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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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 particles over their histories."

It is important to note that in his 1912 paper, Einstein never stated that time is the fourth dimension, but rather he and Minkowski wrote x4=ict, implying dx4/dt=ic: the fourth dimension is expanding relative to the three spatial dimensions, distributing locality and fathering time--the central postuale of Moving Dimensions Theory. Godel et al. had problems with the block universe and frozen time General Relativity implied, so thank goodness this block universe does not exist--thank goodness that time is not frozen and we have free will! MDT has unfrozen time and progress in theoretical phyiscs, liberating us from the antitheory regimes! Thank goodness change--which marks all realms of physics and measurement (as there is no physics without measurement)--has finally been woven into the fundamental fabric of spacetime with MDT's simple postulate and equation: dx4/dt=ic. For without change, how would we be able to change the antitheory establishment, which is enforcing the block universe alongside the anthropic principle to justify their supposed inevitable permanence. Unfortunately, all this is done at the expense of physics and physicists.

You write, "The quasilocality of gravitational energy and momentum is very different to a nonlocality of interactions

in flat spacetime which some physicists occasionally postulate and which is anathema to many, myself included."

Nonlocality is a fact of quantum mechanics. Perhaps I am misreading this. Do you not agree with the nonlocality observed in quantum mechanics, from the foundational double-slit experiments to Aspect et als.' experiments supporting quantum entanglement and action-at-a-distance?

Moving Dimenisons Theory also accounts for nonlocality with a *physical* model--the fourth dimension is expanding relative to the three spatial dimensions, distributing locality. Hence two ageless photons, which shared a common point of origin, can yet be entangled, as they inhabit the same locality in the fourth expanding dimension, no matter how far apart they travel. More on this can be found in my paper "Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics by Elliot McGucken": http://fqxi.org/community/forum/topic/238

And too, MDT's fundamental universal invariant--from where relativity, entropy, time, and quantum mechanics all arise--the fourth dimension is expanding relative to the three spatial dimensions at the rate of c (dx4/dt=ic)--naturally accounts for the gravitational slowing of light and time, as well as the gravitational redshift, as shown in the attached figures. It also expalins why there is no need to quantize gravity, as while the fourth dimension expands as a probabilistic wavefront with a wavelength given by the Planck length, space is continuous. Perhaps this novel, more realistic view of time in General Relativity, which also accounts for time and all its arrows, entropy, and quantum mechanics' nonlocality and entanglement, while providing a deeper universal invariant from where all of relativity can be derived, can help you further your work!

Well, I hope MDT might aid you in your work, as the goal of physics is to unify diverse *physical* phenomena with simple, beautiful, common principles reflecting the deeper truth of our physical reality. And this is best accomplished with simple *physical* postulates representing *physical* realities that can be summed up with mathematical equations: dx4/dt=ic.

Best & thanks for the paper,

Dr. E (The Real McCoy)

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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

http://www2.phys.canterbury.ac.nz/~dlw24/universe/

There
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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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

http://www2.phys.canterbury.ac.nz/~dlw24/universe/

There
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 of an interview on Radio NZ. However, the FAQ does not yet address anything relating directly to the Cosmological Equivalence Principle discussed in the essay, and I probably won't have time to update it for a while.

I will respond to some of the questions/statements of John Merryman and "E Real McCoy"...

---------------------------------------------------
----

John Merryman:

>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... How is it then that the overall universe could be

>expanding?

The statement "the expansion of space and gravitational contraction are roughly equal" is incorrect, even in the standard Friedmann-Lemaitre cosmology. So there is no problem that overall the universe is expanding.

The observation you are in fact referring to is the angular scale of the Doppler peaks in the CMB anisotropy spectrum. What is actually measured is a back body spectrum with tiny fluctuations in its mean temperature. A cosmological model is needed to interpret the statistical spectrum of angular correlations in these fluctuations, as they were laid down at the time of last scattering when the universe was a few hundred thousand years old. The CMB photons have propagated to us over the largest distances possible in the intervening 14-15 billion years (by our clocks). Thus any inferences made from the CMB are always limited by the assumptions of a cosmological model for the expansion history of the universe between last scattering and now.

The "theory" which you are citing is in fact the standard cosmology based on assuming homogeneous isotropic expansion and no structure in the universe. If you assume that the spatial curvature of the universe is the same everywhere, that the universe contains no structure and evolves as a smooth featureless fluid by a Friedmann-Lemaitre solution, then it turns out that the angular scale of the sound horizon, which determines the overall angular size of the Doppler peaks, coincides with the expectation based on a spatially flat universe.

I claim that the 80-90 old standard cosmology presents an oversimplified view of the universe that ignores the actual observations of voids and inhomogeneities. Such observations would suggest realistically that spatial curvature is not the same everywhere. If the spatial curvature is not the same everywhere then you have to redo the calculations to analyse the CMB. For this a specific model universe is required. Constructing such a model is nontrivial and very difficult to study in general, and why most people persist with the standard Friedmann-Lemaitre model. I did the relevant calculations recalibrating the angular scale of the sound horizon for the new model universe I have developed in New J. Phys. 7 (2007) 377. Furthermore, in a paper with my student Ben Leith and former student Cindy Ng, which appeared in Astrophys. J. 672 (2008) L91, we found that the parameter values which best-fit the Riess gold supernovae data also pass the angular scale of the sound horizon test. This is one of the three independent tests I mention in my essay.

>As you point out, it is time which is faster in these voids. Could this

>affect the propagation rate of light across intergalactic space?

One quibble: time independently of specifying an observer does not mean anything, as you can get very different results to that of any given observer by making a local boost at any point. So I would not say "time is faster in voids". That's very Newtonian wording. As Einstein pointed out in 1905, time is always a property of a clock. In the present case it is the time *measured by the clocks of observers who see an isotropic CMB radiation* that is faster in a void.

And yes of course this does affect the interpretation of the propagation of light, and give important the differences to the expectations of the Friedmann-Lemaitre models. There many different predictions which are outlined in my papers. The next paper - cited in the essay as ref [16] "in preparation" will be out before the end of the year, and looks at several distinguishing predictions.

>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.

There are a number of things with your wording here that are problematic. Firstly the "Big Bang" is simply the idea that the universe has expanded over a finite time from a state in which it was much smaller, hotter and denser in the past. So my model, just like the standard model is a "big bang" model, as opposed to universes which have no beginning in time such as the "steady state" and Hoyle-Narlikar models. Furthermore, interpreting very distant redshifts in terms of "recession velocities" can lead to all sorts of interpretational difficulties. Really, a velocity is only something you measure locally at a point. You can interpret any derivative of a distance by time as a velocity, but in general relativity such quantities do not necessarily have to be bounded by the speed of light because they are only formal definitions, not local operational measurements. A locally measured velocity - something whizzing past you - by contrast is always bounded by the speed of light. In cosmology, even with distant events you can formally interpret things in terms of recession velocities if you use the connection of general relativity to parallel transport a local 4-velocity at some distant event to your event along the lightcone. (For a recent non-technical account see Bunn and Hogg arXiv:0808.1081.) These sorts of interpretation issues are no different in my model than in the standard Friedmann-Lemaitre models. What is different is that the expansion history is different, and the interpretation of cosmic acceleration is purely apparent. Furthermore, below the scale of statistical homogeneity there will be an observable variance in the apparent Hubble flow that correlates to observed structures in a particular fashion. This is perhaps the most interesting and definitive prediction of the new cosmology, as it has no counterpart in the Friedmann-Lemaitre models. (See http://arxiv.org/pdf/0712.3984 for the least technical overview of my work.)

------------------------------------------------------
-

E. Real McCoy:

>Nonlocality is a fact of quantum mechanics. Perhaps I am misreading this.

Indeed you are misreading it because I said "nonlocality of interactions" not "nonlocality of quantum states". These are very different things. One part of physics deals with spacetime structure and the interactions that live in the structure: gravity, electromagnetism, weak and strong forces. The propagation of these forces is always subject to the limits of causality imposed by the spacetime structure. Quantum mechanics addresses something else - the fact that there is a fundamental indeterminism in correlations of particular measured properties of matter fields, beginning most importantly with position and momentum (in a given direction). Quantum mechanics will certainly limit our ability to completely determine future spacetime structure from a given matter distribution, but it does not change the rules of causality. It does not give "action at a distance" in the way that Newtonian gravity involves action at a distance.

It is certainly possible to have quantum states which are entangled with each other over spacelike distances, in Einstein-Podolsky-Rosen type experiments such as the ones of Aspect that you refer to. I know that there is a lot of confusion around this, some of it promulgated by Einstein who spoke about "spooky action at a distance" in reference to EPR experiments. Quite apart from the fact that Einstein was wrong in his predictions about the outcome of EPR experiments, I would say that he was wrong in referring to EPR type entanglements as "action at a distance". The fact is that you cannot transmit a message by making EPR type measurements, so you do not violate causality. People who do foundations of quantum mechanics type experiments can do "quantum teleportation" of quantum states by these means. However, such teleportation cannot be used to transmit a message, and so is not teleportation of the sort you see on Star Trek. By making a measurement on one part of an entangled state, you can know some correlated property of the other part of that state at a spacelike separation. E.g., by learning some property of a particle at your location you determine some related property of a distant particle that was entangled with it. This may seem "spooky" as you do have a freedom to determine which property you measure and as classical physicists we are used to thinking of such properties "existing" independently of the observer, when quantum mechanics says that is not so. But if you can get over your classical hangups then what is "spooky" is not so frightening. And whether it is "spooky" or not it is not "action" in the sense of transmission of a message. It is just an inevitable consequence of indeterminism of the sort supplied by our rules of quantum mechanics.

I should mention that the Cosmological Equivalence Principle approaches general relativity from the philosophical viewpoint that laws of physics only have meaning relative to an observer. I view Einstein's equations for the universe as evolution equations which only make sense from the point of view of an observer, and so are limited by the finite size of the past light cone at any event (the "particle horizon" technically speaking). This has important consequence on account of cosmic variance in density fluctuations and their subsequent evolution, as I elaborate in section 9 of my paper "Cosmic clocks, cosmic variance and cosmic averages" [ http://stacks.iop.org/1367-2630/9/377 ]. Furthermore, since a regional "cosmological inertial frame" is one which is conformally flat - i.e., with zero Weyl curvature - and since any "nonlocal" curvature has to be part of the Weyl tensor, the CEP also includes the idea that the only "non-local curvature" we are allowed physically is that which has arisen as a result of local processes within the past light cone at any event - through gravitational collapse and production of gravitational waves etc. In the limit of the earliest times that means we are not allowed Weyl curvature for initial conditions of the universe, giving a fundamental link to Penrose's Weyl curvature hypothesis. I discuss this in detail in a section of the recent paper in which I introduced the CEP: Phys. Rev. D 78 (2008) 084032 [ http://arxiv.org/pdf/0809.1183 ]. Penrose formulated the Weyl curvature hypothesis motivated by understanding the arrow of time and gravitational entropy. These are further aspects of the nature of time which the proposed framework may shed light on; but this has still to be developed.

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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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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 all.

The standard model has an exactly homogeneous isotropic expansion as

described by an FLRW metric. My proposal alters that by the observation

that homogeneity is only true statistically on scales of at least of order

100/h Mpc, and we have to deal with large variances in geometry and

perceived expansion rates below that scale. Statistically on large

scales there is still homogeneous isotropic expansion.

When you say homogeneity and isotropy are problems, and then mention

inflation, I think you are getting confused with the horizon and flatness

problems, which are quite different issues and have nothing to do with

the overall conceptual notion of expanding space. The horizon and flatness

problems arise because in a standard FLRW model with only matter and

radiation as sources, the past history of the universe before the time

of last-scattering is such that points on our CMB sky which are further apart

than about one degree cannot have been in causal contact before the epoch

of last scattering, once you calculate the volume of their past light cones

at that time. However, the evidence of the CMB

radiation is that the universe was at thermal equilibrium with a near

perfect blackbody spectrum at that time, an impossibility if regions more

than one degree apart had not been able to talk to each other. The

inflationary universe scenario has been invented to solve conundrums such as

the horizon problem I have just described. It has nothing to do with the

notion of expansion of space per se; it just means the relative expansion

rate would have been many orders of magnitude faster very early on, changing

the shape of the past light cones in a dramatic fashion that solves the

horizon problem etc.

The rate of expansion of the universe has nothing

to do with the speed of light per se. The speed of light in vacuum enters

relativity as a fixed universal constant, and the whole of spacetime

structure in Einstein's theory depends on the speed of light being

constant. That means we can always choose a local inertial frame at

a point, which is a Minkowski space. The constancy of the speed of

light is built into the physical structure of this Minkowski space. General

relativity is a larger theory in which spacetime overall can bend

and warp, so that the relative calibration of clocks and rods in

widely separated local Minkowski frames can differ markedly, in a manner

than depends on solutions of Einstein's equations. The Cosmological

Equivalence Principle I have introduced is a means of clarifying

the problem of the relative calibration of clocks of widely separated

frames on cosmological scales in the absence of

exact symmetries of the background - a problem which otherwise has no general

solution in general relativity. I believe I do this in a manner which

naturally incorporates an aspect of Mach's principle, consistent

with the rest of the conceptual foundations of general relativity.

It is dangerous to take the "rubber sheet" analogy too far. Empty

space does not have a fabric, and light can never be brought to rest

in any frame - so you should never talk about space "carrying light

along with it". The fact is that in general relativity the laws of

geometry are not Euclidean but pseudo-Riemannian, and the spatial

relationship of objects are dynamical so the curved geometry changes

over time. That's all there is to the "rubber sheet" or the "fabric"

of space. It is just an analogy for us to be able to conceptualise

curved geometries. But the curvature of the geometry is determined

by matter - matter tells the geometry how to curve and the curved

geometry tells matter how to move - so ultimately it is matter telling

matter how to move via the rules of Einstein's equations, and light moving

in this background is part of the matter in the game; light being

matter which can never be

be brought to rest relative to other matter. If you just think of expanding space as being that the distances and

volume of space between distant galaxies is getting larger, then you

will never go wrong. I think your confusions may arise from taking the rubber sheet too

literally. Whatever started the matter rushing apart in a particular

manner at the earliest fractions of a second of existence is something

beyond the laws of general relativity, and part of physics still to be

determined, maybe in quantum gravity. Inflationary models have such properties but depend highly on even earlier initial conditions within their parameter space; and there are hundreds of models of inflation. I regard them as phenomenological decriptions which fit the observations, but do not deserve to be called a fundamental theory until some genuine theoretical insight that has yet to be made picks out some scenario in a compelling fashion, and rules others out. Anyway given initial conditions at the time of last scattering or earlier in the radiation-dominated era,

once the laws of physics as we understand them held sway ordinary matter

will decelerate the expansion rate in a manner consistent with Einstein's

equations.

Finally when you say "there doesn't appear any other reason for redshift

than a form of recessional velocity", that is not correct. Redshift just

arises in comparing the relative frequency of a photon at the emitter

and at an observer's position, and these can change in general relativity

due to a relative calibration of the clocks of those two frames in many

ways. Here are three ways: (i) cosmological redshift - the overall volume

of space has increased; (ii) peculiar velocity redshift - there is

a local boost of the emitter and/or observer frame relative to the overall

cosmological expansion; (iii) gravitational redshift - the relative

distribution of matter between source and emitter and resulting gravitational

"potentials" induces other relative changes of frequency over and above the

first two effects. Now it is true that general relativity is a theory in

which we cannot *locally* distinguish these different types of redshifts,

but that is quite different than your statement.

The Cosmological Equivalence Principle extends the range of these equivalent

circumstances. In particular, even though it is an old textbook statement

that the redshift in a FLRW universe is *locally* equivalent to a peculiar

velocity in Minkowski space, the CEP states that even though the universe

is inhomogeneous - so the FLRW models are not globally valid - nonetheless

regional frames can always be found for arbitrarily long periods during

which the average decelerating volume expansion is conformally equivalent

to a Minkowski frame, and therefore with volume expansion which is

indistinguishable from the equivalent motion of a congruence of particles

in a static Minkowski space by the standard textbook argument. Furthermore,

I establish a new type of gravitational time dilation and gravitational

redshift. We are used to thinking about static gravitational potentials

and the equivalence of a static observer in such a potential (eg someone

at the Earth's surface) with an observer firing rockets. I introduce a

different notion of gravitational time dilation due to cumulative variations

of dynamically varying "potentials". This is to be thought of as relative

deceleration of the expanding average background due to gravity being

equivalent to a regional homogeneous/isotropic symmetry-preserving

deceleration of a tethered lattice of observers in Minkowski space.

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Dimi

>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

>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). Linearity is a property of combinations of variables in equations. In Minkowski space it is the Lorentz transformations which relate inertial frames that are linear. Conservation laws are described by divergence-free currents, which via Gauss's law give conserved charges when doing the relevant integrals on hypersurfaces. The time-direction is hypersurface orthogonal in formulating such conservation laws.

BTW In relativity one also has to be always careful to distinguish between arbitrary coordinate variables, and proper lengths and proper times which are invariants. Your watch measures your proper time. If you are talking about your proper time, say it; "variable" is too vague as it can also refer to non-measurable things.

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

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. Gravitational energy comes into the relative calibration of clocks at widely separated events.

Why is energy conservation difficult in GR? Well, in the absence of exact symmetries one cannot do the same procedure of a Gauss law style integration to extract a conserved 4-momentum and conserved covariant angular-momentum from the divergence-free energy-momentum tensor as you can in Minkowksi space. In general relativity one can in general define conservation laws for completely antisymmetry tensor densities. However, one cannot do this for the rank 2 symmetric energy-momentum tensor. If you try to integrate it, in the manner of Gauss's law there is an extra bit involing the connection which is in a sense "the work done by gravity". In the presence of exact symmetries of the background spacetime, described by a Killing vector, it is possible to contract the Killing vector with the energy-momentum tensor to get rid of the extra bit and get conservation laws. Just symmetries of the background are required, and to get a conserved energy you need a timelike symmetry of the background. What I am talking about here is the "energy of the spacetime" but the same is true for geodesic motion; if you have a timelike Killing vector you can contract it with a particle 4-velocity to get a conserved energy of the particle in motion. Without a timelike Killing vector you cannot do that.

Timelike symmetries describe bound solutions extremely well, but since the universe is expanding there is no absolute time symmetry, and so the time symmetry is approximate. Since the timelike Killing vector is normalised to unity at "spatial infinity", defining the equivalent of the zero of the Newtonian gravitational potential; in the actual universe this non-existent spatial infinity has to be replaced by something like "finite infinity", as first discussed by George Ellis in the 1984.

Since spacetime is intrinsically dynamical in GR, the "work done by gravity" enters into energy-momentum conservation in an inextricable way in general, when there are no time symmetries.

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

Nothing changes because the idea of "introducing quasilocal variables" is not something that is being done on top of general relativity. We are simply disussing known properties of Einstein's theory. It is a property of his theory that propagation of the gravitational interaction is causal. It is also a property of his theory that "the work done by gravity" cannot be separated out in general, making energy conservation an intrinsically different problem from the same problem in flat spacetime. This is a consequence of the equivalence principle; you can always get rid of gravity near a point, and gravity is a property of spacetime structure. Gravitational energy involves the calibration of clocks at different points (via the connection), and can only be regionally refined; so it is at best quasi-local.

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Dimi,

Regarding

>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,

Regarding

>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 evolution of the universe. In his 1984 paper George Ellis described this passively as a "fitting problem" of how do we reconstruct the past geometry of the universe from information on our past light cone using averages on different scales. I think of it as an active problem: how does the universe construct itself from averages of what has happened? I think George's evolving block universe ideas are important; quantum mechanics does introduce a fundamental indeterminacy until things have happened. Thus the future geometry is not absolutely determined. But whereas it might be impossible to predict the precise number of black holes in our galaxy say, just from stochastic rather than quantum fluctuations, once we average on larger and larger scales the universe can only evolve so far away from its state of almost homogeneity at last scattering, given the limits of causal evolution. The question of coarse-graining and averaging in GR to extract something like an average spatial hypersurface is therefore a big unsolved problem of classical GR. Gravitational entropy - and gravitational energy since there ought to be a first law - are both part of the same puzzle as far as I'm concerned.

Now you say

> I am indeed trying to suggest that the proper time, as measured by a clock, can be replaced by a new quasi-local variable

Having read your comments on Dean Rickle's thread I can see intuitively where you are coming from. Let me make the following suggestion: suppose I was to say that you are not trying to replace local proper time as measured by a clock by a quasi-local variable, but the "time evolution parameter" in something like the ADM-formalism by a quasi-local variable. Would you buy that? Because that is exactly what I believe we are trying to do in the coarse-graining/averaging/fitting/evolution problem. What is needed is a clarity of concepts. In GR the proper time on a particle clock is a measurable quantity *at a point on a timelike worldline* which is related to some particular observer; it is an invariant of the metric given a solution of the geodesic equation for free-fall or of a non-geodesic equation if there is some additional forcing term. However, you can have infinitely many different worldlines passing through the neighbourhood of any given spatial point, all with different 4-velocities. So there is no unique proper time at a point. The geometry in some spatial region is an average over all particles and motions via an appropriate average of the Einstein field equations. Thus within an averaging scheme you can describe some coarse-grained cell by some average time parameter. This then is "quasi-local" as it relates to the whole averaging region. It will not exactly coincide with the proper time of every observer within the coarse-grained cell, because these will differ. The average time parameter may happen to be a local parameter for some particular observers, but not all, and the averaging scheme should operationally incorporate an understanding of both the average and the variance of relevant observables.

This is precisely what I am trying to do in the cosmological context. I am using the averaging scheme of Thomas Buchert, but operationally interpreted in a manner quite different to the way in which he conceived it. I am using Buchert's scheme simply because it is developed in a way which makes the most direct contact with the FLRW models, which are remarkably successful. However, I do not think that as it is currently formulated it is ultimately the best scheme for the problem, and I would advocate trying to generalise the "Mach 1 gauge" of Bicak, Katz and Lynden-Bell beyond the perturbative framework. In fact, I think that any averaging scheme that begins with a decomposition into spatial hypersurfaces is part of the problem operationally. 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. My attempt at defining finite infinity is just work-in-progress, and in fact it is not yet well-defined and has many weaknesses. However, one can only make progress by making concrete physical models. Concepts, words and philosophy alone are not going to solve things.

If your motivation for this question is to understand vacuum energy, (judging from links you give) then that is the angle that I came at this from originally, because I think it is likely that there is no vacuum energy. At least from the cosmological model I have proposed one can get a fit to observations, and predict observational differences from the Lambda CDM model that can be tested in future. My essay (and the longer version in Physical Review D) argue that this has its origin in a refined understanding of the equivalence principle in the coarse-graining/fitting/averaging/evolution problem. Ultimately, if this is correct, I think it will be important for quantum cosmology and quantum gravity, and the full "cosmological constant problem". However, rather than try to tackle everything at once, I think one first has to refine the mathematical tools with concrete problems and predictions, while striving for conceptual clarity.

BTW; as I have recently returned home to the southern hemisphere, my location makes a white Christmas extremely improbable, but it wasn't particularly warm either. I wish you a pleasant holiday.

David

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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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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 the obstructions we face with quantum gravity and cosmology". Here you are referring to the proper time of a particle on one hand, an invariant, and a Hamiltonian on spatial surfaces with an orthogonal time parameter, which is gauge dependent.

Let me bring back the equivalence principle, the focus of my essay. The strong equivalence principle tells us that we can always find a local inertial frame, and really we only know how to quantum mechanics and quantum field theories in such Minkowski frames. The question is how do we proceed when we include gravity and go beyond local inertial frames? Although I am not directly addressing quantum gravity, I have worked on quantum cosmology in the past, and it is my view that the canonical Wheeler-DeWitt quantization, or any similar quantization based on something like the ADM formalism, is inadequate. The problem, as you have recognized, is this formalism treats spacetime at the classical level timelessly as a 4-dimensional construct. There is no real evolution built in, and therefore any 3+1 split, as a pretext to quantization, is gauge-dependent. My view, which is maybe similar to George Ellis' evolving block universe, is that we have to formulate the classical cosmological problem already as an evolution problem before we quantize. Since the universe had a beginning the domain of the causal past is limited at any event; and the geometry near any event can only depend on things in its past light cone. Viewed this way it is imperative that we treat cosmological GR as an evolution problem. I am not trying to understand quantization yet, but to understand this other question. How does the universe choose something akin to an average spatial hypersurfaces as it evolves? My answer is to go back to first principles and extend the equivalence principle.

Dimi: I have looked at your website, but in general find it too incoherent to follow. I feel you have some good physical insights, and you certainly ask some probing questions, but you use your own personal idioms and mix so many different problems simultaneously in an intuitive way, that it is very hard often to see precisely what you are aiming at. Writing a focused paper on just one topic to clarify what is meant by some of these personal idioms, with a degree of rigor to make it acceptable for publication in a journal, would greatly help your cause. When you talk about your notion of 'quasi-local time' with two components, "global" and "local" etc, actually intuitively I think this is very much the sort of thing I am doing in my approach. I do not use the words "quasilocal time" because I think it is best to reserve "time" to describe the local proper time that we usually refer to in GR. Something cannot be both local and quasilocal. However, I do have a "volume-average time" which is different to the local proper time of observers at finite infinity. You talk about "negotiation" in the global component of time, without defining what "negotiation" is. I would suggest that averaging and coarse-graining, which are words that you do not like, are the same thing as you are are referring to with your terminology "negotiation". It's just that people who have thought about the problem in other ways come with their own terminology.

How we do averaging is an unsolved problem in GR, and I think George Ellis gave a pretty good statement of the problem to you in his reply of December 21. My essay describes a proposal that we interpolate between the local and "global" frames by extending the equivalence principle to a cosmological equivalence principle. The thing is that the strong equivalence principle already tells us how the average frame is "negotiated" on very small scales. In the standard cosmology we assume an answer to the averaging problem by simply demanding that there is a single global average FLRW geometry for the whole universe, and that matter averages to homogeneous isotropic pressureless dust for all times. That is at odds with observed inhomogeneity below scales of 100/h Mpc. Rather than demanding a single global geometry, I propose that we can always choose regional cosmological frames with the spatial symmetries of Minkowski space, but not the time symmetries - since we are talking about the dynamical regime. Rather there is a regional conformal timelike Killing vector. When George Ellis mentioned the conformal timelike Killing vector on his thread, he was probably referring to the standard FLRW cosmology. In some general cosmology, like various Bianchi models, there is no such timelike conformal Killing vector. I propose (for a variety of reasons that I discuss in the essay and the recent paper in Physical Review D) that even though the universe is inhomogeneous, the manner of the "negotiation" between local and global in the fitting/averaging/coarse-graining problem is that we can always choose average regional frames which are Minkowski up to a timelike conformal scaling; i.e., with spatial symmetries of Minkowski and an additional timelike conformal Killing vector. This is less strong than what we do in the standard cosmology, which assumes this as a global frame. My reasons for doing this - which also lead to a testable model - are stated in the essay, so I will not repeat it here, unless there is some point that is not clear.

On an unrelated issue. Why is there only positive mass? Weak equivalence principle: if negative mass were to have a meaning it ought to violate the principle of uniqueness of free fall. This is an intuitive answer, not a theorem. However, I'm sure the positive mass theorems could be rederived with a suitable definition of finite infinity. It would require a very tight definition of finite infinity first, however.

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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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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 describe the physical world. The strong equivalence principle already tells you how the classical spacetime is negotiated in a very small region near a point; because particles are in motion, and they continue to move in a manner consistent with special relativity including any non-gravitational forces in a very small spacetime region. This is a spacetime region in which no one spacelike 3-surface is singled out as special. So there is no Buridan donkey paradox near a point because there is no special space direction and the local inertial frame (LIF) is regionally "negotiated" from the prior motions of particles in both space and time.

The difficult part of course is patching together these LIFs in general relativity. If the local geometry is dominated by some mass concentration possessing symmetries, then we assume that we can approximate the geometry by an exact solution such as the Schwarzschild or Kerr geometry with the relevant symmetries. Furthermore, this assumption enables us to do calculations and make predictions of the motion of particles and light which agree extremely well with what is observed. The real problem is when we now attempt to patch these almost isolated geometries together in the absence of exact symmetries.

That is the fitting problem that George Ellis spelled out. We do not use the Friedmann equation to solve for the Earth's motion about the sun, or the sun's motion around the galaxy. Yet we naively assume that we can write down solutions of the Friedmann equation and apply the invariant time and distance definitions of this geometry as if it were the local geometry, when it is not. Since homogeneity is only reached by averaging on scales of at least of order 100/h Mpc, this step is not justified by any principle of general relativity. Since the deduction of cosmic acceleration and vacuum energy energy are completed based on this unprincipled assumption, also at odds with the evidence for inhomogeneity from our telescopes, it is prudent to ask whether one can realistically account for the observations in another way. What I have shown in my work is that, to the level of the observational tests I have considered thus far, one can. Furthermore, conceptually it means thinking about quasilocal gravitational energy.

The cosmological equivalence principle is a precisely step towards a "Machian relational ontology". As far as I am concerned neither space nor time has an existence separate from the matter fields that exist within it. A lot of essays in this competition are hung up on the question of the "reality" of the spacetime continuum. Well, what do we mean my "real"? If one supposed that a spacetime could exist independently of material fields, then as far as I am concerned no such entity exists (which to me also makes it perfectly sensible that there is no vacuum energy). However, I do not consider myself an "illusionist" in George Ellis' terminology because the time on my clock is real, just as the length of ruler is real, or the energy of a photon I measure after transmission across a cosmological scales is real. That is the only reality, the rest is a mathematical relational structure.

As a purely mathematical theory of differential geometry, general relativity is I believe, too general. All those crazy solutions with closed timelike curves, wormholes, and the like, are I believe ruled out by physical principles, and the key ones are the class of principles we call the equivalence principle, as they deal with the concept of inertia. The strong equivalence principle already severely restricts our choice of metric connection; placing restrictions on the way you might try to introduce torsion as a physical variable. I am proposing the cosmological equivalence principle as a means of further restricting our choice of background universe - in a way which would make all those solutions with closed timelike curves, or indeed anisotropic Bianchi models, physically redundant.

At the basis of this is a further clarification of the notion of inertia - the centre piece of the Machian ontology. How is the average relational background "negotiated" as an average of all fields and motions? The semi-tethered lattice I have introduced is a Minkowski space analogue of a collective regional deceleration, with conversion of energy from kinetic to other forms, while no net force is felt by any particle in the lattice, justifying the statement that for this regional decelerating frame we have a sense of "inertia". Real energy is extracted from this process just as energy is extracted from gravitational collapse; yet the sense of inertia relates to geodesic motion that maintains a collective average regional homogeneity. This is what makes it quasilocal. My cosmological equivalence principle is to demand that the "negotiation process" is such that in the fitting problem we can always choose such average regional frames with such a quasilocal notion of inertia. At such a level we cannot distinguish the regional deceleration of matter due to its average density from an equivalent semi-tethered lattice deceleration process in a Minkowski space in which an equivalent amount of kinetic energy of expansion is converted to other forms. (That's the equivalent of the work done by gravity.) This is the essence of the regional timelike conformal Killing symmetry. Thus I claim over the scales of regional inhomogeneity we find such frames in the universe, which have decelerated by different amounts, and their local clocks will differ cumulatively.

In saying that something cannot be both quasilocal and local, I am simply stating that you have to be absolutely clear in your concepts and definitions to build a predictive physical model, and if you cannot build a physical model (through such lack of precision) then you are wasting time. "Quasilocal" means something more than local, otherwise we would simply use the word "local". In my case it is a property of regional collective motions, as outlined above. The time measured on a clock will still be a local time always. Different regions will have different average local proper times on their clocks. So there is a sense in which the measurement of an average time parameter is "quasilocal" as in being region dependent. But I would suggest that one does not learn that much from a slogan such as "time is both local and quasi-local" which just hides a degree of precision of clarification. For a physical model one needs a degree of precision in stating just by what amount some average time parameter will differ from one region to another, and for observers defined in what manner. How do we calibrate the clocks in the regional "negotiation"? In my case is it observers who see an isotropic CMB in regions of varying spatial curvature due to strong inhomogeneities consistent with the observed structure of voids and walls below the scale of homogeneity in the observed universe. Furthermore, I quantify the cumulative difference in clock rates of relevant canonical observers.

Personally, I think my cosmological equivalence principle is just a first step, and there is a lot further to go. Think of it this way: individual stars are treated as vacuum solutions and have ADM energies. Yet from all that vacuum and ADM energy we assume that the collective description is described by a dust fluid with a non-zero Ricci tensor, even though the original solutions had zero Ricci curvature (and only Weyl curvature). So, how do we mathematically convert a collection of ADM energies to dust with Ricci curvature? Ricci curvature is so very nebulous a concept in the presence of the equivalence principle. Some of my colleagues such as Thomas Buchert and Mauro Carfora think that this averaging away of Ricci curvature is related mathematically to Ricci flow, as a sort of renormalization process. That may be the case, but I think we are just scratching the surface, and have to ask penetrating physical questions, rather than just playing with mathematical formalism. If we construct such flows mathematically, then there has to be a physical relation to the relative calibration of local clocks and rods in doing the renormalization. I do not pretend to yet have all the answers; what I have tried to do in my essay is to approach the foundational physical questions, to the extent that one can start to build quantitative cosmological models.

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. As someone who has worked in quantum gravity, I think that often we try to do too much at once, while overlooking the fact that some things we think we understand are not really that well understood. The idea that we have to sort out the mystery of some physical fluid in the vacuum of space is grossly premature, when our entire deduction of the existence of dark energy relies on ignoring a "too hard" fitting problem and pretending that our universe has exact symmetries which differ from what we observe. I stopped working in quantum gravity and cosmology because I though I was wasting my time until these other fundamental issues, such as the fitting problem, are not sorted out. Let us first try to sort that out, and maybe it will give us fresh ideas about the other problems, even if the fitting problem is only a classical problem.

Lawrence: at this point I would say that any connection between my "cosmological conformal principle" and conformal structures of strings and branes is not at all obvious. I am only talking about a timelike conformal symmetry for establishing average regional cosmological frames in which it is useful to phrase a quasilocal gravitational energy concept. I am not talking about complete conformal structures per se, but only a timelike one for average regional frames in cosmology. Now, it is true that the unboundedness of the Euclidean action in the naive approach to 4-d quantum gravity can be spelt out mathematically in terms of conformal equivalence classes (I seem to recall though I cannot remember the paper - circa 1979). So maybe there is some relation, who knows. But while strings and branes make for interesting models (which I have worked on too) they are not established observationally as anything to do with the real world. So I am only worrying about classical GR and observational cosmology at present. The essay of David Hestenes - which involves a real world experiment beyond standard particle phenomenology - is indeed very interesting, and I thank you for pointing it out.

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