Subject: A perfectly smooth Cauchy hypersurface
Date: Fri, 4 Nov 2005 12:59:22 +0200
From: Dimi Chakalov <dimi@chakalov.net>
To: Ezra Newman <newman+@pitt.edu>
Cc: Larry Horwitz <larry@post.tau.ac.il>,
Miguel Sanchez <sanchezm@ugr.es>,
Paul Ehrlich <ehrlich@math.ufl.edu>,
Giuseppe Ruzzi <ruzzi@mat.uniroma2.it>,
Jorge Pullin <pullin@phys.lsu.edu>,
Karel Kuchar <kuchar@physics.utah.edu>,
Bob Geroch <geroch@midway.uchicago.edu>


Dear Ted,

Four days ago, on Mon, 31 Oct 2005 16:29:23 -0500, you asked me to give you a very brief idea why I consider your H-space important. I believe the idea is explained on p. 16 and pp. 7-8 from my essay on GW astronomy at

http://www.God-does-not-play-dice.net/gw.pdf

To be specific, I believe one could derive a perfectly smooth Cauchy
hypersurface at each 'end points' of the quantum-gravitational "jump" (cf. p. 16), and your H-space could serve as an additional constraint which would eliminate all pathological behavior of the dynamics of gravitational fields (cf. p. 7). This would be, however, a static, frozen solution valid for a given "snapshot" or quantum-gravitational "jump"; we may think of it as the gravitational analogue of the "collapse" in non-relativistic QM.

It seems to me that Geroch theorem [Ref. 1] does not explain the real dynamics of GR. See the problems from 'diffeomorphism freedom', as explained by Bob Geroch [Ref. 2],

http://www.God-does-not-play-dice.net/Schwarz.html#freedom

I believe some essential constraint condition is missing, and my hunch is that it can be unraveled with your H-space. Otherwise we're destined to the jungle of non-linear field equations of GR, which produce all sorts of artefacts or rather catastrophes that have never happened: Closed Time Curves (CTCs), Cauchy problems and geodesic incompleteness, and spacetime singularities, either shielded by some mythical Cosmic Censorship Conjecture (black holes) or not (naked singularities). Perhaps all these are artefacts from an essentially incomplete description of the dynamics of GR, and the latter can be unraveled with your H-space. Specifically, I believe the Cauchy hypersurfaces and Cauchy time functions of a globally hyperbolic spacetime can be made (i) *perfectly
smooth* and (ii) well-defined over a finite spacetime domain "covered" by the H-space. The linearized spacetime domain, however, is just a frozen snapshot from the global dynamics of fields: it is valid only for one quantum-gravitational "jump" (please see above). More at

http://www.God-does-not-play-dice.net/Horwitz.html

All this is, of course, just a hunch. But if we don't leave for India, how can
we discover America? :-)

I will appreciate your comments, as well as the feedback from your
colleagues.

Best regards,

Dimi
---
References

[Ref. 1] R. Geroch, Domain of dependence, J. Math. Phys., 11 (1970) 437-449.

[Ref. 2] Robert Geroch, Gauge, Diffeomorphisms, Initial-Value Formulation, Etc.
http://fanfreluche.math.univ-tours.fr//notes/geroch/geroch_notes.pdf

R. Geroch: "Einstein's equation as it stands does not admit an initial-value
formulation in the traditional sense, precisely because the gauge freedom
prohibits this." (See pp. 44-46, the vertical vector field as "connecting
vector".)

======

Note: In April 2002, Chris Isham asked me if I know what is Ted Newman's H-space (cf. [Ref. 23] in gw.pdf). Perhaps my email above can provide an answer. I believe Ted Newman's H-space can help us unravel a generic patch to the non-linearity of Einstein's GR. The current "linearized gravity" (cf. [Ref. 10] in gw.pdf) is just an oxymoron. Yes, we can use it in many cases, but only as 'shut-up-and-calculate' tool, just like von Neumann's projection postulate, since it does not make sense to describe the dynamics of gravitational fields: see the ladder on p. 6 from gw.pdf.

In practical terms, the question is this: what can we make from these long, dark, air-conditioned, L-shaped tunnels of LIGO? I suggest we convert them into wine cellars.

Don't tell me you knew nothing about it!

D. Chakalov
November 4, 2005

======

From: Dimi Chakalov <dimi@chakalov.net>
To: Ezra Newman <newman+@pitt.edu>
Subject: Re: A perfectly smooth Cauchy hypersurface
Date: Sun, 6 Nov 2005 15:28:16 +0200

Dear Ted:

> Since i doubt that anyone - at this time - understands Quantum Gravity

A brief outlook from Seth Major,

http://academics.hamilton.edu/physics/smajor/quantgrav.html

"The equations of motion are a list of expressions which must vanish. At each "instant," the degrees of freedom are the coordinate independent part of the metric known as the "3-geometry." This observation effectively solves all but one constraint -- the Hamiltonian constraint."

We can't solve the Hamiltonian constraint without the global mode of
spacetime, however. Please see

http://www.God-does-not-play-dice.net/Nicolai.html#C5

Please also note that the ideas I advocate are very old and widely known,

http://www.God-does-not-play-dice.net/Szabados.html#note

http://www.God-does-not-play-dice.net/Price.html#note

I believe you prefer math instead of philosophy, so please see how Karel
Kuchar explains the global mode of spacetime: the Aristotelian Unmoved Mover,

http://www.God-does-not-play-dice.net/Kuchar.html#1

I believe your H-space can reveal it. Will be happy to elaborate, but please don't expect some advanced math from me -- I'm just an umbrellaless psychologist from Bulgaria.:-)

Best regards,

Dimi

========

Subject: Would you bet your dog's life on noncommutative spacetime?
From: Dimi Chakalov <dimi@chakalov.net>
To: A Connes <alain@connes.org>,
Paolo Aschieri <aschieri@theorie.physik.uni-muenchen.de>,
Marija Dimitrijevic <dmarija@theorie.physik.uni-muenchen.de>,
Frank Meyer <meyerf@theorie.physik.uni-muenchen.de>,
Julius Wess <wess@theorie.physik.uni-muenchen.de>,
Peter Schupp <p.schupp@iu-bremen.de>,
Christian Blohmann <blohmann@math.berkeley.edu>,
Shahn Majid <s.majid@qmw.ac.uk>,
M Maceda <maceda@th.u-psud.fr>,
J Madore <John.Madore@th.u-psud.fr>
Cc: M Gromov <gromov@ihes.fr>,
Steven Weinberg <weinberg@zippy.ph.utexas.edu>


Dear colleagues,

Would you bet your dog's life [Ref. 1], or that of your mother-in-law, on
noncommutative spacetime?

I believe your line of thought is that the infinities in renormalizable field
theories and the singularities of general relativity are signaling a new
structure of spacetime, which can eventually be elucidated by noncommutative geometry. Then you tweak the very heart of geometry: the infinitesimal length element [Ref. 2, Eq. 2]. But what if the latter is the 'end result' from a dynamic process called 'emergence of time and space' [Ref. 3]?

I believe the idea is explained at my web site,
http://www.God-does-not-play-dice.net/download.html

and particularly on p. 16 and pp. 7-8 from my essay on GW astronomy,
http://www.God-does-not-play-dice.net/gw.pdf

Your professional comments will be highly appreciated.

Kindest regards,

Dimi Chakalov
--
[Ref. 1] Steven Weinberg, Living in the Multiverse, hep-th/0511037 v1.

"About the multiverse, it is appropriate to keep an open mind, and opinions
among scientists differ widely. In the Austin airport on the way to this
meeting I noticed for sale the October issue of a magazine called Astronomy, having on the cover the headline "Why You Live in Multiple Universes." Inside I found a report of a discussion at a conference at Stanford, at which Martin Rees said that he was sufficiently confident about the multiverse to bet his dog's life on it, while Andrei Linde said he would bet his own life."


[Ref. 2] Alain Connes, Gravity coupled with matter and the foundation of non commutative geometry, hep-th/9603053 v1.


[Ref. 3] C.J. Isham and J. Butterfield, On the Emergence of Time in Quantum Gravity, gr-qc/9901024 v1.

"The usual tools of mathematical physics depend so strongly on the real-number continuum, and its generalizations (from elementary calculus 'upwards' to manifolds and beyond), that it is probably even harder to guess what non-continuum structure is needed by such radical approaches, than to guess what novel structures of dimension, metric etc. are needed by the more conservative approaches that retain manifolds. Indeed, there is a more general point: space and time are such crucial categories for thinking about, and describing, the empirical world, that it is bound to be ferociously difficult to understand their emerging, or even some aspects of them emerging, from 'something else'."

====

Note: My email above was prompted by the speculations about some gravitational backreaction [Ref. 4], which I think are entirely unfounded. If we knew how to "attach" gravity to quantum fields, the first off task will be to solve the cosmological constant problem. Besides, if we knew the nature of quantum nonlocality, we would first try to reconcile the strong equivalence principle with QM. My impression is that Noncommutative Geometry has indeed captured 'an essential germ of truth', but I wouldn't bet even my old hat on it. To find the common ground of GR and QM, I believe we need (i) continuous Planck lenght spacetime, and (ii) a phenomenon that is both local and quasi-local; see the shoal of fish metaphor on p. 7 and p. 16 in

http://www.God-does-not-play-dice.net/gw.pdf

D. Chakalov
November 8, 2005

[Ref. 4] P. Aschieri et al., Noncommutative Geometry and Gravity, hep-th/0510059 v2.

"... one is strongly lead to conclude that noncommutative spacetime is a feature of Planck scale physics. This expectation is further supported by Gedanken experiments that aim at probing spacetime structure at very small distances. They show that due to gravitational backreaction one cannot test spacetime at Planck scale (footnote 1). Its description as a (smooth) manifold becomes therefore a mathematical assumption no more justified by physics.
--
Footnote 1: "For example, in relativistic quantum mechanics the position of a particle can be detected with a precision at most of the order of its Compton wave length l_C = l/mc. Probing spacetime at infinitesimal distances implies an extremely heavy particle that in turn curves spacetime itself. When l_C is of the order of the Planck length, the spacetime curvature radius due to the particle has the same order of magnitude and the attempt to measure spacetime structure beyond Planck scale fails."