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Mens agitat molem: A proposal for changing the Zeitgeist

April 10, 2008: A possible unification of Quantum Theory and General Relativity, on the basis of a new notion of reality, potential reality, which is missing in QM and in GR, has been proposed. A new structure of spacetime, resulting from a hypothetical 'arrow of spacetime' producing 'potential reality', has been postulated. The crucial prediction of the theory is the so-called 'scale relativity principle' concerning the nature of 3-D space and its time.




It's all about the "spirit" of geometry. Since Hermann Minkowski's lecture, delivered on September 21, 1908, all arguing about the presentation of time by contemporary theoretical physics boils down to the following simple question: can we imagine that the 3-D space can "move"? For time can "move" just as much as space can. No way, according to GR textbooks. Hence the puzzle of time can only be solved by making the 3-D space "move", but in a special, hidden way. Namely, by introducing a faculty of "lapsing" for 3-D space, such that no observable "gaps" would be detectable in principle, thus rendering the (local mode of) spacetime a perfect continuum, with dimensions of physical bodies along the "direction" of the "lapse" of 3-D space being infinitesimal, i.e., tending asymptotically toward zero. For this purpose, we introduce a new parameter of spacetime, called 'global mode', which pertains to a new kind of reality, called 'potential reality'. Both will be used, arguably, to (i) enable the new dynamics of 3-D space (with embedded "dynamic dark energy", as we shall see later), (ii) reveal the dynamics of quantum reality and the mechanism by which the quantum of action and quantum waves are being produced, and (iii) explain the artifacts from "filtering out" the potential reality (obliterating the input from the final cause) by the local structure of spacetime at the scale of tables and chairs (unanimated macro-devices). The underlying idea is to model the whole universe as a brain with a quasi-local structure of spacetime; hence the notion of dynamical determinism. The latter seems to require some non-Hausdorff manifold allowing possible potential states to "branch" into the realm of potential reality. Hence the rock-ribbed assumption that the model of spacetime can be based on Hausdorff manifold is rejected as being inadequate -- it cannot accommodate the potential reality and its global mode of spacetime. Finally, it is suggested that the mechanism of converting the physical reality into pure geometry can be provided only by the dynamics of the whole universe in the global mode of spacetime. Put it differently, the final layer of reality, occupied by the First Cause of Aristotle, is the "spirit" of geometry: Mens agitat molem (Vergil, The Aeneid, Ch. 6, 727). It may look "dark" only to those obstinately opinionated people who endorse the Hamiltonian formulation of GR.

Sunday, October 8, 2006


A personal note: I've been studying the issues mentioned in the abstract above since January 1972. I have to stress that the professional (not emotional) opinion of Britain's leading expert in quantum gravity was formulated in the following fashion: "You do not know enough theoretical physics to help with any research in that area." I think this is a polite way of saying that I am mentally retarded person. If true, I am obliged to issue a cautious note to the reader of these lines: you may be wasting your time.


It is difficult to get a man to understand something when his salary depends upon his not understanding it, said Upton Sinclair. Let me presume that he was wrong.

Let us begin with St. Thomas Aquinas, Summa Theologiae, Part I, Question 85, Article 8: "Punctum non ponitur in definitione lineae communiter sumptae" (‘A point is not included in the definition of a line as commonly assumed’). He didn't use math, however.

Unlike St. Thomas Aquinas, Kurt Gödel was a brilliant mathematician. He addressed the peculiar statement above in "What is Cantor's Continuum Problem?", American Mathematical Monthly, Vol. 54, No. 9 (November 1947), pp. 515-525 (see the first page here), and wrote: "Cantor's continuum problem is simply the question: How many points are there on a straight line in Euclidean space?"

Uncountably infinite, we would say, although we can't prove it. All we could claim is that the alternative option is definitely wrong, for if the "number" of points were 'countably infinite', we wouldn't be tortured by the Thompson's lamp paradox:

Imagine a lamp that is turned 'on' at some instant labeled with  0 , and is left 'on' for 1 min, then turned 'off' for 0.5 min, then 'on' for 0.25 min, etc., ad infinitum. Do we have a limit? Obviously yes: 2 min. Even my teenage daughter can understand the limit. There are obviously finite things around us, which build up the good old 3-D space, such as a table being 2 m long and a time interval of duration 2 min. Fine, but what is the state of the lamp in the instant/point labeled with '2 min'? This is the lesson from St. Thomas Aquinas. But let's go back to Kurt Gödel.

Considering the mass points on a square and on a line segment, "both completely filled with mass points (so that at each point of them exactly one mass point is situated)", Kurt Gödel showed that there is no difference between the "number" (Sic!) of the mass points, because "the mass points of the square can be so rearranged as exactly to fill out the line segment, and vice versa". Therefore, the "number" of these 'infinitesimal points' or  "atoms" of spacetime is anything but a number per se, as we should have gathered from Lucretius.

A 'number' is a static, frozen mathematical entity. It does not necessarily have to be exact (one good example are the irrational numbers), yet we know that under certain conditions we can obtain an exact, rational number from irrational ones (e.g., by multiplying sq. r. from 3 with sq. r. from 12 we obtain +/- 6). The "number" of the atoms of spacetime, which exactly fill out a line segment or a square (such that there are no empty gaps of 'something that is smaller than the atom'), is not a number at all, since it is dynamical: it never stops (I shall elaborate on this issue later). If we "stop" it by imposing some smart recipe on it (as we do in differential calculus or as Archimedes did quite some time ago), we can, of course, obtain a seemingly precise numerical value, but it will already be from the local mode of spacetime -- just a shadow on Plato's cave. The "number" of such "shadows", called  φ , is UNdecidable, since it originates from the global time (an example with four sayings,  φ = 4  , can be read here).

To those interested in math, I strongly recommend Wolfram MathWorld, in which I found a very nice song: "Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat). This non-Archimedean quality of Aleph-null is shared by the just-introduced UNdecidable "number"  φ . It can be any number, hence it isn't a number per se.

Briefly, any finite spacetime domain, regardless of its size, contains an uncountable -- undenumerably infinite -- and undecidable  φ - "number"  of atoms which do not have size, as stressed by Lucretius. In the local mode of time, we can think of the atom of spacetime as being either 'extremely small but finite' or 'exactly zero', hence cannot recover the finite size of physical bodies around us, as known since Zeno's first aporia. To recover the finite size of things and hence the key attributes of 3-D space -- 'large vs small', 'inside vs outside', etc. , -- we need the atom of Lucretius. It does not have any "size" whatsoever, since it lives in the global mode of time. Thus,  φ  can take any number (John's jackets). The simplest case of  φ = 2  (spin-up & spin-down) was presented by Wolfgang Pauli with the famous phrase "eigentümlichen, klassisch nicht beschreibbaren Art von Zweideutigkeit". If we replace Zwei with infinite, we will obtain the full potential power of  φ .

Unlike Aleph-null
φ  can take any positive rational number, the "number"  0  ['the ideal monad' and John 1:1] and Aleph-null included. It rather acts  like an operator, and represents an Aristotelian potentia -- a broader form of reality which may be described as “between the idea of a thing and a real thing.” Or simply 'potential reality'. More here.

We have encountered, since the time of Lucretius, a brand new, holistic, non-Archimedean reality (global mode of spacetime), from which the observable and finite things emerge (C. Isham and J. Butterfield) in the physical world. Just as the mathematical reality arises from the agreement of mathematicians on axioms which, as Kurt Gödel put it in November 1947, "force themselves on us as being true", there exist simple metaphysical ideas which form a metaphysical reality. We cannot reject these metaphysical ideas, because they also ‘force themselves on us as being true’. I will talk here about such metaphysical reality only. The math simply follows the rules and the logic of metaphysical reality, and if we choose to ignore them, the result can only be a devastating mathematical jungle.

As an example for 'metaphysical reality', consider the famous quote from St. Augustine: "How can the past and future be when the past no longer is and the future is not yet? As for the present, if it were always present and never moved on to become the past, it would not be time but eternity." If you ignore these simple ideas, you will be destined to "explore" the jungle of block universe, until you completely and totally retire.

To demonstrate the rules and the logic of metaphysical reality, I will refer the reader to a beautiful paper by Alfredo Macias and Hernando Quevedo, Time paradox in Quantum Gravity, gr-qc/0610057 v1, Sec. 3: the infinitesimal displacement, which induces an infinitesimal change in the "proper time" (the elementary timelike displacement or 'tick of time') goes along with the same infinitesimal change in spatial coordinates (the elementary shift in 3-D space) in some kind of 'proper space', correct? Even teenagers can grasp this "atom" of spacetime with which we build the geometry of the physical world -- 'the grin of the cat without the cat', as observed by Alice.

This "proper time" does not, and cannot exist in GR, however. It is used only in classical mechanics -- we are indeed allowed to use a clock that reads some 'proper background time', because the time measured by such a clock is presumed to be ‘external’, existing independently of it (see the so-called dualistic conception of time here), while in Einstein's GR the dynamics of gravitational field contains this same "clock".

Hence in GR it is not possible in principle for the gravitational field to be parameterized (nor quantized -- read Angelo Loinger) "by itself", like Baron Munchausen. Which is why the founding fathers of GR, Hilbert and Einstein, were searching for some fundamental 'reference fluid' that can be held accountable for the obvious, but unexplained by GR, existence of some elementary timelike displacement. Another reason why the fundamental proper time cannot in principle exist in GR is the fact that GR does not include the cosmological time arrow: no explanation of the dynamic "dark" energy is available in GR. Even if someone manages to upgrade GR by including the driving force of the cosmological time arrow, the dynamics of gravitational fields will still have to be parameterized "by itself", like Baron Munchausen. As stressed by C. Rovelli, "properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable." The price to pay for having a genuine non-linear dynamics is that we can't have any observable time variable which would parameterize an agent that is 'acting on itself', like Baron Munchausen. Only the human brain can act on itself, but the time variable in question is far more complex than the one read by an unanimated wristwatch. The latter can read only a chain of already-completed bi-directional "talks" between matter and geometry (J. Wheeler), since in GR "a spacetime notion can only be recovered a posteriori, once the complete dynamics of the coupled geometry-matter system is worked out" (A. Perez). The human brain -- as well as the universe modeled as a human brain -- can read the full non-linear dynamics of the matter-geometry "talk" in the global mode of spacetime. It is designed to leave absolutely no gaps between those already-completed snapshots of physical reality in the local mode, thus rendering the latter a perfect continuum. Obviously, if you look at the local mode of spacetime only, it would look dead frozen (here could be no dynamics within a perfect continuum itself; see Bob Geroch below), since there are no "gaps" of empty space (L. Krauss) from which the dynamic "dark" energy could spring.  (But I could be all wrong -- see the cautious note above.)

Thus, the dynamic "dark" energy, left on a frozen Cauchy hypersurface, can only be (i) always vanishing small, and (ii) always a constant. Also, you will never be able to answer the question 'dynamic dark energy ... of what?', and will be forced to talk like parapsychologist. It is pointless to introduce new exotic scalar fields 'by hand' and try to resolve their "dynamics"; see T. Padmanabhan.

Thus, the fundamental proper time, as being build from a chain of elementary timelike displacements, is expelled from GR from the outset: there is no "evolving time" in the so-called block universe. Just a dead frozen snapshot of the world: "There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes" (Bob Geroch). Not surprisingly, GR cannot explain "the dynamics within space-time itself" induced from the dynamic "dark" energy of the cosmological time arrow. Moreover, the fact that we actually observe time evolution in FRW cosmology is a mystery: it implies that "either the mathematical formalism ... is inappropriate or we are missing some new physics." (T. Thiemann).

Or both, as Thomas Huxley would have said, stressing that the great "tragedy" of Science is the slaying of a beautiful hypothesis by an "ugly" fact.

For these reasons, the 'true proper time' is nonexistent in GR. Yet many people claim that its counterpart, the 'proper space', is the 3-D space around us!

But if you lose the time, you lose the space as well: "Nobody has ever noticed a place except at a time, or a time except at a place", says Hermann Minkowski. Even A. Ashtekar had to acknowledge that the splitting of spacetime into space and time is "doing grave injustice to space-time covariance that underlies general relativity". This unimaginable error is the cornerstone 'salient point' of the Hamiltonian formulation of GR, as advocated by Dirac-ADM, and the result from this grave injustice to Hermann Minkowski is well-known: "In general relativity, dynamics is entirely generated by constraints. The dynamical data do not explicitly include a time variable." (K. Kuchar). Not surprisingly, there is no solution to the Cauchy problem in GR (J. Stachel), for more than half of a century (more below). It is a fundamental problem of GR, which cannot be solved in the framework of classical GR in principle. You need to accept the full reparametrization invariance of GR, and the invariance of the theory under arbitrary reparametrizations of time (R. Parentani) leaves your hands tied to the 'coordinate time' only. As Hermann Weyl put it, "The introduction of numbers as coordinates ... is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations." Hence non-tensorial quantities are not accepted in GR. Since you are forced to model the dynamics of GR under the limitations from the "diffeomorphism freedom" (R. Geroch), you cannot bring back into the theory the very thing you've expelled from the outset: the flexibility of gravitational field's degrees of freedom (dynamical determinism). (For example, the gravitational energy cannot be a 'local observable' in principle (J. Kijowski), yet there is no general rule for 'quasi-local' interactions; just some custom-made recipes with highly unrealistic applicability -- see R. Penrose.)

Hence the Cauchy problem in GR cannot be resolved in classical GR in principle. Mother Nature is not constrained to tensorial quantities and covariant derivatives only -- it all depends on what Her definition of "is" is. (But of course I could be all wrong -- see the cautious note above.)

Thus, in order to recover 'the proper spacetime' around us, we have only one choice left: let the 3-D space "move" in the global mode, but hide its "move" (the elementary spacetime displacement) in the local mode, like strips from a movie reel. On one hand, we need a perfect continuum of "points" in the local mode to recover 'the proper spacetime' -- no gaps greater than 'the infinitesimal' are allowed. On the other hand, we need the same "gaps" that are non-existent in the local mode to link the "points" in it, in order to apply the notion of connected Hausdorff manifold. Can't use the local mode, obviously, hence we need the global mode. Which in turn means that we have to introduce two kinds of time, pertaining to two kinds of reality. And the mechanism which hides the infinitesimal "gaps" in the local mode is nothing but the so-called speed of light. All we need to do is to zoom on the famous squared interval which, in units such that the speed of light is unity, is "the difference between a squared time increment and the sum of three squares representing the three dimensions of spatial change" (G. Sparling). Just keep in mind that, strictly speaking, the so-called proper time (cf. Ed Witten below) can exist only in the past, i.e., only after the global time has overlapped with the local time. Hence the global time is not explicitly present in the excerpt below.

If we move from STR to GR, the proper time acquires an additional property: it is not fixed in the future anymore. It has become fully negotiable, relational, and utterly flexible. More here.

Last but not least, the elementary or infinitesimal displacement (the first notion you come across in differential geometry) can be rigorously defined only and exclusively only under the conditions of well-defined "boundaries" of spacetime: you need to define the spacetime in consideration en bloc, in order to define the elementary 'tick of time and shift in space' in it. In other words, you need to know 'the whole' in order to unambiguously define 'the elementary part' in it, which subsequently requires the global mode of spacetime.

There is simply no other choice, ladies and gentlemen. Forget about ADM. Drop it. It is a delusion fostered by wishful thinking, and without regard to the actual evidence in front of us -- any time you look at your wristwatch or look around and see objects in 3-D space, you prove ADM speculations wrong: you shouldn't be able to observe neither the cosmological time nor the 3-D space. Put it differently, the requirement from the Hamiltonian formulation of  GR -- the splitting of spacetime into space and time (the "grave injustice to space-time covariance that underlies general relativity", A. Ashtekar) -- and the requirement that any observable is invariant under change of the "coordinate time" (hence everything is constant in this "coordinate time" just by gauge invariance, C. Rovelli), make it impossible to observe any global feature of the universe, such as the cosmological time and the 3-D space. Since we obviously observe them, we're facing a devastating "catastrophe" predicted by Dirac-ADM, which, just like the ultraviolet catastrophe, has never happened.

This seemingly paradoxical conclusion shouldn't be surprising, since we know that Nöther's Theorem requires background-defined time, hence it is applicable only in STR. This fact alone makes GR nothing but a kinematical theory, therefore the evident infinitesimal (Sic!) change of the basis (Kenneth Dalton, physics/9710001 v2, Eq. 4) is outside its scope: "any change brought by curvature -- in speed, velocity, frequency, and wavelength -- will violate the principle of energy conservation. A gravitational exchange term is needed in order to account for the changes in energy and momentum. The theory of relativity neither provides such a term nor defines the energy, momentum, and stress of the gravitational field." (ibid.)

You need to believe in miracles and parapsychology in order to accept ADM "dynamics" of GR. Alternatively, since the principle of energy conservation is not shattered (see also ANEC), seek a new form of "dark" energy from the new dynamics of 3-D space, and try to formulate Nöther's Theorem in the global mode of spacetime. A good starting point could be the Brans conjecture and the vector field Xm introduced by Evangelos Chaliasos (physics/0611117 v1, p. 3).

But of course I could be all wrong -- see again the note above.

There is a very old idea about two kinds of reality and subsequently two kinds of time, which can be traced back from Prigogine and Whitehead to Plato. Namely, potential reality ('time of being') and its explications as fleeting snapshots of physical reality ('time of becoming'). Let us call the former 'global time', and the latter 'local time'.

The potential reality and its 'global time' can be understood by examining the classical limit of QM and the quantum vacuum. Regarding the latter, Shahriar Afshar (cf. Quantum Theory: Reconsideration of Foundations, 3 June 2005, Vaxjo, Sweden, in: A. Khrennikov et al., quant-ph/0610052 v1) offered the following explanation:

"Zero point field and the energy density associated with it are tricky subjects. It is clear that ZPF becomes physically real, or measurable, when there is radiation reaction. But what about when it is not measured in that sense, when it does not contribute to the physical properties of a test particle? It's just an empty space. The treatment is different, because with radiation reaction I have to treat this energy as real, contributing to the dynamics of the system. Otherwise, without its manifestation as radiation reaction, it cannot be seen as real, because the energy density would be too high, leading to numerous problems such as a cosmological constant many orders of magnitude lager than the value supported by observations."

More from John Baez here; try the 'potential reality' with your brain here.

As to the manifestation of the potential reality in GR, start reading the introductory section from B. Dittrich's gr-qc/0507106, until you reach the point at which she refers to her footnote 1 on p. 2. Very elucidating footnote. It ends up with: "Yet there are infinitely many gauge invariant degrees of freedom." Briefly, it isn't possible to find those Dirac observables in principle, because they too "live" in the global time of potential reality (see Charles Torre here). More about the quasi-local nature of those so-called Dirac observables can be read here.

If it were possible for Dirac observables to exist in GR, you would already know how to construct well-defined, localized energy states of the gravitational field (cf. Hermann Weyl here), solve the Cauchy problem for the Einstein equations (cf. John Stachel here), and then proceed easily toward quantum gravity, by reconstructing "a classical spacetime from nonlocal diffeomorphism-invariant observables" (Steve Carlip). In GR, this task is impossible in principle, because it requires something that is literally expelled from it from the outset: a rock solid potential reality. Einstein was fully aware of this generic problem of GR, which is inevitably produced from applying the principle of general covariance to the local mode of time: see the 'active diffeomorphisms' from J. Stachel here. More on the treatment of time from Alfredo Macias and Hernando Quevedo here.

In general, due to the lack of any pre-determined background in the local mode of time, the dynamics in GR (as well as in QM) can be summarized with the saying: "Traveler, there are no paths, paths are made by walking." These brand new, made-by-walking-paths require a brand new kind of determinism, called 'dynamical determinism'. The latter is a fusion of the Aristotelian efficient and final causes, and requires the potential reality and its global mode of spacetime.

Alternatively, if you choose to work with a foliation of spacetime by space-like hypersurfaces (after Dirac-ADM), you split the spacetime into a 3-D universe developing with "time", and introduce double standards for 'space' and for 'time'. All you can achieve, then, is to make Hermann Minkowski spin in his grave like a helicopter.

To sum up on the global mode of spacetime, there are two main reason for its introduction. Firstly, the 'coordinate time' in GR (called here 'local time') requires the undenumerably infinite "dark gaps" of the global mode -- think of these "gaps" as strips separating snapshots from a movie reel. More from St. Augustine. Secondly, the global mode of spacetime is the pre-geometric plenum which, as stressed by Lucretius, does not possess the quality of 'size'. It refers to the state of the whole universe as ONE. Hence it produces a   φ - "number"   of infinitesimal 'atoms' in the local time. People like L. Smolin claim that there are roughly 1099 "atoms of volume" in every cubic centimeter of space, such that there is "nothing" between them (the guiding metaphor is 'there is no water between two adjacent molecules of water'). Trouble is, this "nothing" creates 96 per cent of the stuff in the universe, and all L. Smolin could say is that this "nothing" is "dark".

In practical terms, the so-called "dark" energy is the ultimate 'free lunch', which can be harnessed only with/by the human brain. (But of course I could be all wrong -- see again the note above.)

Now, what is 'local time'? Suppose we were 2-D Flatlanders. We can easily understand the idea of 'local time', depicted with the finite line segment at the end of the movie animation.mov (source here).


It is being created by a   φ - "number"   of infinitesimal snapshots/atoms of spacetime (notice the word completely in the next paragraph, and follow its link). Also, the nature of the local time is teleological (cf. Stephen Leacock), and is complemented by the global time (not shown in the movie), which corresponds to 'the whole universe as ONE' and hence provides numerically finite but physically unattainable "boundaries" for the universe in its local time mode.

Suppose now we're observing a 3-D object, say, a ball, passing through our 2-D space. Before the ball enters our Flatland, we will see nothing, and "then" we'll observe a point, which gradually increases into a circle, then shrinks to a point, and "then" disappears, like the man in the movie above. We, the Flatlanders, can think of these sections of the 3-D ball as some 2-D snapshots that fill in completely our Flatland, and can postulate a local time variable (the line at the end of the movie above) to "explain" the alleged dynamics of the these 2-D sections/snapshots of the 3-D ball. Hence we can build a 2+1-D spacetime of our Flatland, but cannot see the global time of the genuine 3-D ball  [X] . Put it differently, there is no water "in between" two adjacent molecules of water, yet there must exist something that isn't water (called global time), which would separate the two "adjacent" molecules of water, otherwise there will be no 'coordinate time' to describe their local dynamics.

Let me try to explain this with the well-known Balloon Analogy in Cosmology (courtesy of Ned Wright): All Flatlanders, living "on" the 2-D space of the balloon, will claim that there is no need for any empty space that is "beyond the universe" to expand the balloon, because their local time, depicted with the straight line at the end of the movie above, is perfectly continuous. Thus, there is 'no room' in the local time for any "gaps" from the global time, and the "direction" of the master/cosmological time arrow will inevitably overlap with the local time -- see again the straight line at the end of the movie above, and recall that it overlaps with the intrinsic global time of the 3-D sphere. Hence all 2-D Flatlanders will happily use diff geometry in such seemingly time-symmetric 2+1-D universe, but will have to postulate that the Hausdorff topological space is indeed connected. But in order to "connect" two "neighboring" points from the balloon surface, you need the global time.

Philosophically speaking, if we consider the end result from the bi-directional "talk" of matter and geometry (J. Wheeler), we have a frozen snapshot of 'spacetime', which can only allow us to make extrapolations and counterfactual propositions regarding the possible dynamics of gravitational fields, unfolding from this frozen instant. As known since Einstein's Hole Argument, there is no "background" structure of "bare points" to help us define the "next" state of gravitational fields -- the phenomenon of transience is missing in contemporary theoretical physics -- hence in GR we cannot bridge the gap between two successive states of gravitational fields. Therefore, "GR does not admit a description as a system evolving in terms of an observable time variable", as stressed by C. Rovelli.

Specifically, we need the global time for two main reasons:

1. The fundamental step (fundamental timelike displacement) of the cosmological time arrow does not "occur" on some "constraint surface", as defined by the Hamiltonian and momentum constraints. More here and here.

2. The dynamics of gravitational fields requires boundary conditions that cannot be derived from that same "constraint surface" (more here). We need some "boundary of spacetime" or "ideal endpoints". More here and here.

Please follow the links in red, and recall that, in Einstein's GR, what we call 'spacetime' is being defined solely by the physics of matter and fields that "produce" spacetime.

It is just ridiculous to expect that the dynamics of matter would also define (1) and (2) above, as well as the topology of space: no theory can provide the complete basis for itself, by virtue of Gödel's theorem (in the case of Einstein's GR, there is no "background" to define (1) and (2), hence the latter can only be defined in a more general theory, such as quantum gravity, yet (1) and (2) inevitably show up in Einstein's GR, as expected after Gödel's theorem). Recall Aristotle and read Einstein below.

If you're interested in GR, try Einstein's cosmological "constant": What is the dynamical mechanism that creates and sustains a perfectly balanced flat spacetime during an accelerated expansion of its metric?

Now, to explain how the global time overlaps/fuses with the local time, see the drawing below, an expanding line segment presenting 1-D space.




It is "bounded" by two ideal endpoints (hence the interval is open), and because it "expands" like a balloon, an observer inside it will certainly notice changes in her/his 1-D world, hence will introduce a local time variable, and claim that she/he lives in 1+1-D universe in which there is no preferred place or preferred reference system. If we place the 1-D space above along an 'x axis', then the 'y axis' (not shown) will be the local time of the 1+1-D universe. The global time, however, will be along an 'z axis' (not shown either), and will be orthogonal to both y and x. Hence the cosmological time arrow has two opposite "directions", toward The Beginning (the left ideal endpoint of the line segment above) and The End (the right ideal endpoint), as the space expands in these two "directions" by "walking the steps" in the global mode (the 'z axis'). If you "walk down" the cosmological time arrow (deflation time), you walk down the z-axis as well. But you cannot shrink the space to zero and merge the two ideal endpoints, because their common origin is the global mode of spacetime: no preferred place or preferred reference system can be reached from the local mode of spacetime. Ever since The Beginning, the universe has been "wrapped" with 'numerically finite but physically unattainable boundaries', such as the two ideal endpoints above.

I believe this may solve the cosmological time problem: every finite time interval or object refers to a point-like sliding cut-off at The Beginning, as we place this cut-off always in the past. In other words, in every step along the cosmological time "ladder", the universe has, in the local mode, an exact finite size and exact finite values of all its variables, but it cannot 'eat itself up' by reaching (as a Gedankenexperiment) The Beginning, nor can it catch the two ideal endpoints by chasing them in the future ahead; please see 'logical infinity'. Once born with two modes of spacetime, the universe becomes eternal and unbounded in the local, teleological time (cf. Stephen Leacock), in line with Einstein's finite universe without boundary.

Thus, if we "look" at the past, we'll see a time-symmetric universe in which the global time has already overlapped with the local time, and will be hit by the persistent paradoxes of this teleological time (Stephen Leacock): there is no way to accommodate The Beginning in our finite cosmological time segment, yet we need it to define any finite time interval from it (the math is laid out here). The potential future, on the other hand, comes from the global mode of spacetime, and is completely open, up to 'the unknown unknown'. If we look at the past, the sole imprint from the potential future that we can detect is a peculiar "window" of flexibility of the dynamical determinism, which shows up as a genuine Free Will (see the implications from a non-unitary deflation time dynamics here). If we look into the future, all we could "see" from the potential future is a 
φ - "number"  of possibilities. The latter are genuine propensities --  not probabilities -- because (i) propensities do not sum up to unity, and (ii) the  φ - "number"  of propensities --  unlike probabilities (cf. R. Hedrich, p. 18) -- are causally interdependent and connected. Only in the highly contrived and limited case of the world of unanimated tables and chairs we can safely assume that the propensities are, effectively, probabilities, and claim that the principle of unitary dynamics 'something has already happened with probability unit' can be applied to the boring future ahead, in the format 'something will happen with probability unit'. Not surprisingly perhaps, this boring world is just 4 per cent from the stuff in the universe.

Try to find these simple ideas in the present-day research papers on quantum gravity; latest references here.

Likewise, any time we look around in our 3-D space, or make observations with physical devices, we see the physical world in its local time mode only: the end result from a bi-directional "talk" in the global time.

The dynamics of GR

Think globally, move locally

We cannot observe this bi-directional "talk" directly, because it is hidden by the so-called "speed" of light  -- we can observe the world only post factum. For example, if we look at the Sun, we will see its past state, that is, we will see how the Sun has looked like approximately 8 minutes before we looked at it, since it takes nearly 8 minutes for its light to reach us. Physically, we cannot make instantaneous observations, hence the bi-directional "talk" in the global time is always hidden to us. We can, however, explore it with our brains, which is the scope of 'physics of human intention' (PHI).

Let me try to explain all this as a Husserlian noema -- an indefinite whole which becomes gradually filled with concrete content.

The idea of 'global time' can be explained to a 2-D Flatlander as follows (with just a few links). Suppose there is a third dimension perpendicular to your 2-D space, but the dimensions of all physical bodies are exactly zero in that "direction" (like images on a movie screen). Zeno has noticed that it isn't possible to explain the infinitesimal displacement of bodies in Flatland by neither finite nor zero length (see also the Thompson lamp paradox here). Another great philosopher, St. Augustine, has stressed that there should exist a hidden "gap" of non-existence of physical bodies, or else there could be no motion and no 'change in time' in Flatland. On the other hand, as stressed by Heraclitus of Ephesus, the puzzle is not in the flux of reality; it is in the bold fact that the reality is stable ("a central mystery of physics"). Which is the reason why Kurt Lewin, a famous German psychologist, introduced in 1922 the concept of genidentity (Genidentität), to underscore the persistence of objects through time due to their 'sameness' kept in the hidden "gaps" of non-existence. Hence the third dimension "perpendicular" to Flatland should be totally hidden or "dark". Let's call it 'global time', as pictured here. Now we can have a universal (master) time arrow, with 'local time' and a perfect 2-D space continuum in Flatland, and a 'global time' of the Holon state of all potential states of all physical bodies, which springs from the invisible "dark" gaps. If we could somehow freeze this universal time arrow ("stop the bike"), we would perhaps see the "gaps" of the global time being inserted in the local time, like strips from a movie reel. However, in order to "freeze" the universal time arrow, we would have to make observations instantaneously (faster-than-light, with some transcendental tachyon maybe), which is physically impossible.

Hence we can observe the world post factum only, as in the example with the Sun above, and the world in this post factum state is a perfect continuum of facts that have been already fixed and already negotiated by the bi-directional "talk" in the global time of the Holon. Also, there is another reason why we can never freeze the universal time arrow -- there is no direct track from the local time to the global time. We cannot "chase" the global time from the local time (say, by following the Thompson lamp), because the global time is the realm of pure geometry -- 'the grin of the cat without the cat', as noticed by Alice. This is the final, and truly fundamental, layer of physical reality, expressed with 'numerically finite but physically unattainable boundaries of the world' (e.g., the Planck scale). Any time we use probabilities or differential calculus for our mundane purposes, we literally employ the global time, because only the global time can "stop" the infinite process in the local time, and produce a result that can be expressed with a finite number, which is again a "point" from a numerical axis. There is no physical difference between the imprecise and the (seemingly) precise values of the numerical "points" obtained with the global time (an example for 'precise values' is the exact limit obtained in the case of Thompson's lamp paradox, as compared to the circumference of a circle, after Archimedes). Another example can be brought with the golden section rule: if we consider a final line segment, AB, and attach two seemingly precise numerical values to A and B, then we can find a well-defined and unique "point" C within AB, such that  AC:CB = AB:AC . The numerical value of C cannot be made precise, yet there is no difference in the physical content which can be attached to these three geometrical "points". The purpose of this story is to underscore the tantalizing nature of the geometrical "points" which we use as 'the grin of the cat without the cat', after Alice: these points can indeed "display" a fleeting physical content in the local time (which is why we can calculate a tangent vector attached to a dimensionless geometrical "point", for example), but belong to the global time -- the final layer of reality, occupied by the First Cause of Aristotle. It is also the realm of pure geometry: Mens agitat molem (Vergil, The Aeneid, Ch. 6, 727). (How's your Latin? Si hoc legere scis nimium eruditiones habes.) Physically, it is manifested as "dark energy" -- an invisible force that creates the cosmological time arrow and the elementary timelike displacement (the Heraclitian time of the whole universe) of all physical bodies. It is being spread evenly everywhere in the universe: The One is an unbroken ring with no circumference, for the circumference is nowhere (hence no absolute reference frame) and the center (in the local time) is everywhere. There's nowhere where God is not.

Again, there is no direct track from the local time to the global time of this final layer of reality, and we can only infer the existence of the Aristotelian First Cause by carefully examining the world of tables and chairs around us, as Lucretuis did 2060 years ago. Needless to say, all these ideas are not new at all, but have not been recognized in modern theoretical physics, and probably won't be recognized soon.

Why? Max Planck has suggested the following explanation: "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."

Here's the first off challenge. Larry Ford has stressed that "it is very hard to imagine what new physics would introduce a cut off on a scale of the order of 0.01cm" (L.H. Ford, Spacetime in Semiclassical Gravity, gr-qc/0504096 v1). The new physics could come from the incorporation of the generic properties of 3-D space into General Relativity, as outlined here. Perhaps the solution to this task will reveal a new symmetry of space, pertaining to two "directions" in 3-D space, and a new relativity principle. Very briefly, our world of tables and chairs stays 'in the middle' between the Large and the Small ("on a scale of the order of 0.01cm", Larry Ford), hence we see, in the local mode of spacetime, a galaxy that is "large" and an atom that is "small", along the two "directions" of 3-D space. But the "dimensions" of an atom and a galaxy may be the same, as they change proportionally along the two opposite "directions" of 3-D space (more here).

But do we need to understand the nature of 3-D space? Please click here or on the picture below. The origin of the task is from Hermann Minkowski, as stated on September 21, 1908 (Raum und Zeit, in: Vorträge von der 80. Naturforcherversammlung zu Köln, Physikalische Zeitschrift, 10, 104-111, 1908):

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

See the quiz 'flipping a quantum coin' here, and cast your answer. The task is known since 1931 (E. Schrödinger, Specielle Relativitätstheorie und Quantenmechanik, Sitzungsber. Preuss. Akad. Wiss., phys.-math. Kl., Bd. 12, S. 238-247, 1931). A whirlwind tour begins here!

It's all about a new kind of reality, called potential reality. Evidence can be found in both QM (the relativistic "collapse") and GR (the so-called dark energy). Hence a new path to quantum gravity is suggested, by supplying QM and GR with what they do not have: reality.


It is the potential reality which keeps the possible states of all physical systems in a Holon state. Thus, we restore the reality of 'quantum system out there' and 'spacetime point', and explain the origin of quantum and gravitational waves as an effect of the Holon. If you're short of time, click

For a bit more technical outlook, chick here. In a nutshell, I claim that the true dynamical degrees of freedom for gravitational fields cannot be derived in GR in principle, because no genuine Dirac observables can exist in Einstein's GR: the set that is supposed to encode the complete gauge invariant information by Diff(M)-related configurations cannot exist in principle, just as the complete (or global) presentation of a quantum system by a set of all observable (or local) states of that same quantum system (for Hilbert space dimension greater than two) cannot exist in principle, as we know from Ernst Specker. Why? Because these two sets can "exist"  iff  the Marxist-Leninist interpretation of GR and QM were correct. People who are not brainwashed with Marxist-Leninist religion can understand the dictum 'the whole is more than the sum of its parts' (see an example from solid state physics here; general considerations here). This "more" is the potential reality. Check it out with your brain here. Now, all we need is to model the universe as a huge brain, and the holistic (hence "dark") effects from the potential reality -- the Holon state of the Universe -- will show up in present-day theoretical physics studies: 96 per cent from the stuff in the universe is produced by the Holon.

The main idea of the "dynamical" degrees of freedom for GR, as presented in today's textbooks, is two-fold. To begin with, the alleged dynamics of GR is being "generated" entirely by constraints (e.g., the Hamiltonian constraint), which do not explicitly introduce a time variable. This shouldn't at all be surprising, given the fact that the bi-directional talk of matter and geometry cannot be 'temporal' in principle. The constraints act like prescriptions, much like 'you may go for a walk, provided you always keep 2 meters away from the buildings, and always wear a blue t-shirt' (see the condition 'always keep tµv = 0' here). Then you may generate the allowed "dynamical" states for your tentative walk. Think of these states as being generated by shaking up a kaleidoscope: every time you shake up the kaleidoscope and place it on the table, it will display a frozen, static configuration of colored glass pieces, which are 'physically observable states', since the kaleidoscope can display Diff(M)-related configurations only. Now you are ready to ask a simple question: can I compile the complete set of such 'local' (in the sense of Ernst Specker) static states of the kaleidoscope, such that it will present all the possible "dynamical" states of the kaleidoscope, and hence capture its 'global' description? Yes you can, provided that your kaleidoscope is not larger than the size of the Solar system, because if you increase its size, it will begin to encounter holistic (hence "dark") effects of the Holon. Namely, 'the complete set' of all possible static Diff(M)-related configurations of the kaleidoscope will be 'open' for new elements, like a 'potential point', and you'll lose your unitary dynamics forever. Again, if you insist that your walk be governed by the rules of 'unitary dynamics', specified by a set of denumerable and well-defined energy states of the kaleidoscope, you will have to avoid all holistic effects that may smuggle "during" the atemporal shaking of the kaleidoscope (the bi-directional talk). Thus, if you're searching for the 'global' (in the sense of Ernst Specker) set of all possible dynamical states of the kaleidoscope, but also wish to keep the reductionist Marxist-Leninist metaphysics, you may solve your task  iff  your kaleidoscope is not larger than the size of the Solar system. In the general case, however, you will find out that such Marxist-Leninist "set" cannot exist in principle, because your kaleidoscope has acquired a brand new quality, being connected to, and dependent on, the Holon of the Universe. In a bit more technical language, "even if we start with genuine tensorial variables, then certain important physical quantities turn out to be non-tensorial" (Laszlo Szabados). Again, you can verify the "non-tensorial", holistic nature of your brain here. Isn't that simple?

Notice that, in the case of canonical quantum gravity, the dynamics is instructed by the constraints (see above) in a very peculiar fashion: 'you may go for a walk, provided the time read by your wristwatch will be frozen, just as if you were riding a photon'. But this is exactly the global time of potential reality. In Einstein's GR, the potential reality preserves the 'sameness' of the kaleidoscope, manifested with (not 'by') its Diff(M)-related configurations, and also the reality of 'potential point(s)', thanks to which the fleeting physical content of the geometrical points from the local time can be 'moved around' in an 'active sense'. There is no need for any background structure in the local time, because it is provided in the global time as 'context' kept by the potential point(s). It is simply illogical to expect that the reference fluid in Einstein's GR and the Aristotelian First Cause can be found in the local time, since their proper time must be "frozen", just as the proper time of the human self is frozen, thanks to which it doesn't change in the local time read by your wristwatch (please notice that we refer to the Holon state of the brain, and not to its mental reflection called 'human self'.)

As a concrete example, consider the so-called gravitational wave (GW) astronomy. Angelo Loinger stressed that "GW’s are undulations of the metric tensor gjk, which is the “substance” of Riemann-Einstein spacetime, i.e. of a not “fixed” manifold, that does not possess a class of physically privileged reference systems" (reference here). Hence the question of the dynamics of the metric tensor undulations -- the “substance” of Riemann-Einstein spacetime -- cannot be resolved in any physically privileged reference system, simply because the latter does not exist in the local time. There could exist, however, a unique privileged reference system, which isn't strictly physical. It is pure geometry -- the hypothetical 'potential point(s)' -- and is 'absolutely everywhere' in the local time, like some transcendental tachyon or "dark" energy. It is the long-sought reference fluid in GR. To cut the long story short, there are overwhelming evidence that GWs exist, but I claim that they cannot be directly observed with any inanimate device, such as LIGO and LISA, because such devices cannot access the global time of the bi-directional "talk" which ultimately determines the dynamics of the metric tensor -- the “substance” of Riemann-Einstein spacetime. If you look in GR textbooks, you will read that the spacetime itself is dead frozen. Only these textbooks do not, and cannot account for more than 4 pert cent from the stuff in the universe. They present it as a digital tape with a time code written on it from The Beginning, but leave the question of the dynamical mechanism of imprinting this 'time code' unanswered. This whole bundle of issues is rooted on the absence of reference fluid and preferred reference frame in GR in the local time. The founding fathers of the theory of relativity were perfectly aware of it, from the inception of GR. Perhaps the time has come for a change of the Zeitgeist. What is the nature of 3-D space?

If you're interested in GW astronomy, read gw.pdf (217,009 bytes, 16 pages, Wednesday, 26 October 2005, 23:00:00 GMT). See p. 16 for a hypothesis for the origin of quantum-and-gravitational waves. Critical comments and suggestions are welcomed. Again, if you're short of time, click here.

To sum up, let me paraphrase D. Mermin (N. Straumann, Quantenmechanik, Springer, Berlin Heidelberg, 2002), replacing 'EPR and Bell’s Theorem' with 'the Hamiltonian formulation of GR':

Contemporary physicists come in two varieties. Type 1 physicists are bothered by the Hamiltonian formulation of GR. Type 2 (the majority) are not, but one has to distinguish two subvarieties. Type 2a physicists explain why they are not bothered. Their explanations tend either to miss the point entirely (like Ashtekar's loop quantum gravity) or to contain physical assertions that can be shown to be false. Type 2b are not bothered and refuse to explain why.

) In case you are interested in "quantum computing", click here.

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LIGO failed, as expected. The forthcoming failure of the Advanced LIGO (with "improved sensitivity", etc.) is beyond any reasonable doubt. Just look at their L-shaped interferometers: isn't that parapsychology, plain and simple? Let me explain.

The whole idea of detecting Gravitational Waves (GWs) is build on some very deceptive analogies from electromagnetism. If you consider a light beam coming from the Sun, you can define its direction of propagation. Then you can use two Polaroid filters to cancel the phase of the light beam, by positioning the filters in a plane perpendicular to the direction of the beam: all you have to do is to orient/twist one of the Polaroid filters on 90 degrees with respect to the other one, like the two arms of the letter L. Thus, you use two spatial directions from the plane, X and Y, and the third spatial direction, Z, which has been defined by the light beam coming to Earth from the Sun. Obviously, Z is perpendicular (transverse) to the plane defined by X and Y, and you have consumed all three axes of the 3-D space.

Now, the problem with GW astronomy is in the following. Light propagates in spacetime, thanks to which we can use the latter to uniquely define the direction of light propagation in 3-D space, while GWs propagate 'within themselves' (just as the 3-D space is expanding within itself): there is no "extra dimension" to help you define the propagation of GWs, their phase, and amplitude. Since a photon propagates in spacetime, there is always a spacetime domain 'ahead of it', in which the photon is 'not yet', while GWs are spacetime themselves -- there is no "place" ahead of them, in which they are 'not yet', because the spacetime itself does not move. The picture below can be seen here, the articles will be published in September 2005.

This picture is highly deceptive and misleading!
In the real world, there is no boundary of the lake nor any undisturbed surface from the lake, which is patiently waiting for GWs.

(In Einstein's GR, the lake itself does not move toward the "beach", hence the 3-D component of the GW "push" on a fishing rod float (read: LIGO's arms) will be zero: the so-called dark energy is "outside" the 3-D lake hypersurface. Since GWs are "oscillations of the "fabric" of spacetime itself" (Kip Thorne, gr-qc/9506086 v1), you need some global time "parameter" pertaining to the whole lake to define the global "dynamics" of GWs on the lake. Hence GWs cannot be "timed" within the Hamiltonian formulation of GR. The oscillations of the "fabric" of spacetime require an additional degree of freedom, which cannot be found in the static timeless 'block universe'. That is, LIGO doesn't have access to some preferred reference frame in which you can distinguish between the longitudinal quadrupolar mode and the transverse quadrupolar mode, hence you're in fact measuring the initial dipole mode, and of course you get nothing, zero, zilch. Besides, you need the full quantum gravity to discuss 'the other sign of mass' in the dipole radiation; see Macavity. Yes, there should exist effects of GWs -- relative displacements of the fishing rod floats on the 3-D lake -- at the scale of LIGO, but their values would have to match the current value of the cosmological "constant" of the 3-D lake, so forget about 'improving sensitivity', etc. Why? Because you cannot detect the localized portion of GW energy, which (i) enters the 3-D lake from the so-called dark energy "outside" the lake and (ii) creates the intrinsic time interval associated to any timelike displacement of the "fishing rod floats" on the 3-D lake. The misfortunate Hamiltonian formulation of GR can encode the intrinsic time interval "within" an infinitesimal "point" only, hence the 3-D component of the GW "push" on a fishing rod float (see above) will be zero, as explained by Angelo Loinger in physics/0506024 v2.) People talk about "extra dimension" in parapsychology, while the proponents of GW astronomy talk about some "dimensionless amplitude" of GWs, which they label with some "dimensionless number",  h . Sounds like a dumb joke, only it's real and consumes a lot of cash -- taxpayers' money. (How come the most important parameter in GW astronomy turned out to be just a "dimensionless number", h ? And since it is nothing but a "dimensionless number", I suggest that its "dynamics" should be parameterized with another "dimensionless number",  Z , which no physical clock would be able to read, ever. The only device capable of performing direct, online measurements of GWs could be the human brain, but I was not allowed to speak at EPS13 in Bern; more here.)

All this is comprehensible (except the explanations in parentheses) even to my 12-year old daughter. LIGO Scientific Collaboration (395 "distinguished scholars") don't get it, however. Or maybe they do, but don't want to cancel their lucrative projects: GW astronomy is a huge multi-national business, supported currently by USA, UK, Germany, Australia, Canada, India, and Spain.

Just read the excerpts below (emphasis mine - D.C.), from Louis J. Rubbo, Shane L. Larson, Michelle B. Larson, and Kristina D. Zaleski, Gravitational Waves: new observatories for new astronomy, physics/0509201 v1, p. 1 (accepted to The Physics Teacher):

"Gravity can propagate in waves and carry information from one place in the Universe to another, just like photons."

(Well, not entirely like photons, since GWs supposedly travel along null geodesics with just "slowly evolving amplitudes and polarizations", according to Kip S. Thorne; see "The Theory of Gravitational Radiation: An Introductory Review", in Gravitational Radiation, eds. N. Dereulle and T. Piran, North Holland, Amsterdam, 1983, pp. 1-57. Exactly how "slow" the amplitudes and polarizations would evolve "during" their travel along null geodesics is a bit of a mystery, since the GW clock that travels along null geodesics will be dead frozen.)

And from ESA website:

"Gravitational waves are fundamentally different from the familiar electromagnetic waves. While electromagnetic waves, created by the acceleration of electrical charges, propagate in the framework of space and time, gravitational waves, created by the acceleration of masses, are waves of the spacetime fabric itself. (...)

"A gravitational wave passing through the Solar System creates a time-varying strain in space that periodically changes the distances between all bodies in the Solar System in a direction that is perpendicular to the direction of wave propagation."

What is the time parameter of that "time-varying strain"? Have you read The Adventures of Baron Munchausen? Try to pull yourself (and your horse) out of the swamp by your own hair, as explain by Karl Friedrich von Münchhausen.

You have nothing to 'hold onto', because the amplitude of GWs,  h  , is just a dimensionless number. So, if you fail, recall Steven Weinberg here: "The device measuring, say, the displacements of free mirrors in an interferometer would be "stretched and squeezed" as well."

If you live in USA, UK, Germany, Australia, Canada, India, or Spain, you are paying for this "self-measuring" exercise with your taxes. Thank you for reading this.

Download paper.zip from here, or paper.pdf from PhilSci Archive,

Chakalov, Dimi (2005) Are Gravitational Waves Directly Observable?
ID Code: 2367, Deposited on 20 July 2005,

Ten years ago, on August 15, 1999, I posted the introductory section from my first paper, entitled: "Note on the problem of time in quantum gravity". Dead silence. Well, perhaps the subject was too academic, so I decided to apply my ideas to something far more practical: quantum computing and gravitational wave astronomy. I have so far only one response from the established theoretical physics community: "I don't know you and wish you out of my face, my computer." Perhaps this can explain my attitude toward these "established scholars". I think they don't do science but practice their favorite hobby, the so-called gravitational wave astronomy, and waste real money earned with hard labor by millions of people. This is just outrageous. Period.

D. Chakalov

July 20, 2009


Albert Einstein: "The right side (the matter part) is a formal condensation of all things whose comprehension in the sense of a field theory is still problematic. Not for a moment, of course, did I doubt that this formulation was merely a makeshift in order to give the general principle of relativity a preliminary closed expression. For it was essentially not anything more than a theory of the gravitational field, which was somewhat artificially isolated from a total field of as yet unknown structure."

(Albert Einstein: Philosopher-Scientist, Ed. by P. A. Schlipp, Open Court Publishing Company - The Library of Living Philosophers, Vol. VII, La Salle, IL, 1970.)

Albert Einstein: "The representation of matter by a tensor was only a fill-in to make it possible to do something temporarily, a wooden nose in a snowman."

(Albert Einstein's Last Lecture, Relativity Seminar, Room 307, Palmer Physical Laboratory, Princeton University, April 14, 1954, according to notes taken by J. A. Wheeler. In: P. C. Eichelburg and R.U. Sexl (Eds.), Albert Einstein, Friedrich Vieweg & Sohn, Braunschweig, 1979, p. 201.)