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Dimi Chakalov: on 12/31/08 at 14:58pm UTC, wrote David: I left three comments at George Ellis' thread on Dec. 31, 2008 @...

Lawrence B. Crowell: on 12/30/08 at 14:40pm UTC, wrote I agree that connections with string-brane concepts are not well founded at...

David Wiltshire: on 12/30/08 at 8:57am UTC, wrote Lawrence, There may be just a few too many connections in your suggestions...

Lawrence B. Crowell: on 12/30/08 at 1:45am UTC, wrote At the risk of making a monkey out of myself some thoughts have come to...

Dimi Chakalov: on 12/29/08 at 22:56pm UTC, wrote David wrote (Dec. 29, 2008 @ 02:00 GMT): "Of course the question of...

Lawrence B. Crowell: on 12/29/08 at 14:32pm UTC, wrote The connection with D3-branes is something which came to mind. To be...

David Wiltshire: on 12/29/08 at 2:00am UTC, wrote Dimi: I do not believe that the equivalence principle can be derived....

Anonymous: on 12/28/08 at 20:08pm UTC, wrote Larry: Thank you for your efforts. If you wish to comment on my efforts to...


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Steve Dufourny: "The paper of Wilczek of course is very relevant considering the idea about..." in The Noise of Gravitons

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February 6, 2023

CATEGORY: The Nature of Time Essay Contest (2008) [back]
TOPIC: From time to timescape -- Einstein's unfinished revolution by David L. Wiltshire [refresh]
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David L. Wiltshire wrote on Nov. 14, 2008 @ 15:29 GMT
Essay Abstract

I argue that Einstein overlooked an important aspect of the relativity of time in never quite realizing his quest to embody Mach's principle in his theory of gravity. As a step towards that goal, I broaden the Strong Equivalence Principle to a new principle of physics, the Cosmological Equivalence Principle, to account for the role of the evolving average regional density of the universe in the synchronisation of clocks and the relative calibration of inertial frames. In a universe dominated by voids of the size observed in large-scale structure surveys, the density contrasts of expanding regions are strong enough that a relative deceleration of the background between voids and the environment of galaxies, typically of order 10^{-10} m/s^2, must be accounted for. As a result one finds a universe whose present age varies by billions of years according to the position of the observer: a timescape. This model universe is observationally viable: it passes three critical independent tests, and makes additional predictions. Dark energy is revealed as a mis-identification of gravitational energy gradients and the resulting variance in clock rates. Understanding the biggest mystery in cosmology therefore involves a paradigm shift, but in an unexpected direction: the conceptual understanding of time and energy in Einstein's own theory is incomplete.

Author Bio

David Wiltshire did undergraduate studies in his native New Zealand, followed by a PhD in the Relativity and Gravitation Group at the University of Cambridge, UK, in the mid 1980s. After a variety of research and teaching positions in Italy, UK, and Australia he returned to NZ in 2001, where he is now Senior Lecturer at the University of Canterbury, Christchurch. He is known for his work in higher-dimensional gravity, brane worlds, black holes and quantum cosmology. His recent research has turned to the problem of dark energy, the averaging of the inhomogeneous universe and foundational implications for cosmology.

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John Merryman wrote on Nov. 15, 2008 @ 00:37 GMT
Professor Wiltshire,

Your observation about the different clock rates obtained in gravitational voids is extremely interesting. Given your insight into the current cosmological model, may I ask some questions about issues which cause me to be skeptical about the basic model;

According to theory and observations by COBE and WMAP, the expansion of space and gravitational contraction are roughly equal, resulting in large scale flat space. If this is so, then it would seem the overall expansion is negated by gravity, so that while the measure of space is effectively expanding between gravitational structure, it is also collapsing into these gravity wells at an equal rate. How is it then that the overall universe could be expanding?

It just seems more logical to me that there is some sort of process, where these two effects are opposite sides of the same cycle, which would explain why they are equal. Gravity does cause particulate mass to collapse in on itself, into ever greater density, but either through chemical reactions or pressure, this mass ignites and radiates back out across a broad spectrum of energies. Is it possible that the observed redshift and the expansion of measured space is a consequence of this expanding energy, just as gravitational collapse is a property of mass?

Specifically Einstein realized the presence of mass caused space to collapse over time, so for the Cosmological Constant to balance this effect, it would be an expansion of space, logically where it is not being contracted by gravity. As you point out, it is time which is faster in these voids. Could this affect the propagation rate of light across intergalactic space?

If this is so, the more space that light crosses, the more the effect would be compounded, creating the impression that distant sources are receding at increased rates, but still having the base rate of expansion attributed to dark matter and not one slowing at the geometric rate assumed by Big Bang Theory.

At some point light sources would seem to recede at the speed of light and this would create a horizon line for visible light and the sources thereof, but not black body radiation.

I could further speculate, but this is your forum, so I'll leave to you as to whether it is worth your effort to continue along this train of thought.

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T H Ray wrote on Nov. 16, 2008 @ 13:14 GMT
A wonderfully lucid analysis of the relation between general relativity and Mach's Principle, with a clear and natural path to link a quantum universe with quantum gravity.

The paper really does follow the best tradition of Einstein in taking, metaphorically speaking, a God's-eye view of the universe ("...average cosmic rest frame...") which holds promise for a mathematically complete physical theory. I like the treatment of geometrical congruences in context of scale. Also, among the several insights I find particularly appealing: "We can always find regional frames in which the average volume-expanding motion with deceleration is such that we cannot tell whether particles subject to such motion are at rest in an expanding space, or moving in a static space." (I characterize this same phenomenon in my own work as "volume preserving is energy conserving.")

Thank you, David Wiltshire, for a great read and stimulating ideas!


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Dr. E (The Real McCoy) wrote on Nov. 16, 2008 @ 18:00 GMT
Thanks for the wonderful paper, David!

You write, "In 1905 Einstein completely changed our understanding of the nature of time. Rather than being an absolute standard independent of the physical objects in the universe, time became an intrinsic property of the clocks carried by the objects themselves. In comparing two clocks, time could stretch and bend depending on the relative speeds of...

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David L Wiltshire wrote on Nov. 21, 2008 @ 03:58 GMT
Dear Readers

As I may not have time to answer all of the questions of the sort posted here, may I suggest that you also first check out my web pages

is an FAQ, and links to some popular articles on my work, including a New Scientist feature article from March 2008 (which is now open access), and a 37 minute podcast...

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John Merryman wrote on Nov. 22, 2008 @ 00:08 GMT
Professor Wiltshire,

Thank you for taking the time to reply and I'm having to think through much of what you have said. I would like to put one more question forward that I haven't found an answer to which isn't too confusing for me to understand, but you might be able to clarify for me;

When originally proposed, the expanding universe was assumed to be an increasing volume of space. This posed a number of problems. One, that the homogeneity of it was hard to explain and also that all galaxies outside the local cluster were observed to be redshifted directly away from our position, as if we were the center of the universe. To resolve these, it was proposed that the very "fabric" of space expanded, first in the inflation stage and then more slowly. The problem I have with this is that a standard speed of light seems to be still assumed for the post Inflation expansion. Such that if two points are x lightyears apart, if the universe were to double in size, then they would roughly be 2x lightyears apart. It seems to me this is an increasing amount of stable space, not expanding space, since it would seem that if it is space which is being stretched, not just added to, then these two points would always be x lightyears apart, because the measure would stretch with the measured. But if this is so, then the whole idea falls apart because, all things being relative, how can we say it is expanded? It just seems there are three different concepts of space; That which is inflated and carries light along with it. That which is measured by lightspeed. And that which expands according to redshift. It just doesn't pass Occam's razor for me.

I realize there doesn't appear any other reason for redshift than a form of recessional velocity, but if it was in fact some form of optical effect, it would make for a much less complicated cosmology.

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David Wiltshire wrote on Nov. 27, 2008 @ 03:58 GMT
Dear John,

Your question confuses me as I cannot quite make out what your

conceptual grasp of the notion of expanding space is. The volume of

space between clusters of galaxies on large scales increases with time. That

is what we mean by space expanding. The problems you say this introduces

- of homogeneity and isotropic expansion - are not problems at...

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John Merryman wrote on Dec. 2, 2008 @ 01:00 GMT

Sorry to confuse the issue. I do have a reasonably approximate understanding that space is not a fabric. My observations had to do with how it is described by the singularity/inflation/ expanding universe model. Without further adding to the confusion, I'm of the opinion, purely speculative, that redshift being caused by some as yet unexplained optical effect, possibly as a consequence of the expansion of radiant energy, or vacuum fluctuation, balanced by the optical effect of collapsing space caused by gravitational effects, resulting in a form of convective cycle of expanding continuity and discrete collapse, would be far less complex than the Big Bang model. Obviously it's not based on a bottom up construction of known quantities, but a top down application of whole pattern. Rather then space expanding from a singularity, it's an attempt to construct a cosmological model from infinite space expanding due to energetic instability and collapsing into vortices of accreted stability. The connection with time is that the expanding entangled energy moves into the future, as the collapsing mass is order falling away into the past. This last part may make more sense in the context of my own entry:

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Tevian Dray wrote on Dec. 7, 2008 @ 22:46 GMT
Fascinating! I've always thought that "dark energy" would turn out to be an artifact of a not-yet-fully understood theory, and I never did like the cosmological constant. Thanks for providing a possible alternative. Mind you, I'm also partial to toy models as idealizations; if your ideas pan out, my toys will be forced to become more complicated...

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Lawrence B. Crowell wrote on Dec. 17, 2008 @ 01:21 GMT
This is well written. I will say that I applaud the statement:

... the conundrum of dark energy does not involve a fluid in the vacuum of space ...

I have long been disturbed with the identification of the cosmological constant L = 8pi(rho - 3p) which then is a Ricci scalar term which gives R_{ab} = (1/2R + L)g_{ab}. Such spacetimes are source-free, but in this case a source is proposed, the vacuum energy and pressure with some equation of state w = -1.

Your statement about conservation laws also is pretty close to how I think. Cosmological spacetimes are Petrov-Pirani type O solutions which have no global Killing vector fields. As such there is no K_t*E = constant. The inability to define a global isometry means that a conservation law is not applicable for the entire spacetime. I consider this to be the big elephant in the room of physics and cosmology. I think few people seem able to wrap their minds around the prospect that, ahem energy conservation may simply not apply in cosmology. This is tied to your statement on page 3

... Since the universe is expanding, however, no time symmetry exists absolutely.

Later you write:

A universe as inhomogeneous as the one we observe cannot be adequately described by a single global frame.

I have been working on a general approach to gravity or quantum gravity which employs lattice tessellations and quantum error correction codes. This leads to a noncommutative geometry, described by quantum groups, and where there are systems of quantum groups linked by associators. Nonassociators act on elements of a quantum group G and map it to G' A:G ---> G' so that g^{-1}Ag = g^{-1}g, which is nonunitary. However, this does preserve quantum bits, and if one "coarse grains" over associators it leads to thermal states such as Hawking radiation. This is commensurate with your statement that there does not exist a single global frame to the universe. Any such frame, with an underlying quantum group of noncommutative elements, is local (quasilocal?) and linked to other regions with a different underlying noncommutative structure, a different quantum group, which classically corresponds to a different frame.

I wrote #370 on one aspect of this tessellation approach.

Thanks for the informative and I think one of the more illuminating essays here.


Lawrence B. Crowell

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Cristi Stoica wrote on Dec. 20, 2008 @ 19:13 GMT
Dear Professor Wiltshire,

I like the ideas you presented in your essay. I always thought that the anomalies that seem to require dark energy can be explained by a proper account of General Relativity, instead of the quasi-Newtonian approach combined with additional unobserved energy. Your solution, the CEP, is a good candidate for such an explanation, and seems to provide nice arguments in its support.

Best wishes,

Cristi Stoica

Flowing with a Frozen River

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Robert Sadykov wrote on Dec. 23, 2008 @ 13:13 GMT
Dear Prof David L. Wiltshire,

One of variants of the solution of a problem of gravitational energy is presented in the most modest essay The Theory of Time, Space and Gravitation.

Yours faithfully

Robert Sadykov

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Dimi Chakalov wrote on Dec. 23, 2008 @ 19:32 GMT

You wrote at George Ellis' thread (Dec. 23, 2008 @ 10:22 GMT):

"Dimi - should you read my work and have any further questions - then since George has closed his discussion, I guess you should continue over at my not-so-active thread."

Thanks a lot for your suggestion. I downloaded your essay and tried to read it, but was struck a very unclear -- to me -- introduction, and couldn't proceed.

I am asking you to help me understand the following.

In your essay, you wrote: "A simple way to understand this (quasilocal quantities - D.C.) is to recall that in

the absence of gravity energy, momentum and angular momentum of objects obey conservation laws. A conservation law simply means that some quantity is not changing with time."

Let's find out what kind of 'time' is involved in GR. George Ellis did not answer any of my arguments posted at his thread. Hope you can do better.

Please correct me if I'm wrong: The time read by a wristwatch is assumed to be a linear variable, and it is this linear variable that enters the conservation laws in the absence of gravity (Minkowski spacetime).

Q1: What -- if any -- should be the change or alteration to this linear variable, as introduced by quasi-local variables?

Further, you wrote: "General relativity is entirely local in the sense of propagation of the gravitational interaction, which is causal."

Q2: What -- if any -- should be the change or alteration to the propagation of the gravitational interaction, as introduced by quasi-local variables?

For if you mix apples (local theories) with oranges (quasi-local variables in these same theories), the confusion may be enormous, which is perhaps the reason why I couldn't finish reading your essay. Hope you can help.

My tentative answers to the questions posed above were provided in a link to my web site, in my first posting to George Ellis from Dec. 2, 2008 @ 07:02 GMT. Regrettably, your mentor neither replied to my critical comments on his proposal, nor said anything on mine.


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David Wiltshire wrote on Dec. 24, 2008 @ 10:51 GMT

>Please correct me if I'm wrong: The time read by a wristwatch is assumed to be a linear variable, and it is this linear variable that enters the conservation laws in the absence of gravity (Minkowski spacetime).

Individual variables themselves are neither linear nor nonlinear. So time is not "linear" (apart from being a parameter on the real line which is not what I mean here)....

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Dimi Chakalov wrote on Dec. 24, 2008 @ 15:16 GMT

Thank you for your professional reply. I believe we have at least one thing in common: we both want to develop and modify George Ellis' notion of 'finite infinity', but from entirely different perspectives (I will be happy to explain mine, if you're interested).

You wrote above: "Your watch measures your proper time. If you are talking about your proper time, say it."

Yes, I am talking about the proper time in STR, as read by my wristwatch. Glad you agreed that it is a local quantity.

But in GR we have a formidable conundrum: the metric has a "double role" (Laszlo Szabados, private communication), namely, it is a field variable and defines the geometry *at the same time.*

It seems to me -- please correct me if I'm wrong -- that the metric in GR is treated as a field which not only affects, but also -- at the same time -- is affected by the other fields.

If you agree, would you please elaborate on the dynamics of GR, as encoded in the phrase "at the same time"?

In STR, the proper time read by my wristwatch is a local quantity, so it seems impossible to borrow this kind of time for the dynamics of GR. The latter does include the extra "work done by gravity" (which is is absent in STR).

As you put it, energy conservation is "an intrinsically different problem from the same problem in flat spacetime."

What kind of "time" might be implied in GR, if every instant from it (a "point" in Euclidean 1-D space) is a nexus of an *already* completed -- at this same instant -- negotiation between the two sides of Einstein field equation?

I'll come back to you, after Christmas, about your efforts to tweak George Ellis' finite infinity (FI), as presented in New Journal of Physics 9 (2007) 377, and will ask you to test your vision of FI by recasting the positive mass theorems. If you succeed in replacing the conformal infinity with FI, please try to eliminate the geodesic incompleteness and the Cauchy problems for Einstein field equations, as the ultimate 'test of the pudding' for your vision of FI and the dynamics of GR.

Wishing you a nice white Christmas,


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Dimi Chakalov wrote on Dec. 25, 2008 @ 03:41 GMT

David wrote (Dec. 24, 2008 @ 10:51 GMT):

"Since any "quasilocal" quantity is an integrated regional thing, not a local quantity like a proper time measured by a clock, in any formulation one will never replace any proper time by a quasilocal variable. It is gravitational energy that is quasilocal not time; proper time is a locally measured quantity on the worldline of a particle, gravitational energy is not."

I am indeed trying to suggest that the proper time, as measured by a clock, can be replaced by a new quasi-local variable: please see my postings from Dec. 4, 2008 @ 01:30 GMT and Dec. 10, 2008 @ 14:31 GMT at Dean Rickles' thread.

The aim is to bridge GR and QM with a new form of retarded causality, and to open a "window" in GR for the energy density of the so-called empty space. My opinion on GR matches that of Einstein: "... not anything more than a theory of the gravitational field, which was somewhat artificially isolated from a total field of as yet unknown structure."

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David Wiltshire wrote on Dec. 25, 2008 @ 07:52 GMT


>the metric has a "double role" (Laszlo Szabados, private communication), namely, it is a field variable and defines the geometry *at the same time.*

Yes, I agree the geometry both affects and is affected by the other fields. The problems you are alluding to are of course all part-and-parcel of the problem of coarse-graining and averaging in describing the...

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Dimi Chakalov wrote on Dec. 25, 2008 @ 11:29 GMT

Thank you for your time and efforts.

Since you agree that the geometry both affects and is affected by the other fields, please notice that I am trying to suggest, with the so-called Buridan donkey paradox two kinds of time: "global" time for the negotiations of all particles, and "local" time for the end-result of this negotiation. Then the proper time on a particle clock, as a measurable quantity *at a point on a timelike worldline*, is being created (i) dynamically, and (ii) relationally (Machian type relational ontology), and corresponds to its "local" time. The "global" time is something that belongs to 'the whole universe en block'. To define the latter, I am trying to modify George Ellis' Finite Infinity with some well-known ideas from Aristotle.

All this comes from the solution of the measurement problem, which was suggested at the link from my first posting to George Ellis' thread (Dec. 2, 2008 @ 07:02 GMT).

In my opinion, ADM hypothesis is seriously flawed (I've elaborated extensively on my web site, with many references).

I don't like *any* coarse-graining whatsoever, since I can't see how one could approach the Hilbert space problem in quantum gravity. Will be happy to elaborate, by quoting from Claus Kiefer's research and of course Karel Kuchar's articles.

You wrote: "I would not attempt anything like a positive mass theorem based on finite infinity, until I had a better grasp of these sorts of issues."

To me the main puzzle is that we see only one "charge", called 'positive mass'. The positive mass theorems need a precise cut-off at spatial infinity, which is the crux of my efforts to modify FI with some help from Aristotle (cf. my two postings mentioned on Dec. 25, 2008 @ 03:41 GMT).

You say: "I think it is likely that there is no vacuum energy." I suggest the answer YAIN (both yes and no, in German). It's a whole new ball game, as I tried to explain here.

Sorry about my stupid remark about "white Christmas".

Best regards,


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Lawrence B. Crowell wrote on Dec. 25, 2008 @ 14:14 GMT
The ADM approach to general relativity is a calculation of a Gauss' second fundamental form for spatial surfaces. It is what Wheeler called "geometrodynamics," though it does not tell us how coordinate time is a prescription for describing the evolution of a spatial surface. All it gives us are constraints NH = 0, N^iH_i = 0, and the identificaiton of "time" with the diffeomorphisms between surfaces is unclear.

When it comes to coarse graining, this appears to be implicitely what we do with metric back reactions in Hawking radiation. We so far do not have a complete description of how the black hole quantum mechanically responds to the emission of this radiation. This is similar to the description of how a detector responds to the measurment of a quanta.

Happy Holidays, Sol Invictus, Happy Hanukkah or ... ,

Lawrence B. Crowell

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Dimi Chakalov wrote on Dec. 25, 2008 @ 18:29 GMT
I agree with the first paragraph from Larry's comment above (Dec. 25, 2008 @ 14:14 GMT). In addition to Hawking's statement that "the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity", there is a very interesting, in my opinion, paper by Kiriushcheva and Kuzmin, arXiv:0809.0097v1 [gr-qc], pp. 7-9, which brings specific arguments against such "slicing" of spacetime.

My personal (and certainly biased, if not wrong) attitude toward 'spacetime' is that it is *one* object which might be "disentangled" into '3-D space and its time' for illustrative purposes only, while its genuine dynamics -- if any -- is not traceable to anything in this *one* object: we have only constraints, and also the dubious "freedom" to choose the lapse and the shift by hand, since the latter are gauge functions (M. Alcubierre, gr-qc/0412019v1, Sec. 5).

In a way, ADM hypothesis is like showing the moving parts of a piano, but we can't "see" the player. But this is as it should be, since if we were able to "see" the player with the present-day GR (say, the source of the so-called dynamic dark energy), the latter must be some bona fide 'observable in GR', and we would be able to trace back The Beginning or the Aristotelian Unmoved Mover, whichever comes first :-)


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Lawrence B. Crowell wrote on Dec. 26, 2008 @ 14:01 GMT
The ADM approach to relativity in one sense does "spoil" the origiinal perspective of relativity where spacetime exists as whole. In both ADM and more of a block universe perspective the notion of time is strange. In either case general reltivity is not really a dynamical theory, for coordinate time is an element of the field. This element can in either case be imposed freely by the analyst. In the case of standard GR this is fixed by the coordinate condition the analyst chooses, and in ADM this is given by the freedom to choose the lapse and shift functions. These are different ways a gauge-like condition can be imposed on a problem.

The ADM approach is applicable for numerical problems, which is done through either Regge calculus or grid adaptive algorithms. It also has found use in quantum gravity calculations, usually in a Euclidean form, because the Wheeler-DeWitt (WDW) equation and the path integral formulation that results works within the techniques established in quantum field theory.

It is a standard matter that a Hamiltonian is established on spatial surfaces with some fixed "time arrow," and where equal time commutators are established. This is of course why the ADM and WDW approaches have become popular. There is of course the additional issue that gauge-like connections exist in a nonHausadorff moduli space, which means they do not satisfy Cauchy type of convergence conditions. As a result many analysts Euclideanized these problems. The intention is to examine QFT amplitudes with elliptic conditions (Atiyah-Singer indices etc) which are presumed to capture the physics in the Lorentzian configuration. Of course the topological issues of moduli are ultimately being swept under the rug. Outside of this being some sort of instanton for cosmological tunnelling states, I find this to be a far bigger adulteration than a "space plus time" ADM approach to GR.

I could go on at considerable length here, where underneath this, some correspondence between Euclidean and Lorentzian configurations, involves a Bogoliubov map between inequivalent unitary quantum groups.

There are two notions of time at work here. General relativity only defines a physical time according to the invariant interval or proper time of a particle. Coordinate time as an element of spacetime is a gauge dependent (a gauge theory for an external symmetry) quantity, which ultimately has nothing to do with any evolution. Hence the nature of block time. Yet to do QFT, we establish a Hamiltonian (an internal generator of time translations) which is attached to spatial surfaces with some time direction.

The dichotomy between these concepts of time probably lie at the heart of the obstructions we face with quantum gravity and cosmology. It is worth focusing in on this, and after all Einstein said of his annus mirabilus and his publication of relativity that he solved the problem by focusing in on the nature of time.

Lawrence B. Crowell

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Dimi Chakalov wrote on Dec. 26, 2008 @ 17:15 GMT
In connection with the last paragraph from Larry's posting (Dec. 26, 2008 @ 14:01 GMT), it seems to me that the problem of time in canonical quantum gravity should be solved along with the Hilbert space problem en bloc, since the latter is 'the test of the pudding' for the former. More in my latest posting at Claus Kiefer's thread from Dec. 26, 2008 @ 17:01 GMT.

David: Please excuse my violent curiosity. If you prefer, I will quit.

Best - Dimi

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Lawrence B. Crowell wrote on Dec. 26, 2008 @ 19:12 GMT
Before we know the Hilbert space it might be best if we have some idea of the contact structure of relativity. The action is of the form dS = pdq - Hdt, which defines a one-form. The two form d^2S = 0 = dpvdq - dHvdt (v = wedge) tells us that &H/&p = dot-q and &H/&q = -dot-p (dot = time derivative) and we get from the contant form from dS, the tangent bundle = ker(dS), the equations of motion, a'la Frobenius theorem. With ADM relativity we of course have a similar structure, but the contact manifold is not defined properly with the lapse and shift functions. We do not have well defined notion of how x = NH + N^iH_i is a well defined one-form which defines a two-form dx. The restriction of this two-form to a hyperplane defined by xvdx =! 0 (=! means not equal to). Hence in mini-superspace the meaning of a contact structure, or energy surface, of 5 = 3*2 - 1 dimensions is not apparent.

For quantum gravity the trivial O(1) "point" line bundle is replaced with a U(1) bundle by extending the symplectic structure to a Kahler one. The Hilbert space constructed by polarizations of the bundle is not apparent, mainly because the classical energy H(p,q) = E for the expectated value of the quantum langle Hrangle is absent.

There is one hint we might exploit. Fermions obey Y^2 = 0, which extends to supersymmetry (if one wants to consider that) with Q^2 = 0. This is the topology d^2 = 0 "boundary of a boundary = 0". This means that Y = ker(Q)/im(Q), for Q a boundary-like operator in BRST quantization. Hence the field Y is not QX. Thus for spinorial gravity the Frobenius theorem might not apply directly because the tangent bundle must be replaced by a closed form that is not exact.

Lawrence B. Crowell

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Dimi Chakalov wrote on Dec. 27, 2008 @ 03:32 GMT

You wrote (Dec. 16, 2008 @ 16:36 GMT):

"The issue of time is a bit slippery. I am not out to deny the existence of time, but it is something which appears to be geometrical and as such "relational." It relates kinematic entities to dynamical ones. As I see it the important question is not whether time exists, but as a relational quantity "what does it tell us?""

I believe the so-called Buridan donkey paradox mentioned above (Dec. 25, 2008 @ 11:29 GMT and Dec. 25, 2008 @ 03:41 GMT) offers a tentative answer to your very important question: time as a geometrical entity "tells us" that the world is fundamentally relational (relational ontology), in line with the Bootstrap Principle of Geoffrey Chew (Science 161 (1968) 762).

And here at David Wiltshire's thread, you wrote (Dec. 26, 2008 @ 14:01 GMT): "There are two notions of time at work here. General relativity only defines a physical time according to the invariant interval or proper time of a particle. Coordinate time as an element of spacetime is a gauge dependent (a gauge theory for an external symmetry) quantity, which ultimately has nothing to do with any evolution. Hence the nature of block time."

It seems to me that the "block time" and "block universe" (BU) are artifacts from the current incomplete GR. I've been trying to suggest, in my two postings mentioned above, the notion of 'quasi-local time' with two components, "global" and "local". The latter corresponds to 'physical time in GR', each event from which is *already negotiated* in the "global" component of time. Just try to think of this 'already negotiated' as the "duration" of the flight of a photon, from its emmission to its absorption: it is zero. It's like clapping your hands by which you produce one event of joint emission/absorption.

Hence the "dark gaps" of negotiation in the "global" component of time are completely and totally extinguished in the 'physical time in GR' (the "local" component of time), rendering the latter a *perfect continuum* that is being created dynamically and relationally. Hence we may have the "quantization" of spacetime installed from the outset.

I regret that learned about this FQXi Contest too late, on December 2nd, and haven't submitted my essay here. I can only hope that my ideas might be of some interest to David and to you.


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David Wiltshire wrote on Dec. 27, 2008 @ 08:35 GMT
Dimi and Lawrence,

I don't quite have the time to respond to every point you have raised. I shall restrict my reply to those that I think are most pertinent to the subject of my essay.

Lawrence: you have made a number of insightful comments, also at George Ellis' thread, and I concur with your statement that "the dichotomy between these concepts of time probably lie at the heart of...

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Lawrence B. Crowell wrote on Dec. 27, 2008 @ 14:05 GMT

Curiously what you say at the end connects things in an interesting way, but before then ... .

The WDW equation as a constraint equation might be a condition on the target map between a D3-brane and spacetime. I am invoking some of the ideas of Steinhardt about cosmologies associated with D3-branes which interact by their mutual connection with type II strings. Your idea of cosmological conformal principle I think has connections with conformal structure on strings & branes. This is one thing I find interesting about this.

I advise people to read Hestenes' paper. It is fairly simple in its maths and I think makes some valuable points. The zitterbewegung, which is related to what Penrose calls the "zig-zag" in his "Road to Reality" with respect to the 2-component Weyl spinor equations, is a motion of a massless fermion in a trap which confers a mass to it. The force which traps the particle has a gauge-like structure to it. The reason I bring this up is that I have thought that QCD and conformal gravity are copies of a similar structure. The couplings involved with the two theories might have some Olive-Montenen pq = hbar duality to them. So conformal gravity SU(2,2) which contains the dS and AdS spacetimes is dual to an extended QCD ~ SU(4) that breaks down as SU(3)xU(1). So a fermion is trapped in a bubble in much the same way a particle (or black hole) is confined in the hyperbolic AdS. For the electron this confinement is given by the SU(2) struture of the spinor equations, while for quarks there is the extended SU(3) gauge confinement.

To really discuss this requires use of extended Clifford algebras, but I will defer that until later. I will say that the 120-cell of icosian quaternions I work with in my paper works in this direction.

This then segues into the issue of negative energy. If we consider the grand master Dirac, he illustrated how fermionic states of negative energy are completely occupied. Zap a filled negative state with energy and you pull out a particle with negative quantum numbers but positive mass. This is the anti-electron and other anti-fermions. In a spinorial context gravity is I think similar. Negative energy states simply don't manifest themselves because they may be occupied in the same way the Dirac sea is filled. This is related in some ways to the Boulware (sp?) vacuum and energy states near horizons. Curious solutions to the Einstein field equations, such as wormholes, warp drives and Kraznikov tubes, might simply be completely occupied, just as negative mass, positively charged electron states define a "sea."

I an somewhat conservative and doubt that things such as time travel are really possible. So I think that nature in her wisdom has quantum states corresponding to these solutions filled up, which prevents them from becoming real.

Lawrence B. Crowell

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Dimi Chakalov wrote on Dec. 27, 2008 @ 14:56 GMT

Thank you for your precise and thoughtful reply from Dec. 27, 2008 @ 08:35 GMT. In the last paragraph, regarding positive mass theorems, you wrote: "It would require a very tight definition of finite infinity first, however."

You hit the nail on the head. If we employ the Aristotelian First Cause and Unmover Mover, we may have a precise "boundary" in the so-called "global" component of time, while in the "local" time this same "boundary" would look like an ever-sliding horizon extendable to infinity. The underlying motivation here is that we shall sort out the ambiguities with our notion of '3-D space', and then approach the nature of time, pertaining to this 3-D space.

You said: "Something cannot be both local and quasilocal." I believe it depends on how you understand Quantum Theory (please check out my essay on QM). Which brings me to your comment that I talk about "negotiation" in the global component of time, without defining what "negotiation" is. EPR correlations are just one example of "negotiation", but the really difficult task, to me at least, is to *derive* the Equivalence Principle -- the focus of your essay -- from some broader perspective based on Machian-type relational ontology (cf. the Buridan donkey paradox). At the end of the day, we should be able to understand the origin of the positive mass, and the mechanism by which inertial reaction forces are being generated "instantaneously" (in the "global" component of time, perhaps).

As of today, the Equivalence Principle gives us the dubious "freedom" to eliminate the energy-components of the gravitational field *at a point* (Hermann Weyl, Space-Time-Matter, Dover Publications, New York, 1951, 1922, p. 270). I cannot accept this, and neither did Einstein (quote from Dec. 25, 2008 @ 03:41 GMT above).

You are right that I should produce a "focused paper on just one topic". I will do that by the end of 2009, and will comment on your Essay extensively.

Thank you, once more, for inviting me to your thread.


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Lawrence B. Crowell wrote on Dec. 28, 2008 @ 15:28 GMT
Dimi: The quasilocality referenced with mass-energy, or nonlocality of energy, is due to the fact that

p^a = e_bT^{ab}

is a frame dependent quantity. P_a = (E, p) defines an invariant interval (mc^2)^2 = E^2 - (pc)^2, but the specific components E and p are, just as with t and x, coordinates that are not of primary physical importance. Further, the above definition of the momentum-energy component p^a will in a Stokes' law calculation give a de_b = w^c_be_e (w^c_b a connection term) which can be removed by a coordinate condition. It is in this sense that energy is no localizable.

Nonlocality of quantum states means entanglements can exist between states across any distance in either space or time, recall the Wheeler Delayed Choice Experiment. Entanglements do not correlate with causality conditions according to the spacetime variables we use to represent quantum wave functions.

I will confess that I think these two are related in some ways that we don't understand. However, at this time they are distinct concepts. It is worth noting that a quantum spin system and the structure of parallel transport of vectors in GR share a Galois structure GF(4), and are algebraically equivalent. As I say, if you want to understand physics best geometrical structures are best replaced with algebraic ones according to some functor.

Lawrence B. Crowell

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Anonymous wrote on Dec. 28, 2008 @ 20:08 GMT
Larry: Thank you for your efforts. If you wish to comment on my efforts to *think* of gravitational energy as being both localizable and non-localizable (cf. my postings above), please do it at your thread and I'll jump there, with utmost pleasure. I believe all this pertains to the nature of time in GR, since nobody has managed to separate time from energy. Surely in textbook GR there isn't such animal like the one I propose, perhaps because I address this puzzle in GR after proposing a solution to the measurement problem in QM.


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David Wiltshire wrote on Dec. 29, 2008 @ 02:00 GMT
Dimi: I do not believe that the equivalence principle can be derived. Rather it is a principle that limits the possible class of physical theories, just as the principle of relativity limits the possible kinematic relationship of particles near a point. Physics, I believe, proceeds from physical principles that limit the uncountably infinite number of mathematical structures one could imagine to...

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Lawrence B. Crowell wrote on Dec. 29, 2008 @ 14:32 GMT
The connection with D3-branes is something which came to mind. To be honest I have largely been skeptical of some of these ideas about oscillating cosmologies due to brane-brane bound states tied by strings. You come a bit closer to this idea with the mention of Ricc-flow and renormalization. The Hamilton-Perelman theory of Ricci flow centers around conformal theory, which in a string-brane setting might be induced by a target map from the brane-string sector.

As a further comment, your discussion of the nebulous nature of Ricci curvature, EP and energy, this is one reason I suspect there are deep problems with most models which have the cosmological constant as due to a vacuum energy source.

Of course at this point these are just thoughts which I have been kicking around and nothing serious at this time.

Lawrence B. Crowell

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Dimi Chakalov wrote on Dec. 29, 2008 @ 22:56 GMT
David wrote (Dec. 29, 2008 @ 02:00 GMT):

"Of course the question of initial conditions is vitally important on cosmological scales. Dimi, when you talk about "the Aristotelian First Cause and Unmover" then no doubt you are talking conceptually in such terms, though to me me such phrases do not mean anything until you can write down a physical model which somehow quantitatively matches reality."

The challenge I face with the Aristotelian First Cause and Unmoved Mover is first and foremost mathematical: it is not clear to me what particular blueprint from these notions should be sought in quantum gravity, yet I think it should be presented with pure math only, or else the First Cause and Unmoved Mover will be *physically* reachable.

I will be very difficult to provide compelling evidence that the whole physical world may be grounded on some Aristotelian "cutoff" that is nothing but 'pure math'. Not to mention the UNspeakable 'cat per se' (cf. my essay on QM mentioned above), which is also unclear in mathematical terms. But if some day I make progress, will get in touch with you.

Best - Dimi

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Lawrence B. Crowell wrote on Dec. 30, 2008 @ 01:45 GMT
At the risk of making a monkey out of myself some thoughts have come to mind with respect to this matter. Consider the conformal map g_{ab} ---> Q^2g_{ab} for the diagonal flat spacetime. Then for Q = du/dt we get the synchronous time metric

ds^2 = -dt^2 + Q^2(dx^2 + dy^2 + dz^2)

Now set Q = e^{2B} for B = B(r). The Ricci curvatures are

R_{ii} = -B_{ii} + 2(B_i)^2

where i = x, y, z. This leads to the heat equation &B/&t = nabla^2B, and we get the Ricci flow equation for nabla a gauged operator. So with the tethered grid for expansions in different regions, this sort of Ricci flow would suggest that the "equilibrium" condition obtains according to the renormalization process.

The deSitter spacetime is a case where the density of matter and energy are zero. The current state of the universe is one where the density is small, but not zero. With some calculation it is possible to estimate the De Sitter Horizon should be at 89.98 BLY while the current Cosmic Horizon is 46 BLY. This reflects the deviation from equilibrium, which the Ricci flow equations indicate the universe will eventually reach. At that point the cosmological horizon will evolve to a final state. Potentially beyond that stage the horizon will quantum decay, but that is not a classical domain. So these regions where the tethered net expands in different manners then interact or negotiate (mesh etc) in a way which obeys a Ricci flow type of renormalization.

This does I think have connections to strings. A standard string int d^s sqrt{-q} q^{ij}g_{ab} nabla_iX^a nabla_jX^b, for ij a string index and ab spacetime, will reproduce a Ricci-flow like physics for conformal transformations on the string. Now if that string is not embedded in spacetime, but is attached to two 3D-branes, the Chan-Patton factor or end of the string determines a field phi. For the cosmological constant L ~ H^2W the Hubble parameter will then be a function of H^2 ~ phi', for ' = time derivative. This then connects the time derivative of the metric in the Ricci flow equation with the value of the endpoint (field phi) of the string connecting a D3-brane. There is then some form of a target map between the D3-brane and the induced spacetime.

Lawrence B. Crowell

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David Wiltshire wrote on Dec. 30, 2008 @ 08:57 GMT

There may be just a few too many connections in your suggestions to be completely plausible (as direct connections rather than analogies), but I will think seriously about any quantitative suggestions relating to Ricci flow. So thank you for your thoughts on this. Mauro Carfora has already thought quite a bit about the Ricci flow perspective in inhomogeneous cosmology - indeed, he was thinking in such terms before Perelman's proof of the Poincare conjecture.

I guess by the "cosmic horizon" you mean what is usually called the "particle horizon"? And by "de Sitter horizon" maybe a "cosmological event horizon" (of which the one in pure de Sitter space - i.e., vacuum energy no matter - is an example). In my proposed cosmology there is no actual cosmological event horizon since the universe is in actual fact decelerating, rather than accelerating. We simply misinterpret luminosity distances and the like by a naive assumption that our locally measured spatial curvature is the same globally, and local clocks of isotropic observers everywhere are the same as ours on relevant surfaces of average homogeneity. Once one does a relative recalibration of coarsely-grained average frames relative to smaller regional "cosmological inertial frames" then a relative deceleration of regional backgrounds below the scale of statistical homogeneity, at rate typically about 10^{-10} m/s^2 accumulates to large differences in relative calibration of clocks, which we interpret as cosmic acceleration. But there is no acceleration really, and so no de-Sitter-like horizon.

Best wishes,


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Lawrence B. Crowell wrote on Dec. 30, 2008 @ 14:40 GMT
I agree that connections with string-brane concepts are not well founded at this time. I just ponder whether Ricci flow is a way to describe the evolution of cosmology to a final state as a DeSitter cosmology. Then since the central element is conformal spacetimes whether this might have connections to strings and branes. Again this is not something serious at this time, but more in the way of questions and thoughts.

If the universe is FRW then there can be no cosmological horizon, in particular if the universe is indeed decellerating. Of course it seems to me a matter of formalism (eg a sign on the acceleration) to extend what you work with inhomogeneeous regions and tethered lattices for an accelerated case. In that case a Ricci flow associated with these regions will then approach the equilibrium condition, which would be the pure deSitter cosmology.

Whether the universe is accelerating or not is of course not completely settled, which is often the case in science for some time after a discovery. There are issues on whether the "standard candle" as SN1s are as completely calibrated as we think. These happen with diminishing frequency the closer in one looks. As for the spatial curvature of space, this metrology is determined in part by Einstein lenses. Of course this is an active area of research.

We may get better data with the James Webb space telescope. Fortunately the Hubble appears to be given a life extension in the mean time. So these matters will be resolved with more data. Fortunately Obama appears pretty science friendly, a refreshing change from the goofball who has run the US the last 8 years, so we may get further cosmological data through the next decade.

Cheer, L. C.

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Dimi Chakalov wrote on Dec. 31, 2008 @ 14:58 GMT

I left three comments at George Ellis' thread on Dec. 31, 2008 @ 14:32 GMT. Please notice Comment #1, regarding the missing definition of the non-tensorial gravitational energy in a "fraction DT of time", as George put it.

I will appreciate your professional feedback.

Happy New Year.


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