From time to timescape -- Einstein's unfinished revolution by David L. Wiltshire

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.

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!

Tom

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)Attachment #1: 2_MOVING_DIMENSIONS_THEORY_EXAMINES_THE_GRAVITATIONAL_REDSHIFT_SLOWING_OF_CLOCKS.pdf

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"...

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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.)

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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.

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.

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.

David,

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:

http://fqxi.org/data/essay-contest-files/Merryman_Explaining_Time.pdf

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...

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.

Cheers,

Lawrence B. Crowell

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

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

David:

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.

Dimi

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.

David:

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,

Dimi

Addendum

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."

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

David:

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,

Dimi

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

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 :-)

Dimi