Subject: Chimeras of the Differential Spacetime Manifold
Date: Sun, 29 Aug 2004 16:29:37 +0300
From: Dimi Chakalov <>
To: Ioannis Raptis <>
BCC: [snip]

Hi Ioannis,

I understood that you and Tasos Mallios are writing a book (ref. [79] in gr-qc/0408045 v3), which will soon be posted at the gr-qc e-archive (ibid., footnote 229).

You wrote that "the ADG-approach to classical and quantum gravity is truly background independent" (gr-qc/0408045 v3, p. 37), that it operates "without reference to a background 'space(time)', whether a 'continuum' or a 'discretum'" (p. 38 and p. 53), and stressed (p. 2,
footnote 3):

"In the present paper we tacitly identify the physicists' intuitive term 'spacetime continuum' with the mathematicians' notion of a (finite-dimensional) 'locally Euclidean space' ie, a manifold, looking locally like R^n and carrying the usual topological (C^0) and differential (C^[inf]) structure."

I think that the 'chimeras of the differential spacetime manifold' are hidden in the above-mentioned intuitive term 'spacetime continuum',

If you click on the link above, I suppose you'll remember much of our email and private conversations in the past three years.

To cut the long story short, may I suggest a simple test for your ADG-approach to classical and quantum gravity.

Please explain (i) the long-standing puzzle from G. Lemaître and C. Lanczos [Ref. 1], and (ii) the nature of the energy-components of the gravitational field, after H. Weyl (1922),

and T. Levi-Civita (1917),

More references at

To sum up, I believe it is a good idea to test your ADG-approach to classical and quantum gravity with some old puzzles from classical GR. If you can solve them, I believe it will be the 'proof of the pudding' that you're on the right track.

Then of course other challenges follow, known from E. Schrödinger since 1935, but that's a different story. I wouldn't like to elaborate, since a leading expert in quantum gravity stated that I "do not know enough theoretical physics to help with any research in that area."

I suppose you fully agree with this opinion, since you haven't mentioned, in some footnote or as 'private communication', the ideas I've shared with you in the past three years. Hence I presume you can run the 'proof of the pudding' without the conjecture of two modes of spacetime: a local mode (=looking locally like R^n; see footnote 3 above), and a global mode,

If possible, please do your 'proof of the pudding' as a separate paper, and post it at the gr-qc e-archive. I'll eat it all:-)




[Ref. 1] Andre Gsponer, More on the early interpretation of the
Schwarzschild solution, physics/0408100,

Abstract: Lemaître was apparently the first to make an explicit coordinate transformation resulting in the removal of the singularity at r = a = 2m in the Schwarzschild metric, while Lanczos was the first to express doubts on the physical reality of that singularity since it could be introduced or removed by a transformation of coordinates.

"Hence, by introducing rather than removing a singular term, Lanczos did the "opposite" of what Lemaître was apparently the first to have done. Furthermore, generalizing from the case of the Schwarzschild metric, Lanczos concluded his paper by stressing that [6, p.539]:

        "this example proves how little one might conclude from
        a singular behavior of the g_ij functions about a real
        singularity of the field, for it can possibly be eliminated
        by a coordinate transformation."
[6] Cornelius Lanczos, Ein vereinfachendes Koordinatensystem für
Einsteinschen Gravitationsgleichungen, Phys. Zeit. 23 (1922) 537-539.

Note: Some preliminaries can be read in C. Rovelli's Quantum Gravity (draft version, 4.8MB pdf file available from here), Sec. 2.2.5, "the meaning of the coordinates" (p. 47), and Sec 2.2.4, Meanings of time (p. 58). See also C. Kiefer's Quantum Gravity here. More from C. Isham and J. Butterfield here. The puzzle is called 'the emergence of spacetime', and I believe it requires the global mode of spacetime (also called "the power of the universe as a whole to organize itself", L. Smolin).

However, since I "do not know enough theoretical physics to help with any research in that area", I'll patiently wait to hear from I. Raptis regarding the proposed 'proof of the pudding' of his Abstract Differential Geometry (ADG), as discussed below on January 3, 2002.

D. Chakalov
August 29, 2004


Subject: How to hunt with hunters and run with rabbits?
Date: Thu, 03 Jan 2002 16:29:43 +0200
From: "Dimiter G. Chakalov" <>
To: Ioannis Raptis <>
BCC: [snip]

Hi Ioannis,

I believe there is a typo in the third sentence from your recent "Quantum Space-Time as a Quantum Causal Set", gr-qc/0201004 from Wed, 2 Jan 2002 11:52:08 GMT: "anqd" instead of  and .

You also wrote that "no theory can qualify as a physical theory proper unless it is a dynamical theory" and "the macroscopic space-time manifold should be thought of as a coarse approximation of the fundamental causal set substrata". The search for a plausible dynamics of quantum causal sets boils down tofinding a scenario to curve quantum causality (p. 3 from your PDF file), for which you and Tasos Mallios proposed some elementary particle-like entities, called 'causons', "anticipated to be closely related to gravitons, that dynamically propagate in a discrete manner in the 'curved quantum space-timevacuum' represented by the curved finitary spacetime sheaves of quantum causal sets" (ibid.).

Finally, you stressed that one of the benefits we get from applying Mallios' Abstract Differential Geometry (ADG) is that it can provide "an independence of quantum causal set theory (and of a future dynamics for it) from the smooth space-time continuum", which is "more than welcome from the point of view of modern research in quantum gravity", in the context of "the two notorious problems associated with it, namely, the 'inner product problem' [23] and the 'problem of time' [24]."

These two problems are subsequently explained as "the problem of fixing the inner product in the Hilbert space of physical states by requiring that it is invariant under Diff(M)" (the inner product problem), and "the problem ofrequiring that the dynamics is encoded in the action of Diff(M) on the space of states" (the problem of time).

We've been talking on these issues in the last twelve months, mostly by email, but I still can not understand how you're going to make it. The way I see it -- please correct me if I'm wrong -- is that your theory of quantum causality have to obey two mutually exclusive requirements, with Mallios'ADG or with any other theory of your choice:

[1]: it must be somehow "independent" from the smooth spacetime continuum, to avoid all pathologies from classical spacetime manifold (I. Raptis, gr-qc/0110064),


[2]: it must recover the very same smooth spacetime continuum 'at the end of the day', as a coarse approximation of the fundamental causal set substrata, to get the world of tables and chairs, hence solving the two measurement problems, in QM and in GR.

In other words, you have to hunt with hunters and run with rabbits *ab initio*.

That's tough. Why don't you try my ideas about two modes of time, global and local? I believe you can nicely "hunt with hunters and run with rabbits", since the *quantum* spacetime looks "continual" from the local mode of time,and "discrete" from the global mode of time. The two hypothetical modes of time can be explained with a simple metaphor at

Please see also the third case of [delta] approaching infinity, at

I believe this could be a brand new case of effectively ONE entity being 'both ONE and many' at Planck scale. Hence it (not He!) is a natural solution to the paradox 'the set of all sets is a set'. At the other limiting case of [delta] approaching zero, we recover the classical world of tables and chairs. Hence we may have different physics at these *three* scales: classical, quantum, and a brand new physics pertaining to the Planck scale (cf. the third case at the URL above). (I guess there are two more cases, that of [delta] beingstrictly zero and of [delta] being strictly  [infinity] , but I think they are indistinguishable, first, and secondly -- all this is already in the realm of [John 1:1], which could perhaps be of interest to your former high schoolteacher, Panaghiotatos Father Theodoritos.)

In all cases, however, you'd need the two modes of time, local and global. The latter is also needed as a special 'reference object' for any background-independent theory dealing with how gravity affects spacetime,

It seems to me that, by employing the two modes of time, you could solve the two notorious problems of quantum gravity, the inner product problem and the problem of time, as explained above, as well as the two measurement problems, in QM and in Einstein's GR. That's a bundle, isn't it? As a bonus, you may get the physics of the human brain, since everything said above is derived from brain science and psychology.

Please also keep in mind that these issues are by no means purely academic: billions of dollars and euros, taxpayers' money, are scheduled for chasing the putative Higgs boson(s),

Why not try it first on paper? First things first, correct?

I will highly appreciate your opinion on these matters, as well as the feedback from all physicists reading these lines.

Thank you *very* much in advance.

Wishing you all the best for 2002,

I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case I would rather be a cobbler, or even an employee ina gaming-house, than a physicist.

A. Einstein, Born-Einstein Letters, 29 April 1924