Subject: Amal Kumar Raychaudhuri
Date: Fri, 27 Oct 2006 13:01:23 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Jose M M Senovilla <josemm.senovilla@ehu.es>
CC: Naresh Dadhich <nkd@iucaa.ernet.in>,
     Pankaj S Joshi <psj@tifr.res.in>,
     Probir Roy <probir@theory.tifr.res.in>,
     sbp@theory.tifr.res.in,
     Petr Hajicek <hajicek@itp.unibe.ch>,
     Michele Caponigro <michele.caponigro@unicam.it>,
     Monica Dance <mcdance_1@hotmail.com>,
     Jianhua Xiao <jhxiao@hpu.edu.cn>,
     Xiao Zhang <xzhang@amss.ac.cn>,

     Luen-fai Tam <lftam@math.cuhk.edu.hk>,
     Chiu-Chu Melissa Liu <ccliu@math.northwestern.edu>,
     Shing-Tung Yau <yau@math.harvard.edu>,
     Mu-Tao Wang <mtwang@math.columbia.edu>

Hi Jose,

RE your gr-qc/0610127 v1, I believe the singularity-free cosmological solution, sought by the late Amal Raychaudhuri, is outlined at

http://www.god-does-not-play-dice.net/download.html

Can you or some of your colleagues find the math? If you succeed, please write me back.

Regards,

Dimi
--
Whether you believe you can do a thing or believe you can't, you are right
Henry Ford

 
 

================

Subject: Mens agitat molem: Spacetime singularity?
Date: Tue, 02 May 2006 17:15:19 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Jose M M Senovilla <josemm.senovilla@ehu.es>
CC: Lars Andersson <larsa@math.miami.edu>,
     Alan D Rendall <rendall@aei.mpg.de>,
     Naresh Dadhich <nkd@iucaa.ernet.in>,
     Chris Isham <c.isham@imperial.ac.uk>,
     Christian Beck <c.beck@qmul.ac.uk>

Dear Professor Senovilla,

I greatly enjoyed your review of singularity theorems in GR [Ref. 1]. May I share with you and your colleagues my thoughts on this unresolved issue, ensuing from the famous thesis by Vergil "Mens agitat molem" (The Aeneid, 6, 727). I believe it can be translated as 'Spirit moves/animates the mass' or 'der Geist bewegt die Materie'.

It seems to me -- please correct me if I got it wrong -- that the so-called singularity represents the "entry point" toward some holistic state of the whole universe,

http://www.god-does-not-play-dice.net/download.html

On the one hand, "a singularity in general relativity cannot be a point of spacetime, since by definition the spacetime structure would not be defined there", as stressed by Alan Rendall [Ref. 2]. On the other hand,
the singularity is part and parcel of the spacetime, and "one must be
prepared to probe the limits of any particular theory" [Ref. 1].

One way to tackle this paradoxical situation is to contemplate on the
possibility that "the "points on the singularity" are causally separated", as suggested by Lars Andersson [Ref. 3], but, in view of the above paragraph, it is not clear to me what kind of *manifold* [ibid., footnote 1] could possibly accommodate such peculiar 'separation of causality'.

The puzzle becomes even more difficult with the so-called dark energy
[Ref. 2], which springs from "empty space" and drives the
master/cosmological time arrow: we encounter a *logical paradox*,

http://www.god-does-not-play-dice.net/Rendall.html#Frampton

Now, if you look at the first link above, I hope you'll notice the proposition for two kinds of time, global and local. In the local time [Ref. 1],

"One can imagine what happens if a brave traveller approaches a singularity: he/she will disappear from our world in a finite time. The same, but time-reversal, can be imagined for the "creation" of the Universe: things suddenly appeared from nowhere a finite time ago."

Hence the local time is always a finite variable (say, 13.7B years after
The Beginning), and yet it points to the hypothetical holistic state of the whole universe (global time), which is NOT possible to be reached from the local time. Again, please see the first link above, and notice a similar situation with the Thompson lamp paradox (shall I elaborate?).

To cut the long story short, I believe Vergil was right: 'der Geist bewegt die Materie', only it isn't some spirit or ghost but a holistic state of the whole universe. Its "dark" blueprints can be found in QM,

http://www.god-does-not-play-dice.net/Kochen.html

http://www.god-does-not-play-dice.net/Omnes.html

It can even create a whole dark galaxy,

http://www.god-does-not-play-dice.net/Minchin.html

Let's keep our mind open to all possibilities, okay?

Since you and your colleagues are theoretical physicists, let me finish with a quote from your review [Ref. 1]:

"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. Thus the necessity of a rigorous definition of the boundary of a space-time."

Please see the ‘finite infinity’ proposal of George Ellis,

http://www.god-does-not-play-dice.net/Kirchner.html#2

It seems to me that a rigorous definition of the boundary of a spacetime will again require two kinds of time: a local time accommodating finite values which "must not be too far out, nor too far in" (George Ellis), and a global time of the very "end" of the boundary, as well as of the "singularity" itself.

Locally, we would then experience the following: The One is an unbroken ring with no circumference, for the circumference is nowhere (hence no absolute reference frame) and the center (in the local time) is everywhere. There's nowhere where God/singularity is not.

I hope you and your colleagues can put this in math.

Kindest regards,

Dimi Chakalov
--
It is extremely difficult to induce penguins to drink warm water
John Coleman
 

References
 

[Ref. 1] Jose M. M. Senovilla, Singularity Theorems in GeneralRelativity: Achievements and Open Questions, physics/0605007 v1.
http://arxiv.org/abs/physics/0605007

"In spite of all these achievements, as I was saying, Einstein and the orthodoxy simply dismissed the catastrophic behaviours (singularities)as either a mathematical artifact due to the presence of (impossible)exact spherical symmetry, or as utterly impossible effects, scientifically untenable, obviously unattainable, beyond the feasibilityof the physical world -- see e.g. (Einstein, 1939). Of course, this probably is certain in a deep sense, as infinite values of physicalobservables must not be accepted and every sensible scientist would defend similar sentences. However, one must be prepared to probe thelimits of any particular theory, and this was simply not done withGeneral Relativity at the time.
...

"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 ofthe bases, it can happen that all of them vanish and still there aresingularities.

"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 spaceaxes.

"Therefore, the only sensible definition with a certain consensus within the relativity community is by "signaling" the singularities, "pointingat them" by means of quantities belonging to the space-time exclusively.And the best and simplest pointers are curves, of course.

"One can imagine what happens if a brave traveller approaches a singularity: he/she will disappear from our world in a finite time. Thesame, but time-reversal, can be imagined for the "creation" of the Universe: things suddenly appeared from nowhere a finite time ago.

"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 disappearunexpectedly or, respectively, subito come to existence.
...

"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 notpart of the space-time but that, somehow, it is accessible from withinit. Thus the necessity of a rigorous definition of the boundary of aspace-time.
...

"However, all of the singularity theorems share a well-defined skeleton,the very same pattern. This is, succinctly, as follows (Senovilla, 1998a):

Theorem 1 (Pattern Singularity Theorem)

If a space-time of sufficient differentiability satisfies

1. a condition on the curvature

2. a causality condition

3. and an appropriate initial and/or boundary condition

then there are null or time-like inextensible incomplete geodesics.
...

"Let us finally pass to the conclusion of the theorems. In most singularity theorems this is just the existence of at least one incomplete causal geodesic. This is very mild, as it can be a mere localized singularity. This leaves one door open (extensions) and furthermore it may be a very mild singularity. In addition, the theoremsdo not say anything, in general, about the situation and the characterof the singularity. We cannot know whether it is in the future or past,or whether it will lead to a blow-up of the curvature or not.
...

pp. 12-13: "Observe that the whole universe is expanding (that is, [XXX] > 0) everywhere if t > 0, and recall that this was one of the possibilities we mentioned for the third condition in the singularity theorems: an instant of time with a strictly expanding whole universe. (...) Of course, the model is not realistic in the sense that it cannotdescribe the actual Universe -- for instance, the isotropy of the cosmicbackground radiation cannot be explained --, but the question arises ofwhether or not there is room left over by the singularity theorems to construct geodesically complete realistic universes.
...

"All in all, the main conclusion of this contribution is to remind ourselves that it is still worth to develop further, understand better, and study careful the singularity theorems, and their consequences forrealistic physical systems."
 
 

[Ref. 2] Alan D. Rendall, The nature of spacetime singularities,
gr-qc/0503112 v1.
http://xxx.lanl.gov/abs/gr-qc/0503112

"New things can happen if we go beyond the usual framework of the singularity theorems. The cosmological acceleration which is now well-established by astronomical observations corresponds on the theoretical level to a violation of the strong energy condition and suggests that a reworking of the singularity theorems in a more generalcontext is necessary. (...) The study of these matters is still in astate of flux."
 

[Ref. 3] Lars Andersson, The global existence problem in general
relativity, gr-qc/9911032 v4 (June 27, 2004).
http://xxx.lanl.gov/abs/gr-qc/9911032

"Roughly, one expects that the "points on the singularity" are causally separated ... ."

Footnote 1: "All manifolds are assumed to be Hausdorff, second countable
and C^[inf] ... ."
---------------------
 
 

Note: Abhay Ashtekar posted today his latest LQG paper [Ref. 4], in which he again conjectured about "another large, classical universe" hidden behind the singularity: "Rather than following the classical trajectory into the singularity as in the Wheeler-DeWitt theory, the state ‘turns around’."

How does it "turn around"? Easy. All you need is to choose a different math: difference, instead of differential, equations. It's a bit like the old joke about how a mathematician can catch a lion in Sahara. Read about it here.

Just don't miss the confession of Abby Ashtekar here, and my comments on the the Dirac-ADM misunderstanding here. All this mess has been produced by following blindly the math. Recall that math is a language which we use to express our ideas. If these ideas contain just a small fraction from the truth ("the ideas underlying loop quantum gravity may have captured an essential germ of truth", A. Ashtekar), it is very likely that this fraction of the truth will be displayed in a very distorted, almost unrecognizable, way, as it happened with M. Bojovald here. Then we can never reach the stage at which our math will unleash its power and guide us toward the truth.

Put it differently, the “miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics” (Eugene Wigner) can be harnessed if we do not make essential errors.

John von Neumann says: "In mathematics you don't understand the things. You just get used to them." Just four examples: Roger Penrose believes that the "ordinary unitary evolution is "local" simply because it is described by a partial differential equation (the Schrodinger equation), albeit in configuration space rather than in ordinary space." Second example, with Chris Isham, can be read here. The third example is the alleged dynamics of GR, from Stanley Deser here. And the fourth example is kindly provided by Abby Ashtekar [Ref. 4].

In all four cases, these magnificent mathematicians mix apples with oranges, by approaching 'things we don't understand' with 'things we're used to' -- the math that is at the tip of their fingers. How long will it take to recognize that, because "a singularity in general relativity cannot be a point of spacetime, since by definition the spacetime structure would not be defined there" [Ref. 2], we cannot apply the notions of time and space to probe the singularity? Or that we cannot apply the notion of time derived from the world of tables and chairs to the quantum realm, as in the case of Chris Isham?

Well, I could be wrong too. As Chris Isham put it, "You do not know enough theoretical physics to help with any research in that area."

So be it. It's a free world, there are plenty of Barbies for everyone.
 
 

D. Chakalov
May 3, 2006

------------

[Ref. 4] Abhay Ashtekar, Gravity, Geometry and the Quantum, gr-qc/0605011 v1.

"If the singularity is resolved, what is on the ‘other side’? Is there just a ‘quantum foam’, far removed from any classical space-time, or, is there another large, classical universe?
...

"In the Hamiltonian framework, the dynamical content of any background independent theory is contained in its constraints. In quantum theory, the Hilbert space H and the holonomy and (smeared) triad operators thereon provide the necessary tools to write down quantum constraint operators. The physical states are solutions to these quantum constraints.

"Thus, to complete the program, one has to: i) obtain the expressions of the quantum constraints; ii) solve the constraints; iii) construct the physical Hilbert space from the solutions (e.g. by the group averaging procedure); and iv) extract physics from these physical sector (e.g.,
by analyzing the expectation values, fluctuations of and correlations between Dirac observables).
...

"However, then it bounces. Rather than following the classical trajectory into the singularity as in the Wheeler-DeWitt theory, the state ‘turns around’. What is perhaps most surprising is that it again becomes semi-classical and follows the ‘past’ portion of a classical trajectory, again with 
[xxx], which was headed towards the big crunch.
...

"In the distant past, the state is peaked on a classical, contracting pre-big-bang branch which closely follows the evolution dictated by Friedmann equations. But when the matter density reaches the Planck regime, quantum geometry effects become significant. Interestingly, they make gravity repulsive, not only halting the collapse but turning it around; the quantum state is again peaked on the classical solution now representing the post-big-bang, expanding universe.
...

"The suggestion from LQC is that a repulsive force associated with the quantum nature of geometry may come into play and could be strong enough to counter the classical, gravitational attraction, preventing the formation of singularities.
...

"Is there a general equation in quantum geometry which implies that gravity effectively becomes repulsive near generic space-like singularities, thereby halting the classical collapse? If so, one could construct robust arguments, establishing general ‘singularity resolution theorems’ for broad 
classes of situations in quantum gravity, without having to analyze models, one at a time."
 


 

Subject: Off-shell fiducial boundary conditions?
Date: Mon, 05 Jun 2006 13:50:48 +0300
From: Dimi Chakalov <dimi@chakalov.net>
To: Lisa Lowe <lllowe@ncsu.edu>
CC: David Brown <david_brown@ncsu.edu>,
     Ronald Fulp <fulp@math.ncsu.edu>,
     Dae-Il Choi <choi@milkyway.gsfc.nasa.gov>,
     Laszlo Szabados <lbszab@rmki.kfki.hu>,
     Gregory B Cook <cookgb@wfu.edu>,
     Stuart L Shapiro <shapiro@astro.physics.uiuc.edu>,
     Saul Teukolsky <saul@astro.cornell.edu>,
     James W York Jr <york@astro.cornell.edu>,
     Karel Kuchar <kuchar@physics.utah.edu>,
     David Robinson <david.c.robinson@kcl.ac.uk>,
     George Ellis <ellis@maths.uct.ac.za>

Dear Dr. Lowe,

I read with great interest your recent paper [Ref. 1], and am deeply impressed by your results depicted in Fig. 1.

Since you and your co-author Dr. Brown stressed that you've used just initial data that violates the constraints, but have put aside "the very important issue of boundary conditions" [Ref. 1, p. 3], I wonder if you would be interested in developing a technique for fixing custom-made "patches" in terms of off-shell fiducial boundary conditions. It seems to me that the ideas of George Ellis could be very fruitful [Ref. 2], if you choose to use integration to a null surface,

http://www.god-does-not-play-dice.net/Stachel.html#3

You also wrote: "Although we cannot rule out the possibility that the combined system Eqs. (14), (15) is mathematically ill-defined in some sense, ..." [Ref. 1]. Perhaps your model is correct but not complete.
Please see

http://www.god-does-not-play-dice.net/Price.html#infinitesimal

http://www.god-does-not-play-dice.net/Senovilla.html

Regards,

Dimi Chakalov
--
http://www.god-does-not-play-dice.net
http://www.god-does-not-play-dice.net/download.html
 

References

[Ref. 1] J. David Brown and Lisa L. Lowe, Modifying the Einstein
Equations off the Constraint Hypersurface, gr-qc/0606008 v1, 2 June 2006
http://arxiv.org/abs/gr-qc/0606008

"The Einstein equations separate into a set of evolution equations and a set of constraints. The evolution equations are partial differential equations (PDE’s) that determine how the gravitational field variables g_ab (the spatial metric) and K_ab (the extrinsic curvature) evolve forward in time. The constraint equations are PDE’s that the field variables must satisfy at each instant of time. From a Hamiltonian point of view, the evolution equations define solution trajectories in phase space with coordinates g_ab and momenta K_ab. Physical trajectories are those that lie in the constraint hypersurface, or subspace, of the gravitational phase space.

"Einstein’s theory of gravity is a "first class" theory, that is, the time derivatives of the constraints are linear combinations of the constraints. This property implies that, analytically, the constraints will hold at each instant of time if they hold at the initial time. However, for numerically generated solutions of the theory the initial data will not satisfy the constraints precisely and numerical errors will kick the phase space trajectory away from the constraint hypersurface. This is a critical problem for numerical modeling because the Einstein evolution equations, as they are usually written, admit solutions that rapidly diverge away from the constraint hypersurface [1, 2]. Any numerical scheme that evolves the gravitational field data using the evolution equations in one of their traditional forms will eventually fail to produce physically meaningful results. Inevitably the numerical solution will choose to follow a trajectory that violates the constraints.

"A number of strategies have been devised to address this problem. One approach is to modify the theory off the constraint hypersurface by adding linear combinations of constraints to the evolution equations [1, 3, 4, 5, 6, 7]. In this way one hopes to alter the solution trajectories so that they are better behaved away from the constraint hypersurface. We will use the terminology "off-shell" to refer to solution trajectories that lie off the constraint hypersurface.

"The strategy discussed in this paper is of this sort. We add terms proportional to the constraints to the Einstein evolution equations in such a way that the evolution equations for the constraints can be freely specified. In principle we can eliminate all constraint violating modes by demanding, for example, that the time derivatives of the constraints should vanish. The price we pay for this degree of control over the unphysical, off-shell solutions is that the terms added to the evolution equations are nonlocal. They are determined through the solution of an elliptic system of PDE’s.

"Another strategy for keeping a numerically generated solution from diverging away from the constraint hypersurface is constrained evolution. In this scheme the constraints are used in place of certain evolution equations to update some of the gravitational field variables in time. This approach has worked well for spherically and axisymmetric problems [8, 9, 10, 11, 12].

"A closely related idea is constraint projection [5, 13, 14]. With constraint projection one evolves the full set of field variables using the evolution equations, then periodically (perhaps every timestep) solves the constraints to project the solution back to the constraint hypersurface. Both constrained evolution and constraint projection require the solution of the constraint equations during the course of evolution.

"For these approaches to be viable, the constraints must be expressed as an elliptic system of PDE’s. From a computational perspective, our strategy is closely related to constraint projection since we also solve an elliptic system of PDE’s at every (or nearly every) timestep. In fact, the PDE’s that we solve are the linearized Einstein constraints.
...

p. 3: "Thus, for now, we intentionally avoid facing the very important issue of boundary conditions. We use initial data that violates the constraints.
...

p. 4: "The results displayed in Fig. (1) show that we have indeed modified the equations of motion off-shell in such a way that unwanted growth in the constraints is eliminated.

"Ultimately, what we would like to show is the ability to prevent constraint growth in the first place. Our preliminary attempts to demonstrate this ability have not been completely successful, for reasons that we suspect are purely numerical. Although we cannot rule out the possibility that the combined system Eqs. (14), (15) is mathematically ill-defined in some sense, the problems that we have encountered appear to be caused by numerical issues."
 
 

[Ref. 2] George F R Ellis, Cosmology and Local Physics, gr-qc/0102017
v1; 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."

[See T. Padmanabhan, gr-qc/0609012 v2, Sec. 6, and D.C. Robinson]

=========
 
 

Note: To get started, read about the generic ambiguities in the "dynamics" of GR -- which inevitably occur due to the 'diffeomorphism freedom' (R. Geroch) -- in the review article by Gregory B. Cook "Initial Data for Numerical Relativity", gr-qc/0007085 v1, Sec. 2.1. Then compare this misfortunate 3+1 decomposition to the presentation of Einstein equation by M. Frank in gr-qc/0203100, then look at the "balloon" metaphor from Ned Wright here, and ask yourself a simple question: where is the direction of the "normal observer" (G.B. Cook, op. cit., Eq. 7) or the timelike unit vector (M. Frank)? It "points" toward the putative global mode of spacetime and the source of the "dark" energy.

Aren't all distances between the dots on the balloon increasing en bloc? Yes they are, but with respect to what? We need a brand new "background" to explain this new (to GR) dynamics of the spacetime itself, since GR textbooks do not permit any change of the spacetime itself. The global mode of time is designed to explain the global dynamics of spacetime due to its "dark" energy: it "expands" the spacetime in the local time with respect to the global time, and produces the elementary timelike displacement (the "tick" of the film reel) for the whole universe.

Stated differently, the dualistic conception of time [Ref. 3, p. 3] is not applicable to the alleged "dynamics" of GR, since we can only work with an "internal" phase space that is "dependent on the physical attributes of the clock", while the fundamental clock that is presumed to be ‘external’, existing independently of it, is excluded from GR from the outset, by virtues of the basic postulates in GR.

NB: To avoid misunderstandings, let's try to explain the dualistic conception of time [Ref. 3, p. 3] with the following metaphor: imagine you sitting in a movie theater, and watching a static picture on the screen, [X] . It corresponds roughly to the Cauchy formulation of GR, in which you've foliated spacetime into three-dimensional spacelike hypersurfaces of constant "time"  t  (see, e.g., Oliver Rinne, gr-qc/0606053 v1, Sec. 1). You cannot observe any dynamics of  [X]  itself, since the spacetime itself doesn't "move", as we know from Bob Geroch. But you are aware of the fact that  [X]  flickers 24 times per second (in the movie reel), which corresponds metaphorically to the fundamental clock that re-creates  [X]  in the global mode of spacetime. Yet you don't hesitate to use this frozen snapshot of  [X]  to apply the "rules" of Dirac-ADM to "evolve" data on it, in line with the "laws of an instant in canonical gravity" (K. Kuchar) and the constraint equations: "the constraints will hold at each instant of time if they hold at the initial time" [Ref. 1].

Question: Is your Dirac-ADM "dynamics" parameterizable by the fundamental background time [Ref. 3, p. 3] as well? Nope. Your 3-D spacelike hypersurfaces of constant "time"  t  is just one instant, just one "tick" of the fundamental background time. This global/background time provides the fundamental clock that re-creates  [X] -- not 24 times per second, as with the movie reel, but infinite number of time per second, as we know since the time of Zeno. Hence Mother Nature has avoided all the Cauchy problems, Closed Timelike Curves (CTCs), and singularities, either shielded with some "horizon" (black holes) or with metaphysical postulates (Penrose's Cosmic Censorship Conjecture, in the case of timelike naked singularities), while you're stuck with them for over fifty years now, as I tried to explain to my 12-year daughter here. (I couldn't, however, explain to her how the metric field changes "in time", and at the same time determines its own next temporal state in which it will evolve after a fraction of time, that is, "after" completing the bi-directional "talk" between matter and geometry, on some frozen spacelike hypersurfaces of constant "time" [Ref. 3, p. 12]. More on GR dynamics in the context of GWs here, and on the self-acting "dark" energy here.)

I believe all this can be used to provide some custom-made "patches" by means of off-shell fiducial boundary conditions, as suggested above. These fiducial boundary conditions pertain only to a snapshot of "isolated" regions of spacetime, only to some fixed 3-D domain in the local mode, as suggested before. Hopefully, you may be able to reduce the ambiguities of the alleged dynamical evolution and eliminate the pathologies for your particular numerical exercise.

But you should always think of such 'fixed 3-D domain' in terms of global and local modes of spacetime, since it is not possible to reach the "edge" of such domain from the local mode: numerically finite but physically unattainable "boundaries" of spacetime (see below). If that were possible, you would be able to physically enter the perfect fluid of the omnipresent "dark" energy of  [X] . See its "dark" QM counterpart here.

Have questions? Don't hesitate!
 

D.C.
June 6, 2006
Last update: June 16, 2006

-------

I was asked today to give an example for 'numerically finite but physically unattainable "boundaries" of spacetime'. Please read the interpretation of the Planck scale in Robert Wald's textbook General Relativity (University of Chicago Press, Chicago, 1984), p. 471. It is like a wall beyond which the usual continuum description of spacetime cannot survive. It is a numerically finite "boundary" that cannot be reached by any physical object, or else it will slip into some "extra-Planck scale" realm, which marks the end of known physics. Think of the latter as the absolute zero "temperature", which isn't a temperature per se anymore.

Another example: the cosmological time must have a fixed, point-like "beginning" some 13.7 billion years ago, yet the beginning of time, if approached backwards along the deflation time, cannot reach "time zero" for any finite amount of time, as read by your wristwatch.

Why? Because this local time can only tend asymptotically toward "time zero". Hence many prominent physicists are not ashamed to write that the universe has started "asymptotically" from it, although such expression certainly doesn't make sense. (Notice also that there are two kinds of time in the singularity hypothesis: a finite time for the "brave traveler" [Ref. 1], and infinite time for his terrified mom watching him diving into the black hole. It doesn't make sense either, but people have somehow used to it.)

Yet we use "time zero" -- as we have no other choice -- to mark the time interval of the history of the universe in the local time. The history spans inevitably over a finite time interval, hence one of its edges must be fixed in the past, correct? Only we cannot literally place it on "time zero" that marks the end of known physics, because this local time always requires an instant in the past for its determination, while 'time zero' is by definition an instant "before" which there is no spacetime at all. Hence we arrive at the Aristotelian First Cause and the Heraclitean universal time arrow (sorry for repeating this all over again).

So, given the idea expressed before, let's say more on the fiducial boundary conditions. They are fiducial because they apply to some particular case study (custom made patches), and because they fix a case-specific boundary that cannot be reached "from within", just as you cannot reach the state of the Thompson lamp in the instant '2 min' by following the causal changes of the state of the lamp in the local time.

Here of course you might say, 'but if I'm going from London to Brighton, why would I not be able to arrive in Brighton?' Because in the local time, Brighton is infinitely miles away from you, hence your journey can only tend asymptotically toward Brighton. It is effectively at null infinity -- see George Ellis [Ref. 2], as well as the "ideal points" [R.P. Geroch, E.H. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. Roy. Soc. Lond. A 237 (1972), 545-67] defined as "a future (past) ideal endpoint for every inextensible future (past) timelike curve, in such a way that it only depends on the curve’s past (future)" (quoted after Jose L. Flores, The Causal Boundary of spacetimes revisited, gr-qc/0608063 v1; see also Sec. 9, Causal ladder and the boundary of spacetimes, p. 26). 

However, Brighton is at the edge of a finite spacetime domain as well, like the state of the lamp in the instant '2 min' in the Thompson lamp paradox. Notice that the instant '2 min' belongs to the global time, hence it just "looks" like a finite time interval (compare it to the proper time of the observer falling into the so-called black hole [Ref. 5]), while we're confined into the local time, by following the actual state of the system, hence we can never actually reach the instant '2 min', because our local time can never last forever, i.e., be infinite (compare it to the proper time of the distant observer watching his colleague falling into the so-called black hole [Ref. 5]). In other words, the instant '2 min' is an 'ideal endpoint' (see above) in the global time, which is needed as some cut-off (recall G.F.R. Ellis' 'finite infinity' proposal) to define the actual dynamics of matter fields in the local time.

This is the only way to obtain the dynamics of GR, in my opinion. Once we apply it to the cosmological time, the solution to the "time zero" paradox is obvious: replace "time zero" with the instant '2 min' (an 'ideal endpoint' in the global time), such that its "duration" will always be finite (e.g., 13.7 billion years after The Beginning), and then recall that The Beginning can never be actually reached by any physical system in the local time, much like the story told by Dirac [Ref. 5]. Once created, the spacetime of the universe is "bounded" in the past and the future by two 'ideal points', which can never be reached 'from within', that is, from the local time. Hence the universe is both finite and infinite, depending on your point of view.

Hence "singularity" and "time zero" do not exist in the local time (see my email to J. Senovilla above), and we can use the global time as some fiducial "background", without resorting to any fixed structure in the local time (see ‘Strong Background Independence’ in I. Raptis, gr-qc/0606021 v2, and the recent papers by J. Christian [Ref. 3] and L. Smolin [Ref. 4]).

The whole magic, then, is to make the unphysical cut-off of the off-shell boundary "edge" disappear in calculations, after removing all the pathologies from the diffeomorphism freedom (recall how the fixed "time zero" disappears in the simple mathematical example here).

Generally speaking, there is too much freedom in the phase space of GR, and all we can do is to seek some custom-made constraints from the global mode of spacetime. Think of these constrains as some "context" that is supplied from the global mode of spacetime, for the case in consideration only. Look again at the QM counterpart of "context" here.

Can you suggest some 'virtual geodesic path' formulation of GR, by introducing a brand new (the putative global time) degree of freedom to the phase space?

Perhaps the Hamiltonian formulation of GR (Dirac-ADM) needs to be amended by some "context-like" selection mechanism, which would make some geodesic paths "actual", while keeping the rest of them "potential", after which you pick up the "actual" (=pertaining to the particular case) geodesics and decompose them into a localized snapshot of 3+1-D spacetime (3-D spacelike hypersurfaces of constant (frozen) "time"; see, e.g., Oliver Rinne, gr-qc/0606053 v1, Sec. 1). Put it differently, the principle of general covariance (active diffeomorphism) isn't enough. Please see NB above.

To sum up: in the local mode of spacetime, we're observing a finite, in size and duration, "projection" of The Universe, and in this frozen snapshot we're hooked on the "time zero" and the "dark" stuff, but we cannot physically reach them from the local mode, because they exist as potential reality in the global mode of spacetime: 'numerically finite but physically unattainable "boundaries" of spacetime'.

I've left here many points unexplained, assuming that you've read the relevant web pages. Notice also the treatment of 3-D space in Einstein's GR, which I think amounts to introducing a fixed background of bare points with fixed spatial relations (inside vs outside). Do you like the double standards in the treatment of time and space?

Should you have further questions, please don't hesitate.
 

D.C.
June 8, 2006
Last update: August 31, 2006
 

[Ref. 3] Joy Christian, Passage of Time in a Planck Scale Rooted Local Inertial Structure, gr-qc/0308028 v4.

p. 3: "After all, a classical clock is supposed to register the time external to itself, and the phase space of its evolution is an internal space, dependent on the physical attributes of the clock. In other words, the time measured by such a clock is presumed to be ‘external’, existing independently of it, whereas its dynamic evolution, although parameterizable by this background time, is viewed to be ‘internal’, specific to the clock itself. Such a dualistic conception of time has served us well in special relativistic physics.
...

p. 12: "...the ‘block’ view of time [20], which is widely thought to be an inevitable byproduct of Einstein’s special relativity.

"According to this ‘block’ view, since in the Minkowski picture time is as ‘laid out’ a priori as space, and since space clearly does not seem to ‘flow’, what we perceive as a ‘flow of time’, or ‘becoming’, must be an illusion. Worse still, in Einstein’s theory, the relativity of simultaneous events demands that what is ‘now’ for one inertial observer cannot be the same, in general, for another. Therefore, to accommodate ‘nows’ of all possible observers, events must exist a priori, all at once, across the whole span of time [36]. As Weyl once so aptly put it, "The objective world simply is, it does not happen" [37]. Einstein himself was quite painfully aware of this shortcoming of his theories of relativity -- namely, of their inability to capture the continual slipping away of the present moment into the unchanging past [38].

"To be sure, the alleged unreality of this transience of ‘now’, as asserted by the ‘block’ view of time, is far from being universally accepted (see, e.g., [2, 39, 40]). However, what remains unquestionable is the fact that there is no explicit assimilation of such a transience in any of the established theories of fundamental physics."
 

[Ref. 4] Lee Smolin, The case for background independence, hep-th/0507235 v1.

"The background consists of presumed entities that do not change in time, but which are necessary for the definition of the kinematical quantities and dynamical laws.

"The most basic statement of the relational view is that

"R1 There is no background."
 

[Ref. 5] See Ch. 19, "Black Holes", in Paul Dirac's General Theory of Relativity (John Wiley & Sons Inc., New York, 1975), and compare it with G. Lemaître and C. Lanczos here. More from A. Loinger.
 
 


 

From: Dimi Chakalov <dimi@chakalov.net>
To: Vera Rubin <rubin@dtm.ciw.edu>
Subject: Cold Dark Matter & Dynamic Dark Energy
Date: Thu, 15 Jun 2006 14:48:01 +0300

Dear Professor Rubin,

You've been quoted at
http://www.newscientist.com/article.ns?id=mg18524911.600

"If I could have my pick, I would like to learn that Newton's laws must be modified in order to correctly describe gravitational interactions at large distances."

May I ask you for details and reference. Thank you for your time.

I've been trying to suggest a hypothesis on the origin of CDM (VIRGOHI 21 included) and DDE, and would like to quote your opinion.

Best regards,

Dimi Chakalov
-- 
http://www.God-does-not-play-dice.net
http://www.God-does-not-play-dice.net/download.html

------

Note: If neuroscientists were following the "logic" of their colleagues from the established theoretical physics community, they would have to postulate some totally invisible, hence "dark", computer in the human brain, which could correlate some 100 billion neurons and 60 trillion synapses. Such "dark" computer would also have to remain unchanged during the life of all people, since it would be the "neural correlate" of the human self.

Instead, I suggest we model the universe as a human brain, and seek the holistic, hence "dark", effects of the Holon in the global mode of spacetime, as explicated with two opposite remnants: implosion for the cold dark matter, and explosion/expansion for the dynamic dark energy. See also the "spin" of the universe here, which too is an effect/remnant from the Holon (sorry for repeating this again).

In both cases, the universe and the human brain, we have the same holistic effect of spacetime -- 'think globally act locally' --  only the physical stuff through which it is being explicated is very different. Like a song being played with two entirely different instruments, as I tried to suggest to Prof. Chris Isham during our last meeting on March 9, 2006.

It is extremely difficult to induce penguins to drink warm water, says John Coleman.
 

D. Chakalov
June 15, 2006
 


 
 

From: Dimi Chakalov <dimi@chakalov.net>
To: Graham Nerlich <graham.nerlich@adelaide.edu.au>
Subject: Re: Cold Dark Matter & Dynamic Dark Energy
Date: Sat, 17 Jun 2006 15:03:08 +0300

Dear Graham,

Thank you for your reply. I hope you haven't missed the explanatory note 
after
http://www.god-does-not-play-dice.net/Senovilla.html#Rubin

I believe we need new ideas, because nearly 96% from the stuff in the universe is "dark", and GR cannot handle it. Hence I dare to suggest a new kind of causality, which is relevant to 'What GR is all about' ("The Shape 
of Space", Sec. 10.12).

I can demonstrate this (new?) kind of causality with your brain, by offering you some simple exercises. Please tell me if you're interested.

You wrote ("What Spacetime Explains", p. 34): "It is very widely believed 
that a single principle about causality lies at the foundations of the relativity theories. It is the Limit Principle, that nothing outstrips light."

Surely "nothing outstrips light" in the local mode of spacetime, while the 
bi-directional "talk" of matter and geometry "takes place" in the global 
mode of spacetime,
http://www.god-does-not-play-dice.net/download.html

RE your "What Spacetime Explains", my speculations are relevant also to 'Cause and spacetime' (Sec. 7.6, pp. 180-184), 'Cat time be finite?' (Sec. 
10), and 'How to make things have happened' (Sec. 11), as well as to the 
flow of time, the symmetry/asymmetry of time, and backward causation (pp. 228-229).

Which is why I asked you for a copy from your recent draft paper and talk at Cambridge U. -- I suppose you've elaborated on the ideas discussed at my web site. I don't want to invent the wheel.

I wish you a nice white winter.

As ever yours,

Dimi