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.
