| Subject: Upper bound on the volume of 3-D space?
Date: Thu, 16 Oct 2003 17:04:50 +0300 From: Dimi Chakalov <dchakalov@surfeu.at> To: Christina Sormani <sormani@math.jhu.edu>, Steven G Harris <harrissg@slu.edu> CC: Robert Oeckl <oeckl@cpt.univ-mrs.fr> Dear Drs. Sormani and Harris, May I ask you to help me formulate the question in the subject line: is there an upper bound on the volume of 3-D space? I need a precise quantitative formulation of this fundamental question, in the context of your recent studies on exponential length space [Ref. 1] and the causal boundary of spacetimes [Ref. 2]. It is my speculation that there should exist two *physically* unattainable "cut-offs" on the volume of 3-D space, which can not be reached with any physical process. If the cut-off toward the "small" is the Planck length, I suppose there should be another cut-off in the opposite "direction", namely, some numerically finite but physically unattainable volume of space that would set the upper bound on the volume of 3-D space. It should be physically unattainable in the sense that gravity _and_ causality would break down at this scale, and there would be no conditions for spacetime at all. Similar constraints which serve as physically unattainable cut-offs are the absolute zero temperature and the speed of light for tardyons. It seems to me that we do need to understand the nature of 3-D space and its "dark" component [Ref. 3], since current observations suggest that space it is both finite and infinite, http://members.aon.at/chakalov/Cornish.html Please note that the philosophy of the conjecture of physically unattainable cut-offs for 3-D space is very old: we obtain "points" and finite dimensions of physical bodies only because there exists a special unique cut-off for infinite series, as in differential calculus. We call this cut-off 'atom', http://members.aon.at/chakalov/xindex.html#atom It does not have any size nor parts, and contains all classically incompatible possibilities, such as 'both on and off' for the Thompson's lamp, and 'alive and dead' for Schrödinger's cat [Ref. 4]. No physical system can physically reach this ultimate cut-off, I suppose. If so, there should exist an upper bound on the volume of 3-D space as well. I will highly appreciate your feedback, and will keep it strictly private and confidential. With kindest regards, Dimi Chakalov
References [Ref. 1] Christina Sormani, Friedmann
Cosmology and Almost Isotropy, Tue, 14 Oct 2003 21:40:56 GMT,
Abstract: In the Friedmann Model of the universe, cosmologists
assume that spacelike slices of the universe are Riemannian manifolds of
constant sectional curvature. This assumption is justified via Schur's
Theorem by stating that the spacelike universe is locally isotropic. Here
we define a Riemannian manifold as almost locally isotropic in a sense
which allows both weak gravitational lensing in all directions and strong
gravitational lensing in localized angular regions at most points. We then prove that such a manifold is Gromov
Hausdorff close to a length space $Y$ which is a collection of space forms
joined at discrete points. Within the paper we define a concept we call
an "exponential length space" and prove that if such a space is locally
isotropic then it is a space form.
[Ref. 2] Steven G. Harris, Boundaries
on Spacetimes: An Outline, Wed, 15 Oct 2003 01:13:40 GMT,
Abstract: The causal boundary construction of Geroch,
Kronheimer, and Penrose has some
universal properties of importance for general studies of spacetimes, particularly
when equipped with a topology derived from the causal structure. Properties
of the causal boundary are detailed for spacetimes with spacelike boundaries,
for multi-warped spacetimes, for static spacetimes, and for spacetimes
with group actions.
[Ref. 3] Astronomers date Universe's 'cosmic
jerk', by Sharmila Kamat. New Scientist, 17:04 13 October 03,
The point when the repulsive force of dark energy overwhelmed gravity and started the accelerating expansion of the Universe that continues today has been revealed. "It happened five billion years ago," says Adam Riess, an astronomer at the Space Telescope Science Institute in Baltimore, Maryland. "That was when the Universe stopped slowing down and began to accelerate, experiencing a cosmic jerk." Astronomers discovered that the Universe is expanding at an accelerating rate in 1998 by studying exploding stars known as Type 1a supernovae. These supernova always emit the same amount of energy, so their brightness indicates how far away they are. Because the Universe is expanding, the light from the supernovae shifts towards the red end of the spectrum. The 1998 observations revealed that light from such supernovae appeared dimmer than their red shifts predicted - showing that the expansion of the Universe was accelerating. But there was always the worry that other factors, like intergalactic dust, could make the objects seem less bright. Now, Reiss's findings have set aside these doubts. Stop and start Reiss and his colleagues analysed more distant supernovae using the Hubble Space Telescope. They spent two years observing the light from six supernovae which was emitted between about 9 and 11 billion years ago. They found the light was brighter than expected from their red shifts, indicating that expansion was decelerating at that time. "Such a deceleration is vital to allow for galaxy formation," notes Michael Turner of the University of Chicago. So after the big bang, matter was still relatively dense in the Universe and therefore gravity braked expansion. But as galaxies moved farther apart, dark energy began to exert a more significant influence. For a brief period, two forces balanced and "the expansion of the Universe coasted along at a steady rate, like a car in cruise control," says Riess. But then, five billion years ago, dark energy got the upper hand. No-one yet knows what dark energy is, but supernova observations provides some constraints. "It gives a handle on how much there is," says Saul Perlmutter of Lawrence Berkeley National Laboratory in California. The new results were announced at the CERCA-Kavli Future of Cosmology conference at Case Western Reserve University in Cleveland, Ohio. Sharmila Kamat, Cleveland
Sec. 9, Connectedness of the boundary, p. 8: "Pushing
this further we might even "invert" the picture
of Fig. 1 and consider the observer's world line as surrounded by a boundary
outside of which "the quantum mechanics happens". I will not pursue this
point of view here though. Sec. 12, Conclusions and Outlook, p. 11: "For the interpretation
of quantum mechanics conclusions might be drawn, in particular the necessary
"delocalization" in time of the "collapse
of the wave function". It also implies a shift in the interpretation of
quantities in present approaches to quantum gravity (e.g. the "wave function
of the universe"). (...) A first step would be the introduction there of
boundaries with both space- and time-like components. The interpretation of amplitudes should then become clearer
once the quantum mechanics and quantum field theory situations have been worked out."
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