometrical point' (called 'atom'), has a very rich internal structure reflecting the whole universe. The term suggested for the "binding forces", as mentioned by Bernhard Riemann, will be 'the Aristotelian Connection'. It is supposed to act as the reference fluid of GR and Hilbert’s causality (Trevor W. Marshall, arXiv:0707.0201v1, p. 2 and ref. [7] therein). NB: In the non-relativistic (and highly misleading) presentation of spacetime (cf. 'Seeing back into the cosmos' above), the "binding forces" (Bernhard Riemann) would be spread "over" the 3-D surface of the "expanding balloon", and we would be able to pinpoint their location by considering the preferred/absolute axis that connects every point from 3-D space to their common "Big Bang". In the real, relativistic presentation, these "binding forces" reside in the global mode of spacetime, hence are "dark"; more here. Let's go back to classical differential geometry and try to find the exact "blank spot" left from this unsolved puzzle. In modern terminology (courtesy from Alain Connes), the most basic geometric structure in differential geometry is given by a differentiable manifold, which is a topological space equipped with a “differential structure”. (The latter allows to define vector fields/sections of the tangent bundle, and then differential one-forms are introduced as linear maps acting on vector fields, in case you're curious.) But how does the topological space get equipped with a “differential structure” in the first place? To be specific, what entity makes a variable “infinitesimal”, after Lucretius? (A variable is called “infinitesimal” if among its particular values one can be found such that this value itself and all following it are smaller in absolute value than an arbitrary given number, says A. Connes; more from Wolfram MathWorld here.) Notice a very important issue in the description of the “infinitesimal” above: the “infinitesimal” itself cannot display some 'intrinsic numerical value', because an “infinitesimal” does not, and cannot possess such value in principle. NB: On the one hand, it is a dynamic entity which always runs "one step ahead" from an arbitrary given number that is chasing it; notice that, in the epsilon-delta approach, Thus, the “infinitesimal” only serves as a tool for obtaining some limit and hence get a number (whenever that is possible), but is never exposed in the local mode of spacetime. A "point" endowed with some value of some physical variable would only refer to the calculated infinitesimal (much like John Wheeler's 'cloud'), but never to the infinitesimal itself. Otherwise the latter will be exposed in the teleological, local mode of spacetime, and will automatically become bigger in absolute value than 'an arbitrary given number'. See the discussion of Thompson's lamp paradox here, and the new "number" [phi] here. It is a genuine infinitesimal and is rooted on 'the ideal monad', which belongs to the world as ONE, which exist 'without parts', after Lucretius. It cannot be reached from the finite, and infinitely divisible, world of the local mode of spacetime. The latter is a teleological world which always remains "separated" from the ultimate cutoff -- the Aristotelian First Cause residing in the global mode of spacetime. This is the solution proposed to the 2060-year old puzzle of “infinitesimal” (cf. Lucretius). NB: Notice that 'the world as ONE', which exists 'without parts', after Lucretius, cannot be 'large' nor 'small'. If you consider the "width" of the sign | to be 'the atom', then 'the world as ONE' may contain an infinite -- actual infinity and "undenumerable" -- amount of such atoms, which again will be presented as 'the potential reality of ONE', with the same "width" of | . All we can do, in the local mode, is to "chase" 'the ONE' and get 'as close as possible towards it' (cf. "closeness"), and then 'the potential reality of ONE' will sort of 'lose patience with us' and will act as cutoff on our seemingly infinite steps towards it. And this cutoff will finally produce a nice geometrical "point" from 'the grin of the cat without the cat', with a calculable (although not always "precise") instantaneous value of some physical stuff. And the "number" of such 'produced-by-cutoffs physical stuff' is [phi] . Notice also that if we compare (sqrt3 x sqrt12) and (sqrt3 x sqrt11), we have in the first case a "precise", point-like numerical value displayed with a real number (+/-6), while the second case produces an irrational number tending asymptotically towards its “infinitesimal”. Another example: take a line segment (A,B), and imagine a "point" C dividing it in the Golden Mean proportion, AC/CB = AB/AC. The geometrical "point" C is just as real as A and B, yet it can't get "dressed" by a real number. Lucretius would probably say that in all cases there is indeed some hidden limit and cutoff, called “infinitesimal”, which (i) can never be exposed in the local mode of spacetime, and (ii) enables the basic relations of 3-D space, such as 'big vs small', 'inside vs outside', regardless of whether it can be calculated precisely (e.g., +/-6) or not. Again, we can never bridge, from the local mode of spacetime, the one-step "dark" gap to the “infinitesimal”, hence use some poetic expressions, such as "closeness" (cf. Wolfram MathWorld here). Another poetic expression used by theoretical physicists is 'instant': recall that in physics we always imply intervals, and subsequently need three mathematical points to define an 'instant', by instructing a finite interval (t2 - t1) to approach asymptotically zero and reach an 'instant' from the realm of pure geometry (the grin of the cat without the cat), as depicted with the drawing below.
Notice that we can produce "points", by instructing the interval (t2 - t1) to approach asymptotically zero, as well as 'the largest volume of 3-D space', by instructing the same interval to approach infinity, since we are dealing with two Aristotelian "cutoffs", as explained here. NB: Just think of the two Aristotelian "cutoffs" as resembling the 'absolute zero temperature': surely such cutoffs exist, but can never be reached 'from within' the local mode of spacetime. Hence we enjoy 'local mode of spacetime' -- an isolated system with fixed dimensions and well-defined properties (one-at-a-time) of all physical objects, at all length scales. This is possible due to the global mode of spacetime of 'the whole universe as ONE', which has the ontological status of Aristotelian First Cause, and "isolates" the local mode of spacetime from itself: see again the note after Stephen Leacock. Basically, all this can be viewed as Ellis’ 1984 notion of finite infinity, updated from Aristotle. The 'elementary cycle of time' is totally "dark" to all 'passengers inside the train'. The "direction" of the expansion of 3-D space points to all possible directions in 3-D space, hence none of them is special or preferred -- see again the relativistic cosmological interpretation (not the misleading picture!) above. If we use the teleological local mode of spacetime only, the ultimate [dark gap] produces an insoluble metaphysical problem of 'the third point', as explained with the question posed by Robin Le Poidevin: "If between any two points in space there is always a third point, can anything touch anything else?" This 'third point' belongs to the global mode of spacetime, hence in the local mode it is both 'existent' and 'non-existent'. And this is the crux of the puzzle from Lucretius and Thompson's lamp. If we can grasp it, we may understand the nature of continuum. Clearly, we need new math. NB: Notice my prediction from January 9, 2003: The [dark gap] will produce a chain of quarks in Fibonacci sequence, because the so-called "God particle(s)" cannot live in the local mode of spacetime. Moreover, if we think of the Aristotelian First Cause as two 'ideal endpoints', the drawing here may have profound implications for cosmology, by presenting a brand new (to present-day theoretical physics) model of eternal universe: once created with two Aristotelian 'ideal endpoints', the universe becomes effectively eternal to all quasi-localized 'passengers inside the train'. Just keep in mind that any time you look at your wristwatch, you actually "measure" at instant from the cosmological time of this eternal universe. Now, mathematicians are happy people because they can introduce their “differential structure” by hand, but then physicists face a highly non-trivial challenge, because they cannot derive this 'structure' from type I matter fields that are introduced on top of it later. Often they mention this miracle of an already-introduced “differential structure” in footnotes, like Chris Isham did in his Lecture Notes. But we don't accept miracles. Therefore, we need to know how Mother Nature introduces a “differential structure” on a topological space. Here's an explanation for pedestrians:
To be specific, I argue that the phenomenon which "connects" the geometrical points in 'the grin of the cat without the cat' (called Aristotelian Connection) is a unique object -- the whole universe as ONE. It lives in a hypothetical 'global mode of spacetime', and its ontological status is that of the Aristotelian potentia. Viewed from the perspective of the local mode of spacetime (the world of facts), the Aristotelian Connection, performed by the whole universe as ONE, will look like being placed simultaneously at the two edges of the length scale of 3-D space: its spatial dimensions would seem to be both extremely small or "infinitesimal" and "extremely large", as denoted with Vmax above. Hence the 3-D space of the local mode of spacetime is being "wrapped" by two 'numerically finite but physically unattainable boundaries'. One of them is fixed (the Planck length), while the other one, toward the Large, is increasing its "value" (cf. Vmax above) along the cosmological time arrow driven by DDE. Thus, in every instant 'now' from the cosmological time arrow, the local mode of spacetime provides the faculties of 'isolated system', with fixed dimensions and well-defined properties, to all physical objects, at all length scales. This is made possible due to the global mode of spacetime of 'the whole universe as ONE', which "isolates" the local mode of spacetime from itself due to its ontological status of Aristotelian First Cause. Put it differently, the Aristotelian Connection, resulting from the dynamics of the universe along the cosmological time arrow, defines the "boundaries" of the local mode of spacetime and its 'differentiable manifold' by a 'reference fluid': we cannot define any 'elementary step' on the differentiable manifold unless we have defined the latter 'as a whole', which means fixing the "boundary" of spacetime. Notice again that we obtain an infinitesimal "point" by a dynamical procedure, which involves three initial points (cf. M. Spaans above) needed to define an interval, and then we instruct this interval to shrink to zero (cf. David Bohm here). A simple-minded engineer wouldn't ponder on the meaningless expression 'tangent vector at a point', because engineers do calculations and are not concerned about the fundamental objects in geometry. I suppose those interested in modern differential geometry would be far more curious. Recall the derivation of the formula for the circumference (or perimeter) of the circle: you started with two polygons and set the number of their sides to approach infinity. Suppose you used the following recipe: starting at t0 , you began doubling the number of their sides, and instructed them to approach infinity.
Precisely at the moment tX , the two polygons disappeared and were converted into a perfectly smooth circle.
You've reached the final level of the physical world, at which it is converted to pure geometry: the sides of the polygons were converted into "points". How do you know? Because now you can happily add or subtract any number of "points" from the circumference of the circle, and you won't change it a bit. Hence the "number" of these "points" is a very special unique number, denoted with φ . (Recall that the alleged "distribution" of prime numbers, as specified with the Riemann Hypothesis, poses an insurmountable mathematical challenge, which might be solved only with φ .) Nothing can go further than these "points": you have reached the atom of Lucretius. Your wristwatch can, of course, read tX , because you can run along the perimeter for a finite time interval, as every finite time interval is composed of infinitely many elementary steps tX (Zeno had some troubles with understanding this conjecture, however). Now, think of the sides of the polygons as a lamp which goes on and off at every step at which you have increased the number of the sides by a factor of two. The crux of the Thompson's lamp paradox is this: what is the state of the lamp at tX ? Is it "on" or "off"? YAIN -- is the correct answer. You've reached the Aristotelian First Cause, which shows up as 'pure geometry': the grin of the cat without the cat. The local (teleological) mode of time is inapplicable for this final layer of the universe, because it measures changes of states of the type 'lamp on' and 'lamp off', which exist as facts (objective reality out there). The new kind of time pertinent to this unique form of reality (called 'potential reality') resembles that of the extended moment 'now' of the human brain. It is called 'global mode of time'; check it out with your brain here. In the local (teleological) mode of spacetime, there is no "short circuit" or direct connection between the Cheshire cat and its 'pure geometry'. The time of Thompson's lamp is the time of facts (either "on" or "off"), and such teleological time cannot be applied to the transitions: two polygons <--> perfect circle
NB: Recall that the Aristotelian First Cause was derived after recognizing the nature of the teleological time of facts (called here local mode of time), as illustrated with the Thompson's lamp paradox. The local mode of time is such that it provides two alternatives only: either reaches the 'last layer of the physical world', or not. Hence we need to complement the teleological time with a brand new (to theoretical physics community) kind of time to accommodate the First Cause: the global mode of time in which the First Cause can "insert" its "cutoff" on the local, teleological time of facts, but without leaving any possibility to be reached 'from within' the local mode of time. This can only be achieved dynamically: the universe in its local mode of time will constantly "chase" its final boundaries, like the old story about the Dragon chasing its tale. Once created with The Beginning, the physical world of facts becomes effectively eternal, since its is "chasing" asymptotically its two "ideal endpoints". There is no need for any "cyclical" or "re-collapsing universes" to avoid The Beginning and make the local mode of universe "eternal". Aristotle has already taken care of it; all we need is to cats his story in math. The world of 'pure geometry' is very real indeed, but if we try to analyze it from the viewpoint of 'objective reality out there', it will inevitably look "dark".
It doesn't have a proper "force" but only serves as a tool with which the holistic effects from 'the universe as ONE' are entering the local mode of spacetime, via its 'pure geometry': no form of matter is associated with its geometry (e.g., there are no hypothetical "gravitons" or physical gravitational field, as proposed in school of 'GR scientific communism'), hence if we try to trace back the Holon, it will inevitably look "dark". In other words, there is no room for DDE in present-day GR, because people who teach GR are very reluctant to acknowledge that those "twice-contracted Bianchi identities" are valid only for the local mode of spacetime, in which we can indeed enjoy 'local physics', but only "over" an infinitesimal "point". The empirical fact that we do not observe any catastrophic events from neither the 'dynamical dark perfect fluid' nor from the inevitable non-conservation of energy in curved spacetime (Noether's theorem holds only in flat spacetime) can only be explained with the status of the local mode of time as an "isolated system" (see above): each and every "point" is wrapped by, and isolated from, the global mode of spacetime due to the Aristotelian Connection. Hence at each and every instant from the cosmological time arrow, driven by DDE, the local mode of spacetime stands as an "isolated system": dynamically, one-at-a-time. As stated above, Einstein GR provides only the necessary conditions for describing gravity, while the sufficient conditions are delivered by the Aristotelian Connection producing the reference fluid. If the latter were also produced by some "structural quality of the gravitational field", they would belong to the teleological time of facts (local mode of spacetime), hence would inevitably require some further physical stuff and physical laws for its determination, as noticed by Aristotle (for a modern version of the argument for the First Cause, try Gödel's theorem). Briefly, it is not possible to bind the points with their fleeting quasi-local physical content only. This has been made exceptionally clear in the background-free Einstein GR, in which the "points" have the ontological status of Aristotelian potentia, and not some 'objective reality out there'. The simple idea comes from Plato: "observations are mere shadows of some more fundamental entities" (Calude, Hertling & Svozil, p. 372). We need a Potential Reality (PR) interpretation of QM. Physically, every "point" is the center of the universe, and the very presence of one "point" requires the presence of infinite (actual infinity) "points", otherwise The Beginning would be exposed in the local mode of spacetime (see above), and no theory of relativity would be possible. All the rest is (relatively) simple math, which you can get from your local Professor in Theoretical Physics or from Chris Isham's Modern Differential Geometry For Physicists, 2nd Edn. But before you proceed, please read a cautious note here. Back in 1999, Chris Isham kindly suggested to me that I should consult a textbook in differential geometry. I asked him, very politely indeed, whether his famous textbook Modern Differential Geometry For Physicists would be suitable, to which he responded, also very politely, that it may be too complicated. He was right: I was stuck at footnote 1 on p. 61, and couldn't go further. Hence I decided to write a brief note 'for pedestrians', to elucidate this particular footnote, as well as the “medium” in Einstein GR (cf. Landau & Lifshitz below). Reading Chris Isham is a must, and I always pay special attention to his crucial footnotes. The latest example can be found in arXiv:quant-ph/0703066v1, p. 2, footnote 3. It reads: "The ideal monad has no windows." I fully agree, but the math is yet to be discovered. In my opinion, a good starting point is Chris Isham's Modern Differential Geometry For Physicists, 2nd Edn. The basic basics can be read in Michael Spivak's introduction to differential geometry, vol. 2: just zoom on the Levi-Civita connection in Ch. 6, and you'll unravel the Aristotelian Connection introduced there 'by hand'. All you need is to trace back its origin and find out how it has been "inserted" there. Just please do not call it "dark"!
"It is necessary, strictly speaking, to have a set of an infinite number of bodies filling all space, like some “medium”. Such system of bodies together with connected to each of them arbitrarily clocks is a frame of reference in the general theory of relativity".
Subject: arXiv:0704.1291v3 [math-ph] Note: Notice that the Schrödinger cat per se will never undergo any "collapse" by casting one of its possible states (either |alive cat> or |dead cat>) in the local mode of time. If we ask questions about its state as Aristotelian potentia, the answer will always be YAIN. I am unable, however, to avoid the "collapse" of the 'cat per se' by using non-Hermitian operators and cancel all-but-one of its possible explications with "negative probabilities", hence recover the dynamics of a single quantum system in its ever-changing 'quantum phase space' in which [tau] approaches asymptotically zero. See the three Schrödinger equations in Farias & Recami here. Perhaps one can describe the "negotiation" in the global mode of time with retarded and advanced Schrödinger equations, in a way resembling Cramer's Transactional Interpretation of QM. If the human brain were operating with non-Hermitian operators, here's what you would probably get (courtesy from Lewis Carroll): `Twas brillig, and the slithy toves
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There is no math here, and it will never be posted on this web site either. Consider Henri Poincaré's principle of relativity, as stated in 1905 (The Principles of Mathematical Physics, The Monist, 15(1), 1905): "The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion." Do you see any math from Henri Poincaré? It was nothing but clear thinking, and then the math automatically appeared due to its obvious, albeit unreasonable, effectiveness in the natural sciences (E. Wigner). Good luck with the Aristotelian Connection. D.C. =================
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Seriously, GWs and QWs do not carry energy, which is why they are "empty", and do not travel 
and 
are some sort of pseudo-distances that we use to explain poetically the "
