The main task of this website is to suggest a new,
Relative Scale theory of gravity based on a hypothetical
Arrow of Space which springs from God (Luke 17:21).
Its presentation with Maximal Set Theory is due in 2014.

The basics of Quantum Theory are spelled out here,
starting from a well-known task, since 1929.
The latest entry is from 20 October 2013 at 20:05 GMT.




Indefinable Boundary: Point I and Points II




At every instant 'now' the spacetime points are determined by matter, and have dual structure: the spacetime it is both irreversibly fixed in the past and indefinable in the future. At every instant 'now' points emerge, and have structure (FR = 1) exhibited with Point I and Points II.

Every point is emerging -- one-at-a-time -- in the Arrow of Space as dual object: it is both irreversibly fixed in the past by Points II and "open" (indefinable by matter) in Point I (global mode of spacetime). The "separator" between Point I and Points II is the instant 'now'. The Cauchy limit is the final endpoint C (Point I) from which Points II emerge in the irreversible past as perfect continuum (called local mode of spacetime) in which dt/ds is effectively non-existent, while at the same instant 'now' the initial Point I offers the next 'open set' of re-created Points II to be chosen from the potential future of the Arrow of Space in the next elementary step dt/ds. Because every point is dual object, it is also suggested that the logic of propositions and truth statements must be YAIN (Yes And neIN).


FR = 1

The Universe has indefinable boundary at  C .  No function can be defined on the very endpoint  C  (Point I). In the Cauchy limit Ansatz  [ε(..........)ε] , the untraceable endpoint  C  is excluded by using open intervals only. Surely with actual infinity we can think like bartenders and obtain the physical Points II (always in plural), but never the endpoint  C  (Point I) itself.

rx ry = 1 (multiplicative identity)

------>  <------

Perfectly smooth torus-sphere transition via endpoint  C  in the
so-called global mode of spacetime of Point I (the Universe as ONE).

The small red circle contains the Dedekind cut  in the infinite, unphysical, and non-Archimedean spacetime (Point I) of the loop 'now' (see below), obtained with actual infinity. An asymptotically flat spacetime (called flash or slice) corresponds to the local (physical) mode of spacetime. It is made of physical Points II which can be individuated with matter (the Cheshire cat) and hence obtain point-like numbers, included imprecise ones from irrationals.

The four quadrants below are mirror images obtained by replacing (t) with
(-t) and 'left' with 'right' (not shown). The atemporal loop 'now' is nested within Point I (endpoint C) in the non-Archimedean global mode of spacetime.






Atemporal loop 'now'


Spacetime quadrants in Relative Scale
gravity (the favicon of this website is
inserted as decoration only)



In Relative Scale (RS) gravity, the emergence of asymptotic boundaries of spacetime in the Large (B) and the emergence of physical points in the Small (B) are produced en bloc by Point I , with Points II. We shall introduce Point I: a non-Archimedean, uncountably infinite, purely geometrical (a grin without the Cheshire cat), and potential (yet-to-become physicalized) entity inhabiting the so-called global mode of spacetime from which the Cauchy limit and Dedekind Schnitt (C) are projected in the local (physical) mode of spacetime by Points II -- one-at-a-time along the Arrow of Space. In RS gravity, the whole local (physical) mode of spacetime (called also flash) is being re-created en bloc in two directions, toward the Large (B) and the Small (B), starting from A (multiplicative identity) in null "directions".

In a nutshell, our Ansatz explains the limit/cutoff (C) by replacing the options 'either finite or zero' in Archimedean geometry with emergence (always with unit probability) of unique flashes from the global mode of spacetime -- one-flash-at-a-time along the Arrow of Space. With Archimedean geometry only, the Cauchy limit and Dedekind Schnitt (C) bring two alternatives: either (i) always finite (hence never zero) dt/ds increments in spacetime, or (ii) always zero. The solution is to include non-Archimedean geometry as well, and use the instant 'now' in the Arrow of Space as separator: option (i) belongs to 'potential reality' (Point I), while option (ii) pertains to ever-increasing past (Points II).

Stated differently, Point I is yet-to-be-physicalized Macavity state (Adam Helfer) of potential negative-positive mass pairs (Belletête and Paranjape, pp. 6-7), called here pure dark energy, while Points II are individuated only by positive matter (Brill and Jang, 1980; Hans Ohanian).

Thus, the Universe remains in indefinable ONE state at Point I, to allow for its potential future, and at the same time (Sic!) is fixed by Points II in its ever-increasing past.

This is the only possible solution to the problems of set theory and Continuum Hypothesis: the "carrier" acting within dt/ds has been set to zero (perfect continuum) by the "speed" of light, hence producing an ever-increasing past by Points II, while at the same time the potential, yet-to-become physical state of the universe is presented with uncountably infinite (no metric can be defined on null surfaces) and purely geometrical Point I (the grin without the cat) residing in the indefinable non-Archimedean global mode of spacetime.


The Aristotelian Connection (AC) along the w-axis of
the Arrow of Space


Taking the risk to be terribly boring again, I will introduce an example for 'potential reality' from General Relativity (GR): the reference fluid and 'individuating field'. For reasons which I haven't been able to understand in the past 40 years, people frantically believe that GR were 'classical theory'. But it isn't, because it can't. Surely GR is not quantum theory, but is not classical theory either. In addition to the arguments from Erich Kretschmann (Über den physikalischen Sinn der Relativitätspostulate, Annalen der Physik 53 (1917) 575-614), in GR "fixation of a frame of reference and gauge transformations are intertwined in a manner not encountered in any other area of physics" (Peter Bergmann, 1988), which brings insurmountable problems to the reference fluid and 'individuating field'. As John Stachel explained in 1993 (pp. 139-140), "there is no structure on the differentiable manifold that is both independent of the metric tensor and able to serve as an individuating field", in order to uniquely identity "the points of the manifold by some property or properties that characterize(s) each of the points."

So, where and how does 'potential reality' fit in this century old debate?

As Clifford Will et al. put it, "the principle of general covariance, upon which general relativity is built, implies that coordinates are simply labels of spacetime events that can be assigned completely arbitrarily (subject to some conditions of smoothness and differentiability). The only quantities that have physical meaning – the measurables – are those that are invariant under coordinate transformations. One such invariant is the number of ticks on an atomic clock giving the proper time between two events."

The first two sentences from the excerpt above are clear: an object will remain 'the same' if we look at it from different directions, just as a house remains invariant under different coordinates from different maps, say. These are invariants. But are they 'observables'?

NB: Not in GR, ladies and gentlemen. The invariant objects in GR resemble Platonic ideas, which are UNspeakable and physically indefinable. If we say, for example, 'when it rains it pours', we apply particular "coordinates" (words) to express an entity that can be equally well expressed with many different "coordinates" (languages), because it will always remain an invariant object, called here 'potential reality'. In GR, the same phenomenon is called 'reference fluid' and 'individuating field', thanks to which we have an exact 'one meter' and exact 'one second' as invariant objects. Just like Platonic ideas, these invariants cannot be directly observed -- we can physically observe only their "shadows" cast with different "coordinates", and of course require that "coordinates are simply labels".

But look at the last sentence in the excerpt above: "One such invariant is the number of ticks on an atomic clock giving the proper time between two events." I strongly disagree: the phenomenon which creates time as  dt  cannot be temporal. Same tallies to space.

We can only try to reproduce these invariants in metrology, and inevitably use a finite number of physical constituents. We cannot use physical Points II cast from the invariant 'one second' residing as 'potential reality' at Point I. The claim that an atomic clock "gives" the proper time is tantamount to saying that your morning coffee is hot because it contains many tiny little and very hot "particles".

These invariants produce the physical spacetime of Points II (local mode of spacetime). In Relative Scale gravity, we further postulate that these invariants are dual. Namely, they "expand" toward the Small (B) and "contract" toward the Large (B), starting from A in null "directions", yet a co-moving observer will always observe one and the same 'meter', be it an electron or a galaxy; see below.

To cut the long story short, gravity does not produce "curvature". It only "shrinks" the invariant 'one meter', after which bodies moves by the principle of least action, and hence are "attracted" until they become neutralized by the opposite centrifugal force: dynamical equilibrium. At scales larger than our solar system we encounter gravitational "dark" effect and further at Hubble scale its mirrored effect, called "dark energy".

This is how gravity builds up the physical universe. Simple, no?

D. Chakalov
August 6, 2013
Last updated: 7 October 2013, 12:37:00 GMT

Download printable copy, Indefinable.pdf

Read online at



Whether you believe you can do a thing or believe you can't, you are right,
said Henry Ford. Hence the only way to "predict" the future is to create it;
see my first paper about biocausality from January 1990.

Please follow the links below, and feel free to ask questions.



Asymptotic boundary

Origin of light


Wilson chamber

Torsion & rotation



Relative Scale (RS) Theory of Gravity


Download PDF copy, Relative_Scale.pdf (June 8, 2013, 13:25:26 GMT)
See for the record, 19 July 2013



The so-called Scale Relativity Principle was announced on 21 September 2008; the main idea is to remove the background structure in GR, introduced with 'metric of space', as explained on 25 January 2005. Very briefly, the current model of 3-D space is based on absolute relations such as 'inside vs. outside' and 'small vs. large volume of space', which are not acceptable. We need to eliminate all absolute notions pertaining to 'space', to match the absence of explicit "time parameter" in GR (Karel Kuchar), and suggest a new metric theory of gravity, in which the spacetime itself is an emergent phenomenon explicated from a metaphysical pre-geometric quantum-gravitational "dough"; see the Kochen-Specker Theorem and Wilson chamber.

To avoid confusion with other theories, the terms have been changed to Relative Scale Principle (RSP) viz. Relative Scale (RS) theory of gravity. It is a theory of quantum gravity in which the "quantization" of spacetime is introduced from the outset with two modes of spacetime, local (physical) and global, and the "verdammten Quantenspringerei" (Erwin Schrödinger, 1926) are interpreted as artifacts of the macroscopic measuring devices: Dead matter makes quantum jumps; the living-and-quantum matter is smarter.

According to RSP, the geometry of spacetime is effectively Archimedean only at macroscopic length scale, at the lowest part from the drawing above (fixed observer A), while in the directions toward the two ends (10-35 m and 1026 m) a special kind of fusion (also known as 'mutual penetration of Small and Large') occurs, relative to co-moving observers B. Namely, a macroscopic table with length 1 m will be "shrunk" or "expanded", relative to a fixed observer A, along the "opposite" directions toward the two ends, yet the metric of spacetime will also change accordingly: relative to co-moving observers B "travelling" toward the ends of the drawing, a table with length 1 m will always keep its RS-size of 1 m, although the same table will be observed by the fixed macroscopic observer A as 'small like an electron' or 'large like a galaxy'. That is, relative to the fixed observer A at the length scale of tables and chairs, the spacetime is indeed Archimedean, while relative to the co-moving observers B the same spacetime undergoes mutual fusion by keeping invariant length of all objects toward the ends of the above drawing.

Who has 'the right scale'? Nobody, according to RSP, because all contradictory estimates of length, relative to observers A and B, are in fact correct -- the scale itself is dual. Namely, an object at Hubble scale will indeed be "large" and an elementary particle will be "small" to observer A and its Archimedean geometry, while at the same time the "two" (in fact, one) object(s) B will be entangled and will keep its invariant RS-size 'one and the same' in their respective domains pertaining to "two" (in fact, one) observer(s) B. In their respective domain(s) 'out there', an RS-large object does not contain many RS-small objects: both a galaxy and an electron are made of one and the same uncountably infinite "number" of geometrical points (Georg Cantor). Stated differently, the "two" RS-templates, cast in opposite "directions" with respect only to observer A, are in fact one entangled object with invariant RS "size". Hence the universe itself does not change its size but its metric, and evolves along the Arrow of Space as ONE bootstrapped self-regulated entity endowed with self-acting faculty from its Aristotelian First Cause. The fact that we can think about our brain, by our brain, makes our brain 'self-acting'. Physically, we cannot observe our mind in the brain -- just a self-acting brain. In the case of 'the universe as ONE brain', simply replace self-acting brain with 'sufficient conditions for spacetime'.

According to Relative Scale (RS) theory of gravity, there is one and only one "direction" which begins at the macroscopic world with Archimedean geometry (fixed observer A), with two dual presentations toward the Small and the Large, cast in the local (physical) mode of spacetime. This dual "direction" is denoted in the Arrow of Space with w-axis, and is being nullified by the "speed" of light -- one-instant-now-at-a-time. There is no background resembling some "canvas" (John Baez), but a re-created "back bone" of the whole universe at all length scales, made by matter itself -- one-point-at-a-time.

The main applications of RSP are to suggest two presentations of entanglement, quantum and gravitational, which do not occur in the spacetime of a fixed macroscopic observer A -- the underlying phenomenon of entanglement occurs in the fused spacetime with non-Archimedean geometry toward the Small and the Large, and the dual RS-distance controlled by entanglement remains one and the same. And secondly, the fundamental object which facilitates the entanglement is 'the universe as ONE', as explained with Kochen-Specker Theorem and Wilson chamber. Thus, the RS theory of gravity explains the phenomenon of curvature-and-rotation as alteration of spacetime metric by 'the universe as ONE' (cf. sufficient conditions for spacetime): the so-called "dark matter" and "dark energy" are interpreted as tug-of-war and time-symmetric presentations of RS gravity, corresponding to "shrinking" and "expanding" of the metric with respect to a macroscopic observer A, while the actual distance between all point in the fused spacetime remains one and the same with respect to observers B. Stated differently, the cases of RS-shrunk or RS-expanded metric (viz. positive or negative curvature-and-rotation) are explained without any localized physical "dark matter" or "dark energy", just as in the case of 'spin'.

Bear in mind that the current interpretation of gravity inevitably leads to "the worst theoretical prediction in the history of physics!" (Wiki). In particular, the radius of the universe “could not even reach to the moon,” as calculated by Wolfgang Pauli. To understand how we produced such staggering reductio ad absurdum, recall the tacit idea 'only matter can interact with matter'. Namely, if we observe "anomalous" gravitational effects at length scales exceeding the size of our Solar System, we claim that such effects can only be caused by matter. Fine, but what matter, and how?

The phenomenon of transience, exhibited in the Heraclitean flow of time (cf. option YAIN (iii) below), can exist only and exclusively only in a self-acting universe: it is produced by 'the universe as ONE' (cf. sufficient conditions for spacetime), which acts on itself by virtue of the Aristotelian First Cause. In RS theory of gravity, this self-acting action is called Aristotelian Connection, and is presented with an infinitesimal "displacementdt  in the Arrow of Space. Its source (Luke 17:21) cannot be traced back from its physical effects, just as we cannot detect the human mind in its brain -- physically, we can only observe a self-acting brain. And because the universe is modeled as 'ONE brain', its self-action is facilitated by alteration of geometry and topology of the whole universe, which affects the distribution and dynamics of matter localized at all entangled spacetime points, en bloc. Which is why the so-called "cold dark matter" and "dark energy of (you-name-it)" do not exist, just like the physical basis of 'spin'. Such Machian effects may be caused by 'all matter in the whole universe', but their quasi-local implementation by 'the universe as ONE' is with two purely geometrical cases of curvature-and-rotation, such as RS-shrunk or RS-expanded metric. We should just forget about supermassive black holes, "280 million solar masses per particle," etc., and focus our efforts on Einstein's theory of 'total field of as yet unknown structure'.

Last but not least, the alteration of spacetime metric, according to RS theory of gravity, is an effortless phenomenon, because it requires alteration of the phase of gravitational waves. The energy release (e.g., 1054 ergs/pulse in GRBs) is produced by the "engine" of the universe -- the Arrow of Space. Similar to the "quantum waves", the gravitational waves do not possess intrinsic energy and can be manipulated effortlessly, just like we drive our thoughts in our brains. The same effortless action is performed jointly by 'the universe as a brain' and its complementary (Wolfgang Pauli) Universal Mind (Henry Margenau). As Max Planck stated in 1944, "All matter originates and exists only by virtue of a force. We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter". Or simply God's thoughts,” Albert Einstein.

D. Chakalov
Easter 2013, 17:50:34 EET


Online at

PDF file (June 8, 2013, 13:25:26 GMT):




Outline of RS theory of gravity


1. Introduction: The total field of Einstein

Albert Einstein: The present formulation of General Relativity (GR) is "merely a makeshift in order to give the general principle of relativity a preliminary closed-form expression. For it was essentially no more than a theory of the gravitational field, which was isolated somewhat artificially from a total field of as yet unknown structure."

The main unsolved task in Einstein's unfinished GR is the presentation of matter ("timber"):


Firstly, the density of matter in the energy-momentum tensor (Erik Curiel; Babak and Grishchuk) is presented with some continuous "dough", ignoring its quantum structure, which in turn leads to "the worst theoretical prediction in the history of physics!" (Wiki). Secondly, in metric theory of gravity there is no physical gravitational energy obeying conservation law (Jose Geraldo Pereira): the conversion (Hans Ohanian) of "marble" into "timber" (Hermann Bondi) must be presented in such way that only the "timber" can do work, but not the "marble" itself. The latter must not obey Newton's third law, as the inertial mass of an accelerating particle is not "a back-reaction to its own gravitational field" (Wolfgang Rindler, p. 22). And thirdly, the conversion between the "marble" and its physicalized "timber" is a bi-directional "talk" (cf. below), which makes matter ("timber") self-acting. Why? Because the other party ("marble") is hidden by the "speed" of light.


Yes, the gravitational waves are real, but they do not transport energy or momentum. They do not obey Newton's third law, cannot perform work, and must not be localizable (MTW, p. 467). Just like the pre-quantum Kochen-Specker "state" (never in plural), the "marble" itself must be wegtransformierbar (cf. below). It can only cast its "jackets" under "measurements" -- one-at-a-time -- made by asymptotically flat spacetime, in terms of
"a parametrization of the gravitational field and nothing else" (Günter Scharf). What we call "marble" actually belongs to a wider form of reality: potential reality, endowed with "pure" and intangible (Hermann Bondi) energy of 'the universe as ONE'.

NB: As anticipated by Albert Einstein, the solution to "marble-timber" relations can indeed be derived "from a total field of as yet unknown structure": it could be a hypothetical General Platonic Idea (GPI) field of the joint phenomenon 'entanglement & flow of time', under the stipulation that 'the universe as ONE' functions as a brain as well. Physically, we cannot observe the mind inside its brain -- just a self-acting brain.

In the case of 'the universe as ONE brain', we replace 'self-acting brain' with 'sufficient conditions for spacetime', to recover a holistic quantum-gravitational phenomenon which determines -- one-at-a-time -- the quasi-local mass of quasi-local fish, bootstrapped (Geoffrey Chew) by their school of fish. The end result from 'entanglement & flow of time' is a wave pattern of the "timber", like the wave-like holomovement of centipede's legs. That's how we see "waves" without any physical source that would otherwise have to jitter or pulsate in space, due to energy loss. (Forget about dimensionless "strain" h.)

Yes, the gravitational waves and quantum waves are real, because they belong to  potential reality, yet their physicalized "timber" can only display a self-acting universe.

Which means that there are no "carriers" of some biological "field", no "physical basis" of spin, no "particles" for gravity, and of course no "Higgs boson".

NB: This fundamental phenomenon is widely known from life sciences; we simply translate it to the language of theoretical physics with entanglement & flow of time -- Arrow of Space from emergence of spacetime -- and suggest the Relative Scale (RS) gravity.

To be specific:

1.1. GR has unlawful background: absolute size of macroscopic objects viz. absolute relations such as 'inside vs. outside' and 'small vs. large volume of space'. An immediate problems comes from entanglement: if Archimedean geometry were fundamental phenomenon, an EPR-like correlations of quantum and gravitational objects would imply, or even require Geistfelder (spooky "fields" devoid of tangible energy) and various "ghost fields" introduced from "the infinitesimal gauge transformations of quantized gauge fields" (Günter Scharf, p. 1).

1.2. Spacetime topology is not fixed in current GR (Alan Macdonald; MTW, p. 837); the Einstein-Hilbert action is derived from the "dynamics" of values taken by the 3-metrics on a frozen (Robert Geroch) "slice" with fixed spatial topology -- no explicit time variable. Solution: Time requires dynamics of spacetime topology in terms of Arrow of Space -- an infinitesimal 'change of space' (not coordinate change within space).

1.3. Ansatz: Gravity is not quantum phenomenon, for the same reasons why quantum entanglement is not producing "dark" gravitational effects (rotation & curvature). Instead, the underlying phenomenon is entanglement & flow of time in Relative Scale (RS) gravity.

2. Entanglement & flow of time: Arrow of Space

2.1. The emergence of spacetime (Isham & Butterfield) is made by an Arrow of Space along null "direction" (w-axis) -- one-at-a-time (Phoenix Universe). Namely, a global, atemporal, and non-Archimedean realm of the universe (global mode of spacetime) is complemented by a quasi-local, physical, and teleological realm of the universe (local mode of spacetime). The global, non-Archimedean mode keeps an intact potential reality separated from its fleeting "jackets" (Plato), while the local, Archimedean mode is produced as re-created "back bone" of the whole universe at all length scales, made by "measurements" executed by the self-acting universe on itself -- one-at-a-time.

The instant 'now' separates the two modes of spacetime, which evolve along null "direction": one-instant-now-at-a-time. Hence the topological dimensions of spacetime are being accumulated during the Arrow of Space, and because all "dark gaps" of the global mode of spacetime are made zero by the "speed" of light, we observe a perfect spacetime continuum and try to explain the dynamics of spacetime with 'time read with a clock' due to change of coordinates within spacetime. Hence Einstein's total field and the 'thoughts of the Universe' are completely missing in current GR and QM textbooks.

3. Potential reality: Quantum, astrophysical, and cosmological implications

3.1. Quantum form of entanglement: The Kochen-Specker Theorem and Wilson chamber.

3.2. Astrophysical form of entanglement: No "dark" basis, just as in the case of 'spin'.

3.3. Quantum cosmology: Dual age of the universe and The Gospel.

4. Discussion of RS gravity and outlook

4.1. The theory is indirectly falsifiable: every alternative theory of quantum gravity must necessarily be wrong.

4.1.1. Specific errors in alternative theories: localization and "boundaries" of spacetime.

4.2. Outlook: Asymptotic "boundaries" of spacetime, creation of mass one-at-a-time (elevator metaphor), and atemporal "Macavity".



Frequently Asked Questions (FAQ)


Q1: "GR works perfectly well and I can't agree with your ideas."
A1: This is a statement, not question. Recall that the current version of GR is based on "miracles": you can't have any geometry at Planck scale. T
here is nothing resembling law and order in the "spacetime foam" to raise a robust Lorentzian metric within 10-30 seconds "after" the "big bang" and inflation of universe's "size" (with respect to what?) by a factor of 1078, and keep the Lorentzian metric for at least 13.77 billion years rooted on Planck length at which "points" are fuzzy and locality has lost its meaning 13.77 billion years ago.

Q2: "What do you intend to achieve?"
A2: A model of bootstrapped universe, in which every geometrical "point" is determined by states of mater ("jackets") that are pre-correlated with 'the rest of the universe'. Which will be impossible if Archimedean space were fundamental phenomenon. Hence RSP. Notice that at Planck scale the equation  SL = 1  (Small is denoted with S, Large with L), which holds for observer A, is again valid, but now it describes 'the non-Archimedean universe as ONE' of observer(s) B. Then we use this unique ONE entity as Reichenbach's Third Cause to explain the relational ontology produced by entanglement during the flow of time (cf. Escher's drawing hands below).

Q3: "How do you explain the mind-body problem?"
A3: With 'potential reality' in the Arrow of Space; but please see first Gottfried Leibniz, Wolfgang Pauli, and the Eskimo metaphor. We encounter two forms of potential reality: one explicated under macroscopic conditions (BrainMonad), and the GPI field of quantum-gravitational world (observer B). The first interacts with the brain and the physical world along the Arrow of Space, and can be metaphorically explained as a 'steering wheel' of 'the car' (brain and 'the rest of the universe'). Namely, the BrainMonad is neither 'matter' nor 'mind', but a macroscopic form of potential reality. One might suggest that it has a dual nature, because it acts like a "filter" through which the Psyche can enter spacetime, a bit like images displayed on a TV screen (not located inside TV). So, if the BrainMonad is a 'steering wheel', the 'driver' is the human mind endowed with Free Will. We speculate that the GPI field may also act as a "filter" for Universal Mind (Henry Margenau), but cannot prove such claim. Point is, the 'driver' (human mind) may have access to the joint 'steering wheel' (BrainMonad & GPI field), and alter the propensities (not "probabilities") for future potential events, almost like learning a new motor skill. Math is a crucial issue, too. The first off task here is to explain the physics of binding phenomenon -- how all sensory
"data" are combined into a single experience, derived from their joint amodal presentation (BrainMonad). In my opinion, the only way to approach the challenge is to assume that the whole universe as ONE works as a "brain". Hence RS gravity.

Q4: "What is this all about, Dimi?"
A4: Tough question. Actually, it's all about cat astrology. I'm also selling cat food from my website, with special discounts for theoretical physicists like you. Aren't you interested?

Feel free to submit your questions. Just please don't forget that RS theory of gravity is an alternative to all multidimensional theories in which gravity would operate in some "extra dimensions" with some hypothetical "gravitons".

For example, people try to bridge the "desert between the weak and Planck scales" with "extra compact spatial dimensions" leading to "(4 + n) dimensional theory" in which "particles cannot freely propagate in the extra n dimension, but must be localized to a 4 dimensional submanifold", while "the only fields propagating in the (4 + n) dimensional bulk are the (4 + n) dimensional graviton" (N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, arXiv:hep-ph/9803315v1, p. 2). Then comes this (ibid., p. 3):

"As within any extension of the standard model at the weak scale, some mechanism is needed in the theory above mEW to forbid dangerous higher dimension operators (suppressed only by mEW) which lead to proton decay, neutral meson mixing etc. In our case, the theory above mEW is unknown, being whatever gives a sensible quantum theory of gravity in (4 + n) dimensions! We therefore simply assume that these dangerous operators are not induced."

But every sensible quantum theory of gravity should solve the most widely known public secret in theoretical physics -- localization. First things first. Then comes QM and GR, and the new Quantum Geometry in which the geometrical points possess quasi-local structure due to the global mode of spacetime of 'the universe as ONE', shown with red in the drawing below.



One can introduce "boundary" on spacetime only and exclusively only with RSP. This kind of "boundary" is the only possible logical option for gravity. We are macroscopic observers, and in the case of Archimedean geometry the entanglement of space  Espace , expressed with "fusion" of Small and Large, is effectively zero, yet it takes values in an open interval

Espace Є (0, ∞).

Also, Baldy's Law, according to which “some of it plus the rest of it is all of it,” is strictly valid only for an inanimate macroscopic world with Espace effectively zero, but does not hold for Quantum Gravity. More in A2 above.


D. Chakalov
May 7, 2013
Last updated: June 8, 2013, 13:25:26 GMT








Today, 14 March 2013, Albert Einstein (b. 1879) would be 134 year old.
My efforts to unravel his
'total field of as yet unknown structure' and
the nature of gravitation and quantum entanglement are posted here.




Happy Birthday, Albert!







The so-called 'God's thoughts' refer to a web of correlations
of all physical systems, which occur in a hypothetical
'global mode of spacetime' produced by the Arrow of Space.
Relative to a physical (inanimate) clock, the global mode
of time will inevitably look "frozen".

0.47-0.52: "Relative to the platform,
time on the train completely stops."

This "frozen" time pertains to 'the whole universe as ONE'
and to the so-called BrainMonad. The "speed" of gravity
is dual: both "instantaneous", in the global mode of
spacetime, and finite, in the local mode. The end
result is a quasi-local mass and quasi-instantaneous
correlations, resembling those in a school of fish.

Relative to the local mode of spacetime, the global mode
is at 'absolute rest', and serves as the reference fluid of
General Relativity. It is located ]between[ any two
neighboring "points" from the spacetime manifold, and
renders its local mode a perfect continuum: due to the
so-called speed of light, the "separation" of the spacetime
points (local mode of spacetime) is in fact zero.

There is no direct link between the local and global
modes of spacetime, i.e. between 'physical reality' and
'potential reality': the UNdecidable pre-quantum KS state
must not be included in the set of its color-able explications.

The 3-D projection(s) of the global mode of spacetime are
an infinitesimal "point" and 'the infinitely large universe'.
It is a dual object which wraps up the local (physical) mode
of spacetime, and produces finite 'templates' for spacetime.

The cosmic vacuum/quantum aether, called here
 global mode of spacetime, is an absolute reference
frame pertaining to the whole universe as ONE:
the physical world there is in absolute rest, in the
sense that its proper time is zero, as "seen" from
such luxonic reference frame.

Thanks to the Arrow of Space, 'the universe as ONE'
is not "frozen". In the quantum realm, it exists as
'potential reality' or UNdecidable pre-quantum KS state.
As to the current GR, 'the universe as ONE' is the
absolute reference frame in which space "expands".
It is also the atemporal medium for bootstrapping
the physical world and generation of Machian inertia.

More on the errors in GR literature here.


The current GW detectors are manifestly blind and deaf to
the ripples of spacetime. Their proper detectors must
be endowed with the faculty of 'self-acting', resembling
the human brain. Ditto for the so-called "dark energy".

LIGO tunnels should be converted to wine cellars. Any other ideas?





Latest entries on GW "astronomy":




"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

"In the first place, we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception."

Albert Einstein, Relativity: The Special and General Theory, 1920

"According to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time."

A. Einstein, Äther und Relativitätstheorie, May 5, 1920

(Lisa M. Dolling et al., The Tests of Time: Readings in the Development of Physical Theory, Princeton University Press, Princeton, 2003, p. 346)



Panta rei conditio sine qua non est

Your Global Time is ZERO

It is suggested that the spacetime manifold is a dynamical entity re-created at every step of a hypothetical spacetime arrow; the latter is due to the "expansion" of space by the dynamic dark energy (DDE). Two modes of spacetime are postulated in this Phoenix Universe: local mode, in which we have point-like events cast on a perfect continuum, with a "carpe diem" unit probability (cf. the measurement problem in QM here), and a global atemporal mode, in which a Machian-type negotiation of every next step is being processed. The effects from the global mode of spacetime, which literally build up '3-D space', begin from the macro-scale of classical physics in two "directions": quantum effects toward the Small, and dark matter & dark energy effects toward the Large. Briefly, we model the universe as a huge brain which 'thinks' with its global-mode state by following the rule 'think globally, act locally'. The implications for quantum gravity are explained by revealing the two modes of spacetime in Quantum Theory and General Relativity, and by suggesting conceptual solutions to the problems and paradoxes hindering the quantum gravity of He Who Does Not Play Dice.


We haven't the money, so we've got to think!
Lord Rutherford, 1962 Brunel Lecture, 14 February 1962

Overfunded research is like heroin: It makes one addicted, weakens the mind and furthers prostitution.
Johann A. Makowsky, The Jerusalem Post 19.4.85


Does a fish need a bicycle?


Latest update: November 26, 2009

Printable copy (current version) from

Check out 'Quantum Mechanics 101' here, my detailed reproach upon wasting taxpayers' money with LIGO here, and my efforts toward quantum gravity here.

Regarding Quantum Mechanics (QM), the aim is to avoid the incomprehensible paradoxes and artifacts in it (watch the double slit experiment here), which originate from its textbook interpretation (Niels Borh's belief that quantum world can only be "seen" through classical "glasses"). We can indeed understand the quantum world (but not the current QM textbooks; cf. Richard Feynman), by changing the "glasses" through which we "see" and construe the quantum world: the universe modeled as a brain.

Regarding the artifacts in QM, we may be in a situation similar to an Eskimo trying very hard to comprehend the notion of "trunk". In our case, we encounter an incomprehensible wave-particle complementarity, which could be just an artifact from our wrong thinking, like the "nose-arm complementarity" in the case of the Eskimo observing elephant's trunk. Surely 'the quantum system' doesn't live in any relativistic space, so one should expect all sorts of headaches and artifacts (e.g., "diese verdammte Quantenspringerei", Erwin Schrödinger) from imposing wrong "glasses" onto the quantum world 'out there'. (A typical example is Franco Selleri's quest for detecting de Broglie waves.)

The prerequisites for the interpretation of QM suggested here originate from Schrödinger. Back in 1935, Erwin Schrödinger stressed the following:

"The rejection of realism has logical consequences. In general, a variable has no definite value before I measure it; then measuring it does not mean ascertaining the value that it has."

And in a letter to Einstein dated 18 November 1950 (quoted after J. Bub, p. 115), he wrote (emphasis added):

“It seems to me that the concept of probability is terribly mishandled these days. Probability surely has as its substance a statement as to whether something is or is not the case — an uncertain statement, to be sure. But nevertheless it has meaning only if one is indeed convinced that the something in question quite definitely is or is not the case. A probabilistic assertion presupposes the full reality of its subject.”

We are obviously dealing with a new form of reality: a probabilistic assertion, and the Hilbert space itself, cannot accommodate the Kochen-Specker case in which "the something in question" is beyond an unequivocal "is or is not" state (cf. "an unequivocal true-false value" in Isham and Butterfield, p. 3; emphasis added):

Check out the implications of KS Theorem to the Precise Value Principle (PVP) and the statistical interpretation of QM from R.I.G. Hughes, p. 164.

Going back to Ernst Specker's tripod, if there are states of the tripod in which one of its legs has UNdecidable color, then not just this particular leg, but the whole tripod will be UNdecidable. It's a package.

The phrase "an incomplete Kochen-Specker colouring" (Helena Granström, p. 2) has no meaning whatsoever; it is the result from imposing wrong "glasses" onto the quantum world. And if you subscribe to the modern quantum mysticism -- "the quantum state is not a physical object, it is a representation of our state of knowledge, or belief" (Itamar Pitowsky, p. 28) -- your brain will wind up in a schizophrenic state of, say, 68% "knowledge" of the quantum state, and 32% of "[what da heck is that uncolored KS sphere?]".

No mental concepts, such as 'knowledge' or 'imagination', are admissible in the ontology of quantum reality. We must never mix apples with oranges (Res Extensa and Res Cogitans).

Karl Svozil refers to this UNdecidable faculty of the quantum world as "ambiguity" (p. 4), and stressed: "This ambiguity gets worse as the number of particles increases." If you think about the quantum world with classical concepts, it will get from bad to worse, until you end up with the (old) cosmological "constant" problem (more on that from Alan Guth).

As Erwin Schrödinger might have said in 1935, the same "variable" that has had no definite value before you measured it will continue to keep its UNdecidable nature after you "measure" it as well. And you can't fit it in any Hilbert space, of course (what is the dim(H) for 32% "uncolored" and 68% colored KS sphere?).

Let's give it a name: potential reality. In the quantum realm, it (i) offers its context-dependent explications (a.k.a. "observables"), (ii) keeps the sameness (Genidentität, Kurt Lewin) of particles of the same type, which MTW regard as "a central mystery of physics" (p. 1215), and (iii) facilitates the ultimate quantum phenomenon: entanglement. It may be difficult to grasp, but is much simpler to the juggling with the possible implications from Bell's inequality and their loopholes (e.g., Ghim and Zhang).

As to quantum gravity (notice the opinion of an expert here), the 'potential reality' is introduced to revive the physical objectivity of spacetime "points", by making the spacetime manifold itself an emergent phenomenon: "The requirement of general covariance takes away from space and time the last remnant of physical objectivity" (A. Einstein, Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 49 (1916) 769-822). The current formulation of GR can only determine "the mutual relations that exist between the gravitational field and the matter fields (i.e. the value the gravitational field takes where the matter field takes such and such value)", but not "the proper time between spacetime points" (Wiki; more from Butterfield and Isham), and the field equations "cannot even uniquely determine the topology of a manifold" (Alan Macdonald, Einstein's Hole Argument, p. 4).

How can we fix these problems? By introducing two connections, geometric (local mode) and torsion (global mode of spacetime). The torsion connection is completely vanished (Hehl and Obukhov) in the local mode, hence its effect are considered "dark" (see Alex Murphy).

Notice also that a hierarchy of 'potential reality' (never in plural), resembling the structure of cognitive concepts, is postulated (application here). In metaphysical terms, it supports the views of Aristotle and Spinoza: no "parts" of the infinite can exist, as the infinite Substance is indivisible. It's not like Russian dolls.

In the case of a human brain (not mind or consciousness), the UNspeakable potential reality can be explicated with, for example, three (and many more) sayings, which produce "measurements" (if you prefer QM jargon) on it:

1. All are not hunters that blow the horn.
2. La robe ne fait pas le médecin.
3. Es ist nicht jeder ein Koch, der ein lang Messer trägt.

None of these "measurements" can "collapse" the potential reality (Platonic idea) explicated with these sayings. And if our brain can work with 'potential reality', so can the universe modeled as a brain.

Regarding the brain above your neck: its 'potential reality' is to its human self what EM radiation is to your subjective experience of 'color' -- no direct action of the human self on its brain is allowed in science. Hence it may be conceivable that the potential reality entwined with the human brain could be bridged to that of the quantum vacuum (BCCP). All you may need is an arrow of spacetime. (Notice, however, the 'two rules of success' here.)

Notice also that 'potential reality' is an intrinsically holistic phenomenon, so it will be very misleading to call it "dark", just because it is not possible to be traced back from any of its physical explications in the r.h.s. of Einstein field equation.

Forget about "energy conservation in GR". It's an oxymoron (details from Denisov and Logunov). What kind of "time" is implied in the non-linear dynamical cancellation of the two "fluxes" (Merced Montesinos), such that the ether would not "come back"? Can your wristwatch read it?

The sole "explanation" of this ultimate puzzle of GR, offered by Chris Isham, was that, "after all, general relativity does seem to work well as a theory, and yet I can certainly read the time on my wrist watch!" But your wristwatch should not be able to read more than one instant from this (global) non-linear "time". Complex problems have simple, easy-to-understand wrong answers (Murphy's Law No. 15).

We need quantum gravity, to uncover "the proper time [tau] along spacetime trajectories" (Carlo Rovelli; drawings here) and the genuine 'time variable' associated with the expansion of space due to DDE. Once we achieve this formidable task, we will (hopefully) find out what -- if anything -- might remain unchanged/conserved in this particular (global mode of) time. In the current 'GR without DDE', the dynamics is "entirely generated by constraints. The dynamical data do not explicitly include a time variable" (Karel Kuchar), and one can only hope that "the energy momentum tensor which is the source of gravity" might be conserved "due to the Bianchi identities" (E. Guendelman, p. 9). Such hopes are (i) utterly murky even in the textbook GR, because any observable of the gravitational field is "necessarily quasi-local" (Laszlo Szabados) and (ii) not applicable to the new dynamics of GR due to DDE. For example, Noether's Theorem holds only for flat spacetime, and the Bianchi identities are applicable only for 'GR without DDE'.

At this point, I got an emotional response from Eduardo Guendelman, saying that the Bianchi identity is a mathematical theorem, so there is no ambiguity involved here. Well, I'm just a psychologist, so let me quote Matt Visser, p. 3:

"... the Einstein equations of general relativity are local equations, relating some aspects (notice the poetry -- D.C.) of the spacetime curvature at a point to the presence of stress-energy at that point. Additionally, one also has local chronology protection, inherited from the fact (Sic! -- D.C.) that the spacetime is locally Minkowski (the Einstein Equivalence Principle), and so “in the small”  (that's the sole application of those 'twice-contracted Bianchi identities', cf. George F R Ellis and Henk van Elst, Eq. 2 -- D.C.)  general relativity respects all of the causality constraints of special relativity.

"What general relativity does not do is to provide any natural way of imposing global constraints on the spacetime — certainly the Einstein equations provide no such nonlocal constraint."

Which is precisely the missing element needed to address the 'global properties of spacetime' in the presence of DDE. As R. Rakhi and K. Indulekha acknowledged (p. 5): "Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence (the same phrase was used by Sean Carroll -- D.C.), more of this energy-of-space would appear. (...) So the mystery continues."

And so does the confusion about it -- click here.

The intrinsic dynamics of the phenomenon of 'more space comes into existence' could only be detected in a reference frame from which this phenomenon is excluded, like 'not acting there'. But because DDE acts on the whole 3-D space en bloc, without any irregularities, there is no place in the universe in which we could install a clock and say -- look, this clock does read the (global mode of) time pertaining to the omnipresent phenomenon of 'more space comes into existence'. It is the theory of relativity itself, which cannot "detect" it. Which is why some people call it "dark".

Although the mathematical meaning of the phrase "more space comes into existence" is not clear, I don't think Kurt Gödel would have called the ultimate source of spacetime points "dark". Once we move from physics to geometry, it's a whole new world there. Pity Mike Turner called it "dark"; that's so wrong!

One cannot insert the "dark" energy of  X  into its consequence -- accelerated expansion of space. The source  X  does not belong, and cannot be fitted into the same 3-D space (Cauchy hypersurface) which is being created by  X . You can do this only in GR textbooks that deal with 'GR without DDE'. Capiche?

Notice also that Matt Visser (see above) considers the metaphysical assumption that the spacetime were "locally Minkowski (the Einstein Equivalence Principle)" to represent a fact. But we only have a mathematical fact that  locally , "over" a point, one can indeed eliminate the gravitational "field" by hand (Hermann Weyl). But because nobody has so far explained the resulting quasi-local nature of the gravitational field's observables (see Laszlo Szabados above), nor the origin and the mechanism of inertial reaction "forces", I think we should be very cautious and open-minded in interpreting such mathematical facts and theorems, like the above-mentioned Bianchi identity.

If you are looking for a genuine quantum-gravitational measuring device, your wristwatch (as well as the one of Kip Thorne) fits the bill, because it reads an already-linearized (see the explanation of 'already' below) time variable obtained from the "the proper time [tau] along spacetime trajectories" (Carlo Rovelli) and the (global mode of) time associated with the expansion of space due to DDE.

This is to me the ultimate puzzle in present-day GR. The sole "explanation" of Chris Isham was that, "after all, general relativity does seem to work well as a theory, and yet I can certainly read the time on my wrist watch!"

I will desist to comment on C. Isham's observation, and will instead take the liberty of being (again) deadly boring, by explaining the difference between 'GR without DDE' vs 'GR with DDE'.

First, a simple example from STR, with a trajectory of a Frisbee, on the fixed background of Minkowski spacetime: we can calculate the instantaneous state of the Frisbee at each point from the trajectory, and attach to this dimensionless point a well-defined vector. My teenage daughter couldn't understanding how it is possible to attach a vector to a "point", and I explained the puzzle by saying that the information from a finite interval from the history of this infinitesimal "point" is encapsulated in it -- we instruct this interval to shrink asymptotically toward zero -- so the vector is indeed well-defined. All this is possible under the premise of the fixed "grid" in STR. But once we move to 'GR without DDE', the "grid" is gone (Emilio Elizalde): at each and every "point" from the trajectory, the non-linear mutual determination of matter and space (John Wheeler) takes place. But in what time (see Chris Isham's wrist watch above)? There is no background grid or "ether" w.r.t.w. one can define the dynamics of 'GR without DDE'. Yet people don't ask such questions and prefer to just do calculations with the linearized approximation of 'GR without DDE'. As another expert explained to his undergraduates, "one begins by introducing the notion of a tangent vector to describe an infinitesimal displacement about a point  p " (Bob Wald, p. 4). But again, in order to recover the true dynamics of 'GR without DDE', you need some "ether" or rather 'reference fluid' w.r.t.w. one can describe the fundamental phenomenon of transience (Abner Shimony): see the so-called Aristotelian Connection here.

At this point, the 'GR with DDE' comes to rescue the Hamiltonian formulationof 'GR without DDE': we have a brand new, global degree of freedom of spacetime en bloc, hence can recover the transience of spacetime, as driven by the source of DDE, along the arrow of spacetime -- see above.

To identify this same source in the quantum realm (called 'potential reality'), let's go back to the interpretation of QM here.

In a nutshell, the so-called PR2 interpretation of QM offers a solution to the non-unitary "collapse" by replacing the alleged "U" and "R" processes with a new (at least to people like Ed Witten, Steven Weinberg and Gerardus 't Hooft) form of reality, known since Plato, called here 'potential reality'. Its quantum presentation is ubiquitous and has zero entropy; hence the familiar notion of time, which pertains to monotonic increase of entropy "in time", is not applicable to 'potential reality'. It may produce "shadows" in terms of 'quantum observables', yet these "shadows" cannot be traced back, to reveal any evidence of such events (or "quantum information") ever having emerged from 'potential reality'. It may act, yet not experience any backaction from its "shadows". Hence it is the ultimate "background" for QM and GR. It evolves along the arrow of spacetime (resembling the "memory" of the universe), by unfolding from 'the ideal monad without windows'. In the local mode of spacetime, every "point" is filled with an already explicated value of its 'potential reality', and as the latter evolves and becomes enriched, the "number" of its localized explications increases accordingly: more and more things come into existence in the universe. As John Wheeler put it, "Time is Nature's way to keep everything from happening all at once". Only with 'potential reality' there is no need nor place for any 'unitary dynamics' in the local mode of spacetime. This is the metaphysics of 'the universe modeled as a brain', viz. the interpretation of 'expansion of space' along the arrow of spacetime.

From this perspective, if we follow the deflation time arrow in the local mode of spacetime, things will gradually fade away in a strictly non-unitary fashion, but will only approach asymptotically The Beginning, because it is logically impossible to reach It from/within the local mode of spacetime (cf. the paradox here). The solution is 'dual age cosmology'.

As M. Gell-Mann and J. B. Hartle have noticed, "quantum mechanics is best and most fundamentally understood in the framework of quantum cosmology" (quoted after Claus Kiefer, p. 1). And vice versa.

The current situation with inflationary cosmology was presented by Alan H. Guth, the winner of the 2009 Isaac Newton medal, on 13 October 2009. He speculated that our galaxy could be an amplified "quantum fluctuation" (Part 2, 3:30 - 6:16),


... and then confessed his "Nightmare of Dark Energy" (Part 2, 11:36), on which the "eureka" of inflationary cosmology is grounded.

Perhaps Alan Guth should have entitled his talk 'The Dark Energy Nightmares of
Inflationary Cosmology', as encapsulated in the [?] area in V. Mukhanov's paper:

With the so-called
dual age cosmology, the "first" Plankian time "after" The Beginning is stretched toward infinity in the local mode of time. This is otherwise being explained as 'the universe started asymptotically from time zero'. In this local mode of time (read by your wristwatch), the universe is effectively eternal and its age is indecisive, while in the global mode of time it has a finite age. Another nice feature of dual age cosmology is that The Beginning is an atemporal phenomenon, and is always present (global mode of time) in the instant 'now'.

Let's go back to QM. Think of a trajectory of a Frisbee: at each "point" from the trajectory (the latter could be a perfect continuum of such "points"), the state of the Frisbee is a perfectly well defined fact, with a 'carpe diem' unit probability. I use this simple idea to introduce a new kind of cancellation mechanism, pertinent to 'potential reality', for removing the non-unitary "R" process (historical account from October 2002 here); hence the motto of this web site, since July 1997: Dead matter makes quantum jumps; the living-and-quantum matter is smarter.

Notice that not only the "R" process is being removed (as did Hugh Everett III in his relative state interpretation of QM), but the alleged "U" process and Hilbert space as well.

The established theoretical physics community, particularly Niels Bohr, ridiculed Everett's interpretation to the extent to which Hugh Everett left physics (and became millionaire).

NB: Notice that if the PR2 interpretation of QM is correct, then all tentative solutions proposed for the measurement problem and the classical limit of QM, based on either probabilistic or "toposification" (Chris Isham), ‘neo-realist’ interpretation of QM (Cecilia Flori, p. 211), will necessarily be wrong.

The next step is to elaborate on the continuum of such explicated quantum states (local mode of spacetime), which emerges from the back bone of the whole physical world (global mode). Here we enter quantum gravity, particularly the emergence of time and space from "something else" (Isham and Butterfield): check out the arrow of spacetime here, and think of 'the whole universe' as a huge brain which self-determines its consecutive quantum-gravitational states in line with the Bootstrap Principle of Geoffrey Chew. Also, the alleged quantum "fluctuations" are interpreted as flexibility of 'the quantum state' to offer potential states, one of which to be chosen (one by one) by 'everything else in the universe'. Hence the speculation about some "spacetime foam" and the insoluble problems from it (how do you tell apart spacelike from timelike at Planck scale?) are avoided from the outset. Possible practical implications are outlined here.

As in the case of Hugh Everett, many theoretical physicists jumped to play the role of Niels Bohr, chief among them was Britain's leading expert in quantum gravity, who declared seven years ago (
Wed, 23 Oct 2002 19:24:15 +0100):

"You do not know enough theoretical physics to help with any research in that area."

To the best of my knowledge, nobody has so far offered some new cancellation mechanism as a joint solution to the measurement problem of QM and the cosmological "constant" problems. Such cancellation mechanism is built in the UNdecidable quantum state (quantum presentation of Platonic ideas) from the outset: instead of dealing with some |alive cat> & |dead cat> from the "U" process, we encounter an UNdecidable 'cat per se', so if we happen to observe an 'alive cat' as a fact, with unit probability, the 'dead cat' will happily live undisturbed in 'the UNdecidable quantum state'. No "collapse" nor dead cat doppelgänger à la Everett are needed.

The animated cat above is very deceptive, because its "evolution" prior to the "collapse" seems fully deterministic. If you think about the quantum "evolution" of such superposed states in terms of 'energy eigenstates', notice the imaginary unit in S. Carroll's essay, Eq. 4: "all of the time evolution is encoded in the phases [XXX]" (ibid., p. 6). But how do you encode 'time evolution' in complex phases (
Chen Ning Yang)? Recall that, after you "collapse" the cat, the alleged "time parameter" in the Schrödinger equation turns into some non-relativistic [you-name-it], but the equation itself doesn't say anything whatsoever about the nature of this "time parameter" prior to the "collapse", as introduced by hand (along with the Born rule) in QM textbooks.

To understand the origin of this whole mess, read the second sentence from the excerpt below (R.I.G. Hughes, The structure and interpretation of quantum mechanics, p. 77):

This "general principle", however, does not cover the case of UNdecidable, hence not-yet-physical, KS quantum state, as explained by R.I.G. Hughes here. One can hardly overestimate the enormous confusion from this huge blank spot in current QM textbooks -- just recall the ongoing quest for "quantum computing" and the alleged "decoherent histories" (J. Halliwell; cf. Franck Laloë, Sec. 6.4 and footnote 47 on p. 81).

All these problems are resolved in the PR2 interpretation of QM from the outset. Moreover, the new cancellation mechanism is introduced to explain the ongoing, as-we-speak mechanism of 'the flatness problem' (asymptotically flat spacetime conjecture), producing an extremely precise balance between the two tug-of-war effects, CDM & DDE, of the geometry of spacetime at cosmological scales.

Otherwise we have to inject up to 96% "dark stuff" (with all sorts of "ghosts") into the current theoretical physics, as calculated under the assumption that 'potential reality' doesn't exist. As Evalyn Gates put it (p. 196), the detection of DDE was "like finding an elephant on top of a table impeccably set with the finest china and silver (...). We stare in shock at the uninvited guest and demand to know where the elephant came from -- and how it got into (the) room."

Regarding the arrow of spacetime: notice that the Frisbee-like sequence of explicated facts (local mode of spacetime) is inherently background-free, because the background (the reference fluid of GR and the UNdecidable quantum state) is 'not there', being placed  ]between[  the "points" of the continuum of the local mode of spacetime. In other words, the "dark gaps" of the global mode are not like the real gaps between the tiles in R. Penrose's bathroom: regardless of how small tiles you choose, if you decrease the size of the tiles in a Fibonacci sequence, you will never ever reach the "gaps" from/within the local mode of spacetime (more on Fibonacci here).

This proposal makes the local mode of spacetime a perfect 3-D continuum with dynamical topology of 'asymptotically flat spacetime'. The "dark" gaps of the postulated global mode of spacetime are completely sealed off  by the arrow of spacetime -- the mechanism by which the "dark" gaps of the global mode are made nonexistent in the local mode is the same that makes the "speed" of light a fundamental constant (and also hides the mirror tachyonic world). This renders the Schlaefli conjecture (L. Schlaefli, Ann. di Mat. 5 (1873) 170), as well as all "branes" and other multidimensional superstitions in GR and string hypotheses (Lisa Randall; see also A. Vilenkin below) redundant, to say the least.

In the context of GR, the global mode of spacetime is located "within" each and every point from spacelike hypersurfaces, as it "lives" exclusively on null hypersurfaces. (Unlike the geometry of spacelike hypersurfaces, the geometry of null hypersurfaces is not metric (D. C. Robinson), which leaves a challenging opportunity to introduce an additional, to the Christoffel connection, global torsion connection.) To be precise, the "duration" of the global mode, recorded with a physical clock (local mode), matches the "duration" of the atemporal "handshaking" transaction in Cramer's interpretation of QM: it has been already completed at each and every instant we "look" at it (see below).

Hence in the local mode of spacetime, the dimensionless GW amplitude is zero, nonexistent, zilch. (Another case of reining a dimensionless amplitude, the mythical "quantum computing", is examined here.)

Notice also that the hypothetical global mode of spacetime cannot be read by a physical clock (it will "stand still"). It is introduced to replace the "external time parameter" in H.-D. Zeh (p. 13) and the “auxiliary internal time” (cf. Macias and Quevedo, p. 8) by 'the reference fluid of GR'. The latter can "act" upon matter without being affected in turn by matter. In this unphysical "absolute" reference frame, an electromagnetic radiation field can indeed "stand still" (recall that EM radiation field cannot stand still with respect to any physical observer, Bahram Mashhoon, p. 14).

The next metaphysical idea is straightforward: 'time' does not originate from 'change in space' but from 'change of space'. Only if you have the latter (global mode), you may introduce the former, as 'time read by a clock' (local mode). Why? Because one cannot insert the "dark" energy of  X  into its consequence: accelerated expansion of space. The dynamics of 'the change of space' is defined relative to the "omnipresent ether  X " (global mode of spacetime). The latter is located "within" each and every point from the local mode of spacetime, and is wrapping the local mode by two (in fact, one) 'numerically finite but physically unattainable Aristotelian boundaries'.

All we can physically observe is that the local mode of spacetime is being 'acting upon itself'. Such self-action will of course look "dark" to all local sub-systems (see Alex Murphy).

Hence we can bridge QM and GR, and understand the origin of quantum and gravitational "waves": EPR-like correlations (global mode) will inevitably induce wave-like holomovement of physical stuff along the arrow of spacetime. Such wave-generation effects can be found in our brains and in many living organisms. Perhaps it determines the inertial reaction "forces" as well (don't bother to ask Criss Angel, he knows nothing about it).

Recall what William Kingdon Clifford claimed in his paper ‘On the Space-Theory of Matter’, presented to the Cambridge Philosophical Society on February 21, 1870 (quoted after Domenico Giulini, p. 2):

I believe Clifford's idea in (2), about "the manner of a wave", is amended here with the proposed origin of quantum-and-gravitational "waves": the continuous passage of "curved or distorted" from one point (not "portion", as in (1) above) to the next one is what the arrow of spacetime does on the perfect continuum of the local mode of spacetime (the "dark" gaps of the passage are being completely sealed off  by the arrow of spacetime).

But why 'arrow of spacetime'? Because our good old 3-D space is not like a huge static warehouse, in which we would notice some redshifted light from moving objects, receding from us in line with the Hubble Law (some balloon metaphors may be highly misleading). It is the other way around: distant galaxies are not "speeding up" with respect to us in some absolute static space, but the very metric of space is "expanding" (I firmly disapprove of this notion of "expansion", and have suggested the so-called 'relative scale principle').

Hence the space itself is endowed with dynamics, but then we need some ether w.r.t.w. such 'global dynamics of 3-D space' can be formulated. Then the only possibility -- trust me, there is no other option -- is to place the omnipresent ether, as 'the source of the "dark" energy', in the global mode of spacetime. Otherwise you will have to define the dynamics of space w.r.t. itself, and will look like Baron Munchausen. That's why we need an arrow of spacetime, in my opinion (but notice the opinion of Chris Isham above).

NB: If this is the case chosen by Nature, then any approach to quantum gravity, based on the "splitting" of spacetime (Brett Bolen), will necessarily be wrong.

The speculations of Roger Penrose will necessarily be wrong as well: "The fuzzy idea of where and what is infinity was clarified and made more specific by the work of Penrose [45, 46] with the introduction of the conformal compactification (via the rescaling of the metric) of spacetime, whereby infinity was added as a boundary and brought into a finite spacetime region." (Ted Newman et al.)

Perhaps we may have to develop new mathematical theory of 'potential reality', such that the "state space" of Margenau's Onta (quantum presentation of Platonic ideas) would match the structure of cognitive concepts; notice that in the "cheating on 20 questions" the answer 'cloud' was explicated by a Baeysian learning rule (not the Born rule). For comparison, the categorification of Feynman diagrams requires "black boxes with many wires going in and many wires going out" (Baez and Lauda, p. 16), while in our case all wires are "instantaneously" (global mode of spacetime; see the Escher drawing below) keeping track of all virtual 'black boxes' as well (relational ontology), in order to dynamically adjust to the changing context of the game, until they jointly select the final, explicated 'black box': 'cloud' (see also the four dice here).

NB: In the local mode of spacetime, the "duration" of the total negotiation with 'everything else in the universe' (relational ontology) is zero. Hence a wave pattern is being created, without any source of these "waves" being present in the local mode, and a new form of retarded causality (biocausality; see below) can be postulated -- a revitalization of Leibnitz' harmonia praestabilita, Jung's Synchronicity, and Einstein's Überkausalität.

We definitely need mathematical theory of 'potential reality'. The task is highly non-trivial, but once we unravel the correct mathematical theory, the astonishing effectiveness of mathematics (Eugene Wigner) may drive us closer to the true quantum gravity of He Who Does Not Play Dice -- the world is not deterministic but flexible, and the 'chooser' of one possibility (one at a time) amongst infinitely many is 'the whole universe' in its state of ONE.

All I've been getting so far is either dark silence or insults (some of them really harsh).

Perhaps the situation will improve in 2010, after the sixth consecutive failure of LIGO Scientific Collaboration (LSC) to detect GWs with the so-called "enhanced LIGO". GW energy transfer is fundamentally non-linear phenomenon (Hermann Bondi), but is wiped out with the "linearized approximation" adopted by LSC. Also, GW energy is intrinsically quasi-local, in the sense that GWs do not propagate exclusively "in one direction only" (“when the waves are all moving in the same direction”, cf. P.A.M. Dirac, Ch. 33, p. 64), as they also have a holistic global component (atemporal "handshake"), which covers the whole 3-D space en bloc.

LIGO Scientific Collaboration (LSC) stubbornly refuse to acknowledge that there are no bans whatsoever on the dipole radiation, simply because conservation of gravitational mass-energy and momentum, in a world dominated by an evolving cosmological "constant", is a wishful thinking. They fear to even mention the implications of DDE to their project, and have never tried to address the (old) cosmological "constant" problem, which inevitably occurs if you trust the unwarranted assumption that the "dark energy" from the quantum vacuum can only spring from Lorentz-invariant stuff with positive energy density.

Is it possible to detect some 'elementary shift' of the expanding metric -- the "intrinsic time interval associated to any timelike displacement", T. Jacobson, pp. 18-19 -- due to the omnipresent and perfectly smooth DDE? In what reference frame? Notice that you're dealing with some "fluid" that "has zero inertial mass! It can be accelerated with no cost, no effort" (B. Schutz, p. 255) and "provides an all-pervading energy density and negative pressure that are the same to all observers, at all places, and at all times in the history of any universe model, even the expanding ones." (p. 257)

Similar rhetoric questions apply to the dynamics of the metric, producing inflationary gravitational waves on the 3-D "balloon" hypersurface. It's a bundle.

Yet the same kind of waves, only much "weaker", are expected to be detected by LIGO (see below).

Forget it. There is no need for "precise calibration" of a dead turkey.

The insane efforts of LSC remind me of the old joke about a drunken man, who has lost his key somewhere in the dark, but is searching for it under the street lamp, simply because it is brighter there. Only LSC's "key" costs billions. Which is why I accused LSC members of aggressive professional negligence, and offered them to review my White Paper. They responded with dark silence, as usual.

Check out the communist censorship of Paul Ginsparg's "moderators" here.

Since all this points to the unknown dynamics of GR, consider this: similar to the case of particle-wave duality, the splitting the spacetime into two "components" is allowed for educational purposes only. In fact, we are dealing with one object, as stressed by Hermann Minkowski (recall again the elephant's trunk). Hence if you split the spacetime into two "components", and let one of them ('time read by a clock') to "evolve" with respect to the other one (3-D space) -- as Paul Dirac and ADM did in the misfortunate Hamiltonian formulation of GR -- you will end up with a dead frozen snapshot of the arrow of spacetime, in which all GW effects (the positivity of mass and its inertial reaction "force") have already been completed by the "instantaneous" non-linear Machian-type negotiation (global mode of spacetime; see the Escher drawing below) with 'everything else in the universe' (the so-called biocausality).


To explain "already", let me quote from Thomas E. Phipps, Should Mach's Principle be taken seriously? Speculations in Science and Technology, 1(5) 499-508 (1978), p. 504:

"Gravity is a different beast from radiation of any kind. Being mediated by virtual
particles, which may be considered to be kept permanently virtual by the physical
non-existence of gravity shields or absorbers, gravity can act (nonlocally) with infinite speed -- in effect, with precognition. That is exactly what it does, if Mach's principle has any substance. The fixed stars "know" the subway is going to jerk, because they have sent their virtual spies forward in time to find out about it."

The crux of the idea of biocausality is well-known. Back in 1953, Wolfgang Pauli
suggested that the concept of finality ("the end (telos), that for which a thing is done", Aristotle, Physics 194b33) should be considered as a complement to causality in deterministic and statistical laws. The stipulation here is that Aristotle's effective cause & final cause determinate jointly the next state of all material constituents of the universe relationally, in line with the bootstrap principle 'think globally, act locally' (global mode of spacetime; see the Escher drawing below), but in the local mode the resulting biocausality is retarded, along the arrow of spacetime.



From this perspective, detecting GW effects requires "online" access to the global mode of spacetime, in which the dynamical determination of spacelike and timelike directions (hence Lorentzian metric) is being produced -- one-at-a-time, along the arrow of spacetime. In simple words, this means that the proper GW detectors must be endowed with the self-acting faculty of the human brain, to match the "interaction of spacetime with itself" (C. Kiefer, p. 2; cf. also J. G. Pereira et al., arXiv:0909.4408v1 [gr-qc], p. 10, Eq. 7.2).

As of today, nobody cares. Nobody.

I can take it. I'm psychologist, and don't need quantum gravity to practice PHI.

Why would a fish need a bicycle?

D. Chakalov
Thursday, 26 November 2009



[click the image to enter the web site]



Some history of the project outlined above. In January 1972, shortly after my demobilization from BG army (age 19, with the lowest possible rank), I decided to study psychology and "sort out" the physics of the human brain, to explain and eventually explore its amazing latent abilities. It took me sixteen years to realize that the task is unfeasible with the established Weltbild, and on 22 May 1988, at 23:45 local time, I decided to try the project 'the other way around', namely, to develop a model of the universe as a 'brain', such that there will be a natural explanation of the physics of the (small) brain from the outset. After many trials and errors, the first (relatively) encouraging evidence appeared in July 1998, but the project is still far from being completed. I am trying to find any weak points or inconsistencies in it, just as I would do with a brand new parachute, which I would have to put on ultimate test by jumping from a helicopter. I do like the unfolding of the whole project, since it seems to me that all pieces of 'the bridge' snap to their places effortlessly. Yet there is no room for contention, and I never go into the mood of 'chi si contenta gode' (a contented mind is a perpetual feast; he who contents himself, enjoys).

Hopefully, if we join our efforts and knowledge (included at the level of our collective unconsciousness, cf. Jungian Kollektives Unbewusstes), some day we may succeed.

Meanwhile, please keep in mind the prediction of Robert Millikan, Nobel Prize in Physics (1923): "There is no likelihood man can ever tap the power of the atom."

This web site is my feedback to all people, who are helping me, one way or another, with improving the theory. Feel free to download it (app. 11.8MB) from

By unzipping, a new folder, !Einstein_PHI , will be created on your hard drive. Find there START.html.lnk and open it (it is linked to this front page (index.html) in the website folder).

If, for some reason, you believe we don't need new approach to quantum gravity, try to understand Sean Carroll's speculations (448 pages, January 2010):

"... wavefunctions appear to collapse in one direction of time but not the other is not an explanation for the arrow of time, but in fact a consequence of it. The low-entropy early universe was in something close to a pure state, which enabled countless "branchings" as it evolved into the future." Yet he acknowledged: "we can't, once again, define a conserved total energy in any reasonable way." Consequently, the main speculation of S. Carroll about some "low-entropy early universe" is just as unclear as is its gravitational energy which would evolve "into the future". (Also, there may be a critical low geometric entropy state of the 'extremely early' universe, which may blur the timelike and spacelike directions themselves, and turn them into some primordial quantum dough from which nothing could possibly "decohere", ever.) As one of his senior colleagues summarized, "the magnitude of the entropy of the universe as a function of time is a very interesting problem for cosmology, but to suggest that a law of physics depends on it is sheer nonsense."

... Alex Vilenkin's arXiv:0908.0721v1 ($63,000 FQXi Grant):

Alex Vilenkin, $63,000

... Laura Mersini-Houghton's arXiv:0909.2330v1 [gr-qc] ($50,000 FQXi Grant):

"... when treated in a multiverse framework, fundamental time is directionless and consequently physical laws inherit its time-reversal symmetry. Despite that reversal symmetry is broken for the local time by the bubble nucleation, the bubble still inherits laws of physics at birth from the multiverse, without modification. Thus the emergent time’s arrow in the bubble does not affect the time-reversal symmetry imprinted onto the physical laws that the bubble inherits from birth in the multiverse. (...) An emerging time in the multiverse does not appear plausible since the emergence adds information on the multiverse that wasn’t there prior (... but we face the same kind of emergent non-unitary phenomenon with DDE -- an evolving cosmological "constant" that springs from the quantum vacuum - D.C.)."

... and the obstinate belief of Andrei Linde ($164,179 FQXi Grant):

"During the last 25 years a new scientific paradigm gradually emerged. (...) My main goal is to learn how to make scientific predictions in this complicated framework."

Sure enough, Andrei Linde tried to answer the first off question of how many universes are in the "multiverse" (arXiv:0910.1589v2):

Coincidently or not, his calculation matches the number of angels (mostly blond) that can fit on the head of a pin, as suggested by A. Linde's medieval colleagues.

If these people were here in the 21st century, I suppose they would gladly join LIGO Scientific Collaboration (see above), and speculate that, shortly after the "big bang", the whole universe was still small enough to fit on the head of a pin, with strong ripples of the spacetime metric (see their picture above). They would love to feel again like an 'absolute observer' with an absolute clock paired with an absolute measuring rod, to tell the dimensions of the universe and its current cosmological age, as read by their absolute wristwatch. But instead of arguing about angels (either blond or not), they would certainly prefer to measure the "ripples of the metric" (albeit very weak), which again can only be seen from the same standpoint of 'absolute observer'. Needless to say, they will easily publish tons of articles on "GW astronomy", and will ignore any alternative viewpoint on what can be "seen" by such absolute observer: nothing but "gauge-dependent" stuff.

That's how 'potential reality' is being camouflaged in present-day GR, simply because if it were possible to detect the source of DDE as 'gauge invariant observable', the omnipresent and perfectly smooth ether (the source of DDE) will be exposed to direct physical observations, as 'spacetime acting upon itself'.

Back in April 1986, Yakov Zel'dovich wrote in a letter the following (private communication): "Long time ago, there was a period of time during which there was still no time at all." Of course he was joking.

Yet the 'global mode of time', pertaining to potential reality, cannot be read by any physical clock, because the poor inanimate clock will read it as 'stand still' or "no time at all", as Yakov Zel'dovich put it. Maybe The Beginning, which lives in "no time at all", is always with us (dual age cosmology).

Perhaps in the next twenty-five years a new paradigm will emerge: the universe modeled as a brain. The driving force of its arrow of spacetime cannot spring entirely and exclusively from the "brain" itself, which brings us to Virgil's statement: Mens agitat molem (The Aeneid, Ch. 6, 727).

In German, it reads: Der Geist bewegt die Materie. Physically, Der Geist may look like 'the ideal monad without windows'. However, no scientific predictions can be made about it, or else we will conflate religion with science. Thank God, this is impossible.

D. Chakalov
October 31, 2009


Subject: "best of all possible worlds."
Date: Fri, 8 Oct 2010 21:09:20 +0300
From: Dimi Chakalov <>
To: Robin G Jordan <>

Dear Professor Jordan,

It was a great pleasure to read your essay on Newton vs. Leibniz,

"Leibniz thought the idea of God as an astronomical maintenance man as absurd. He believed that God had carefully chosen among an infinity of possible worlds, the one He felt the most suitable. So that although we may not have a perfect world, it was the

"best of all possible worlds."

I intend, Deo volente, to talk on a similar subject on 25 November 2015,

My web site isn't encrypted (like Newton's 6accdae13eff7i3l9n4o4qrr4s8t12ux), and if you have some spare time, I will highly appreciate your comments on my efforts.

Kindest regards,

Dimi Chakalov


Subject: New limit on the mass of Father Christmas' beard (a.k.a. WIMPs)
Date: Fri, 8 Oct 2010 21:55:02 +0300
From: Dimi Chakalov <>
To: Joe Silk <>,
Cc: M Angeles Perez-Garcia <>,
Jirina R Stone <>,,,,,,,,,,,,,,,

Dear Joe,

I am really surprised that you took part in the calculations of a new limit of WIMPs (arXiv:1007.1421v2; Phys. Rev. Lett. 105 (2010) 141101).

May I use this opportunity to invite you and your colleagues to my talk on quantum gravity,

All the best,

How do we know that Father Christmas has a beard?
We know it, because snow falls when he shakes his beard.

Old Tanzanian saying


Subject: Spherical cows
Date: Wed, 12 May 2010 14:42:46 +0300
From: Dimi Chakalov <>
To: Richard Gray <>
Cc: Jim Hough <>,
Sheila Rowan <>,
Ralph Cordey <>,
Keith Mason <>,
Bcc: [snip]

RE: Largest scientific instrument ever built to prove Einstein's theory of general relativity, by Richard Gray. The Daily Telegraph, 8:30 AM BST, 09 May 2010,

Dear Mr. Gray,

I trust you are familiar with the anecdotal story about a 'spherical cow',

Suppose someone claims that *the real cows are indeed round*, because cows might be approximated as spherical objects, and then ask your government to allocate a significant portion from your taxes for detecting the unique pattern of 'real spherical cows'.

Likewise, you were told by a number of people that, after applying their spherical-cow approximation to Einstein's theory of general relativity, they might eventually detect gravitational waves (GWs): "we haven't been able to detect them yet because they are very weak" (Jim Hough).

However, their persistent optimism is rooted on artifacts due to their spherical-cow (=linearized) approximation of GR,

I fact, they ignore all problems due to their approximation. For example, Hermann Weyl proved in 1944 that such spherical-cow approximation implies the existence of a tensor that, except for the trivial case of being precisely zero, does not otherwise exist,

Regarding Ralph Cordey at Astrium UK and the so-called LISA Pathfinder: How much this spherical cow will cost to UK taxpayers, I wonder.

Yours sincerely,

Dimi Chakalov
35 Sutherland St

Note: The failures to detect GWs were "explained" by Jim Hough with a very misleading statement: "we haven't been able to detect them yet because they are very weak". In fact, GWs are immensely powerful phenomena, but nobody -- Jim Hough and Sheila Rowan included -- can offer a non-linear theory of GWs. All they can do is to imagine that, by the time GWs reach LIGO or LISA, they will be "very weak", such that their spherical-cow approximation to Einstein's GR would be correct. But again, they don't have any non-linear theory of strong GWs, from
which some "weak limit" can be derived.

All they do is asking for more taxpayers' money for detecting spherical cows, instead of doing their homework first on paper, to demonstrate such "weak limit" to initially strong GWs. Pity nobody cares.

May 12, 2010




Subject: arXiv:1005.1614v1 [gr-qc]
Date: Tue, 11 May 2010 05:14:25 +0300
From: Dimi Chakalov <>
To: Robert Geroch <>
Cc: Alexander Vlasov <>,

Robert Geroch, Faster Than Light? arXiv:1005.1614v1 [gr-qc]

R. Geroch: "I am not sure that this is the right perspective — or even whether “right” makes much sense in this context."

Bob, I think you can have your cake and eat it -- the key word is 'quasi-local',

You and your Russian colleague are so good in math ...


To explain quasi-local, and 'what is going on' in QM, check out Feynman [Ref. 1, 12-1]: 

"The first question we have to answer is: What are the base states for the system? Now the question has been put incorrectly. There is no such thing as “the” base states, because, of course, the set of base states you may choose is not unique. New sets can always be made out of linear combinations of the old. There are always many choices for the base states, and among them, any choice is equally legitimate. So the question is not what is the base set, but what could a base set be? We can choose any one we wish for our own convenience. It is usually best to start with a base set which is physically the clearest. It may not be the solution to any problem, or may not have any direct importance, but it will generally make it easier to understand what is going on."

There is such thing as “the” base state (never in plural) -- the UNdecidable KS state, as explained below.

For example, in the case of two spin-half particles, everything you insert in brakets

|1> = |++>, |2> = |+->, |3> = |-+>, |4> = |-->

... are just possible physical manifestations of  “the”  base state, much like the three sayings above, emanating from their UNspeakable potential-reality state of your brain.

Thus, the system {“the” base state & |whatever>} evolves along the arrow of spacetime in a strictly quasi-local fashion: see Fig. 1 and Fig. 2 below.

Feynman also stressed [Ref. 1, 12-2]: "That’s the question: How do the amplitudes change with time in a particular (fixed) base?"

The amplitudes change along the global mode of time: see again Fig. 2 below. Don't be befuddled by the anti-relativistic "time parameter" in the Schrödinger equation, because you need “the” base state to eliminate the measurement "problem" in QM and reconcile QM with STR, as well as solve the puzzle of quantum vacuum and gravitation: “the” base state does not gravitate.

It would be nice if Robert Geroch writes up a sequel to his 30-year old book General Relativity from A to B, entitled "General Relativity from A to A+ds", to elucidate the concept of 'interval' in GR -- professionally. But he wouldn't. Trying to discover new math is a tough challenge.

The last time I heard from Robert Geroch was eight years ago, only to require his email to be removed from my web site. I will gladly do that, if only he writes a serious paper on GR, or at least reply professionally. Here are two questions:

Do you believe that the "points" from the underlying manifold can be connected only and exclusively only by their physical content that is invariant under "active" diffeomorphisms? If your answer is 'no', what could be “the” base state in GR (a.k.a. the reference fluid of GR), which binds the "points" by one single unique bare UNdecidable matrix?

If your answer to the first question is 'yes', you are ready to teach GR and enjoy its generic pathologies [Refs 2 and 3].

It is generally believed that (i) one can picture the spacetime in GR as a manifold that can be "locally modeled" on some fictitious flat Minkowski space, but (ii) this picture should break down at short distances of the order of the Planck length. Neither of these ideas are needed, however. There is no need for any limitation in the possible accuracy of localization of spacetime events either. Why is that? Because the so-called Planck length may possess an inner geometrical structure.

Ignore it at your peril.

May 12, 2010
Last update: May 14, 2010

[Ref. 1] Feynman Lectures on Physics. Volume III : Quantum Mechanics, Addison-Wesley, Reading, MA, 1965; ISBN: 9780201020144

[Ref. 2] José M.M. Senovilla, Singularity Theorems in General Relativity: Achievements and Open Questions, arXiv:physics/0605007v1

"The problem of the definition of the concept of singularity in General Relativity is very difficult indeed, as can be appreciated by reading on its historical development (Hawking and Ellis, 1973; Tipler, Clarke and Ellis, 1980). The intuitive ideas are clear: if any physical or geometrical quantity blows up, this signals a singularity. However, there are problems of two kinds:

• the singular points, by definition, do not belong to the space-time which is only constituted by regular points. Therefore, one cannot say, in principle, “when” or “where” is the singularity.

• characterizing the singularities is also difficult, because the divergences (say) of the curvature tensor can depend on a bad choice of basis, and even if one uses only curvature invariants, independent of the bases, it can happen that all of them vanish and still there are singularities.

"The second point is a genuine property of Lorentzian geometry, that is, of the existence of one axis of time of a different nature to the space axes.

"All in all, it seems reasonable to diagnose the existence of singularities whenever there are particles (be them real or hypothetical) which go to, or respectively come from, them and disappear unexpectedly or, respectively, subito come to existence.

"And this is the basic definition of singularity (Geroch, 1968; Hawking and Ellis,
1973), the existence of incomplete and inextensible curves. That is to say,
curves which cannot be extended in a regular manner within the space-time
and do not take all possible values of their canonical parameter.

"Singularities in the above sense clearly reach, or come from, the edge of space-time. This is some kind of boundary, or margin, which is not part of the space-time but that, somehow, it is accessible from within it."

[Ref. 3] Lars Andersson, The global existence problem in general relativity, arXiv:gr-qc/9911032v4

Footnote 1: "All manifolds are assumed to be Hausdorff, second countable and C∞ (maximal differentiable atlas, cf. Michael Spivak, Vol. 1, Ch. 2 - D.C.), and all fields are assumed to be C∞ unless otherwise stated."

Id., Notes on Differential Geometry, 1, p. 8:

"A differentiable manifold is a topological manifold M together with a differentable structure, i.e. a way of defining differentiable functions on M. The natural way of doing this (forget about this "natural way" - D.C.) is to use the charts [X] to transfer the definition of differentiable functions from Rn to M."



Subject: The global existence problem in general relativity,
Date: Wed, 6 Apr 2011 03:54:55 +0300
From: Dimi Chakalov <>
To: Lars Andersson <>

Dear Dr. Andersson,

I suppose if you (i) do not assume upfront some space-like Killing fields, (ii) don't confine your analysis to some hypothetical vacuum spacetimes, and (iii) wish to avoid poetry [Ref. 1], you may have to start from scratch,

Perhaps the tasks are strictly mathematical.

Yours sincerely,

Dimi Chakalov
[Ref. 1] José M.M. Senovilla, Singularity Theorems in General Relativity: Achievements and Open Questions, arXiv:physics/0605007v1

p. 6: "This is some kind of boundary, or margin, which is not part of the space-time but that, somehow (Sic! - D.C.), it is accessible from within it."



Subject: Positive-mass conjecture in the case of "more and more space appears" ?
Date: Wed, 2 Jun 2010 05:28:23 +0300
From: Dimi Chakalov <>
To: Shing-Tung Yau <>,
Richard M Schoen <>,
Niall Ó Murchadha <>,
Claus Gerhardt <>,
Adam Helfer <>

Dear colleagues,

As of April 2010, “over 2250 papers with the words ‘dark energy’ in the title have appeared on the archives since 1998, and nearly 1750 with the words ‘cosmological constant’ have appeared” (Shinji Tsujikawa, arXiv:1004.1493v1, p. 39).

I believe Michal Chodorowski explained, in arXiv:astro-ph/0610590v3, the meaning of "more and more space appears",

It is totally unclear to me how one could define an isolated system in GR, in which "more and more space appears", to start thinking about some new positive-mass conjecture and the total energy in such "isolated system". Perhaps if one can demonstrate that _no_ asymptotically flat spacetime can be stable under the conditions of such (accelerated or not) "flux" of "more space", we will face a paradoxical situation similar to the ultraviolet catastrophe of late 19th century, after which some bright mathematician will sort out this whole mess.

Please advise.

With kindest regards and admiration,

Dimi Chakalov


Re: Positive-mass conjecture in the case of "more and more space appears" ?
Date: Mon, 25 Oct 2010 15:11:29 +0300
From: Dimi Chakalov <>
To: Shing-Tung Yau <>,
Shing-Tung Yau <>
Cc: Chiu-Chu Melissa Liu <>,
Mu-Tao Wang <>,
Xiao Zhang <>,
Lau Loi So <>,
Hsin Chen <>,
Fei-Hung Hoa <>,
Chih-Hung Wang <>,
Hwei-Jang Yo <>

Dear Professor Yau,

I hope my email from Wed, 2 Jun 2010 05:28:23 +0300 has been safely received.

I believe some young and hungry grad student might crack the puzzle of quasi-local mass,

If you and/or some of your colleagues know such person, please pass her/him the link above. The task is highly non-trivial and strictly mathematical, and also requires guidance from Shao Yong.

I also believe the future of new energy sources belongs to your country, but if you wish to extract energy from 3-D space, you should always follow the advice of Shao Yong. Since I haven't heard from you so far, it seems the latter turned out to be too difficult for you.

Yours sincerely,

Dimi Chakalov

Examine the objects as they are and you will see their true nature; look at them from your own ego and you will see only your feelings; because nature is neutral, while your feelings are only prejudice and obscurity.

Shao Yong, 1011-1077

Fooling around with alternative current is juts a waste of time.
Nobody will use it, ever.

Thomas Alva Edison, 1889



Subject: arXiv:1107.1374v3 [math-ph] and arXiv:0912.2886v4 [math-ph]
Date: Fri, 29 Jul 2011 17:37:03 +0300
From: Dimi Chakalov <>
To: Bert Schroer <>,
Cc: Carlos Perelman <>,
Robert M Wald <>,
Stefan Hollands <>

Dear Bert,

You consider Haag's 1957 idea "of interpreting the spatial extend of a measuring device and the duration of its activation as an observable localized in the corresponding spacetime region fulfilling Einstein causality and an appropriately formulated causal propagation was (and still is) metaphoric if not to say naive" (arXiv:1107.1374v3 [math-ph], p. 8). His latest views are summarizer in [Ref. 1].

There is no such thing as "local covariance principle" (arXiv:0912.2886v4 [math-ph], Sec. 6 and p. 32).

Regarding the nonobservance of the holistic aspects of QFT (arXiv:1107.1374v3 [math-ph]), see a quasi-local approach to GR & QM at

I think the opposition to such proposals is not related to our current Zeitgeist. People just don't care. They live in total socialism and love to play with their hobbies.

Names? They're all at my web site.

All the best,


[Ref. 1] Rudolf Haag, Questions in quantum physics: a personal view,

"In simple cases the event may be regarded as the interaction process between a particle and a detector. But the notion of ‘particle’ does not correspond to that of an object existing in any ontological sense. It relates to the simplest type of global state and describes possibilities, not facts. The notion of ‘partial state’ demands in addition that we ignore all possible events outside some chosen region and thus ignore possible correlations with outside events."



Subject: 260037 PR Spezialisierungsmodul Mathematische Physik und Gravitationsphysik
Date: Sun, 26 Aug 2012 17:45:27 +0300
From: Dimi Chakalov <>

Dear colleagues,

I learned that the Spezialisierungsmodul offers "guidance to scientific research on open problems in mathematical physics and general relativity", which I hope to discuss with you during the ESI Workshop in December this year.

Meanwhile, please check out the Ansatz for 'necessary and sufficient conditions for spacetime', which can (hopefully) eliminate 'dieser verdammten Quantenspringerei',

As I mentioned in my email to Dr. Robert Beig (Sun, 8 Apr 2012 20:04:57 +0300), the task is strictly mathematical. I will be happy to explain it under the roof of Erwin Schrödinger Institute.

Kind regards,

Dimi Chakalov
Wenn es doch bei dieser verdammten Quantenspringerei bleiben soll, dann bedauere ich, mich mit der Quantentheorie überhaupt beschäftigt zu haben.
Erwin Schrödinger




Printable version is available from here; mathematical details here.




Space inversion

Finite 3-D space (depicted with circle) obtained by snapping the
inversion of points along the  w  axis (sphere-torus transitions, Fig. 5).
Space becomes both "infinite" (local mode of spacetime) and bounded
from  S  and  L  (global mode of spacetime).




To introduce "boundaries" on the physical spacetime at all (timelike, spacelike, and null) directions, I will consider an ideal dimensionless point of 'pure geometry', and will postulate a structure of such point: a dual presentation of 'the universe as ONE', interpreted along an axis (w) as both 'an infinitesimal' (S) and 'arbitrarily large volume of 3-D space' (L). Then I will suggest topological transformations of points (called 'space inversion') in an infinite-dimensional Euclidean space to obtain a 'collapse over infinity'-reduction of this unphysical Euclidean space to 'asymptotically flat 4-D spacetime' endowed with "boundaries" placed at the dual object, 'the universe as ONE'. The spacetime of 'the universe as ONE' is called 'global mode of spacetime', with equation  L = S = 1  (Eq. 2), while the equation of the Arrow of Space, generating perpetual emergence and re-creation of an asymptotically flat spacetime ('collapse over infinity'), is postulated as  LS = 1  (Eq. 1), in line with Virgil's dictum Mens agitat molem or Der Geist bewegt die Materie.

I will also outline the so-called 'relative scale principle' (RSP) aimed at removing an absolute structure of space known as 'size of objects': nobody has 'the right meter'. Relative to an observer placed at the macroscopic length scale (the "middle" of  w ), objects in 3-D space would look like "large" in the direction toward  L  and "small" in the opposite direction toward  S , while a co-moving observer will not notice any chance in her 'one meter and one second', and will always experience the same "speed of light". Since 'space' is interpreted as an emergent phenomenon along the Arrow of Space, I will introduce finite templates for 'size of objects', and will argue that their scale-dependent alteration resolves the paradox of the (accelerated) "expansion" of space toward  L  and the (non-accelerated) "shrinking" of space toward  S , as seen by an observer placed at the macroscopic length scale (the "middle" of  w ), while their local alteration recovers the correct geometrical manifestation of gravity (not "curvature"). Hence one can eliminate all "dark" effects of gravity such as "black holes", "cold dark matter", and "dynamical dark energy", and amend Einstein's General Relativity with the "
total field of as yet unknown structure" from the Arrow of Space.



The quest for Finite Infinity has a long history, starting from Gunnar Nordström (Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereiningen, Phys. Z. 15 (1914) 504-506). It is an age-old problem of General Relativity. Nothing could be more important than understanding the topological manifold of the Universe, and its dynamics.

Traditionally, experts in GR start with what I hope to derive at the end of this project: asymptotically flat spacetime with quasi-local positive mass. For example, Rick Schoen would presuppose a "smooth manifold", which has already been equipped with a "Lorentz signature metric", "asymptotic flatness", and "appropriate falloff" conditions, and then ask question like: 'why do we see positive mass only' ? A short answer: because we have an Arrow of Space. The detailed answer requires careful analysis of all initial presumptions in Rick Schoen's talk and in present-day GR textbooks.


A New Mean Curvature Proof of the Spacetime Positive Mass Theorem
By Richard Schoen, November 13, 2011, at
Celebrating Jim Isenberg's 60th birthday

Pacific Northwest Geometry Seminar, Corvallis, OR, November 12-13, 2011

A rigorous definition of 'isolated gravitating system', which would ensure an "asymptotic regime such that all gravitational effects are localized inside of it" (Adam Helfer) and proper boundary conditions, is still an unresolved task: there are no physically motivated boundary conditions in the case of the Einstein equations; for example, "we do not know how to build a mirror for gravitational waves" (A. Rendall), nor can we resolve the paradox of geodesic incompleteness and "black holes". Moreover, how can we define an 'isolated gravitating system' and its (obviously) positive mass if the space itself is endowed with a new, dynamical "dark" energy? The calculating machinery of ADM, suggested half a century ago, doesn't work anymore. Enter the Finite Infinity (FI).

In a nutshell, the idea of FI is to suggest a mechanism for obtaining a finite volume of Archimedean 3-D space. The very notion of 'finite 3-D space' implies the existence of two distinguishable volumes of 3-D space, separated by a "trapped" surface (cf. lion's cage below), such that we can always define the notions of 'inside vs outside' (hence "large" vs "small") and 'left vs right'. (In order to eliminate the absolute structure of 'size', we will introduce later the so-called Relative Scale Principle, RSP.) Now, how can we introduce some process and mechanism by which 'finite space' can be fixed at all length scales, in such way that 'physical space' will never actually reach zero nor infinity but will always remain 'finite' ? We will use an infinite-dimensional Euclidean space and will introduce smooth sphere-torus conversions in it, along a new axis  w  , such that these sphere-torus conversions snap and fix all Archimedean volumes of 3-D space from both "below" and "above" (cf. Fig. 5). Hence space becomes both "infinite" (local mode of spacetime) and bounded from  S  and  L  (global mode of spacetime). The latter is physically unobservable, because we're stuck in the Archimedean 3-D space and don't have access to 'actual infinity'. Only Chuck Norris has been there (twice).

As in a good crime novel, all will become crystal clear at the end. Well, eventually.

Some history. The notion of 'finite infinity' (Fi) was suggested by George F R Ellis in 1984; please see:

George F R Ellis, gr-qc/0102017v1, Sec. 5, "Finite Infinity and Local Physics",

"This led me some years ago to ask the question: ‘How far away is an effective ‘infinity’ to use in discussing boundary conditions for local physical systems of this kind?’ (...) Then incoming and outgoing radiation conditions can be imposed on that surface F, rather than at infinity or conformal infinity I as is usual [57]. (...) Furthermore the famous positive mass theorems [64] should also be generalized to this case.
"This may also be the best setting for numerical calculations for ‘isolated systems’, which often talk about ‘integrating to infinity’, but in most cases do nothing of the sort. As in the rest of theoretical physics, it would be advantageous to have a theoretical framework that corresponds more closely to actual calculations - namely an integration to a surface at a finite distance from the centre of coordinates. It is usual to make that surface a null surface; the suggestion here is that it would be better to make it timelike, corresponding to the region in the real universe where the exterior is physically separated from the local system.

"So the obvious proposal [54] is that we should put boundary conditions on all fields at that distance, rather than at infinity itself, leading to the concept of a 'finite infinity' FI ... "
[54] Ellis G F R, Relativistic Cosmology: Its Nature, Aims and Problems, in: General Relativity and Gravitation, Ed. B. Bertotti et al., Reidel, Dordrecht, 1984, pp. 215-288; cf. Sec. 5.2 and Fig. 11(c).

See also: Ehlers J (Ed) (1979). Isolated Gravitating Systems in General Relativity. Proc Int School Enrico Fermi Course LVII (Academic Press, New York).


By using exclusively the notion of 'potential infinity', Stephen Leacock posed the dilemma of 'infinite space' in the following fashion:

"We cannot imagine that the stars go on forever. It’s unthinkable. But we equally cannot imagine that they come to a stop and that beyond them is nothing, and then more nothing. Unending nothing is as incomprehensible as unending something."

In order to resolve the conundrum of 'ending something', notice that the nature of the local Archimedean mode of spacetime (cf. below) is determined by 'potential infinity': every step toward the Finite Infinity provides the necessary and sufficient condition for the next step, just as in the Thompson's lamp paradox. If we go in the "direction" toward 'the infinitesimal' (S), one can (with some luck) work out a cutoff and end-point, but it will belong, again and always, to the local ("colored") mode of spacetime (see the story about 'John's jackets' below).

If we wish to work out a proper Finite Infinity, the obvious choice is to introduce a new 'cutoff and end-point' conjugated with the infinitesimal,  S , and to employ the two forms of 'infinity': potential and actual. The latter is 'already completed', in the sense that (i) it does not entail any "dynamics" that can be recorded with a physical clock, and (ii) applies only to the global non-Archimedean mode of spacetime.

We begin with postulating an uncountably infinite "number" of points packed in any finite -- arbitrarily "large" or arbitrarily "small" -- line segment, plane, or volume of space in the local (physical) mode of spacetime, in such way that 'there is nothing ]between[ these points'. Then we will "insert" the global mode of spacetime ]between[ the points from the local mode, and will make sure that the global mode is non-existent in the local mode, with the sole exception of the instant 'now' from the Arrow of Space, in which the two modes of spacetime coincide (cf. Fig. 1 below).

Notice the crucial difference between the two modes of spacetime: in the local mode, a test particle equipped with a clock reading its proper time will need a finite Archimedean time interval to pass through a finite Archimedean volume of space, even if the test particle travels with the speed of light. It cannot pass through a finite volume of space for 'zero time', even if the volume of space is 'tending asymptotically toward zero', as in the case of the empty set (R). Thus, it will always need 'more Archimedean time' to pass through all uncountably infinite points from the perfectly smooth continuum of the local (physical) mode of spacetime. This is the essence of 'potential infinity', and it is valid only for the local mode of spacetime.

Hence the interpretation of 'the infinitesimal' as the empty set (R): in the local mode of spacetime, it is and will always remain a finite entity, tending asymptotically toward zero, being comprised from uncountably infinite points. In the global mode, however, it is a purely geometrical, dimensionless point -- just one single geometrical point (Fig. 1).

NB: This one single geometrical point cannot be reach from/within the local (physical) mode of spacetime, for any finite duration of time, as read with a physical clock. Why? Because its physical content (jacket) is UNdecidable, after the Thompson Lamp paradox.

Thus, the empty set (R) is endowed with a structure, as hinted in Fig. 1 below, and with 'space inversion' (Fig. 1.1). It is the flashmob for the two modes of spacetime, at which they "coincide". It is also the instant 'now' (now-at-a-distance) from the Arrow of Space. And thanks to the so-called "speed of light", it has zero duration and zero spatial extension.

Compared to the local mode, the global mode of spacetime is ontologically different: it is a non-Archimedean realm of 'potential reality' (GPIs), which can traverse any finite -- arbitrarily "large" or arbitrarily "small" -- line segment, plane, or volume of space from the local (physical) mode of spacetime for zero time, as it would have been read by the clock attached to a test particle from the local mode. Namely, the global mode is equipped with 'actual/completed infinity': it can traverse the uncountably infinite points of any finite object for zero time (as it would have been read by the clock attached to the test particle from the local mode), and endows the local mode with a web of "instantaneous" correlations (as they would have been read by an inanimate clock from the local mode). With the Arrow of Space, such kinematics gives rise to a new dynamics resembling a living organism: a 'school of fish'-type bootstrapping of physical systems in their quantum and gravitational regimes, and quasi-local geodesics, in line with the rule 'think globally act locally' (hence one can introduce a background-free, relational reality and Machian quantum gravity).

The question of what kind of 'global time' runs in the global mode of spacetime, equipped with "instantaneous" actual infinity, can be answered by explaining its corresponding 'global space' and the non-Archimedean structure of the purely geometrical GPIs in it.

In general, the quest for amending Finite Infinity with an Arrow of Space is highly non-trivial, because we should also consider the following tasks:

1. The two modes of spacetime are separated by the fleeting instant 'now' from the Arrow of Space, namely, the global mode is placed in the realm of 'potential reality' (GPIs residing in the potential future in the Arrow of Space), while the local mode refers to the irreversible world of explicated GPIs in terms of facts, placed in the steadily increasing 'irreversible past' from the Arrow of Space. Such 'ever-increasing past' is what makes an 'arrow', due to irreversible 'information gain': every instant 'now' pertains to an explicated physical universe, which contains more information that its immediate predecessor in the 'ever increasing past', and at the same instant 'now' the physical universe is offered an enriched spectrum of potential states to choose from for its next instant 'now', just as in the cognitive cycle of Ulric Neisser (Fig. 2 and Ch. 2 and 4).

Metaphorically speaking, the Arrow of Space is depicted with the Dragon devouring its tail (Ouroboros), from the Chrysopoeia ('Gold Making') of Cleopatra during the Alexandrian Period in Egypt. The enclosed words mean 'The All is One.'

The ultimate source for such information gain is 'the true monad without windows' which remains at absolute rest within the instant 'now' (see below), depicted with Fig. 1 below.

1.1. The only "meeting point" of the two modes of spacetime is the instant 'now', in which they coincide (Fig. 1). The result is an already completed and already negotiated physical universe, spanned across the absolute instant 'now' from the Arrow of Space, with 3-D space and zero "thickness" along  w  (cf. below), in which “there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields” (Sean Carroll); check out the Photoshop layers metaphor below and the resulting non-linear dynamics here.

2. The unique "boundaries" on 3-D space in all (timelike, null, and spacelike) directions, introduced with Finite Infinity, should make the universe an 'isolated system', which (i) contains 'absolutely everything', included its Aristotelian First Cause, (ii) is self-enclosed (Albert Einstein), and (iii) is "bounded" by some ambient unphysical spacetime (called here 'global mode of spacetime') which is part and parcel from the same 'isolated system'.

2.1. To describe such 'universe as ONE' -- self-wrapped (cf. (iii) above) and endowed with the faculty of 'self-acting' due to its "gravitational field" -- one needs to place its source "inside" the instant 'now' from the Arrow of Space (Fig. 1), and make sure that the so-called "dark" energy of 'the universe as ONE' is unobservable in principle.

3. Last but not least, the ultimate source of 'the universe as ONE' should be interpreted as 'zero nothing', that is, a special kind of "zero" opposite to the mundane case of 'zero something'. Perhaps the only way we could grasp the notion of 'zero nothing' or 'the true monad without windows' is by recalling the relation between the content and volume of concepts: the larger the volume, the smaller the content; hence the source of 'absolutely everything, the unknown unknown included' should possess zero intrinsic content (cf. the undefinable matrix).

I haven't been able to find suitable mathematical formalism to describe these widely known ideas. All I can offer is a simple (but incomplete) geometrical explanation of Finite Infinity.

Firstly, there should exist a maximal volume of 3-D space (L), at which we place the Finite Infinity (FI), such that any finite volume of space, no matter how large, is identified as an Archimedean sub-volume smaller than the 'maximal space volume',  L . Likewise, there should exist a minimal volume of 3-D space (S), at which we place the same Finite Infinity (FI), such that any finite volume of space, no matter how small, is identified as an Archimedean volume larger than 'the minimal space volume',  S .

Secondly, the two physical, finite, Archimedean volumes of space tend asymptotically toward  L  and  S  in the local (physical) mode of spacetime, but cannot reach them for any finite Archimedean duration of time due to the "structure" of the empty set (R) at which the global mode and the local mode coincide (Fig. 1).

The 'no-go' axiom about the empty set (R) ("cannot reach them for any Archimedean duration of time") stems from our belief (not 'fact') that there exist, in the local mode of spacetime, an 'uncountably infinite' "number" of points between any finite Archimedean volumes of space and their "two" cutoffs, L and  S : the empty set (R) cannot be bridged by any finite Archimedean system for any finite duration of time, even if it jumps over the points with the "speed of light". That is, in the local mode of spacetime the empty set (R) is 'finite' entity, and will always occupy a finite, albeit "very small", volume of space packed with 'uncountably infinite' points arranged by 'potential infinity'. On the other hand, the same empty set (R) is consumed/traversed instantaneously in the global mode of spacetime, which has non-Archimedean nature and is endowed with 'actual/completed' infinity.

Thus, we shall place the "two" cutoffs, L and  S , within the red point in Fig. 1 below, stressing again its UNdecidable nature (Thompson Lamp paradox) of 'potential reality': pre-quantum non-colorizable General Platonic Ideas (GPIs).

FI Postulate: Due to the Thompson Lamp paradox, any finite, arbitrarily "large" or "small", Archimedean volume cannot physically reach  L  and  S  residing in the global mode of spacetime, equipped with actual infinity.

A precise explanation of 'physically reach': in the local (physical) mode of spacetime, we always reach/nullify the empty set (R) at the point of '2 min', which is why the state of Thompson's lamp is always defined by the rules of 'bartenders', but it is just a fleeting "jacket" cast by John from the global mode of spacetime. The latter is endowed with actual infinity. In other words, in the global mode of spacetime the interval [0, 2] is closed, while in the local mode the same interval is open -- (0, 2).

Thanks to the Arrow of Space, the 'large finite volume' will chase  L  in the local mode indefinitely; in the local mode of spacetime the empty set (R) can only tend asymptotically toward zero. The same holds for the opposite case of the separation of any arbitrarily small Archimedean volume of space from  S  by the same empty set (R): in the local mode, it would require an infinite -- actual infinity -- amount of time for the elimination of the empty set (R) and reducing it to zero, at which point the Arrow of Space would stop, and the universe would cease to exist.

Notice that the FI Postulate introduces new structure of spacetime at the level of 'differentiable manifold' prior to any matter; example here.

The very notion of 'space' requires that all physical objects acquire finite Archimedean size; a table with length 'one meter' fills in a template for 'one meter'. In the case of an one-meter template, we observe a finite table with length one meter, which is again a 'sub-volume of finite space'. The purpose of Finite Infinity (FI) is to define the largest template and the smallest template for all finite volumes of 3-D space.

The maximal volume of 3-D spaceL , and minimal volume of 3-D spaceS , belong to the global mode of spacetime for which the actual infinity holds ('already completed'), and are related by

LS = 1  (Eq. 1).

Thanks to the empty set (R), no physical, Archimedean stuff can 'physically reach'  L  and hence go "beyond"  L  , and also no physical stuff can 'physically reach'  S  and hence go "below"  S  (compared it to the conformal recipe): L & S  is ONE object which belongs to the non-Archimedean global mode of spacetime.

In a nutshell, the modified Finite Infinity satisfied two conditions: in the local mode of spacetime the universe approaches asymptotically its causal boundary (cf. Eq. 1 above) within an open interval, while at the same time the universe is being (present continuous) permanently wrapped by itself in the global mode of spacetime. Hence 3-D space can be literally wrapped by itself, exactly as Albert Einstein required.

Again, there is no path whatsoever to the global mode of spacetime from the local mode of spacetime; hence the FI Postulate above. The difference between  S  and  L  is that in the case of the former we can find some 'numerically finite but physically unattainable boundary/cut off' (the Planck length), but for  L  we have only an eternally expanding 'sub-volume of finite space', which can only chase  L  but can never physically reach it: L  and  S  belong to the non-Archimedean global mode of spacetime.

It is the Arrow of Space which creates such asymmetry in the treatment of   L  and  S . We have a numerically finite but physically unattainable "bottom", while space "expands" in the local mode toward  L  indefinitely: at each and every instant 'now' from the Arrow of Space, the universe occupies a finite sub-volume, being literally wrapped from both "below" and "above" with Eq. 1 above. Depending on the direction we look at the edge of the universe in the local mode of spacetime, either toward  L  or toward  S , we see "two" edges, while in the global mode they are ONE -- the universe as ONE, as stressed by Lucretius some 2060 years ago. We don't have such structure of spacetime in differential geometry textbooks.

Notice that we face here a kind of 'logical infinity': no physical, Archimedean volume of space can reach  L , because  L & S  belong to the 'the universe as ONE' (Eq. 1 above). In other words, one could logically reach something only if this "something" is 'not yet reached', while in our case it is logically impossible to "reach" something (L) which is being eternally residing "inside" us (S) from the outset. Thus, the notion of 'logical infinity' refers to the statement that if the Arrow of Space is terminated, the universe will cease to exist and will return to its initial state of pure light and cognition, known as [John 1:1].

Without the global mode of spacetime, the universe would be governed exclusively by Archimedean geometry, conformal recipes would have worked, tessellation of space with 3-D "tiles" would have been possible, and we would calculate the exact finite number of "atoms" filling a finite volume of space without any gaps from the empty set (R): we would hit an "end-point" beyond which "is nothing, and then more nothing" (S. Leacock). Thank God, this is impossible.

Again, we cannot physically "see"  L & S , because we don't have access to 'actual infinity'. Only Chuck Norris has been there (twice).

Now, let me try to explain geometrically the empty set (R), which would