Can we trace the cosmological time back to its dead-end?
Subject: Re: Gravitational
waves remain out of reach
In the current issue of Nature, section "News Feature", your Washington Correspondent Geoffrey W. Brumfiel wrote [Cosmology gets real, by Geoff Brumfiel, Nature, 422(6928), 108 (13 March 2003)]:
"By clarifying the age and make-up of the Universe, researchers have ushered in an era of precision cosmology. Now they are preparing to probe the mysteries of dark matter and dark energy."
In my previous email to you from Wed, 19 Feb 2003 23:40:26 +0200, I tried to calm down the optimistic expectation from such projects. Please see
I'm afraid we are still miles away from clarifying the age and make-up of the universe,
Moreover, the road toward this highly ambitious goal is not at all clear.
The essential issue which we need to resolve is whether the universe bears a degree of freedom that has not been captured in our current Weltbild.
It can not be ruled out, I believe, that the effect from this unknown parameter can be pinpointed in the structure of spacetime,
Its presentation could be vanishing small at the scale of tables and chairs, but it could grow up simultaneously toward micro and mega scales, producing quantum effects and dark matter & dark energy. The latter constitutes 96 per cent from the observable stuff in the universe,
As to the effect toward the Planck scale, please see
Additional arguments in favor of the quantum nature of this hypothetical parameter of spacetime (called 'global mode of spacetime') can be found in examining the puzzling "cosmic equator" of the universe,
The cosmic equator may be due to *rotation* of the universe. It was Mário Everaldo de Souza who suggested in "The General Structure of Matter" (hep-ph/0207301, p. 14) four units that fill in spacetime: primon, nucleon, atom, galaxy, and universe. "The first four have angular momenta, and thus, we expect that the last unit, which is the Universe itself should always have an angular momentum." (M. de Souza, private communication, Wed, 12 Mar 2003 18:15:17 -0300)
Simple and clear. All we have to do is to recall that quantum parameters such as 'spin' elude any classical visualization. They are simply 'klassisch nicht beschreibbare Zweideutigkeit' (Wolfgang Pauli).
In other words, the "timing" of the rotation of the universe and its physically observable constituents is kept in the putative global mode of spacetime. What we can observe, in the local mode of spacetime, is only a localized snapshot of The Universe in which everything is ONE. (I believe this should sound very familiar to the readers of Nature, and will refrain from quoting from The Gospel.)
Last but not least, there are many models of the universe with "retrodiction" from the future (e.g., Vladimir Mashkevich, "Cosmological Quantum Jump Dynamics II. The Retrodictive Universe", gr-qc/0303046). Surely the snapshots from The Universe do not evolve in the local mode of spacetime, they are "frozen" there.
But what if we interpret the "retrodiction" from the future as anticipation in the global mode of spacetime,
What if all constituents of The Universe evolve like a holistic living system along a hypothetical universal time arrow?
I believe the ultimate mystery we have to solve is what happens "between" the snapshots of the alleged universal time arrow. Then we will understand the structure of spacetime and cosmology will get real. As of today, it isn't.
Regarding the hunt for some gravitational "waves" [Ref. 1], may I bring to your attention the research monograph by Prof. Angelo Loinger, entitled: "On Black Holes and Gravitational Waves" (La Goliardica Pavese, Pavia, 2002; ISBN 88-7830-371-2). You will find there rigorous mathematical proofs of the non-existence of gravitational waves and black holes. Perhaps it would be a good idea if you provide a reference to it in your 'Box 1 General relativity for beginners'. [Ref. 2]
Given the fact that we can speak of only 4% of the physical stuff in the universe, while the nature of the remaining 23% of dark matter and 73% dark energy is totally unknown [Ref. 3], perhaps the time has come to recall Confucius: "The hardest thing of all is to find a black cat in a dark room, especially if there is no cat."
[DENVER] An ambitious US project to detect gravitational waves presented its first results this week at the American Association for the Advancement of Science's annual meeting in Denver, Colorado. The data were collected over 17 days in September by the two facilities of the Laser Interferometer Gravitational-Wave Observatory. No waves were seen, but the measurements will help astronomers predict the power and frequency of the high-energy events that are thought to create detectable gravitational waves.
The observatories, based
in Livingston, Louisiana, and Hanford, Washington, have been trying to
reduce noise picked up by their detectors (see Geoff Brumfiel, "General
relativity: To catch a wave", Nature 417, 482-484; 30 May 2002), and it
is unclear whether the current designs will be able to detect gravitational
waves. Plans for an upgrade costing US$100 million to US$150 million were
unveiled this week.
No one has ever detected
a gravity wave directly since they were first postulated 90 years ago —
although physicists concur that the waves do exist. Albert Einstein's general
theory of relativity, published in 1916, predicts that large masses in
rotating systems should produce ripples in space-time (see 'Box 1 General
relativity for beginners').
The new data imply an age for the Universe of 13.7 billion years, and a distribution of mass and energy in which 4% of the Universe is normal matter (atoms), 23% is dark matter, and 73% is dark energy.
Fact: Black holes do not exist
RE: Gravity wave detector
all set, by Jonathan Amos, BBC News Online science staff in Denver, Tuesday,
18 February, 2003, 01:24 GMT,
"Professor Kip Thorne, one of the lead researchers on the project, told the BBC: "It'll be wonderful. We'll have the joy of discovering things about the dark side of the Universe that cannot be discovered in any other way.
"We'll have the joy of testing
theories about black holes that cannot be tested any other way. We'll be
seeing the Universe and the fundamental laws of space-time in a manner
that we have never seen before."
The factual error in the story above is that black holes and gravitational "waves" do not exist.
Please see the reference at
I think it's time for Kip Thorne to get professional. The sooner, the better.
Gamma-ray burst mystery solved?
RE: Gamma-ray burst mystery
Dear Professor Matheson,
I'm wondering if you believe that have solved the puzzle of GRBs engine.
I will also appreciate your opinion of the hypothetical gravitational waves. In the first LIGO scientific results (April 9, 2003), no gravitational wave events were observed,
I will appreciate feedback from your colleagues too.
... approaching the planned level of sensitivity?
I'm afraid you will never observe directly those empty waves that you call "gravitational waves". [Ref. 1]
In addition to the email by S. Weinberg of 25 Feb 2003, which you quote as ref.  in your paper [Ref. 1], please see
I will appreciate your critical comments, as well as those from your distinguished colleagues.
Please do not hesitate.
1] Leonid Grishchuk, Update on gravitational-wave research,
"We are now at a special
and decisive junction.
"By all counts, gravitational-wave
research is now one of the most exciting and promising areas of physical
"It is difficult to detect
gravitational waves because they carry their energy
practically without scattering and absorption.
"The logic seems to be impeccable. If "space" expands by a factor of 2 in 10 billion years, why would not the Earth or an atom expand by 10% in 1 billion years? The gravitational-wave research is plagued in a similar fashion. It is often stated that "gravitational waves are oscillations of space-time itself".
"The next phrase seems to be logically unavoidable: "gravitational waves act tidally, stretching and squeezing any object that they pass through". If this phrase were correct, we would never be able to notice gravitational waves. The device measuring, say, the displacements of free mirrors in an interferometer would be "stretched and squeezed" as well. In this situation, we can probably find comfort in the wise observation : "I agree that much of what one reads in the literature is absurd. Often it is a result of bad writing, rather than bad physics. I often find that people who say silly things actually do correct calculations, but are careless in what they say about them."
" I am grateful to S. Weinberg for the permission to quote his e-mail message of 25 Feb 2003."
Re: ... approaching the planned level of sensitivity?
P.S. My previous email can be read at
Regrettably, in your recent gr-qc/0309058, "Gravitational Lorentz force in the field of a gravitational wave", you wrote:
"The large-scale interferometers
for the detection of gravitational waves are approaching their planned
level of sensitivity .
"A laser interferometer monitors
distances between the central mirror and the endmirrors in the interferometer's
arms. In the existing instruments, the length l of the interferometer's
arm is significantly shorter than the wavelengths X of the
gravitational waves which the instrument is most sensitive to. The evaluation
of the change [d]l of the distance between two mirrors is usually
formulated as [d]l/l =
I'm afraid it isn't.
Let me remind you of the critical remark by Steven Weinberg, sent to you by email on 25 February 2003:
"I agree that much of what one reads in the literature is absurd. Often it is a result of bad writing, rather than bad physics. I often find that people who say silly things actually do correct calculations, but are careless in what they say about them."
I think there are at least 368 'people who say silly things',
Why? Because the "recoil" from the physical bodies to the gravitational wave is zero. Zilch.
Why is zero? Because the gravitational waves are *bona fide* empty waves. They are only endowed with a false (pseudo) tensor (i.e. a non-tensor) of energy-momentum, as proved by T. Levi Civita in 1917,
Angelo Loinger, "Quantum
gravity": an oxymoron,
Hence h , "the characteristic amplitude of the incoming wave" (cf. above), the so-called curvature of spacetime, and the whole "talk" between matter and geometry (John Wheeler) are defined only and exclusively only on ONE single mathematical point that you call 'event'.
NB: The reason why you and your colleagues can not, even in principle, detect any *physical* effect of the gravitational waves is that you can not, even in principle, go BETWEEN any two successive points from the spacetime continuum,
The "talk" between matter and geometry takes place *between* any two successive points/events from the spacetime continuum, hence determine *unique* values of *physical* quantities at these points, one-at-a time,
This is the missing "chooser" (P. Pearle) in QM,
The whole story goes back to Lucretius, some 2060 years ago,
Ignore it at your peril.
On Wed, 14 May 2003 14:11:49
+0300, Dimi Chakalov wrote:
Explanatory note: Nobody is concerned anymore with the detection of empty waves or de Broglie waves propagating in the 3-D space of special relativity theory, but the fuss about another kind of empty waves, called "gravitational waves", is far from being settled.
Most importantly, it is still not clear whether this task can be achieved in principle. Maybe this is impossible. For example, if the detection of GR waves can be achieved only and exclusively only in the absolute reference frame left from The Beginning, then we would have plenty of hints from GR that this might -- just might -- be possible, since any physical clock reads the cosmological time as well, but, on the other hand, this absolute reference frame is "spanned" over a mathematical point only. It can not be employed for any physical measurements performed with any inanimate measuring device whatsoever. If this were possible, we would have never encountered the problem of time in canonical quantum gravity, the principle of diffeomorphism-invariance would be disabled, and we would have to trash GR. Better not, it's too early.
What if gravitational waves can be "detected" on a mathematical point only? What do gravitational waves carry so that their "shipment" could be physically detected?
To detect this "shipment", we need to stay confined in some reference frame for a finite time interval while making two consecutive measurements of the "shipment", say, at t_1 and at t_2 . We also need a cutoff, t_0 , to define the duration of the "shipment", (t_2 - t_0) - (t_1 - t_0) = t_2 - t_1 .
What we utterly need, however, is some clear understanding of what the heck this "shipment" could be, how it could change at t_2 compared to t_1 , and how we could possibly detect the change of the "shipment", say, by finding out that we have suddenly shrunk because the Coca Cola bottle in the fridge has suddenly gone bigger.
All these "stretching
and squashing" can be beautifully explained with Leonardo da Vinci’s
man, from Barry Barish' talk Gravity -- Studying the Fabric of the
Universe (AAAS Annual Meeting, Denver, Colorado, 17 February
2003, LIGO-G030020-00-M, slide #13):
Barry C. Barish: "The effect is greatly exaggerated!!
"If the man was 4.5 light years high, he would grow by only a 'hairs width' LIGO (4 km), stretch (squash) = 10-18 m will be detected at frequencies of 10 Hz to 104 Hz. It can detect waves from a distance of 600 106 light years."
Note that a physical clock is supposed to measure four consecutive snapshots, and to compare them to the undisrupted Vitruvian man kept in a safe place (only we don't know where). Besides, the same physical clock, as defined in General Relativity, should not encounter any problems in localization of gravitational energy along the trajectory of the four snapshots (contrary to what we know from textbooks in GR).
Only we don't know whether gravitational waves carry their energy practically without scattering and absorption, or theoretically without scattering and absorption.
Not quite. NASA have formed a joint Task Force to investigate these elusive gravitational "waves",
I wrote to all members of this special Task Force (Kip Thorne and Saul Teukolsky included) on Sun, 23 Feb 2003 02:17:17 +0200, but no answer has reached me so far. I also asked Hans-Jürgen Schmidt, the Editor of General Relativity and Gravitation, the following question (Thu, 20 Feb 2003 12:05:02 +0200):
"Isn't it true that the *necessary* condition for detecting gravitational waves is that the distribution of energy-momentum of a physical system in the gravitational field must be uniquely defined on a continual trajectory, not just in one instant of measurement?"
Professor Hans-Jürgen Schmidt did not reply to this question, regrettably. I can only hope that some of the readers of my feedback to Nature will provide the answer. It will certainly shed some light on another mystery, the so-called black holes and naked singularities, which have never happened in the past 13.7 billion years, just like the ultraviolet catastrophe discovered by Lord (John William Strutt) Rayleigh in 1900.
The task of detecting some gravitational "waves" can be explained for pedestrian as follows. Imagine a wave in a 2-D lake (space), the existence of which can be inferred only and exclusively only from measuring the blueprint from this wave left on the lake surface. We are 2-D creatures and firmly believe that the "thickness" of our lake is zero. Why? Because we successfully use differential and tensor calculus and have found out that there is no need whatsoever for any extra-dimension in which the wave could propagate as well: the mathematics of tensor calculus allows us to handle these concepts "intrinsically", as John Baez put it. Besides, our 2-D measuring devices can only measure the effects of the wave in our 2-D space/lake. And yet we can not detect this elusive wave since 1916. Are we missing something?
Let me try to zoom on the issue which has been waiting for its conclusive resolution since November 1915.
Take, for example, A.P. Balachandran's article "Classical Topology and Quantum States", quant-ph/0002055. He wrote:
"Thus we see that instantaneous measurements are linked both to classical topology and to classical measurement theory. But surely the notion of instantaneous measurements can only be an idealization. Measurements must be extended in time too, just as they are extended in space. But we know of no fully articulated theory of measurements extended in time, and maintaining quantum coherence during its duration, although interesting research about these matters exists [J. Hartle, Phys. Rev. D51 (1995) 1800; M.D. Srinivas, Pramana 47 (1996) 1]."
Let me pin down the sentence "Measurements must be extended in time too, just as they are extended in space."
Well, they aren't extended in time nor in space. The fact that we use 3-D space in Einstein's GR is warranted only by our subjective perception of 'space'. If we are to follow strictly the rules of GR, we must admit that the 3-D space is a persistent illusion due only to the gaze of our consciousness, just like Hermann Weyl "explained" another illusion, that of our subjective flow of time: "The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time" (Hermann Weyl, Philosophy of Mathematics and Natural Science, Princeton: Princeton University Press, 1949, p. 116).
It seems to me that if some mathematical physicists can not explain time and space, they prefer to label it as 'illusion' and drop it in the basket of introspective psychology. Let's see if we can do better.
Recall that in canonical GR we are faced with a static, frozen world, the infamous 'block universe' of timeless 'now'. We might be tempted to think of the instant 'now' as a light spot of a torch illuminating step-by-step that timeless 'block universe'. We might be allured to compare it with playing a song with a musical instrument (or like playing a song with Windows Media Player, say), only we can run it backwards as well, but then both we and and the song will disappear into Nothing from which the song has been created. However, this would be a highly misleading picture since the so-called block universe has no intrinsic (=belonging to a thing by its very nature) time itself, as we know from textbooks.
Again, the subjective experience of 'time' and 'space', which we are fully aware of, do not exist in GR. They have been completely wiped off. As stressed by J. Butterfield and C.J. Isham in "Spacetime and the Philosophical Challenge of Quantum Gravity", gr-qc/9903072: "In particular, these definitions of time are in general unphysical, in that they provide no hint as to how their time might be measured or registered." (If we had the choice of some preferred foliation and hence some preferred reference frame, we wouldn't have the Hamiltonian constraint and problem of time from the Wheeler-DeWitt equation. Then we would be able to deny rigorously the notion of 'closed system' --there would be no such animal in the universe. We would know exactly when and where the universe was created, just by following the time-reversible physics of the universe in that unique preferred reference frame. Would that life were so simple!)
The same applies to the 3-D space. In order to make measurements extended in spacetime, we need to 'move up' into some global perennial spacetime à la Kuchar where we could keep simultaneously at lest two successive frozen snapshots/foliations of spacetime hypersurfaces, and hence determine the so-called ripples of spacetime metric due to the propagation of the gravitational "waves" (recall the old joke from Martin Gardner here).
Indeed, we need a whole "trajectory" of such instantaneous snapshots, which was the point in my question to Hans-Jürgen Schmidt above. Each and every one of these snapshots has been created in full compliance with the principle of general covariance (Diff(M) invariance) of Einstein's GR. Then we must somehow build up a 3-D space, not just take it for granted. This, however, requires new physics. I can only hope that NASA's special Task Force will discover it soon.
Surely Einstein was aware of the criticism by David Hilbert (Die Grundlagen der Physik, Mathematische Annalen, Heft 92, S.1-32, 1924): "I assert that for the general theory of relativity, i.e., in the case of general invariance of the Hamiltonian function, energy equations corresponding to the energy equations in orthogonally invariant theories do not exist at all. I could even take this circumstance as the characteristic feature of the general theory of relativity."
And yet in the year 2003 we are comfortable with the facts that, because of the Diff(M)-invariance, (i) the quantity proposed by Einstein for the gravitational field stress-energy turned out not to be a tensor, (ii) the gravitational field energy is not localizable, in the sense that there is no uniquely defined energy density, (iii) "moving points around" leads to peculiar non-localities but, strangely enough, nobody suffers from them, and (iv) "time is gone" due to the absence of fixed background metric, but ignore the fact that the good old 3-D space is gone as well, due to the full reparametrization invariance of general relativity. What a wishful thinking!
We intuitively keep some 'canvas' of 3-D space, some global perennial 3-D space that would not change along the changes of the unphysical "time" pertaining to a given spacetime foliation only, as warned by J. Butterfield and C.J. Isham above. This constant perennial 3-D space shows up in each and every case of "local" time, one and the same, like a ghost. Where does it come from? From the gaze of our consciousness, along with the subjective flow of time? Not very likely.
"If you realize that all things change, there is nothing you will try to hold onto", says Lao-tzu. Sure, but don't try to hold onto the 3-D space in GR, for it's not there. Never been, never will.
The spacetime in GR is modeled as a smooth differentiable manifold, and hence the main puzzle of what is a "point" endowed with some physical content -- a typical Clintonian question of what is 'is' -- remains unsolved. This fundamental question is outside the scope of GR -- it entertains "points" and provides calculation tools only. Nothing more. As explained by Robert Wald (R. M. Wald (1984), General Relativity, University of Chicago Press, Chicago, p. 14), "It is the notion of 'infinitesimal displacements' or tangent vectors which lies at the foundation of calculus on manifolds." A similar view on the infinitesimal can be found in the explanation of Einstein's equation for the general audience, provided by Matthew Frank: set a parameter to run toward zero, and "at the end of the day" you will get a "point". However, you will be inevitably locked into a frozen, static world of timeless 'now': there is no room for the phenomenon of transience in present-day theoretical physics.
Why? Recall that the phenomenon of transience implies moving between two points. We need a genuine gap between these points, in which the physical object is non-existent, or else there will be not two but one single timeless point, as is the whole block universe. This is one of the problems of continuum, which can be bypassed in classical mechanics only. For example, we can calculate an instantaneous velocity at any given point from a ball's trajectory thanks to the metric of spacetime in Special Theory of Relativity. This metric provides a fixed grid, a firm background on which we can display a time-reversible trajectory. Hence we can claim that the temporal behavior of the ball is sort of encoded in the instantaneous spacetime point at which we have calculated its instantaneous state.
Once we move to GR, however, things are not at all simple. There is no fixed external background, no fixed metric, and anything we can say about an instantaneous state of the ball at a given spacetime point is relational, as explained eloquently by John Baez. There is nothing you can try to 'hold onto', as stressed by Lao-tzu, and subsequently there is no possibility for a genuine transition between two spacetime points: in present-day theoretical physics, we claim that the spacetime itself does not "move", does not change in some other "time and space". There are no intrinsic gaps in the smooth differentiable manifold which is used to model the spacetime in GR. What we might be tempted to call "time" in GR is only valid for a case-specific physical field, not for a bare spacetime point (recall the famous hole argument), but this notion of "time" is only valid for the particular point at which we have calculated Diff(M)-invariant values of the physical field. Hence we can not, even in principle, compare this particular "time" with some other "time" obtained with the same Diff(M)-invariant recipe for the same physical field to find out which one happened first. We can do this only in classical mechanics, as in the example with ball's trajectory above. There is no global perennial entity in GR, which we can use as background, define its "spacetime" metric, and say that this "instant" of a physical field has always been before/after another "instant" pertaining to the same physical field. We can not, even in principle, refer to some metric of some global perennial background in GR to identify some "temporal order". Everything is relational, included the before/after distinction. There is nothing genuinely temporal in the intrinsic structure of the block universe: there is no background metric from some independent unmoved global perennial substance covering the whole block universe. Time is gone, again.
But so is the 3-D space. To explain time, we don't refer to time but to space, and vice versa. Everything we say on the disappearance of time leads to the disappearance of space.
If we trust Einstein's GR, we should not see the good old 3-D space. If the "thickness" of spacetime is truly zero, as is a dimensionless mathematical point, then we should lose both time and space. But we do see and use the 3-D space, as acknowledged by Karel Kuchar. We have obviously missed something very important. What do we do then? Here are two options.
First, follow the trend set by Hermann Weyl and pronounce the 3-D space as illusion due to the gaze of our consciousness. Surely we can not build up a 3-D entity out of some zero-dimensional one such as a "point", so everything we see around us should be an illusion. (It may not be quite clear how could we detect some gravitational waves, but that's a different matter, and hopefully NASA's Task Force will take care of it.) We're safe.
Second, acknowledge that we are faced with an utmost paradox, and try to find out where we went wrong. Perhaps by neglecting the phenomenon of transience which creates these "points", but on the other hand entertaining the product of this same phenomenon of transience: the 3-D space created by a perennial, trans-temporal entity (see again Karel Kuchar on perennials) called here Holon. It does not change and facilitate the temporal identity of particles (Kurt Lewin's Genidentität thesis). As always, the Angels are in the details, this time in the details of the infinitesimal and the nature of continuum.
In order to 'have our cake and eat it' -- employ the continuum and explain the emergence of its points "between" them -- we need two modes of spacetime, called here global mode (also 'mode of being') and local mode (also 'mode of becoming'). Philosophically, we can support both substantivalism (global mode of spacetime & lattice of non-existence) and relationalism (local mode of spacetime & perfect continuum). The nature of spacetime is dual.
The only possible way to have a perfect continuum of "points" is to place them in the irreversible past along a universal time arrow: everything we observe is already in our past light cone and is an already created local mode of spacetime. The global mode in which this creation "takes place" is a discrete 'mode of being' which is physically unobservable due to the so-called speed of light. Just as riding a bike, you can have a stable creation process only if you constantly move forward, along the universal time arrow. In physics, however, measurements are instantaneous, and they can provide only a frozen snapshot from this universal time arrow. Hence the illusion of some 'block universe', wave function collapse, and the like. In the inanimate world at the scale of tables and chairs, the effect of the global mode of spacetime is vanishing small, and so is the volume of the potential future of the universal time arrow: it can accommodate effectively only one propensity for the future possible states, and hence the laws of classical mechanics apply flawlessly. The case of the living and quantum matter, in micro and mega scales, is quite different. Perhaps the human brain has access to the potential future of the whole universe, and can alter the evolution of any physical system "while" the latter resides in the global mode of spacetime. This conjecture is not new at all, it belongs to Leibnitz and was substantiated by Pauli more than half a century ago. The only new element in it is the hypothesis of a universal time arrow, and subsequently two modes of spacetime. Briefly, we need quantum gravity to explain the emergence of time and space from the Holon.
Crucial to this task is identifying a new kind of symmetry transformations of the 3-D space in the putative global mode of spacetime. Just as all dynamical laws are time-reversible, which I interpret as a blueprint from the global mode of spacetime, I suppose there should be a similar space-symmetry as well. What is the generic property of 3-D space? In the local mode of spacetime, we always have a boundary in the 3-D space that separate things, hence we have "inside" and "outside, "big" and "small", and can invert a left glove into right glove. In the global mode of spacetime, we should be able not only to change the direction of time, as we do by hand thanks to T-invariance (and CPT-invariance in general), but to perform the putative space-inversion as well. I suppose that the 3-D "axis" along which we perform this transformation, inverting the 3-D space and exchanging the points like inverting a left glove into a right one and back, will point "toward" the global mode of spacetime. It is a very special entity which defines the "thickness" of 3-D space along the "axis" above, and the space-inversion in it might resemble the non-smooth transformation of a left 3-D "glove" with positive curvature into a right 3-D "glove" with negative curvature, much like a non-smooth transition of a sphere into torus and back. Hence I believe we might have the chance to understand the 3-D space and the dual nature of spacetime.
Mind you, all these transformations "take place" in the gaps of the phenomenon of transience, but we can not detect these gaps due to the so-called speed of light. That's how the global mode of spacetime is postulated. In the local mode of spacetime, the "thickness" of 3-D space is truly zero, and we enjoy a perfect continuum: in the breaking point of torus/sphere transition, the global mode of spacetime is compacted into a 3-D point with truly zero "thickness". Hence we successfully use differential and tensor calculus, get "points" to which attribute real numbers, and behave like some happy little 3-D Flatlanders searching for gravitational waves.
Anyway, maybe by adding the putative symmetry over space-inversion to the group of transformations of Diff(M)-invariance we will learn how to build a 3-D space.
There is a lot more to say on this hypothetical extended Diff(M)-invariance, but I will stop here, hoping to learn the opinion of Professor Karel Kuchar on the disappearance of space and the inner product problem. Again, the problems of Einstein's GR are very old, they have been raised in the crucial month of November 1915. Einstein has never considered his theory complete: "The right-hand side includes all that cannot be described so far in the Unified Field Theory, of course, not for a fleeting moment, have I had any doubt that such a formulation is just a temporary answer, undertaken to give General Relativity some closed expression. This formulation has been in essence nothing more than the theory of the gravitational field which has been separated in a somewhat artificial manner from the unified field of a yet unknown nature." It's about time.
Ceterum censeo Carthaginem delenda
Dimiter G. Chakalov
Information about this proposal has been sent by email, subject "Is there 3-D space in GR?", on Sat, 12 Apr 2003 06:41:39 +0300, Sat, 12 Apr 2003 23:39:04 +0300, Sat, 12 Apr 2003 23:55:16 +0300, Tue, 15 Apr 2003 04:17:43 +0300, and Wed, 16 Apr 2003 00:09:53 +0300, to:
Adolf Grunbaum <firstname.lastname@example.org>
There are good reasons to assume that nature cannot be represented by a continuous field. From quantum theory it could be inferred with certainty that a finite system with finite energy can be completely described by a finite number of (quantum) numbers. This seems not in accordance with continuum theory and has to render trials to describe reality with purely algebraic means. However, nobody has any idea of how one can find the basis of such a theory.
A. Einstein, Grundzüge der
Relativitätstheorie, Vieweg, Braunschweig, 1956, p. 163.