|Subject: The super/mysterious time
Date: Wed, 26 May 2004 22:26:19 +0300
From: Dimi Chakalov <email@example.com>
To: Graham Nerlich <Graham.Nerlich@adelaide.edu.au>
CC: William G Unruh <firstname.lastname@example.org>, email@example.com,
Dear Professor Nerlich,
I don't know why is the speed of light the same in every reference frame, and am searching for some hints. I will highly appreciate feedback from you and your colleagues.
I wonder whether your idea of supertime [Ref. 1] can be related to the so-called mysterious time introduced by Bill Unruh,
Philosophically speaking, the question of persistence over time and the concept of genidentity (Genidentität), introduced by Kurt Lewin, suggest that there could exist some 'supertime', provided we elaborate a common ground for the two current views on time [Ref. 2], and examine them as complementary (not alternative) views on the nature of time. If we believe that an object *must* exist for more than one instant, then we have to consider some gap of alleged "non-existence", known as the Gap of Zen,
Well, I speculate that this gap is related to your supertime and Unruh's mysterious time. If we unravel it in today's Weltbild, perhaps we'll get closer to the questions of why is the speed of light the same in every reference frame, and what is the *shape of spacetime* [Ref. 3].
"The first of these metaphysical beliefs is well known,
in philosophy at least, as the Myth of Passage of Time. The image of time's
passage or flow is the image of an ontically preferred time (the present)
moving along a space-like dimension in which events lie ordered (which
I will call supertime)."
[Ref. 2] Christian Wüthrich, Quantum
Gravity and the 3D vs. 4D. Wednesday, May 12, 2004, 11:00-11:45,
International Conference on the Ontology of Spacetime, Concordia University, Montreal, Quebec, Canada, May 11-14, 2004, http://alcor.concordia.ca/~scol/seminars/conference/index.html
"An object is said to *endure* just in case it exists at more than one time. If we foliate four-dimensional spacetime and compare the set of enduring objects in any two spacelike hypersurfaces, the intersection of the two sets will be, in general, non-empty. In this view, thus, we expect an overlap between different times in the sense that one hypersurface shares some enduring objects with another.
"On the other hand, there are those who maintain that
temporally extended objects consist of temporal parts just as spatially
extended objects are comprised of spatial parts. According to this conception,
objects *perdure* by having different temporal parts at different times
with no part being present at more than one time. Perdurance implies that
two hypersurfaces as above do not share enduring objects but rather harbour
different parts of the same four-dimensional object."
"(W)ithout the affine structure there is nothing to determine how the [free] particle trajectory should lie. It has no antennae to tell it where other objects are, even if there were other objects (...). It *is because space-time has a certain shape that world lines lie as they do*."
I meant something pretty simple by what I wrote in Shape of Space. Consider a space which has the structure only of a differential manifold. Then, so far, no affine structure, no geodesics, no curvature, no Christoffel tensor. The transition from this to affine structure is not given by or extruded from Christoffel symbols or the 3-tensors which they represent. The affine structure is a further primitive (not definable from mere differential structure) structure which you can postulate using some representation or other of it. You can postulate it as a covariant derivative, a connection, or a tensor which can be represented in coordinates by a Christoffel symbol. But that representation makes sense only if the affine structure is already there, so to speak. True, in GR, the fundamental equation tells us (among other things) that the curvature and the "matter distribution" are co-determinate. That doesn't mean that the curvature is caused by the matter tensor. A simple analogy shows the catch in that way of thinking. The distance relations between London, New York and Sydney entail that the cities aren't on a flat surface. But the distances don't cause the shape of the surface. These places couldn't have those distances if the surface wasn't curved in the first place. The basic equation of GR places a mutual constraint on the tensors on each side of it.
I guess you know that the tensor as represented by a Christoffel symbol isn't straightforwardly like other tensors. If you don't, B. Schutz A First Course in General Relativity sec. 5.5 gives a clear account of it.
Note 1. I believe Graham Nerlich makes a very important point by stressing that "the affine structure is a further primitive (not definable from mere differential structure) structure which you can postulate using some representation or other of it."
This can be understood by recalling the ultimate puzzle of 'time per se' and 'space per se', which we try to describe (not explain) by introducing some concrete physical stuff that 'lives in time and space'. Surely there is no other way to describe time and space but by describing some concrete physical stuff that lives in time and in space. The old philosophical puzzle is that the 'time per se' and 'space per se' cannot be fully reduced to the representation of some concrete physical stuff that we place 'in time' and 'in space'. True, there is no spacetime without some concrete physical stuff. Stated poetically, there is no forest without two or more trees. The point is that the 'forest' cannot be reduced completely to the properties of the trees and their physical presentation: the 'forest' is more than the sum of the trees. It's something different that emerges by bootstrapping two or more trees into a forest. I'm sure this sounds very familiar, but try to unravel the mathematical correlate of 'the forest', and you'll see that the task is very tricky. Very. See my email to Prof. E. Rosinger here.
Subject: Re: The shape of space
Dear Professor Nerlich,
Thank you for your email from Thu, 23 Dec 2004 12:38:15 +1030.
> I meant something pretty simple by what I wrote in Shape
To introduce the affine structure, I suggest to "break" that smooth differential manifold, and insert two "modes of spacetime", for reasons outlined at
> I guess you know that the tensor as represented by a
Thank you. I personally haven't discover any straightforward presentation of any tensor in GR in the past 33 years, and I certainly cannot and do not understand the theory,
BTW David Hilbert didn't agree with GR either.
I wish you a very merry Christmas and all the best for 2005.
Download the whole web site, 4.4MB, from
Note 2: Graham Nerlich mentioned above the peculiar derivation of Christoffel symbol. His in fact replied to my 'shoal of fish' speculation, which was prompted by his clarification of the nature of the affine structure.
To me this is a very complicated issue. Recall that in Einstein's GR the gravitational "force" -- a crude and perhaps misleading analog from Newtonian mechanics -- exerted on a particle is expressed by Christoffel symbols which are obtained from the first order derivatives of the ten potentials of the gravitational field represented by ten components of the metric tensor. Now, Friedwardt Winterberg explained why these Christoffel symbols can be considered as 'pure gauge fields' [Ref. 4], and I went further by speculating that these 'pure gauge fields' are the inevitable "surplus structure" [Refs. 5 and 6] that is being introduced in Einstein's GR. Hence all I could speculate is that these pure gauge fields [Ref. 4] could be an effect of the Holon in Einstein's GR. On the one hand, they can be considered physical to the extent to which they facilitate the "force" of the gravitational field -- the 'grin of the cat' -- exerted on a particle. On the other hand, they "emerge" only if gravity is present, and totally disappear in the limit to Minkowski spacetime. Pretty spooky stuff, and I haven't been able to make it clear, as I confessed earlier.
Besides, it's Christmas and my brain
is very lazy, after a considerable amount of gintuition.
"Physical theories of fundamental
significance tend to be gauge theories. These are theories in which the
physical system being dealt with is described by more variables than there
are physically independent degrees of freedom. The physically meaningful
degrees of freedom then reemerge as being those invariant under a transformation
connecting the variables (gauge transformation). Thus, one introduces extra
variables to make the description more transparent and brings in at the
same time a gauge symmetry to extract the physically relevant content."
[Ref. 6] M. Redhead,
The interpretation of gauge symmetry, in: K. Brading and E. Castellani
(eds.), Symmetries in Physics: Philosophical Reflections, Cambridge
University Press, Cambridge, 2003, pp. 124-139.