Right About Time? by Sean Gryb and Flavio Mercati

Dear Sean and Flavio,

Interesting and well-written essay. One quick question: What exactly are the elements of the shape space of your model? Special types of 3-manifolds, I presume? Thanks,

Ben Dribus

Dear Ben,

first of all thanks for the interest in what we wrote!

To answer your question, points of shape space are "conformal 3-geometries" or "conformal 3-manifolds", that is 3-geometries (which are described by 3-metrics modulo 3-diffeomorphisms), modulo local conformal transformations. A conformal transformation preserves only angles but not

vector's lengths, so you can describe a conformal 3-geometry with just the angle-determining

part of a 3-metric. There is quite some mathematical literature on them:

http://en.wikipedia.org/wiki/Conformal_geometry

In our next paper (it's coming soon on the arXiv, stay tuned...) Sean and I found how to describe shape dynamics in the simplified case of 2 spatial dimensions with a Cartan geometry in which

the structure group is the conformal group. You have then a description of the gravitational field

in terms of "conformal frame fields", which are useful for several reasons (in GR it's easier and somewhat more natural to couple fermions to frame fields, and they provide the best known reformulation of GR as a gauge theory). Written in this way, shape dynamics looks like a Chern-Simons theory of the conformal group.

Something about Cartan geometry can be found here:

http://en.wikipedia.org/wiki/Cartan_geometry

Cheers,

Flavio

"If you have ever used a GPS you have benefited from special and general relativity." Really?

Yep.

It couldn't work just with calculations based on Newtonian dynamics.

Look at this:

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

Flavio

Btw Ben,

I've seen your essay: interesting!

Are you familiar with Rafael Sorkin's causal sets?

http://en.wikipedia.org/wiki/Causal_sets

Cheers,

Flavio

Flavio/Eckard

The assumption of SR for GPS is not verified by many of those directly involved. It was 'fudged' and rather hijacked by hard line relativists. But I agree Newton also doesn't hack it. I've written one paper on it but there are also more comprehensive ones (link by request).

Ref MMX, remember it was 'trivially' non zero. Also Millers altitude results varied. A rational solution then emerges from my essay (and paper about to be published). Einstein certainly seemed to have accounted for it - from his later paper;

"The aether-theory brought with it the question: How does the aether behave from the mechanical point of view with respect to ponderable bodies? Does it take part in the motions of the bodies, or do its parts remain at rest relatively to each other? Many ingenious experiments were undertaken to decide this question. The following important facts should be mentioned in this connection: the "aberration" of the fixed stars in consequence of the annual motion of the earth, and the "Doppler effect", i.e. the influence of the relative motion of the fixed stars on the frequency of the light reaching us from them, for known frequencies of emission. The results of all these facts and experiments, except for one, the Michelson-Morley experiment, were explained by H. A. Lorentz on the assumption that the aether does not take part in the motions of ponderable bodies, and that the parts of the aether have no relative motions at all with respect to each other. Thus the aether appeared, as it were, as the embodiment of a space absolutely at rest. But the investigation of Lorentz accomplished still more. It explained all the electromagnetic and optical processes within ponderable bodies known at that time, on the assumption that the influence of ponderable matter on the electric field - and conversely - is due solely to the fact that the constituent particles of matter carry electrical charges, which share the motion of the particles. Concerning the experiment of Michelson and Morley, H. A. Lorentz showed that the result obtained at least does not contradict the theory of an aether at rest. 1952

In spite of all these beautiful successes the state of the theory was not yet wholly satisfactory, and for the following reasons. Classical mechanics, of which it could not be doubted that it holds with a close degree of approximation, teaches the equivalence of all inertial systems or inertial "spaces" for the formulation of natural laws, i.e. the invariance of natural laws with respect to the transition from one inertial system to another. Electromagnetic and optical experiments taught the same thing with considerable accuracy. But the foundation of electromagnetic theory taught that a particular inertial system must be given preference, namely that of the luminiferous aether at rest. This view of the theoretical foundation was much too unsatisfactory. Was there no modification that, like classical mechanics, would uphold the equivalence of inertial systems (special principle of relativity)?

The answer to this question is the special theory of relativity. This takes over from the theory of Maxwell-Lorentz the assumption of the constancy of the velocity of light in empty space. In order to bring this into harmony with the equivalence of inertial systems (special principle of relativity), the idea of the absolute character of simultaneity must be given up; ..."

"Relativity and the Problem of Space" Albert Einstein (1952)

Peter

I called your attention to two falsehoods (slips?) in your essay but you did not say anything:

Falsehood 1: "...an experiment by Michelson and Morley that measured the speed of light to be independent of how an observer was moving."

Falsehood 2: "...the Michelson-Morley experiment led to the development of relativity."

Are you afraid to comment?

Pentcho Valev

Dear Sean & Flavio:

I enjoyed reading your well-written and intuitive essay describing the challenges in the fundamental understanding of time. You rightly point out to the two major problems that constitute the unexplained 96% of the universe - "....(dark energy) the accelerated expansion of the Universe, which is some 120 orders of magnitude smaller than its natural value.....and the dark matter problem..."

Then, you rightly state that - "...how can we do science on the Universe as a whole?

We will not directly answer this question but, rather, suggest that this difficult issue may require a radical answer that questions the very origin of time. "

My paper -" From Absurd to Elegant Universe" provides answers to the questions you raise and forwards a mathematical model of the universe as a "Whole" to resolve the well-known paradoxes of the modern science. Julian Barbour also concludes in his paper in this forum -

"....it may be impossible to understand key features of the universe such as its pervasive arrow of time and remarkably high degree of isotropy and homogeneity unless we study it holistically - as a true whole. A satisfactory interpretation of quantum mechanics is also likely to be profoundly holistic, involving the entire universe. The phenomenon of entanglement already hints at such a possibility.."

My paper demonstrates that following a holistic approach wherein the whole universe is considered as a continuum of mass-energy-space-time, a very simple mathematical model of the missing physics (hidden variable) of the well-known spontaneous decay/birth of particles can be developed that explains the observed quantum as well as classical behaviors. The holistic model also successfully predicts the observed data at all scales from below Planck scale to beyond cosmological scales. The proposed model not only resolves black hole singularities but also the unresolved paradoxes of physics and cosmology including the dark energy and dark matter. The holistic model also explains the inner workings of QM and eliminates its paradoxes and inconsistencies with relativity. It also vindicates that time is not a fundamental entity since the observed universe and galactic expansion can be predicted without any explicit consideration of a cosmic time.

I would greatly appreciate your comments on my paper. You can contact me at avsingh@alum.mit.edu.

Best Regards

Avtar Singh

Don't worry. Soon Brendan Foster will delete both my provocations and your absurd replies. Like Julian Barbour, you will be one of the winners.

Pentcho Valev

Pogge's propaganda for relativity is not precise and not convincing to me. Every layman knows that the accuracy of a clock cannot be given as time deviation unless one knows what timespan this value refers to. Perhaps Pogge meant 1 ns per day but he did not write that. He did not even bother to tell further details like Sagnac effect and the open secret that the corrections were made empirically. He didn't give references to the opinion of true experts.

How to judge censorship? How to judge the necessity of a task force making sure that Einstein's relativity is not seriously questioned in Wikipedia?

Shouldn't we welcome the insight that we need not focussing on Einstein if already Michelson's expectation was wrong? I consider Roger Schlafly's remarks on Einstein possibly well founded but I would like to avoid hurting the feelings of believers.

Eckard

Dear Sean,

I appreciate the response. As a matter of fact, you'll find that my own physics ideas are somewhat removed from my algebraic geometry work, but that doesn't mean I wouldn't be delighted to find out about physics applications for the math stuff I have been doing.

I think you might also be interested in the essays by Torsten Asselmeyer-Maluga and Jerzy Kroll about exotic smoothness structures, even though the approach, like mine, is quite different.

By the way, I briefly mentioned a number of prominent approaches in the first part of my essay; I certainly would not have left your approach out if I had been properly aware of it at the time. Take care,

Ben

Dear Sean/Flavio,

Do mind telling me an email address or sending me an email at bdribus@math.lsu.edu? I notice you don't have addresses listed on the paper. Thanks,

Ben Dribus

Sean and Flavio,

I found your essay to be interesting. It takes me a while to digest the more interesting and solid works here, so it has taken me a bit of time to get to your paper. You make a number of very interesting points. I can only this morning discuss a few of these. I will try to follow up later today or tomorrow.

The relationship between shape dynamics (SD) and the extended configuration space (ECS) is one of conformal symmetry. SD has no explicit reference to scale, and scale independence is one aspect of conformal symmetry. The relationship between the two, or a decomposition SD --- > ECS is one where conformal symmetry is broken. General relativity has the group structure SO(3,1) ~ SL(2,C), and the extension of general relativity to conformal spacetime is to SO(4,2).

In my essay I work with the Britto, Cachazo, Feng, Witten (BCFW) recursion relationship. This is a Feynman diagram procedure in twostor space. Twistor space is the symmetry group SU(2,2) ~ SO(4,2), and so is a quantum or prequantum description of conformal relativity. The recursion relationship is a scaling principle which is conformal. In fact it leads to Yangian symmetries, which are a form of enveloping Lie algebras for gauge theory. Yangians consist of a gauge (or gauge-like) theory plus a dual theory related by a conformal symmetry, or Mobius symmetry and T-duality.

The BCFW recursion formula is a twistor theory, which has been used in the HopHat algorithm for computing gluon amplitudes at the LHC. Gravitation is in one sense the square of QCD gauge theory. After reading Giovanni Amelino-Camelia , which has regrettably and I think wrongly fallen down the community ranking, I suggested a connection between the boosts employed in κ-Minkowski and twistor theory. This might be a connection between string theory and the more loopy or triangulated theories like LQG. The Wheeler DeWitt equation has not time variables. Physically this means there is no Gaussian surface one may arrange in spacetime to localize energy. So HΨ[g] = i∂Ψ[g]/∂t = 0. The time variables is a coordinate time, used in QFT equations, which is not a proper variable in general relativity. Hence this equation is a constraint equation, classically NH = 0 and N^iH_i = 0. String theory on the other hand requires some external background field from which gravitons as closed strings are represented. This has been a problem the LQG folks like to point out --- never mind LQG has failed to produce even a first order renormalizable calculation. I speculate that somehow the two views of quantum gravity might connect, where LQG provides the background or constraint for string theory, and the field calculations of strings makes LQG more tractable.

Gravity as a dynamic force is conservative. The force in the Newtonian limit is given by F = -∇Φ(r), which is conservative. This means the force evaluated around a closed loop, such as an orbit, is zero. Thermodynamics gives nonzero evaluations for such forces. This is related to the matter in differential geometry that a p-form ω is closed if dω = 0, but a subset of them are exact when ω = dσ, or d^2 = 0. There is some cohomology behind this. The force is determined by the coboundary operator on a 0-form and we have by Stokes law

∫F•dr = ∫∫∇xF•da. da evaluated in the region enclosed by the closed loop.

Yet we know that ∇x∇Φ(r) = 0 (curl-grad = 0 or d^2 = 0) and so the force is conservative.

Verlinde's entropic gravity does not involve the dynamics of a particle in a gravity field. It involves the dynamics of an event horizon or holographic screen. The main idea is that the force on the screen over some unit distance is equal to the work

∫F•dr =W,

and this work is equal to the increase entropy of an event horizon. This by the Bekenstein theorem is S = k A/4L_p^2, for L_p = sqrt{Għ/c^3} --- the Planck length. So the entropy is a measure of how many Planck units of area there on the horizon. So the Verlinde hypothesis is

∫F•dr =TS,

or a force that displaces the horizon some increment gives

F•δr = TδS.

As a result some input of mass-energy into a black hole increases entropy, and this force is what evolves the event horizon, or equivalently the holographic screen.

Event horizons and screens have units of area, and in naturalized units with c = ħ = 1 the gravitation constant G is an area. So this measures the amount of information entangled with the black hole, or the entanglement entropy. As a result the theory you have built up with moments of inertia most likely applies on a holographic surface.

More later & cheers,

LC

Sean and Flavio,

Beautifully written essay. I just have a comment and a question.

Comment: You claim that the problem of time stems from the different way that time and space enter into the formalism of general relativity, and that the evolution equations cannot be solved starting from nonspacelike initial data. The latter is not known to be the case. Certainly for the wave equation, one can begin with data on a nonspacelike hypersurface and evolve in the spacelike direction. (The Holmgren-John theorem guarantees uniqueness, and the work of Craig and myself shows that a solution exists if one begins with data satisfying a nonlocal constraint.) As for the problem of time, I would think that it has something more specific to do with general relativity, since it doesn't arise in the quantization of other relativistic fields.

Question: I see where coarse-graining is important for the fascinating result you get for the Shape formalism, but where does renormalization come in?

Steve