|Subject: The history of the cosmological constant problem
Date: Wed, 14 Aug 2002 18:59:58 +0300
From: Dimi Chakalov <firstname.lastname@example.org>
To: Norbert Straumann <email@example.com>
CC: "R. Tumulka" <firstname.lastname@example.org>,
Shelly Goldstein <email@example.com>,
Chris Isham <firstname.lastname@example.org>
Dear Professor Straumann,
I quoted your recent article "The history of the cosmological constant problem", gr-qc/0208027, at
You wrote: "Most probably, we will only have a satisfactory answer once we shall have a theory which successfully combines the concepts and laws of general relativity about gravity and spacetime structure with those of quantum theory."
Perhaps the problem stems from the lack of understanding of what 'quantized manifold' stands for. I personally don't see some real progress made in the past 60 years; see the report by Steve Carlip at
and the recent work by Chris Isham,
It seems to me that we need some brand new ideas about how to quantize a manifold,
Please drop me a line if you're interested.
Dimiter G. Chakalov
Note: Two years ago, on 14 August 2002, I wrote to Prof. Norbert Straumann, but haven't heard from him so far. I'll post some excerpts from his latest paper [Ref. 1].
Pay attention below to the words "almost exactly", and follow the link. Generally speaking, there is a very important idea in the doctrine of tralism: the two possible worlds, residing in the global mode of spacetime, can overlap only at the apex of Minkowski's cone. Recall also the two blueprints from these two worlds, which are always "dark" and produce two "opposite" dark components in local mode of spacetime: implosion for the "dark matter", and explosion/expansion for the "dark energy". Just click here and here. General remarks here, proposal of 3 November 2002 here.
Some people (unfortunately) called the two blueprints from the global mode of spacetime Dark Energy and Dark Matter. In the context of brain science, other people (unfortunately) postulated some "psychons". We do know that the mind-brain problem requires new physics, and we're fully aware of the problem with the energy of the gravitational field, which is needed to facilitate the bi-directional "talk" in Einstein's GR. Well, it might be complicated.
I was very much eager to talk on
these issues at GR17, but it didn't work out.
I wonder why.
[Ref. 1] Norbert Straumann, Cosmological Phase Transitions, astro-ph/0409042 v1; Invited lecture at the third Summer School on Condensed Matter Research, 7-14 August 2004, Zuoz, Switzerland.
p. 3: "The various changes of the
free energy density in cosmological phase transitions reinforces the cosmological
constant problem, a deep mystery of present day physics. In the last
part of my talk I shall address this topical issue, and then summarize
the current evidence for a dominant component of what people (unfortunately)
call Dark Energy in our Universe. Whether this is a tiny remainder
of an originally huge vacuum energy in the very early Universe remains
open, but observations may eventually settle this issue.
47-50: "It is a complete mystery as to why the two terms in (117) should
exactly cancel. This is -- more precisely stated -- the famous
"This illustrates that there is something
profound that we do not understand at all, certainly not in quantum field
theory (so far also not in string theory). We are unable to calculate the
vacuum energy density in quantum field theories, like the Standard Model
of particle physics. But we can attempt to make what appear to be reasonable
order-of-magnitude estimates for the various contributions. All expectations
(some of which are discussed below) are in gigantic conflict with the
facts. Trying to arrange the cosmological constant to be zero is unnatural
in a technical sense. It is like enforcing a particle to be massless, by
fine-tuning the parameters of the theory when there is no symmetry principle
which implies a vanishing mass. The vacuum energy density is unprotected
from large quantum corrections. This problem is particularly severe in
field theories with spontaneous symmetry breaking. In such models there
are usually several possible vacuum states with different energy densities.
Furthermore, the energy density is determined by what is called the effective
potential, and this is dynamically determined. Nobody can see any
reason why the vacuum of the Standard Model we ended up as the Universe
cooled, has -- for particle physics standards -- an almost vanishing energy
density. Most probably, we will only have a satisfactory answer once we
shall have a theory which successfully combines the concepts and laws of
general relativity about gravity and spacetime structure with those of
"I hope I have convinced you, that
there is something profound that we do not understand at all, certainly
not in quantum field theory, but so far also not in string theory."
concordance universe conceived as a brain
Dear Professor Starumann,
I read again your hep-ph/0505249 v2, and feel the need to thank you for your extremely clear and precise Lecture Notes. Reading your papers is just a joy.
Since you mentioned again the "dark" matter & energy puzzle, may I draw your attention to my modest efforts to suggest a conceptual resolution to this "dark" puzzle: it may be an effect of the Holon. The latter can produce real physical effects in the human brain, only neuroscientists would never speculate that 96 per cent from our brains are in the form of some "dark" computer or "dark side of the brain", nor would they suggest to detect "the speed of thought" (A. Eddington) with LIGO, LISA, or The Big Bang Observer,
The same effect of the Holon (the putative global mode of spacetime) can explain another puzzle: the gravitational waves exist but cannot be detected with any inanimate device, such as LIGO, AIGO, TAMA, GEO600, and VIRGO. I believe we need to trash the Hamiltonian formulation of GR, since it does not, and cannot explain the genuine dynamics of GR in the presence of 96 per cent "dark" stuff. It's just too much.
I regret that will not attend EPS13 in Bern, but will be happy to elaborate, in private, on the issues above, should you find them interesting.
With deep admiration,
Subject: Re: The concordance universe
conceived as a brain
Dear Professor Straumann,
Thank you for your kind reply.
> I have put an english version of this on the net: physics/0504201.
Thank you; I read your lecture from April 27th. I'd be very pleased to read an English translation of your paper dedicated to 100th anniversary of Wolfgang Pauli's birthday, physics/0010003 v1. I've elaborated on Pauli & Jung at
The putative global mode of spacetime (elephant's trunk, in the metaphor at the link above) is supposed to be a generalized version of Pauli's "eigentümlichen, klassisch nicht beschreibbaren Art von Zweideutigkeit" (physics/0010003 v1, p. 7), in line with Pauli's statement:
"It would be most satisfactory if physics and psyche could be seen as complementary aspects of the same reality" (C.G. Jung and W. Pauli, eds., Synchronicity, Princeton, NJ: Princeton University Press, Bollingen Series, 1973; Originally published as Naturerklärung und Psyche, Zurich: Rascher Verlag, 1952).
I believe 'the same reality' is presented in a Holon state, in the hypothetical global mode of spacetime. It is very much like the Platonic ideas: utterly real but UNspeakable, due to their 'klassisch nicht beschreibbaren Art von Zweideutigkeit',
Please see the "dark" effects on the physical world at
> As an old man
Accidentally, I hit 53 today, but I find comfort in the old saying that there are no old eagles: they either fly and fight, or die while flying. Which brings me to the issue of the GWs,
All this started in your country: first Einstein, then Pauli & Jung. It's a pity that the LIGO mafia didn't allow me to speak and show the real effects of the Holon,
It is very nice to be entangled with you, and I do hope we'll meet physically by November 2015, celebrating Einstein's GR by discovering the true dynamics of gravity in the presence of 96 per cent "dark" stuff.
physicists these points are mere ‘potential events’...
Dear Dr. Giulini,
As to the issue in the subject line, would you agree to insert the word "directly" in the last sentence: 'absolute structures carry no directly observable content'? Perhaps absolute structures "do not have an obvious individuality beyond an actual, yet unknown, event that realizes this potentiality". Please see
[Ref. 1] Domenico Giulini, Some remarks on the notions of general covariance and background independence, gr-qc/0603087 v1.
p. 4: "Transition functions relabel
the points that constitute M, which for the time being we think of as recognizable
entities, as mathematicians do. (For physicists these points are mere ‘potential
events’ and do not have an obvious individuality beyond an actual, yet
unknown, event that realizes this potentiality.)
Note: Yesterday, Domenico Giulini and Claus Kiefer posted a paper on canonical quantum gravity [Ref. 2], in which they wrote (p. 1):
"Hence, contrary to the Newtonian picture, in which spacetime acts (via its inertial structure) but is not acted upon by matter, the interaction between matter and spacetime now goes both ways (emphasis and links added - D.C.).
"Saying that the spacetime is ‘dynamic’ does not mean that it ‘changes’ with respect to any given external time. Time is clearly within, not external to spacetime. Accordingly, solutions to Einstein’s equations, which are whole spacetime, do not as such describe anything evolving.
"In order to take such an evolutionary form, which is, for example, necessary to formulate an initial value problem, we have to re-introduce a notion of ‘time’ with reference to which we may speak of ‘evolution’. This is done by introducing a structure that somehow (notice the poetry - D.C.) allows to split spacetime into space and time."
So, Giulini & Kiefer acknowledged that the Newtonian time cannot accommodate the non-linear, bi-directional "talk" of matter and spacetime: the Newtonian time is a linear time, and is therefore inadequate to the task of describing the non-linear "talk".
But then they re-introduce again a linear time variable, "by introducing a structure that somehow allows to split spacetime into space and time" (see above). However, in order to observe the alleged time parameter that is "clearly within, not external to spacetime", you have to take the stand of some meta-observer, who has a bird-eye view of the whole spacetime, and can distinguish three successive instants from this "block universe": some provisional instant 'now', which is defined with respect to an instant that is in its past, and another one, which is in its future. If you are locked inside this "block universe", you are dead frozen, and can't observe any evolution whatsoever, GWs included (Angelo Loinger).
As explained eloquently by Bob Geroch (emphasis added), "There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes. (...) In particular, one does not think of particles as "moving through" space-time, or as "following along" their world-lines. Rather, particles are just "in" space-time, once and for all, and the world-line represents, all at once the complete life history of the particle." Put it differently, there is nothing like a cursor or light spot from a torch, which would "move" from one place to another, highlighting different states of physical systems in this "block universe", hence reading the "time dependence of the metric" [Ref. 2, p. 2]. It is a frozen block of one single instant stretched to infinity, as St. Augustine would have said, since he knew nothing about the Cauchy problems and the rest of the generic pathologies of this so-called block universe.
I hope Prof. Domenico Giulini and Prof. Claus Kiefer are fully aware that this linear time variable t [Ref. 2, p. 2] is (1) unobservable, and (2) crap.
1. It is unobservable, in the sense that it cannot exist: with a linear time variable, we cannot, even in principle, observe the non-linear "talk" online, as it evolves along the cosmological time arrow. Professionally speaking, the so-called 'linearized gravity' is an oxymoron.
2. The linear time variable t is total crap, because it cannot, even in principle, describe the self-acting faculty of matter fields coupled to gravity: recall Baron Munchausen and read the explanation below the red line. More from Carlo Rovelli and John Stachel.
In summary, I claim that the alleged linear variable t , which refers to the "time dependence of the metric" [Ref. 2, p. 2], is an unobservable crap. There is no such thing as 'linear time parameter' in GR -- see again Carlo Rovelli. If you wish to work in GR with a linear time, as you do in Newtonian mechanics, you need some absolute and external background time parameter. It's a bundle. And since you don't have the latter, you should drop the former. Hope that is simple enough.
The inevitable fact that the energy-components of the gravitational field are non-tensorial quantities (cf. Hermann Weyl) requires that the proponents of Hamiltonian GR should either "detach" energy from time, in such a way that their linear time variable could faithfully label "instants in a continuous fashion" [Ref. 2, p. 2], or discover some tensorial presentation of the energy-components of the gravitational field. Neither of these challenges is possible. To be specific, recall that the "dynamics" in Hamiltonian GR, after Dirac-ADM, is being "derived" from a frozen, kinematical snapshot -- "an instant of time" (Karel Kuchar) -- as if they could employ some background Newtonian time to define the energy-components of the gravitational field at this point-like 'instant of time', and then derive the Dirac-ADM dynamics from it, and in it (see the dualistic conception of time here). Which, in turn, means that the proponents of Hamiltonian GR have to produce at least one well-defined state of an isolated gravitational system, from which they can derive the dynamics of GR. Trouble is, they can only speculate about some 'localizable energy-momentum complex' (references here), but cannot -- even in principle -- discover such 'isolated gravitational system', and thus cannot -- even in principle -- obtain/recover even one well-defined instant of time with the Hamiltonian GR: time & energy are a bundle. It is not possible to squeeze the energy-components of the gravitational field into one point only; you might be able to define them only within a domain of 3-D space, but you cannot define rigorously the "boundaries" of this domain. With the Hamiltonian GR, you simply cannot define any 'isolated gravitational system'. It is an oxymoron, too.
If that were possible in Hamiltonian GR, the collection of such well-defined instants, chained "in a continuous fashion" [Ref. 2, p. 2], would display the successive end results from the matter-geometry "talk" (that is, the final and definite results from the matter-geometry negotiation), which would have been already placed in our past light cone, so that we can observe them (cf. #1 above). But in GR we don't have any background or external time whatsoever, firstly, and secondly -- such a chain of end results, which we can only observe post factum, as being already placed in our past light cone, cannot be read by any (inanimate) clock, because the latter cannot read online the non-linear and quasi-local matter-geometry negotiation. Such clock can "read" no more than one instant, as being already placed in its past light cone. It cannot read the next one. Hence the so-called block universe. If you really believe in the "block universe", you may have serious problems with your neocortex (cf. Petr Hajicek). Capiche?
The alleged Dirac-ADM "dynamics" is simply ridiculous. It can only produce some "block universe" (think of it as a rubber stamp printout) marred with inevitable pathologies. Besides, how would such a printout expand due to its "intrinsic" dynamic dark energy? And how do you know that the whole 3-D space "expands" en bloc? With respect to what? Only Baron Munchausen can perform such self-acting miracle with the linearized Hamiltonian GR.
To make the initial idea clear to my teenage daughter, I offered her the following explanation. Imagine this: every time you look around, you see a new state of the whole universe, which has been "inserted" into your world line "just before" you looked at it. Hence your past changes/expands along the cosmological time arrow, but you can detect these changes only post factum, due to the so-called speed of light. Every new state of the whole universe is the final product/end result from the bi-directional "talk" of matter and geometry, hence this chain of new states is non-linear, and its topology (here she looked at me with big eyes) is non-trivial: it is neither a line nor a circle, but both-line-and-circle, just like in Ulric Neisser's cycle (here I again got the same look from her, but didn't pay attention). Again, because of the so-called speed of light (here she looked smart, as if she knew what I was talking about), what you see "at the end of the day" is a perfect continuum of already-accomplished matter-geometry negotiations: the local mode of spacetime. I've been trying to explain this since December 1999. See below.
This is the essence of the linearization procedure in the local mode of spacetime (in the vein of the ‘bare’ charge and mass in QED), and the first step toward unification of GR with QM. Many experts in GR are manifestly silent about it, while in QM textbooks one can read the amazing misconception that the time parameter in Schrödinger equation -- the same "already-linearized" time -- is identical to the Newtonian time. This highly misfortunate confusion in QM and GR textbooks can perhaps be explained with the mechanism proposed for the fundamental time asymmetry: the "already-linearized" time is being cast in the absolute past of the universe, as 'local mode of spacetime'.
The ultimate paradox is in the fact that we observe the cosmological time and the 3-D space -- the first-off challenges to Einstein's GR and canonical quantum gravity, respectively. What we observe around us is precisely what we cannot derive from the theory. Just as the Hilbert space/inner product problem cannot be solved by employing a Hilbert space, since the latter requires "the presence of an external time (with respect to which the probability is conserved)" [Ref. 2, p. 18], the energy conservation paradox cannot be resolved with any intrinsic "time variable", because you would again need the presence of an external time. More on the energy conservation paradox from Kenneth Dalton [Ref. 6].
The proposed solution is very simple: instead of "intrinsic" linear variables, use 'local mode of spacetime'; instead of "external time", use 'global mode of spacetime'. See also the famous Diff(M)-invariant 'constant of motion' (a.k.a. Kuchar's Perennials), presented as 'the quantum state', here. Can't go more "intrinsic" than that.
I wonder if Prof. Domenico Giulini and Prof. Claus Kiefer would elaborate on the issues mentioned in the first two pages from their recent paper [Ref. 2]. The former never replied to my email, while the latter sent his last reply in 2003.
But I am sure some day they will respond. After all, this is their professional work, and not a hobby. If they were collecting paper napkins and I was suggesting to switch to, say, bottle labels, they would have certainly ignored my proposal and ideas. People exercise their hobby because they just like it. Not because they respect it.
If Prof. Dr. Domenico Giulini and Prof. Dr. Claus Kiefer treat canonical quantum gravity as nothing but a hobby, they wouldn't respect it, and may not reply at all. Hope to hear from them, since there are many other cute points to discuss [Ref. 2], most notably the long standing problem of providing physical meaning to the coordinates [Refs. 3 and 4] by employing a perfect fluid as 'the reference fluid' [Ref. 5].
NB: Isn't it agonizingly clear that we should not use any physical entity (cf. John Baez here) for identifying events in spacetime [Ref. 4]? Every physical stuff in GR is intrinsic or "within" spacetime (cf. Emilio Elizalde, p. 38), hence it cannot in principle 'define itself' and 'move itself' [Ref. 5] along a linear time variable, like Baron Munchausen. Moreover, in a purely relational theory, the dynamics cannot be modeled with any linear parameter -- see the Buridan donkey paradox applied to GR. More on the non-linear dynamics of 'relational reality' here.
Ever since the inception of GR in 1915, we are confronted with the paradoxical situation that some phenomenon is evidently performing as 'the reference fluid in GR': it discerns the points of space from one another, like a genuine 'pre-geometric plenum', but does not disturb the geometry [Ref. 5]. Only such reference fluid, which 'acts but is not acted upon' (cf. John Baez), cannot be accommodated in GR in principle. Karel Kuchar called it "hidden unmoved mover", but then what? The last time I heard from him was on January 28, 2003. Maybe he too is collecting paper napkins.
To sum up, here is a brief list of things that are impossible in the framework of the linearized Hamiltonian GR. Should the readers have any doubts, please write me back and I shall elaborate extensively, with utmost pleasure.
1. It is not possible, not even in principle, to reveal the fundamental timelike displacement in GR. The reference fluid, as sought by Hilbert and Einstein [Ref. 5], is missing in GR. This is as it should be, because it isn't there. It is being produced by the Aristotelian First Cause. The task of introducing the effects from the reference fluid requires brand new mathematical notions. Then we can expect “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics” (Eugene Wigner).
2. It is not possible to introduce any physical entity for identifying events in spacetime (cf. NB above), provided that such physical entity could be defined only and exclusively only by considering the spacetime en bloc, by taking the stand of some meta-observer, who can "embrace" the whole spacetime, from some arbitrary (that is, chosen by her) event 'now' up to the "boundaries" of spacetime. Such physical entity would (supposedly) identify a spacetime "point" (no bare "points" are allowed, due to the Hole Argument), but will ultimately require some meta-observer, who has the ability to identify and choose three arbitrary "points": a reference/zero point, call it x0, and two "subsequent" points, x1 and x2, such that the interval (x2 - x1) will be well-defined. Then, by instructing her interval to approach asymptotically zero, she would define the limit at which she would obtain a "point" Y, as being identified by its physical content (again, no bare point are allowed). Firstly, notice the peculiar nature of Y : it bears just an "imprint" (much like 'a tangent vector pertaining to a point') from the three x-points used for its derivation, but is not directly related to the initial x-points. Secondly, Y has been derived with three x-points by applying a procedure which expels the new Y point outside the applicable limits of GR: Y is a singularity. And thirdly, notice that the meta-observer should have the ability to look over the whole spacetime, from any provisional x0 up to its "boundaries". But because you cannot have the luxury of such meta-observer, everything which can be derived only and exclusively only by her presence and intelligent actions is prohibited: you do not have Y points nor the initial x-points. An brief list of corollaries follows below.
3. There is no such thing as 'an instant now' in the linearized Hamiltonian GR.
4. Nothing can "read" the states of gravitational system in a "succession" of instants, resembling the flow of cosmological time due to the expansion of the metric of 3-D space.
5. It is not possible to define any 'isolated gravitational system' whatsoever, not even under some highly unrealistic conditions: read Roger Penrose here.
6. It is not possible to define any 'causal boundaries of spacetime', 'ideal endpoints' (Geroch-Kronheimer-Penrose), nor 'finite infinity' (Ellis) for any realistic gravitational system. An operational definition of the latter is 'a system which can be validated by passing the tests of Robertson-Walker cosmology'.
7. It is not possible to define rigorously the dynamics of gravitational waves (GWs) either, because they propagate 'within themselves' and 'with respect to themselves'. Likewise, it is not possible to design a time-oreintable spacetime, because it is impossible to "time-orient" it with respect to itself.
8. It is not possible to define the energy balance of any 'instantaneous state', from which one could try to derive the alleged dynamics of gravitational systems, because the only way we can fix the tµv = 0 balance for a point-like instantaneous state is by introducing some external background time, as in Newtonian dynamics. In order to define an intrinsic linear time variable, you again need an external background time parameter, pertaining to the intelligent meta-observer (cf. 2 above). It's a bundle. And since you don't have the latter, you should drop the former.
9. In general, it is not possible in GR to define anything by employing a linear time variable, because in the theory of relativity we have 'relational reality', and the dynamics of relational reality cannot in principle be defined with any linear time variable. This is the ontology of our relational reality. Ignore it at your peril.
Thus, we should start with resolving the first off task: the reference fluid in GR. Not only it defines the global properties of spacetime -- the elementary timelike displacement, the "atom" of spacetime, and the "boundaries" of spacetime -- but also provides an answer to the question 'with respect to what?'. The fact that your wrist watch does read the cosmological time arrow is the greatest mystery of GR: in its local reference frame, your wrist watch shouldn't be able to read such global parameter pertaining to the whole spacetime en bloc. Yet it does, thanks to the reference fluid. More from Aristotle.
If, for some reason, the readers (if any) of these lines do not like my stand on GR and mathematical ideas, they shouldn't be concerned at all -- I could be totally wrong. Or perhaps 'not even wrong'. Read the opinion of Britain's leading expert in quantum gravity here, and an eloquent reply from a member of LIGO Scientific Collaboration (LSC), Prof. Dr. Dick Gustafson: "I don't know you and wish you out of my face, my computer."
[Ref. 2] Domenico Giulini and Claus Kiefer, The Canonical Approach to Quantum Gravity: General Ideas and Geometrodynamics, gr-qc/0611141 v1, 27 November 2006. http://arxiv.org/abs/gr-qc/0611141
p. 2: "Only after solving the dynamical equations can we construct spacetime and interpret the time dependence of the metric... "
p. 2: "... t is so far only a topological time: it faithfully labels instants in a continuous fashion ... ."
Footnote 3, p. 2: "It can be shown that the Einstein equations do not pose any obstruction to the topology of E , that is, solutions exist for any topology. However, one often imposes additional requirements on the solution. For example, one may require that there exists a moment of ... ."
p. 16: "Equation (42) is called the Wheeler-DeWitt equation in honour of the work by Bryce DeWitt and John Wheeler; see e.g.  for details and references. In fact, these are again infinitely many equations (one equation per space point)."
p. 17: "Another central problem is what choice of Hilbert space one has to make, if any, for the interpretation of the wave functionals. No final answer to this problem is available in this approach ."
p. 18: "It is well known that the notion of Hilbert space is connected with the conservation of probability (unitarity) and thus with the presence of an external time (with respect to which the probability is conserved). The question then arises whether the concept of a Hilbert space is still required in the full theory where no external time is present."
p. 20: "A final, clear-cut, derivation remains, however, elusive."
"Unfortunately, such a standpoint makes it difficult to view the reference fluid as physical matter."
p. 13: "One of the main problems of the
Hamiltonian approach to GR is to pick out the global variable which can play the
role of “time-like variable”. (...) Separation of real dynamical variables from
nondynamical ones is the crucial step in
extracting the relevant physical information from the gauge theories."
Joseph Katz, Gravitational
Luis Lehner, Numerical
Relativity: A review.
Note: So, if you wish to do numerical relativity and detect those gravitational waves, make a set of "space-times", "after solving the dynamical equations" [Ref. 2], from each and every possible numerical value pertaining to each and every possible "coordinate" that can "play the role of ‘time’ with respect to which the dynamical evolution" may be referred to (Luis Lehner). Please don't tell me there's no need to do that. Make your task as easy as possible (cf. R. Penrose), and you'll see what kind of "dynamics" and "local Dirac observables" you will get from this set of "space-times", and whether all members of this set would indeed describe the same observed reality, as you or T. Thiemann probably expect. This exercise will verify the tacit zeroth hypothesis "numerical relativity works". It may not.
Recall that there is no preferred reference frame in theory of relativity, hence if you wish to "split" the 4-D continuum into "space-time", you may, of course, choose by hand one coordinate, say, a0, to play the role of ‘time’ with respect to which the dynamical evolution will be referred to (cf. Luis Lehner above), but then you are obliged to verify your "numerical relativity" by choosing the remaining "coordinates", b, c, and d, to play the same role. Thus, if you wish to "split" the 4-D continuum into "space-time", you need to produce a set of four (presumably equivalent) "space-times". Here people say -- 'but once I solve the dynamical equations and design a "space-time" [Ref. 2, p. 2], I cannot use that same "space-time" to derive the remaining three "space-times".' Well, too bad. For if you can't shuffle the components of Lorentzian signature, your "numerical relativity" is for the birds, because it will presuppose a preferred reference frame, in which "space" and "time" are already split by Mother Nature (not by you) at the very Riemannian manifold, in such a way that your metric would be imposed on the world ab initio. But suppose you're very good in math and discover some principle in GR, resembling the linearity of QM, such that you can design a (presumably equivalent) "superposition" of four (or more) "space-times". Again, feel free to make your task as easy (or physically unrealistic) as possible (cf. R. Penrose). Would such set of "space-times" describe the same observed reality, as you or T. Thiemann probably expect? I bet you will discover that what is "observable" in one "space-time" is "gauge" in the rest.
To understand the task of numerical relativity and GW astronomy, consider the following simple question: Can the metric gµv , which is the “substance” of Einstein spacetime, serve as a potential of any field studied in classical and quantum physics? Since you work with linear time only, you have two alternative choices: either 'yes' or 'no'. Both lead to insurmountable problems. The only available option is to introduce a linear time variable "after" the completion of every step from the bi-directional negotiation of space and matter, as I tried to explain to my teenage daughter above.
Notice that the "direction" of the space-matter negotiation, presented
with a timelike vector, is being inevitably
obliterated by calculating any observable value of physical quantities in
the local mode of time: see Mathew Frank here.
Notice also that the introduction of a linear time variable, with which you may
wish to tackle the dynamics of "linearized gravity", requires solving the
Einstein equation for each and every space point, just as in the case of Wheeler
- DeWitt canonical quantum gravity: "In fact, these are again infinitely many
equations (one equation per space point)" [Ref. 2, p. 16].
The problem originates from the treatment of the (local) time in GR, as
explained eloquently by
As for quantum theory, one of the basic principle in QM is the superposition of states (e.g., |cat> & |dog>, cf. E. Joos, quant-ph/9908008 v1, Sec. 3.1). It is based on the assumption of linearity of QM (e.g., Bei Jia and Xi-guo Li, math-ph/0701011 v1, Sec. 3). But a closer inspection with the geometric formulation of QM reveals that this "linearity" is a myth, advertised in non-relativistic QM textbooks only (cf. A. Ashtekar and T.A. Schilling, gr-qc/9706069 v1). It's time for a change.
But of course I could be all wrong -- see a cautious note here.
From: Dimi Chakalov <email@example.com>
From: Dimi Chakalov <firstname.lastname@example.org>
Note: If some day Stanley Deser shows respect to the founding fathers of GR (not just typing some obscure humorous remarks in the subject line of his email), I will immediately offer him my apology for my email from Fri, 8 Dec 2006 13:16:48 +0100 (printed above). At this moment, I believe he performs at the General Relativity Trimester at the Center Emile Borel at the Institut Henry Poincaré in Paris. Yesterday, December 8th, he was supposed to deliver his last lecture, bearing the mysterious title "Gravitations in D=3",
Not 'gravity in the 4-D continuum', as you'd probably expect.
See also my email from October 17th to the organizers of the General Relativity Trimester (printed below). I haven't yet heard from them. Qui vivra, verra.
I never make predictions -- never have, and
Subject: General Relativity Trimester
From: Dimi Chakalov <email@example.com>
Note: I reread the ADM paper of '62 that's on GR-qC in '04, gr-qc/0405109, as kindly advised by Stanley Deser. Just consider the following excerpts (emphasis mine - D.C.):
"In Lorentz covariant field theories, general
techniques (Schwinger, 1951, 1953) (valid both in the quantum and classical
domains) have been developed to enable one to disentangle the dynamical from the
gauge variables. We will see here that, while general relativity possesses
certain unique aspects not found in other theories, these same methods
may be applied. "5.1. Expressions for the energy and momentum
"5.1. Expressions for the energy and momentumPμ of the gravitational field. The canonical formalism developed in the previous section has brought out the formal features of general relativity which have their counterpart in usual Lorentz covariant field theories. As a consequence, the physical interpretation of the gravitational field may be carried out in terms of the same quantities that characterize other fields, e.g., energy, momentum, radiation flux (Poynting vector).
To understand how this whole mess has been cooked up, read the intro:
"The general coordinate invariance underlying the theory of relativity creates basic problems in the analysis of the dynamics of the gravitational field. Usually, specification of the field amplitudes and their first time derivatives initially is appropriate to determine the time development of a field viewed as a dynamical entity. For general relativity, however, the metric field gμν may be modified at any later time simply by carrying out a general coordinate transformation. Such an operation does not involve any observable changes in the physics, since it merely corresponds to a relabeling under which the theory is invariant."
And how ADM tackle the diffeomorphism freedom? Easy:
"Thus it is necessary that the metric field be separated into the parts carrying the true dynamical information and those parts characterizing the coordinate system."
So, ADM suggest two separable parts in the metric field: one carries "the true dynamical information", while the other part is just "characterizing the coordinate system" and "merely corresponds to a relabeling under which the theory is invariant." According to ADM, this second part is not interesting to the dynamics of GR, since it "does not involve any observable changes in the physics", although it captures the essential and unique nature of GR: "the metric field gμν may be modified at any later time simply by carrying out a general coordinate transformation." Do you smell a rat? See p. 24 and refs. 99-101 from Karel Kuchar's paper.
Why would Mother Nature exercise in diffeomorphism freedom? What is the physical necessity behind the requirement that "the points occurring in the base sets of diffrerentiable manifolds with which general relativity models spacetime should not be reified as physically real" (Butterfield & Isham)? What is the physical necessity which requires that eigenvalues must not be physically real (context-independent, cf. Kochen-Specker and Conway-Kochen theorems), like the state of the Sun 'out there'?
That's how the potential reality is being manifested in GR and QM. Surely Stanley Deser wouldn't agree, only he has nothing else to suggest, because ADM tells us that an observer can observe and measure only invariants of group of diffeomorphisms of the Hamiltonian dynamics, which includes reparametrizations of the coordinate time (=local mode of time), while the non-invariant "time", considered as 'the time of evolution', is "unobservable" (=global mode of time).
But if it were observable, "the points occurring in the base sets of diffrerentiable manifolds, with which general relativity models spacetime" (cf. Butterfield & Isham above), will inevitably be physically real. Thus, the non-invariant global mode of time got to be "unobservable" to ADM and the like. Once we consider the possibility for a master/universal time arrow, we introduce a unique object -- the global mode of the Holon -- to be the absolute reference frame for this universal time arrow. It is never exposed in the local mode of the physical reality: The One is an unbroken ring with no circumference, for the circumference is nowhere (hence no absolute reference frame in the local time) and the center (in the local time) is everywhere. Isn't this simple and clear? More from Mathew Frank.
Further, ADM wrote:
"In this respect, the general theory is analogous to electromagnetic theory. In particular, the coordinate invariance plays a role similar to the gauge invariance of the Maxwell field."
I don't want to comment on that. But if some
day Stanley Deser decides to get professional, I most certainly will.
Back in nineteenth century, a Hamburg physician and amateur astronomer, Friedrich Wilhelm Olbers, drew attention to the fact that the night sky is dark. He too was an "alien" to the established theoretical physics community, but was talking to astronomers, and his arguments were acknowledged. I've been trying, since December 1999, to do something similar, by pointing to the paradoxes of GR, most notably the simple fact that the linear time variable, as read by your wristwatch, is nonexistent in GR. Nobody seems to care, however. People still do their calculations in GR just like in QM, by 'dividing Tuesday by 11'.
Now, forget about the joke. Look at your wristwatch, and imagine it reading a succession of instants, which form a line (1-D Euclidean space). Try to time the negotiation of the two parties in Einstein equation with this linear time, and you'll see that neither of them can "move": before the left-hand side can evolve into its next state, it needs the next definite state of the right-hand side; but this next definite state of the right-hand side can be fixed only after the left-hand side has fixed its next definite state. The same applies to the other "hand". They are tied. They can't talk to each other with such linear time [Ref. 2, p. 2].
At this point, a renowned expert in classical and quantum gravity said: "And, after all, general relativity does seem to work well as a theory, and yet I can certainly read the time on my wrist watch!"
Of course you can read the time on your wrist
watch! But can you derive it from GR? My daughter is not quite sure, and
neither I am. Since "the metric is treated as a field which not only affects,
but also is affected by, the other fields" (John
Baez), we have a genuine non-linear
dynamics, the topology of which cannot be modeled with a line (1-D
Euclidean space), simply because each and every "point" from this line
is the nexus of two "opposite" negotiations: [metric field affecting
all physical fields] & [all physical fields affecting the metric field].
Hence we need two kinds of time: one for the non-linear
talk (called 'global time') and another one, for the chain of
already-accomplished negotiations (called 'local time'), which your wrist watch
reads (cf. the linearization procedure above).
"the points occurring in the base sets of diffrerentiable manifolds with which general relativity models
spacetime should not be reified as physically real" (Butterfield &
Isham). They do not refer to 'the
same building', and therefore cannot qualify as 'objective reality out there'.
Albert Einstein was fully aware of this problem long
before he and David Hilbert designed General Relativity.
How do you tackle this? Obviously, you need to 'hold onto' something, so you introduce a dead frozen background, which you call "space-time", and which does not change -- see Bob Geroch above. But as noticed by Lao-tzu, "if you realize that all things change, there is nothing you will try to hold onto." And surely "all things change": see Luis Lehner above.
Thus, your "dynamics" is self-contradictory from the outset. All you can achieve is to make Hermann Minkowski spin in his grave like a helicopter. Ninety-nine years after his seminal talk in September 1908, people are still splitting the spacetime and searching for some "Dirac observables", but cannot find such animal even with the most unrealistic perturbative approach, by using "flat Minkowski background" [Ref. 8].
Perhaps the readers (if any) of these lines can help. But before you proceed, please read a cautious note here.
[Ref. 8] Bianca Dittrich and
Johannes Tambornino, A perturbative approach to Dirac observables and their
gr-qc/0610060 v1; Class. Quantum Grav. 24 (2007) 757-783.
elementary proper-time duration: Chronon
Note: To understand the origin of the Chronon and the resolution of the problem with the classical limit of QM, recall the renormalization procedure in QED [Ref. 2] and compare it with the renormalization procedure in GR suggested above. In our case, the "bare" state of quantum-gravitational observables is presented with 'potential point(s)', as explained here. The "duration" of the transition from the bare/potential state (global mode of time) to the renormalized/localized state in the local mode of time is the elementary step of the cosmological time arrow (the elementary timelike displacement, Ted Jacobson). In the quantum realm, this 'elementary proper-time duration' is called Chronon -- see the web site of Erasmo Recami above and read his papers. To unravel the Chronon in GR, perhaps all you may need is to recover the true dynamics of Einstein's GR. Good luck.
[Ref. 2] A.N. Mitra, Einstein
And The Evolving Universe,
Subject: gr-qc/0703035 v1,
It seems to me that you have made an unjustified stipulation on p. 40: "We
assume that all E_t's are spacelike and that the foliation covers M (cf.
Fig. 3.1)", resulting in Eq. 3.3. The latter produces that "timelike and
future-directed unit vector n ", which is indeed "somewhat arbitrary
choice of a time coordinate", to say the least. More at
"From the mathematical point of view, this procedure allows to formulate the problem of resolution of Einstein equations as a Cauchy problem with constraints.
"From the pedestrian point of view, it amounts to a decomposition of spacetime into "space" + "time", so that one manipulates only time-varying tensor fields in the "ordinary" three-dimensional space, where the standard scalar product is Riemannian. Notice that this space + time splitting is not an a priori structure of general relativity but relies on the somewhat arbitrary (notice the poetry - D.C.) choice of a time coordinate."
Let me ask a pedestrian question: Is it logically possible for matter fields to produce their "time" (see ADM "dynamics" below), while at the same time evolve in that same time? Sounds to me like the familiar story from Baron Munchausen, yet Stanley Deser has the feeling that understands his "dynamics" of GR.
NB: Moreover, the drawing above is the non-relativistic (and highly misleading!) presentation of spacetime, because it makes sense only with respect to some 'ideal observer' who can "watch" the dynamics of the elementary transition, z_init --> z_final , and hence the 'proper time' along spacetime trajectories (C. Rovelli).
My understanding of how we wind up with the Fig. 3.1 above is as follows. I will try to talk about the 'global properties of spacetime'.
Imagine a stone block in front of you, painted in changing nuances of blue, such that it is light blue on the left, and dark blue on the right. That would be a "time-reversible block universe", in which the cosmological evolution has produced a pattern of gradual changes of blue color (cf. Thomas P. Sotiriou et al.). If you pick any point from this stone, you can verify the consistency of nuances of blue with their localization on the stone, along some pattern of 'blue color change', such that no dark blue spot would appear among the light blue area, say. Notice that these are just correlations, as there is no dynamics at all involved with this stone block: see Bob Geroch. According to GR, all reference frames or viewpoints on the stone are 'equally good', in the sense that there is no "preferred point" on the stone; more from Butterfield and Isham. (There is a 'cosmic equator', however: see Craig J. Copi et al.)
Notice also that the stone block itself does not possess any background of bare/blank spacetime points, on which Mother Nature "paints" some physical content. We obtain such 'painted stone block' only after we solve Einstein's equations and produce 'spacetime'. Because of the principle of active diffeomorphism, there is no pre-existing constellation of "bare" points (colorless stone block): both the stone block and its concrete physical content/color are being created en bloc (see above), so you can't tell which was preceding what: you don't have 'color' without 'stone block', nor 'stone block' without 'color'.
But then where did this 'cosmological time' come from? We encounter the end result from of an 'evolution', but the "creator of blue color" hasn't left any "fingerprints", neither on the blue color pattern nor on the stone block.
It looks like a completely background-free and
self-made, by self-acting, end result of ... itself .
However, Fig. 3.1 above doesn't make sense from the outset, because it has been confirmed since 1998 that there is a "dark water" spring under the lake, which makes the lake dynamical, by "expanding" the distance between all "points" from the surface of the lake, in such a way that the energy needed to perform such "accelerated expansion" cannot be derived exclusively from the lake: if we wish to derive it exclusively from the lake, it will have to violate ANEC. So, the very fact that we are alive and well to read this web page requires a total rethinking of GR and its perfectly smooth "dark energy" of "empty space".
To be specific, any time you write the seemingly innocent expression dt (see Fig. 3.1 above), you invoke two phenomena: (i) the incremental shift of the lake "up", along a "transverse" axis z orthogonal to the whole lake (not shown), and (ii) the elementary timelike displacement or 'tick of time' of all clocks living on the lake surface. The first shift is being produced by the Dynamic Dark Energy (DDE) of the "dark spring" under the lake, but we cannot observe it due to the 'speed of light' limitations imposed on the lake surface. Thus, we can only observe the second shift, and only post factum, after the first DDE shift has already expanded the whole lake.
NB: The second or "horizontal" shift on the lake surface (on the hypersurface) contains an embedded holistic (or "dark", if you prefer) input from the first, "vertical" shift: it is the feedback from 'everything else' (the whole lake) directed on the local "point" dt&ds , with which the shift dt&ds gets completed "on" the lake surface (on the hypersurface), in line with the rule 'think globally, act locally'.
The newly proposed connection from one hypersurface to the neighboring one (called The Aristotelian Connection) is being performed by a very special object -- the universe as ONE -- which is simultaneously "outside" the lake (to make its "boundaries") and ]between[ any two points from the lake surface. It a way, in the local mode of spacetime 'the ONE' is simultaneously infinitesimally close to us, and infinitely away from us. (I trust the reader is familiar with this very old metaphysical idea; all we have to do is to cast it in math, which shouldn't be a problem, because our mathematical thinking is literally guided by such metaphysical ideas.)
The concrete task is to construct a continuum of such points on the lake surface, in line with the principle of relativistic causality and locality. This is the challenge of the continuum in GR: we still don't know how to embed the "dark water" into our continuum of points on the 3-D lake surface. It is a global property of the 3-D lake, which is projected/inserted ]between[ any adjacent points from the lake surface. The result is a "web" that covers the whole 3-D space and defines dynamically its "boundaries" (see the two "ideal endpoints") with the ultimate cutoff from Aristotle. Admittedly, the task is highly non-trivial.
Notice that the "dark water" and the "lake shore" correspond to the white area in Fig. 3.1 above, and read about the pathologies of the classical spacetime manifold in Ioannis Raptis' gr-qc/0110064 v1, Sec. 2, pp. 2-3. All these problems are well known to Eric Gourgoulhon and Thibault Damour, but they are Type 2 physicists, and don't care.
The solution proposed was explained at the first link in my email to Eric Gourgoulhon from Tue, 17 Oct 2006 22:56:37 +0300; see an alternative to Fig. 3.1 here. As to 'GR and experiment' part from their "General Relativity Trimester", the main problems also widely known, and are listed here. General considerations from quantum theory here.
Briefly, the exact meaning of M , as obtained from Eq. 3.3 above, is a total mystery. You can use Eq. 3.3 only to calculate the volume of a squared barrel with fixed dimensions: see Fig. 3.1 above. As I predicted on 17 October 2006, "by Christmas you all may learn exactly what you *want* to learn, but nothing more."
You can bring a Type 2 horse to the water but you cannot make him drink.
Alternatively, if you are Type 1 physicist, recall the task of introducing an ideal boundary on a spacetime: embed the spacetime (conformally) into a "larger" spacetime domain (all the white area "outside" the bluish slices of M in Fig. 3.1 above). Notice a very important prescription (emphasis added), from Steven Harris [Ref. 1]: "use the boundary of the embedded image, with topology and causal properties induced from the ambient (unphysical) spacetime, as the boundary of the physical spacetime. This is the origin, for instance, of the usual picture of the boundary of Minkowski space (...) derived from its standard embedding into the Einstein static spacetime... ."
Thus, the task boils down to discovering a new recipe for embedding the Einstein static spacetime into 'something else' (=an ambient (unphysical) spacetime), bearing in mind that (i) M isn't static, due to its Dynamic Dark Energy, (ii) the topology of the "ambient (unphysical) spacetime" may be highly non-trivial, because it could be a "superposition" of open and closed topology of 3-D space of M , and (iii) the causal properties, induced from the "ambient (unphysical) spacetime", could result into a new form of retarded causality and determinism (dubbed 'dynamical determinism'), which is utterly needed to accommodate the "quantities expressing the degrees of freedom and the equations governing their evolution", which are "highly non-local", as stressed by John Stachel here, regarding solving the Cauchy problem for the Einstein equations in M .
All this pertains to the global properties of M (and of course the "waves" of its metric), since its topology and causal properties should be induced from the "ambient (unphysical) spacetime": see again the general considerations from quantum theory here. All this web site is about theory of relativity, in which we need a reference object to make sense of 'what the universe looks like from the inside'. We need a new reference object -- call it "ambient (unphysical) spacetime" or global mode of spacetime, your choice -- to understand the "boundaries" of spacetime. The latter is a highly non-trivial task, since our thinking is inevitably teleological, which means that we have to both eliminate all physical or accessible boundaries from our 4-D spacetime and keep the mechanism producing these "boundaries" well hidden in 4-D spacetime, in such a way that it could impose the "boundaries", but not expose itself in 4-D spacetime. Briefly, this mechanism of fixing 'numerically finite but physically unattainable boundaries on spacetime' should be utterly "dark", as seen from all inertial reference frames constituting 'what the universe looks like from the inside'.
Can we have our cake and eat it? Namely, can we impose a cut off or "boundary" on spacetime, such that it cannot be reached "from within"? Take as example the ever-expanding cosmological horizon: if it was reachable to an observer placed inside 4-D spacetime, that would imply some "outer space" available 'out there', into which the physical universe has "not yet" expanded, which is of course parapsychology. So, we have to hide the "ambient (unphysical) spacetime" [cf. Steven Harris in Ref. 1], or the global mode of spacetime, in some very smart fashion: take lesson from Mother Nature and recall how the infinitesimal is being dynamically hidden from the teleological time; see the Thompson's lamp paradox here.
To get started, see Matthew Frank's "Einstein's Equation in Pictures" here. We begin with "a unit timelike vector v at a point p" (ibid.), which points to ... where? To the white area "outside" the blueish slices of M in Fig. 3.1 above, of course. This crucial "white area" involves both the infinitesimal displacement dt and the "boundaries" of M ; more from Jose Senovilla here. To obtain Einstein's equation, Matthew Frank shrinks the footing of his "timelike" vector -- not the "timelike" vector itself but just its 3-D space footprint with diameter 2r -- by instructing it to 'get down to zero'. Hence we obtain a "dark" vector, or rather a remnant from the whole white area from Fig. 3.1 above. This 'remnant' keeps a perfectly physical, albeit "dark", component "inside" M , and in the same time stays "orthogonal" to it, and directly inaccessible from it: measurements across the "rubber band" are unphysical, says Bernard Schutz (cf. Fig. 24.3 here).
Thus, the "boundaries" of
M originate from the smooth
omnipresent "dark" stuff that was expelled from it
from the outset, by the very construction of
the "rubber band" M in
Einstein's GR. See the idea about "boundaries" of the ever-sliding cosmological horizon,
produced with "negative" mass, here. It is
from 3 November 2002. In the local
mode of time, the expansion of 3-D space, bounded by two 'ideal endpoints',
would look like
It is chasing both The Beginning and The End. All we can say, by taking a look 'from inside', is that the universe has started "asymptotically" from The Beginning, and is currently chasing The End/Beginning with some DDE. We cannot observe the two 'ideal endpoints', hence the ever-expanding cosmological time is "bounded" within an open interval. But if we take a look 'from outside', from the perspective of the putative global mode of time (cf. the white area in Fig. 3.1 above), the story looks quite different:
Once created with two ideal endpoints, the 3-D space can happily expand as much as it likes. Namely, once unleashed from [John 1:1], it might have jumped vigorously by what is called "inflation"; we just don't know. What matters here is that it is logically impossible for the 3-D space to trace back its origin at Time Zero in the local mode of time: see the Vacuum Cleaner Paradox here and notice that at Time Zero the two ideal endpoints would have to be squeezed/fused into an "interval" with zero duration,  , which is impossible to be reached 'from within' 3-D space.
Why? Because the local mode of spacetime is composed from infinitely many (actual infinity) "centers" of the universe (more here), hence the fusion of the two ideal endpoints of the Aristotelian First Cause is LOGICALLY IMPOSSIBLE: it would eliminate the whole universe, that is, all explications of [John 1:1]. Likewise, it is impossible to reach physically the two endpoints with any Vmax in the future.
On the other hand, we can understand, from the global mode of time, the old joke by Yakov Zel'dovich: "Long time ago, there was a brief period of time during which there was still no time at all." Only this peculiar 'no time at all' is always present: it is "between" any two adjacent points from the Hausdorff topological space, thanks to which the latter is "connected". Perhaps all we need is new math.
As to the omnipresent [John 1:1], let me quote from A. Döring and C.J. Isham (Topos Foundation for Theories of Physics: IV. Categories of Systems, quant-ph/0703066 v1, p. 2, footnote 3): "The ideal monad has no windows." We cannot find it in the white area of Fig. 3.1 above, because the "white area" or 'global mode of spacetime' is also subject to non-unitary evolution from 'the ideal monad'.
If you are Type 1 physicist, please feel free to write me back. If you are mathematical physicist, please notice the following preliminary considerations regarding the ideal monad.
The so-called 'ideal monad' is a concept "opposite" to that of 'zero', since the latter is being inevitably explained by referring to some concrete concept, or 'nothing-something'. Say, 'zero bananas' means 'absence of any bananas'; or 'quantum vacuum' means absence of any real particles, etc.; more here and here. The ideal monad, however, has the unspeakable quality of being defined as 'nothing-nothing', since everything and anything we can say about it pertains inevitably to either presence or absence (zero) of some object through which it is being explicated. Thus, we cannot, not even in principle, grasp the power of the ideal monad, as the "core value" of the newly introduced "number" [phi]. The only way we can grasp the meaning of the ideal monad is relationally, by referring to something it is not, and in our case it is denoted symbolically by 'nothing-nothing' or 'hidden zero'.
Also, notice that the 'white area' (=global
mode of spacetime) in Fig. 3.1 above represents the
'potential reality'; see again the "number" [phi]
, and the discussion of non-Archimedean reality
here. Think of the potential reality as some virtual phase space of possibilities
of the type 'if A then possibly B[phi].' The term 'virtual' refers to the
counterfactual nature of this phase space, which presents us only with options
of the kind 'what could happen if we choose
A': there are no facts in the potential reality, just propensities
(though not directly observable,
Karl Popper claims that propensities are no more mysterious than forces or
fields). From the perspective of
the local mode of spacetime (the set of already-actualized
possibilities that are fixed in the past), the global mode of spacetime does not
allow for any construction of a proper 'set', because in the realm of 'potential
reality' all sets are inseparable, hence fuse into 'the set of all sets', and
become ONE entity.
To quote again from Wolfram Mathworld: "Paradoxically, there are exactly as many points c on a line (or line segment) as in a plane, a three-dimensional space, or finite hyperspace, since all these sets can be put into a one-to-one correspondence with each other." All these "points c" originate from [phi], which is in turn rooted on the unspeakable 'ideal monad'. Please recall Lucretius and Thompson's Lamp paradox. Regarding the latter, see Robin Le Poidevin [Ref. 1], and think of [phi] as a special "third" point that is being inserted between any two neighboring "points c" from the continuum, thanks to which all the "points c" can preserve their individuality (hence form a 'set'), and allow for the existence of the phenomenon of transience between the members of this set (hence we can speculate about its cardinality). But a straight question about whether [phi] itself is a member of the continuum yields an undecidable answer: YAIN or 'both yes and no' (cf. Gödel and Cohen in 1a here, and the discussion of non-Archimedean reality here). See also the song "Aleph-null bottles of beer on the wall" from Wolfram Mathworld here.
To cut the long story short, the global mode of space cannot be a proper 'metric space', since it would look, from the local mode of spacetime, like 'one single point stretched to infinity', or simply ONE entity. It is 'infinitely connected' in the sense of Whitehead's theorem, but is not 'contractible' nor 'totally vicious' due to its counterfactual and virtual nature. It offers to all physical systems in the local mode of spacetime an 'open spectrum' (I don't want to use 'set') of possibilities for their future states, while the 'chooser' of one of these possibilities is again the ONE state of the whole universe in its global mode of spacetime. The 'negotiation' with 'everything else' goes in the global mode of spacetime: see the bi-directional "talk" here, and recall the 'dynamical determinism' here. The idea of such self-determination is not original, since it is inspired from Geoffrey Chew (I started working on these issues in 1972, after studying his bootstrap philosophy) and from the catchword 'think globally, act locally'.
In three words: God is flexible. The ultimate task is to embed the 'chooser' into the spacetime manifold from the outset, since this 'chooser' -- the whole universe as ONE -- can only be 'pure geometry', as stressed by Leibnitz. Strangely enough, people hesitate to talk about 'the ideal monad', but feel comfortable to speculate extensively on its "dark" effects.
Last but not least, one may ask, what are the practical implications from all this heavy metaphysics. Well, since we definitely have brains, we may be able to detect the Gravitational Wave (there is only one GW), as well as convert the so-called "dark energy" into real energy. Not renewable energy, no. We're talking about the ultimate 'free lunch' from DDE or 'the ideal monad', whichever you prefer.
This story goes back to March 1994. Have you noticed the global climate change? Thirteen years have been wasted. In case you wonder why, read a typical reaction from an established scholar here, and my outline here.
As to my interpretation of 'the ideal monad' above, please notice that I don't endorse Robinson's Non-Standard Analysis (cf. Philip Apps web page), but instead suggest a "microscope" to zoom on Leibniz' ideas on the infinitesimal and Thompson's lamp paradox, since I cannot accept the primitive metaphysics of epsilon-delta approach to limits suggested by Cauchy and Weierstrass. This task has been on the table since the time of Titus Lucretius Carus, 96 BC - 55 BC. There should exist some sort of "cut off" at the level of 'atom of geometry', as Lucretius would have said nowadays, or else there will be no difference between large and small, inside and outside, and hence no 3-D space that can accommodate things with finite dimensions. The crux of the puzzle of 3-D space is that these 'atoms of geometry' or "points c" (see Wolfram Mathworld above) could only be transient flashes from [phi] , and hence from its "core value" called 'the ideal monad'. They are not 'physical reality out there', as we know after the Hole Argument in GR. Namely, if the atom of geometry (or "point c", if you prefer) were 'physically real', it would have been reachable by 'potential infinity' in the Thompson's lamp paradox, and then there would be countable atoms of geometry in every finite volume of 3-D space: 1099 "atoms of volume" in a cubic centimeter of space, 2.1099 "atoms of volume" in two cubic centimeters of space, n1099 "atoms of volume" in every n cubic centimeters of space, etc., as suggested by many philosophers of quantum gravity (see, for example, Lee Smolin). The problem wouldn't be resolved if you use any fixed number, googleplexes included (Leonard Susskind). 2060 years after Lucretius, we are still baffled by the obvious fact that 3-D objects have finite dimensions, but their elementary atoms of geometry are 'uncountable infinite', as elucidated by the their "number" on a line segment, a three-dimensional space, or finite hyperspace (cf. above). The only solution to this 2060- year old puzzle could come from the dual nature of [phi] : it is a member of the continuum, to the extent to which each and every "point c" is being produced by it, and at the same time does not belong to the continuum, to the extent to which it is "outside" it, being an entity "without parts". It is neither "big" nor "small". It will certainly look "small" or "large" if approached from within the local mode of space, as "bounded" by 'numerically finite but physically unattainable boundaries', be it the Planck scale or the current diameter of the expanding universe: see again the cosmological considerations of the two 'ideal endpoints' above. But if we look at [phi] from the global mode of space, it is simply ONE entity.
As Lucretius put it:
Convinced thou must confess such
things there are
Things that "have no parts" are ONE -- the Holon of the universe, kept in its global mode of spacetime. It is the ultimate 'chooser' as well, since it chooses one particular explication of [phi] out of 'uncountable infinite': one-at-a-time, without inducing any "collapse" of the main source [phi] .
Hence the continuum is what it is only because it is being created dynamically: Panta rei conditio sine qua non est.
The only entity that does not change is of course 'the
ideal monad', as it eternally stays hidden "inside" [phi]
, as 'hidden
zero'. Notice that this "unchanging" quality of the ideal monad is also the
crux of the Aristotelian First Cause or Unmoved
Mover, and it is being "transferred" to the Holon in the global mode of
spacetime: the ONE entity that captures the so-called actual
infinity. See again the song
"Aleph-null bottles of beer on the wall" from
Mathworld here, and check out the Holon
with your brain here; general considerations from Quantum Theory
In rebus mathematicis errores quan minimi non sunt contemnendi, says Bishop George Berkeley.
L.B. Szabados, Causal boundary for strongly causal spaces, Class. Quantum Grav. 5 (1988), 121-134.
R. Penrose, Conformal treatment of infinity, in: Relativity, Groups and Topology, ed. C.M. de Witt and B. de Witt, Gordon and Breach, New York, 1964.
R.P. Geroch, E.H. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. Roy. Soc. Lond. A 237 (1972), 545-567.
Robin Le Poidevin, Travels in Four Dimensions, Oxford University Press, New York, 2003, p. 121:
If between any two points in space there is always a third point, can anything touch anything else?