Subject: Einstein's Hole Argument Date: Fri, 13 Nov 2009 23:26:02 +0200 From: Dimi Chakalov To: Alan Macdonald Cc: John Stachel , Mihaela Dorina Iftime Dear Professor Macdonald, I mentioned your name and provided link to your article at http://www.god-does-not-play-dice.net/#NB As you said in "General Relativity in a Nutshell" (p. 84), "We must be prepared for changes, even radical changes, in our model." Since all these radical changes originate from Schrödinger (see the link above), I will be pleased to have your professional opinion. I would also expect to see the impact from Schrödinger's legacy in the second volume of John Stachel's "Going Critical" (The Practice of Relativity). Kindest regards, Dimi Chakalov  ================== Subject: "Going Critical" (in press ?) Date: Mon, 4 Feb 2008 15:31:51 +0200 From: Dimi Chakalov To: John Stachel , John Stachel Cc: M Iftime , M Iftime Please check out http://www.god-does-not-play-dice.net/Petkov.html#ADM Dimi Chakalov ============= Subject: The spatio-temporal structure of our universe is not "underdetermined" Date: Tue, 23 Sep 2008 19:24:37 +0300 From: Dimi Chakalov To: M Iftime , miftime@gmail.com Cc: John Stachel Dear Dr. Iftime, I mentioned your latest arXiv:0809.3596v1 [gr-qc] at http://www.god-does-not-play-dice.net/Szabados.html#Hilbert More at http://www.god-does-not-play-dice.net/Szabados.html#Angelo I wonder when "Going Critical" will be published. Kindest regards, Dimi Chakalov ============== Subject: ... until the dynamical fields are specified. Date: Tue, 24 Oct 2006 16:48:23 +0300 From: Dimi Chakalov To: M Iftime CC: Reiner Hedrich , Wasley Krogdahl , Michael B Mensky Dear Dr. Iftime, Since you used "until" (gr-qc/0610105 v1, p. 8), I wonder if you imply two kinds of time, similar or identical to those suggested at http://www.god-does-not-play-dice.net/download.html Sincerely, D. Chakalov -- Mihaela Iftime, The Hole Argument. Physical Events and the Superspace, gr-qc/0610105 v1 p. 8: "A background independent theory is a physical theory defined on a base manifold M endowed with no extra structure, such as geometry or fixed co-ordinates. If a theory does include any such geometric structures, it is called background-dependent. Theories like QED, QCD are theories on a fixed (flat or curved) background space-time metric. "GR or in any general relativistic theory on the other hand are distinguished from other dynamical field theories by invariance under "active" diffeomorphisms; its field equations are invariant under all differentiable diffeomorphisms (the group Diff(M)) of the underlying manifold M, which have no spatio-temporal significance until the dynamical fields are specified. ... p. 11: "... the stability problem of space-time solutions, which is one of the most important unsolved problems of GR." ----------   Note: I expect some day, hopefully by November 2015, John Stachel or his younger colleague Mihaela Iftime to reply to my arguments, in which case I will spell them out in full details. This hasn't happened in the past five years, which is why I will post below only two additional references. Regarding Geoffrey Chew's bootstrap program, notice the emergence of ONE entity in my bootstrap program -- 'the chooser' in the quantum realm 'out there'. Hence there is no need for God to "play dice". He is flexible. Simple, no? D. Chakalov October 25, 2006   Wasley S Krogdahl, Cosmology in Flat Space-Time, gr-qc/0402016 v3. http://arxiv.org/abs/gr-qc/0402016 We might say, therefore, that is a "time-dependent constant". It has a fixed value for all present calculations, but for convenience’ sake is adjusted as the universe ages. The epoch of observation is therefore a parameter of the universe. ... Intuitively, the human mind balks at the notion of a universe of finite extent. If there is a boundary, what is beyond? This is not an illegitimate question. Humorist Stephen Leacock posed the dilemma this way: "We cannot imagine that the stars go on forever. It’s unthinkable. But we equally cannot imagine that they come to a stop and that beyond them is nothing, and then more nothing. Unending nothing is as incomprehensible as unending something." --- Note: Notice that the questions by Stephen Leacock are wrongly formulated, because they emerge only in the non-relativistic (and very misleading) cosmological picture here. In the proper relativistic case, based on the Cosmological Principle, the ultimate "cutoff" is produced by the Aristotelian First Cause: the local mode of spacetime is wrapped by, and isolated from, the global mode of spacetime. Think of Ellis' 1984 'finite infinity' as resembling the absolute zero "temperature": a numerically finite yet physically unattainable "boundary". --- Reiner Hedrich, String Theory - From Physics to Metaphysics, physics/0604171 v1. http://arxiv.org/abs/physics/0604171 [Geoffrey Chew introduced in the sixties the bootstrap program] p. 7: For the bootstrap program, the only requirement for our description of nature is self-consistency. Its central assumption is that nature is determined completely and uniquely by internal coherence: "In the broadest sense, bootstrap philosophy asserts that nature is as it is because this is the only possible nature consistent with itself." (Chew (1968) 762) .... Footnote 31, p. 18: "an ensemble as causally unconnected (italics added - D.C.) realizations of possibilities"   ================================ Subject: Request for opinion Date: Mon, 21 Jan 2002 13:33:42 +0200 From: "Dimiter G. Chakalov" To: John Stachel CC: shimony@bu.edu, piotr@gargan.math.univ-tours.fr, psj@tifr.res.in, nkd@iucaa.ernet.in, roy.maartens@port.ac.uk, poisson@terra.physics.uoguelph.ca, krolak@panim.impan.gov.pl, parenta@celfi.phys.univ-tours.fr, geroch@midway.uchicago.edu, gary@cosmic.physics.ucsb.edu, moffat@medb.physics.utoronto.ca, cjsc@maths.soton.ac.uk, kip@tapir.caltech.edu, rmwa@midway.uchicago.edu, shapiro@astro.physics.uiuc.edu, saul@spacenet.tn.cornell.edu, adler@stars.sfsu.edu, Steven.Weinstein@Dartmouth.EDU, lehner@einstein.caltech.edu, norbert@physik.unizh.ch, jayant@iucaa.ernet.in, tsauer@gwdg.de BCC: [snip] Dear Professor Stachel,I am respectfully soliciting your opinion on the link and the common source of the following two issues: 1. There is no uniquely defined energy density of the gravitational field in Einstein's GR. 2. The non-existence of a gravitational field stress-energy in the right-hand side of the field equations, and subsequently the purely geometrical interpretation of gravity as some "curvature" of spacetime. Could you please shed some light on their common source? I suspect -- without being able to prove it -- that the roots of these tantalizing issues are in the absence of some 'back bone' of Einstein's GR, as hinted by the following two observations: (i) you can eliminate the gravitational field 'by hand', by a simple coordinate transformation, but this can not solve the conflict of energy conservation and general covariance [Ref. 1];  and (ii) the hole argument (or rather puzzle): "There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field" [Ref. 2]. In short, I have a gut feeling that Einstein's GR is essentially incomplete in the sense that it does not possess some 'back bone' that can remain 'out there' in the case of removing matter. Unlike STR with its fixed background,there is no 'primordial grid' in GR, which can remain 'out there' and can*guide* the bi-directional talk between matter and geometry: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" (John A. Wheeler). There is no 'third party' here, which is probably the reason why Einstein did not consider his theory complete, I will be happy to learn how would you assemble the jigsaw puzzle and trace down the link and the common source of the two problems mentioned above: no uniquely defined energy density of the gravitational field, and the non-existence of a gravitational field stress-energy in the right-hand side of the field equations. Please see also the opinion of Professor 't Hooft at I will also appreciate comments and opinions from all physicists reading these lines. Thank you very much in advance. Sincerely yours, Dimiter G. Chakalov http://members.aon.at/chakalov   References [Ref. 1] Tilman Sauer. The Relativity of Discovery: Hilbert's First Note on the Foundations of Physics. Thu, 26 Nov 1998 12:11:54 GMT, http://xxx.lanl.gov/abs/physics/9811050 (See Sec. 3.2, "The Concept of Energy".)   [Ref. 2] John Stachel. The Many Faces of the Hole Argument: From Space-Time to Quantum Statistics. Wednesday, February 7, 2001, http://www.phys.psu.edu/news/online_cgpg/118.html   ====== [Tue, 22 Jan 2002 05:12:05 +0200] P.S. Regarding my previous email from Mon, 21 Jan 2002 13:33:42 +0200, which you can read at my web site, let me try to outline a similarity between the hole argument (please see the URL above) and the instrumentalist interpretation of QM, for which the notion of 'determination' [Ref. 3] fits perfectly, I believe. If you don't have a context set by the 'determination', as defined by Bill Unruh [Ref. 3], you have a genuine 'hole': you can say nothing about the state of the quantum system. This does not mean that the quantum realm exists in our imagination only, but that we may not have captured its essence. Likewise, by removing all fields, we are left with a 'hole' or "empty space", but this peculiar feature of GR is not a *proof* that the essence of spacetime is determined exclusively by the structural quality of the field [Ref. 2]. It may be nothing but a requirement that the essence of spacetime, as some sort of 'primordial grid', must *not* exist in time, as read with a physical clock. Same in the case of QM: the essence of a quantum system must not exist intime, as read with a clock. If that is the case chosen by Mother Nature, then the only option left, the way I see it, is to elaborate a new kind of quantum spacetime, which would look "continual" from the local time mode, and "discrete" from the global time mode, Thus we might have, in the putative *global* time mode, a perfect 'remnant' from the quantum system 'out there', which will not disappear between determinations [Ref. 3], as well as a 'back bone' for the determination of spacetime in the local time mode, "during" the bi-directional talk between space and matter (J.A. Wheeler). Anytime we make observations in QM and in GR, we're confined in the local time mode, and can see only the final product of this truly creative process, post factum only, in our past light cone. Theputative global time mode, the common denominator of QM and GR, does not exist there. Just a wild guess. Please see my web site for details at the URL below, if you're interested. I will be very happy to learn your professional opinion, as well as those of all colleagues of yours. I am a layman in your field and all I need is to get closer to the truth, whatever it is. Dimi -- Dimiter G. Chakalov http://members.aon.at/chakalov   --- [Ref. 3] Bill Unruh. Varieties of Quantum Measurement. Proceedings of the Conference on Quantum Measurement held at Baltimore, June 19-22, 1994. http://xxx.lanl.gov/abs/hep-th/9410168 "However, the word "measure" brings with it the image of a physical process. Measurements are performed by means of measuring apparatuses.  As aspects of the physical world, such measuring apparatuses should themselves be describable by quantum mechanics itself. But it is difficult to have a systemin which at the same time a concept is an axiomatic feature of the theory, andone describable by the theory. I would therefore suggest that the word"determine" be used instead for this axiomatic feature of the theory. Thus I would rephrase the above sentence as "When one determines a quantity, and the knowledge (or conditions) underwhich one wishes to determine that quantity are represented by the vector $|\phi >$ , then the determination of a quantity represented by {\bf A} gives one of eigenvalues of {\bf A}, say  $a$  with probability  $|{|}^{{2}}$ ."  "Determination, in this axiomatic sense, says nothing about how the determination was made. It is simply a statement of a mapping from the theory to our experience, in which some knowledge sets the conditions on the questions we wish to ask, and some knowledge represents the answers to the questions we want to ask." =========== Subject: The principle of maximal permutability & null hypersurface quantization Date: Fri, 09 Dec 2005 16:30:47 +0200 From: Dimi Chakalov To: John Stachel ,      John Stachel CC: Chris Isham ,      David Robinson ,      Joshua Goldberg ,      Carl Hoefer ,      Graham Nerlich ,      Decio Krause ,      Jeremy Butterfield ,      Michael Redhead ,      Mihaela Iftime ,      Oliver Pooley ,      Szabados Laszlo Dear Professor Stachel, It is a great pleasure to read your papers. Let me please comment on your principle of maximal permutability [Ref. 1], which I mentioned two days ago at I will try to comment on your principle by following the advice of Kurt Lewin [Ref. 2]: "If you want truly to understand something, try to change it." Your feedback, as well as the comments from your colleagues, will be greatly appreciated, and will be kept private and confidential. I'll mark my comments with 'C'. C1. You suggested that "one would expect the principle of maximal permutability of the fundamental entities of any theory of quantum gravity to be part of a theory in which these entities are only individuated dynamically" [Ref. 1, p. 10]. It seems to me that your principle requires the ontological principle of relational reality [cf. NB in Ref. 2]. If true, we can explain how the entities can be individuated *dynamically* only if we can solve the paradox of Buridan donkey for 'relational reality', C2. You wrote [Ref. 1, p. 21: "I see no reason why a quantum theory of gravity should not be sought within a standard interpretation of quantum mechanics (whatever one prefers)." I am not aware of any interpretation of QM, which could solve the puzzle of the 'peaceful co-existence of QM and STR' (I believe Prof. Shimony can elaborate extensively). The way I see it, the problem here is that we cannot trace the "quantum event" back to its origin, The problem very much resembles geodesic incompleteness, in the sense that we have to stop tracing back the quantum beast just "before" its entry point in the Minkowski cone. If true, it seems to me that the 'entry point' could only be the apex of the cone, Which brings us to the next issue, the null hypersurface quantization [Ref. 3]. C3. You wrote [Ref. 1, p. 18, footnote 36]: "The only "natural" foliation would be a family of null hypersurfaces, and null hypersurface quantization has had many advocates, starting with Dirac." I wonder if you can formulate your principle exclusively on null hypersurface [Ref. 3], which is perhaps the only viable framework for addressing the Buridan donkey paradox (please C1 above and NB in [Ref. 2]). All this is still very academic, so let me try to suggest something related to your very important note on the measurement of gravitational field quantities [Ref. 1, Footnote 46, pp. 21-22]. C4. It seems to me that the measurement of the strain of GWs requires a brand new experimental setup, to account for its quasi-local nature, Please see the 'shoal of fish' metaphor on p. 7 in my essay on GW astronomy, I believe Prof. Goldberg can explain why the conservation laws for energy-momentum and angular momentum do not have a local meaning, and even their global definitions are confusing at null infinity (e.g., John Stewart, Advanced General Relativity, CU Press, Cambridge, 1993, Ch.3.12). It seems to me that LIGO and all other ground- and space-based devices are manifestly blind and deaf to the quasi-local nature of GW energy, firstly, and secondly -- they cannot provide the essential *shielding* from GWs [Ref. 1, Footnote 46, pp. 21-22], which is needed to fix a *reference object* with UNdisturbed metric. Thank you for your time. Kindest regards, References and notes [Ref. 1] John Stachel, Structure, Individuality and Quantum Gravity, gr-qc/0507078 v2, 19 July 2005. http://arxiv.org/abs/gr-qc/0507078 p. 9: "Generalizing from the previous examples, I maintain that the way to assure the inherent indistinguishability in of the fundamental entities of the theory is to require the theory to be formulated in such a way that physical results are invariant under all possible permutations of the basic entities of the same kind (same quiddity).20 I have named this requirement the *principle of maximal permutability*. (See [63] for a more mathematically detailed discussion.) ... p. 10: "In both the case of non-relativistic quantum mechanics and of general relativity, it is only through dynamical considerations that individuation is effected. In the first case, it is through specification of a possible quantum-mechanical process that the otherwise indistinguishable particles are individuated ("The electron that was emitted by this source at 11:00 a.m. and produced a click of that Geiger counter at 11:01 a.m."). In the second case, it is through specification of a particular solution to the gravitational field equations that the points of the space-time manifold are individuated ("The point at which the four non-vanishing invariants of the Riemann tensor had the following values: ..."). So one would expect the principle of maximal permutability of the fundamental entities of any theory of quantum gravity to be part of a theory in which these entities are only individuated dynamically. ... p. 18: "General relativity is an inherently four-dimensional theory of space-time -- even more so than special relativity. There is no "natural" breakup of spacetime into spaces and times, such as the inertial frames provide in special relativity. There are no preferred timelike fibrations or spacelike foliations.36 --- Footnote 36: "The only "natural" foliation would be a family of null hypersurfaces, and null hypersurface quantization has had many advocates, starting with Dirac. For a survey, see [?]. ... p. 21: "I see no reason why a quantum theory of gravity should not be sought within a standard interpretation of quantum mechanics (whatever one prefers)." ... Footnote 46, pp. 21-22: "(...) There is one big difference between the Maxwell field and the gravitational field: the non-universality of the electromagnetic charge-current vector versus the universality of gravitational stress-energy tensor. Because charges occur with two signs that can neutralize each other, a charge-current distribution acting as a source of an electromagnetic field can be manipulated by matter that is electrically neutral and so not acting as a source of a further electromagnetic field; and one can shield against the effects of a charge-current distribution. Because mass comes with only one sign, all matter (including non-gravitational fields) has a stress-energy tensor, no shielding is possible, and any manipulation of matter acting as a source of gravitational field will introduce an additional stress-energy tensor as a source of gravitational field. A glance at Bohr and Rosenfeld 1933 shows how important the possibility of neutralizing the charges on test bodies is for measurement of the (averaged) components of the electric field with arbitrary accuracy, for example. This difference may well have important implications for the measurement of gravitational field quantities."   [Ref. 2] Kurt Lewin (September 9, 1890 - February 12, 1947) was a German psychologist. He introduced the concept of genidentity (Genidentität) in 1922, http://www.psicopolis.com/Kurt/lewinsintesi.htm The notion of genidentity is close to the Russel's notion of a 'causal line', a temporal series of events so related that, given some of them, something can be inferred about the others whatever may be happening elsewhere (Bertrand Arthur William Russell, Human Knowledge, its Scope and Limits, George Allen, London, 1948, p. 477). There is an important difference: Russell's 'causal line' is an epistemic relation, whereas genidentity is an ontological one -- the persistence of objects through time due to their 'sameness', The topic is particle identity. Take the electron: there are about 10^80 electrons in the universe, which are exactly 'the same'. (By relying on math only, Paul Dirac proposed that there is really only one electron, and John Wheeler suggested that this 'one electron' is zig-zagging in time.) NB: In the context of John Stachel's principle of maximal permutability, the 'sameness' of objects is expressed *dynamically* [Ref. 1, p. 10], and in full compliance with the ontological principle of 'relational reality', The ontological principle of 'relational reality': The necessary and sufficient condition for an object, call it A, to *exist* at a given instant, is its already-fixed relation to not-A. The union of [A] and [not-A] form a set which (i) can have infinite cardinality and (ii) might refer to non-Archimedean reality, to allow for the emergence of brand new elements of 'the same set': "Time is Nature's way to keep everything from happening all at once." (J.A. Wheeler, "Time Today", in Physical Origins of Time Asymmetry, J.J. Halliwell, J. Pérez-Mercader, and W.H. Zurek, eds., Cambridge UP, Cambridge, 1994, p. 1).   [Ref. 3] D.C. Robinson, Geometry, Null Hypersurfaces and New Variables. In: Revisiting the Foundations of Relativistic Physics. Festschrift in honour of John Stachel, Ed. by A. Ashtekar, R. Cohen, D. Howard, J. Renn, S. Sarkar and A. Shimony, Kluwer Academic Publishers, 2003, pp.349-360. http://www.mth.kcl.ac.uk/staff/dc_robinson/newvariables.pdf [Sec. 1, Null hypersurface canonical formalisms] "A key reason for the successful use of null hypersurfaces in the study of field theories is that constraints arising in the relevant characteristic initial value problems can be integrated more easily than those arising in corresponding Cauchy problems. The true degrees of freedom can be exposed more explicitly when null hypersurfaces, rather than space-like hypersurfaces, are used. "Amongst the geometrical features which distinguish null hypersurface based formulations from space-like hypersurface based approaches to Hamiltonian gravity, are the following. First, the geometry of null hypersurfaces, unlike that of space-like hypersurfaces, is not metric."     Note: The idea behind John Stachel's principle of maximal permutability is the principle of invariance. David Mermin has explained it as follows (N. David Mermin, It's About Time: Understanding Einstein's Relativity, Princeton University Press, Princeton, 2005, Ch. 1): "The principle of relativity is an example of an invariance principle. There are several such principles. They all begin with the phrase "All other things being the same." Then they go on to say: it doesn't matter where you are. (Principle of translational invariance in space) it doesn't matter when you are. (Principle of translational invariance in time) it doesn't matter how you are oriented. (Principle of rotational invariance) The principle of relativity fits into the same pattern: All other things being the same, it doesn't matter how fast you're going if you're moving with fixed speed along a straight line. (Principle of relativity) "It doesn't matter" means "the rules for the description of natural phenomena are the same." For example the rule describing Newton's force of gravity between two chunks of matter is the same whether they are in this galaxy or another (translational invariance in space). It is also the same today as it was a million years ago (translational invariance in time). The law does not work differently depending on whether one chunk is east or north of the other one (rotational invariance). Nor does the law have to be changed depending on whether you measure the force between the two chunks in a railroad station, or do the same experiment with the two chunks on a uniformly moving train (principle of relativity)." The way I understand the principle of maximal permutability (see Sec. 6.3, The Hole Argument for Relations, in J. Stachel, M. Iftime, gr-qc/0505138 v2, which appeared as ref. [63] in [Ref. 1]), it seems to me that it holds for background-independent theories only. See John Baez here, and recall the famous saying of Lao-tzu: If you realize that all things change, there is nothing you will try to hold onto. In fact, the only thing we can try to 'hold onto' is the famous 'relational reality' (cf. NB in [Ref. 2]). Mihaela Iftime and John Stachel have explained it (gr-qc/0512021 v1, p. 12) as follows (emphasis mine): "In a background independent theory ( e.g. general-relativistic theories), if we start from some natural bundle, the base points may be characterized as such independently of the particular relations in which they stand; but they are entirely individuated in terms of the relational structure given by cross-sections of some fibered manifold. "In our invariant formulation, the hole argument can be easily further generalized to the case of sets and relation with some important implications to problems in quantum gravity. [7]. Elementary particles are similarly individuated by their position in a relational structure and this suggests the following viewpoint: Since the basic building blocks of any model of the universe, the elementary particles and the points of space-time, are individuated entirely in terms of the relational structures in which they are embedded, only "higher-level" entities constructed from these building blocks can be individuated independently. "Therefore, the following principle of generalized covariance should be a requirement on any fundamental theory: The theory should be invariant under all permutations of the basic elements, out of which the theory is constructed." The problem is, how can we think of a background-free theory, which obeys the principles of 'relational reality' and 'maximal permutability'? We need a very special kind of background, since "one can dig oneself into a hole by trying to do physics without any background structure - it's a bit like trying to paint a painting without any canvas" (John Baez). Let's think of this special kind of background as a Platonic idea. Consider, for example, the meaning of 'point'. It certainly has very complex Neural Correlates (NC) in all brains of all people, and these NC are physical stuff that does change in time. The meaning remains invariant, however. If we place the meaning of 'point' in the "space of states" of its NC, we shall break the principle of maximal permutability, because the meaning of 'point' will become a physical stuff which has fixed a background in the space of possible states, and subsequently there will be at least one transformation in that space of states, which could break the principle. What, then, could be the physical correlate of the meaning of 'point'? It is just a potential reality. In the case of [Ref. 1], the new kind of "background" is the 'potential point'. It is not a mental phenomenon but a broader kind of reality -- potential reality. Its fleeting content (filling) is invariant under all active diffeomorphisms [Ref. 1, footnotes 20 and 21]. In other words, we have a 'potential point' for all solutions to the gravitational field equations in which "the points of the space-time manifold are individuated ("The point at which the four non-vanishing invariants of the Riemann tensor had the following values: ...")" [Ref. 1, p. 10]. Thus, the 'potential point' does not, and cannot have any intrinsic content. It is merely an empty shell, or matrix, which is being filled with transient physical content. In the case of the human brain, you can verify your ability to create a 'potential point' by reading the following text, compiled on the basis of the principle of maximal permutability: Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae.  The rset can be a total mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn biran deos not raed ervey lteter by istlef, but the wrod as a wlohe. Pritie amzanig huh? "It doesn't matter" (cf. David Mermin above) in what order the letters in a word are, you'll create the proper "meaning" (potential point) as long as the first and the last letters are at the right place. So, where is the "background" that provides the meaning of all words? If your brain can create such background of 'potential points', Mother Nature should be able to do the same. Perhaps much better. To sum up, I fully agree that there is no spacetime without matter: "Space-time does not claim existence on its own, but only as a structural quality of the field" (A. Einstein). Fine. But once we have spacetime, there is something more to it. I call it 'potential points'. They are both "one" and "many", hence can qualify as a non-Archimedean reality [Ref. 2]. They are also completely "dark": see the discussion of the dynamic "dark" energy here and here. In Einstein's GR, the main conjecture is that the gravitational potential is being "produced" from a massive body in space, and we happily point to some massive body 'out there', and say -- look, this is what makes the so-called spacetime curvature. Fine, but for scales larger than the Solar system, there are additional, and totally "dark", holistic effects from the potential points (recall the Pioneer Anomaly). The potential points also show up as 'perennials' (Karel Kuchar). As explained by Rodolfo Gambini and Jorge Pullin (gr-qc/0512065 v1, pp. 5-6): "To try to get more meaningful information one would like to concentrate on "observables". In the continuum theory, these are quantities that have vanishing Poisson brackets with the constraints (also sometimes known as "perennials"). Knowing these quantities as functions of phase space allows to know any type of dynamical physical behavior of the system. (...) It should be noted that in the continuum theory perennials as functions of phase space are defined up to the addition of multiples of the constraints. There are therefore infinitely many versions of a given perennial." We have no other choice but to enter "all the doors at once", as Karel Kuchar argued on April 8, 1993. The potential points are also the 'back bone' of recovering the continuum limit to the classical world, since they provide a smooth and reversible quantum-to-classical and classical-to-quantum transitions without "observers". The problem can be traced back to the non-relativistic QM: no time operators, hence no possibility to model the very source of the real, explicated events or "flashes" (John Bell) represented mathematically by "spacetime points" (J.S. Bell, Are there quantum jumps?, in Speakable and unspeakable in quantum mechanics, Cambridge University Press, Cambridge, 1987, pp. 201-212). If you believe that theoretical physics should consider only the real, explicated events/flashes constituting the "spacetime points", read Sean Carroll's Invited Review for Nature, hep-th/0512148 v1: "... for some reason, the early universe was in a state of incredibly low entropy. ... "The fact that the initial proto-inflationary patch must be smooth and dominated by dark energy implies that it must have a very low entropy itself; reasonable estimates range from SI ~ 100 - 1020. Thus, among randomly-chosen initial conditions, the likelihood of finding an appropriate proto-inflationary region is actually much less than simply finding the conditions of the conventional Big Bang model (or, for that matter, of our present universe). It would seem that the conditions required to start inflation are less natural than those of the conventional Big Bang. ... Footnote 5: "The fact that entropy tends to increase is of course just the Second Law of Thermodynamics, and makes sense once we assume that the initial entropy was low. The puzzle is why the early universe should differ so dramatically from the late universe, despite the intrinsic time-reversibility of the microscopic laws of physics; this is known as the "arrow of time" problem." And how do you solve this puzzle? "But even empty space can have energy" (Sean Carroll and Jennifer Chen, gr-qc/0505037 v1, p. 3). Welcome aboard! But can't you see that 'unitary evolution' is an oxymoron in quantum cosmology? It just doesn't make sense. Last but not least, the potential point(s) can be revealed in the primordial affine structure of space, as argued by Graham Nerlich here. The quantum version of this phenomenon can again be elucidated with an example from non-relativistic QM. As stressed by Larry Horwitz (quant-ph/0507044 v4), "the Hilbert space of the quantum theory is constructed of a set of wave functions satisfying a normalization condition based on integration over all space, e.g., for a single particle, [XXX] = [inf.], for each value of the parameter \t. There is therefore a distinct Hilbert space for each value of the parameter \t." There is a distinct Hilbert space for each and every "flash" (cf. John Bell above), but there is no Hilbert space for the potential point(s) from which these "flashes" emerge. Again, once we use a Hilbert space, there is something 'more' to it: the potential point(s). See also the 'shoal of fish' metaphor on p. 7 and p. 16 from gw.pdf. This whole issue is very old: "The requirement of general covariance takes away from space and time the last remnant of physical objectivity", wrote Albert Einstein in 1916. I believe the "remnant" of physical objectivity can be recovered only with the potential point(s). Needless to say, the ultimate potential point is beyond human comprehension, since it should give meaning to 'absolutely everything': the state of the whole universe as ONE. Any other ideas? I'm fully open to suggestions, like a potential point! Back in 1772, on the occasion of the fall of meteorites, the French Academy of Sciences adopted a resolution categorically rejecting such anomalous phenomena. The obvious reason had been that rocks cannot fall from the sky, simply because there are no rocks there. In our case, having no reasonable explanation for nearly 95 per cent of the stuff in our Universe, we better keep our mind open to all possibilities.     D. Chakalov December 12, 2005 Last update: December 23, 2005 -- 'Twas brillig, and the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe. Lewis Carroll,Jabberwocky