Hi Peter:
I've now taken my time reading through your essay, and I think I've got a much better idea of what you were driving at in your above two comments. In fact, I think that apart from the physical mechanism for achieving a constant speed of light in all frames (I believe it's purely geometric; could you please write something here to help me better understand your thoughts?), I think our thoughts are actually very similar. If we could discuss this a little, I think it would give me a much better understanding of your essay, which is still somewhat unclear for me.
To begin with, when you mention different local background frames within a greater cosmic frame (as Einstein's local "s" within "S"), I think I've got almost the same thing in mind. It's really in the way I think the local frames should be "patched" into the global frame that I expect we differ. I'd like to explain how I think this should be done, which is different from the common method of embedding local solutions into cosmological ones. I hope you consider:---
Please think back to the barograph apparatus in my essay. Space-time emerges as a 1-dimensional absolute space uniformly evolves, so that a global simultaneity-relation exists, whereby only events that occur at *t*=const can be said to have *truly* occurred "at the same time". Special relativity is recovered by imposing Lorentzian metrical structure on the coordination of the emergent space-time continuum of events, with "photons" travelling along (invariant) null lines, so that the physical descriptions will be equivalent in all frames---particularly with light having the same constant speed in every one *by necessity*. The difference between mine and the standard interpretation of special relativity is that whereas it's usually thought that the events that occur at the same (coordinate) "time" in all coordinate systems occur simultaneously in those frames, it's clear from the example that the events that occur at the same *t* are the ones that are *truly* simultaneous, in every coordinate system, and space is tilted in the local coordinate frame of every observer that moves through the universe.
In my essay, I've considered how this elementary (special relativistic) "universe" would be represented in the frames of inertial observers only---but what about one who is accelerated in some way? Clearly, at each point on that observer's worldline, the slices of space-time of constant *t* still have to represent "the universe"---that absolute simultaneity-relation---at that time, as opposed to the slice of constant coordinate time (whatever that might be). Furthermore, the slices of constant cosmic time *t* will have to be successive and never intersect one another---which would be identically impossible, according to the setup of the whole thought experiment. For example, if an "observer" would oscillate through the universe, the slices of constant *t* would teeter-totter in the local coordinate frame, without crisscrossing; the scale of ordered events on one side of the oscillating observer would contract while the scale on the other side would extend. In any case, the slices of constant *t* obviously won't be straight lines in the local space-time geometry of an accelerated observer; the emergent space-time would be curved.
There's an important distinction in this, resulting from the fact that not only the basic Minkowski metric, but the particular foliation along *t* has been made significant a priori in the definition of what's *really* going on. For, according to general relativity such a warped space-time geometry may be described equivalently as being induced by gravitational mass; in the example of our oscillating observer, e.g., it might be said that, instead, a mass to the right of the observer and one to the left alternately go in and out of existence, and space-time is warped accordingly while the observer always follows a geodesic of the space-time metric. General relativity builds on the principle of general covariance and assumes that the connection that describes parallel transport is the Levi-Civita connection of the metric tensor, so that the space-time geometry is fully described by the metric alone [1,2]. Einstein's theory of relativity therefore fully embraces the implication that the coordinates themselves can have absolutely no metrical significance; the theory is totally observer independent.
But this gives a poor description of what's *really* going on in my example. The "universe" is always represented by slices of constant *t* in any frame; space-time emerges with the inertial and causal structures described by the Minkowski metric; there is no *real* curvature of space; and of course space-time itself, which is just the graduating Minkowskian map of events that occur in the enduring 3D Euclidean universe, is not real. It seems, therefore, that the tangent space is very important to the description of what's actually going on. It appears, therefore, that the mathematically equivalent theory of teleparallel gravity would describe this scenario much more faithfully. However, this doesn't change the fact that the local *space-time* geometry in some non-trivial gravitational field (thus attributed to non-vanishing torsion) will be some solution to Einstein's equation (since the two theories are equivalent).
This is really how my research has led me to conceive of space-time. I no longer think of it as a real four-dimensional field in which different solutions to Einstein's equations should be patched together. I think of it as an emergent map of events that happen in a three-dimensional universe, which is described differently in different frames due to different states of motion through actual enduring space. I think that the local form of the SdS solution should do a pretty good job of describing the local geometry of a galaxy cluster, at least near its perimeter where the gravitational repulsion due to Lambda finally overcomes the gravitational attraction due to the mass contained within, and galaxies can't remain gravitationally bound. I also don't think that local mass-energy really contributes to the overall expansion of space, which I think is purely a metrical property, so that while gravity acts between bodies, space (by which I might only mean the metric that describes any instantaneous distribution of matter, and not necessarily *substantive* space (although I don't rule that out as a possibility)) quite simply *will be the size that it is* at any time, while local matter arranges itself dynamically therein.
This is how I think the descriptions of small "s" should fit into the evolution of big "S". If I've done an alright job of explaining myself, and you've followed my reasoning, I think the obvious outstanding question should be, "But why should the fundamental emergent space-time metric be Lorentzian and expanding?" The Lorentzian signature of the emergent space-time metric is absolutely crucial to the way special relativity falls out of the description in my essay, with "photons" travelling along null lines, and therefore at the same invariant rate in all frames. But why shouldn't the space-time geometry just be Euclidean, for example? In that case, the speed of light wouldn't be the same constant value in all frames and space wouldn't expand.
Since you asked to begin with if I've considered any real mechanisms for achieving this, I'd like to give you my answer, which I think you'll find interesting; but since it depends on some understanding of what I've written here, I want to make sure you've followed my reasoning up to this point, and that you're still interested in knowing the answer to your question, before I give it to you. So, provided that I do think I've got an adequate answer to this, do you have any questions or objections regarding what I've written here so far? Even if you're only provisionally alright with what I've written above, as in you get my reasoning but maybe you prefer another possible explanation, I'd be happy to continue.
Best, Daryl
