PeterDonis said:
I haven't been sure, because some of the things you've been saying seem to imply that there is at least a correspondence between the models; for example, this:
If the Milne model is really supposed to be a "different model", then where does the "horizontal plane in the FWD is mapped to the light-cone" come from? That correspondence between the FWD and a "light cone" diagram, for example the "SR" diagram on Ned Wright's page, *only* applies if we are talking about the FRW model with k = -1 and zero density (as Wright says on his web page). In fact, that correspondence is the basis for the claim that the Milne model is a special case of the FRW models (the case with k = -1 and zero density), which you reject. If you are not talking about the FRW model but about some other model, then I don't see how you're coming up with a correspondence between your "light cone" diagram and the FWD.
If I recall correctly, yes, the Milne model has k=0, a=1. It is Minkowski Spacetime with a density that approaches infinity as you go out from the center toward r=c*t, and it has a density that approaches infinity as to go toward t=0. That is not at all compatible with a zero density.
If anyone thinks they have correctly modeled the Milne universe with a zero density, they are just fooling themselves, and using some kind of circular reasoning or a straw-man argument.
There are two variables in the Friedmann Walker Diagram, the horizontal variable is a "space-like" variable, and the vertical is a "time-like" variable. To map to the Comoving Particle Diagram, I'm not sure exactly how it is done, but I
think the vertical "time component" is just mapped straight over, while the horizontal "space component" is some form of velocity * distance. I may be wrong, but I *think* it is the integral of the changing scale factor with respect to the "cosmological time."
The horizontal variable is SPACE, and the vertical variable is TIME; some kind of "Absolute" or "cosmological" time, which really doesn't exist in the realm of Special Relativity.
What the Milne model does is treats the horizontal variable in the FWD as sort of a "rapidity-space" The mapping from the FWD to the MMD assumes that the meaning of the FWD "space-like" variable is distance = rapidity * proper time. For a set of particles all coming from an event (0,0), giving the rapidity and proper time for a particle uniquely defines its position in space and time. To map from the FWD to the MMD, you are simply mapping: (d'=rapidity*proper time,t'=proper time) to (d=space, t=time).
To map from the FWD to the CPD, you are mapping (d'="Stretchy" Velocity * Cosmological Time, t'=Cosmological Time) to (d=Space,t=Cosmological Time).
I'll see if I can express this as mathematically and unambiguously as I can, so that if I'm wrong it can be corrected.
\begin{matrix} FWD \mapsto CPD \text{ as }(d\int a(\tau)d\tau,\tau)\mapsto(d,\tau) \\ d=Proper Distance = Cosmological Distance \\ \tau=Proper Time=CosmologicalTime \\ a(\tau)=ScaleFactor \end{matrix}
On the other hand, the Milne mapping looks like this:
\begin{matrix} FWD \mapsto MMD \text{ as }(\varphi \cdot\tau,\tau)\mapsto(v \cdot t,t) \\ \varphi=rapidity \\ \tau=proper time \\ v = velocity \\ t = time \end{matrix}
As you can see, the Milne mapping is linear; there's no changing scale factor. The relation between rapidity and velocity and distance, time, and proper time is the same as is usually given in Special Relativity.
Rapidities between -infinity and +infinity map to velocities between -c and +c. So the horizontal plane (representing infinite rapidity) in the Friedmann Walker Diagram maps to the light-cone in the Milne Minkowski Diagram.
So after any corrections to the FWD to CPD mapping, I wonder if you are yet convinced that we are talking about two different mappings? Can you see that the rapidity=infinity line is included in the Milne Model? Can you see that the distance vs. time relation is fundamentally different? Can you see how an infinite density naturally results from this mapping? Can you see how the Milne Model has only one event at x=0,t=0, while the CPD has an infinite number of events at x=0,t=0?
Only if you also assume that spacetime can be flat with a non-zero stress-energy tensor. See next comment.
First of all, I assume you mean "globally invariant under Lorentz Transformation," since any distribution which is proportional to the metric (e.g., a cosmological constant or a scalar field) will be *locally* invariant under Lorentz Transformation.
Yes, I mean "globally invariant"
Second, and more important, your model assumes that you can have a non-zero distribution of matter without affecting the spacetime geometry. Or, if you insist on avoiding any "geometric" interpretation, your model assumes that you can have a non-zero distribution of matter without any tidal gravity effects--that any pair of freely moving particles in your model will have the same constant relative velocity (each one as seen by the other) for all time. This is known to be false--for example, because of the curvature in the Hubble diagram that we discussed before.
I may have implied that "any pair" of freely moving particles would have the same constant relative velocity, but I must back off on that. But it is not "any pair." It is a specific set of particles which are completely undisturbed after the moment of the Big Bang which will maintain a constant relative velocity.
You have the "Relativity, Gravitation, and World Structure" e-book; check section 112, the list of properties, item 10, Milne says:
"Every particle of the system is in uniform radial motion outward from any arbitrary particle O of the system, and the acceleration of every particle in the system is zero. But the acceleration of a freely projected particle, other than the given particles, is not zero."
I think what is happening is that every particle that is
at the center of mass remains at the center of mass, but if you move
away from that center of mass, you'll be attracted toward it.
The unusual thing is that accelerating toward the center of mass actually causes the center to move
away from you rather than toward you. That results in inflation.
To be clear, I am not talking here about "local" effects such as bending of light by the Sun, but about "global" effects, about the relative velocity of "freely moving" particles on a cosmological scale (the ones whose worldlines are straight lines radiating out from the Big Bang event in your diagram). In GR, such particles, even though they are freely falling (they feel no acceleration), can have relative velocities that vary with time. This shows up in our observations as a variation in the "Hubble constant"--the slope of the curve in the Hubble diagram--with time. According to the Milne model, this is impossible--this should be obvious from the fact that, as you say, the Milne model is based on logical deductions from SR, since in SR there can be no such variation with time in the relative velocity of freely falling objects (i.e., objects moving on inertial worldlines).
This is the elephant in the room that I wanted to talk about earlier. Acceleration.
The Big Bang is one explosion, but particle decay should lead to secondary explosions. If you have secondary (and tertiary, etc.) explosions of matter, this naturally leads to variations in the "Hubble Constant" based on logical deductions from SR. Our local Hubble Constant is based not on the age of the universe, but on the time since the most recent explosion. Only the most distant Hubble Constant tells you the age of the universe.
This is why GR was necessary--because in the presence of gravity (i.e., when the effect of mass-energy on the behavior of inertial worldlines is significant), freely falling objects can change their relative velocity with time (in other words, tidal gravity is present), and SR cannot account for that.
You can see this "curvature of freely falling worldlines" in Ned Wright's diagram of the "critical density" case (the "FPD" version). Notice that in that diagram, the worldlines radiating out from the Big Bang curve inward towards each other--unlike the "zero density" diagram, where they are straight. This is the effect of non-zero mass-energy (i.e., gravity) on freely falling worldlines (or "spacetime geometry" in the usual terminology). The usual pop-science way of describing this is that "the gravitational attraction of the mass-energy in the universe causes the expansion of the universe to slow down." (This terminology was invented before we discovered that, for the last few billion years or so, the expansion has actually been "speeding up", which is why dark energy has been added to the "standard" cosmological model--Ned Wright's diagrams don't cover that case, although I believe he discusses it elsewhere on his cosmology site.)
The pop-science way of describing these things is not nearly sufficient to give any hint to me about what they are actually measuring. I'm interested in whatever data they've gathered to determine that the "expansion of the universe is speeding up" but I think it more likely that whatever the effect, the cause is much more likely to be our galaxy's acceleration toward the receding center of mass, rather than some universal cosmological scale factor increasing at a faster rate.
Mathematically, it's fairly easy to construct transformations that do weird things like this. For example, the transformations used to construct Penrose diagrams map various points or lines at "infinity" to finite coordinate values (see the Wikipedia page here:
http://en.wikipedia.org/wiki/Penrose_diagram). There's nothing inconsistent about them; you just have to get used to how they work.
As far as other metrics are concerned, yes, there are transformations often used in GR that have similar effects. For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point, as you can see by transforming to Kruskal coordinates, where that entire line becomes the single point at the center of the diagram. (Here I've been ignoring the angular coordinates; when we put them back in, the "point" is actually a 2-surface.) This transformation also maps the "point" at t = infinity in Schwarzschild coordinates to an entire null line (the 45-degree line between regions I and II in the diagram with a yellow background on the Wikipedia page here:
http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates); this null line, the "future horizon", is where all the interesting physics at the horizon actually happens, and it is "invisible" in Schwarzschild coordinates, which often leads to confusion if those coordinates are taken too literally.
The fact that stuff like this can happen is a big reason why physicists are hesitant to attribute too much meaning to coordinates; you always have to check the physical invariants to see what's really going on. For example, I asserted just now that the apparent "line" at the horizon in Schwarzschild coordinates is actually just a point--or, if we include the angular coordinates, what appears to be a 3-surface is actually just a 2-surface. How do I know this is right? (Put another way, how do I know that the description in Kruskal coordinates is the "right" one physically?) Because I can compute the physical 3-volume of the apparent 3-surface, using the metric, and find that it is zero (because the metric coefficient g_{tt} is zero at r = 2M in Schwarzschild coordinates). A similar computation in FRW coordinates shows me that the initial singularity is, physically, a point (because a(t) is zero there, so the spatial metric vanishes), even though it looks like a line (actually a 3-surface, if we include the angular coordinates) in the "conformal" diagram. (Here I do really mean a literal point--zero dimensions--unlike the horizon of a black hole, which is physically a 2-surface--we can compute its area and find that it's non-zero, because the spatial part of the metric doesn't vanish completely. In the FRW case, the entire spatial metric vanishes at the initial singularity.)
I'm afraid I'm at a loss for following most of this. I'm not entirely convinced you have anywhere a single event which is mapped to multiple places (except for the FRW case where a(t) shrinks to zero).
If we could focus on the nature of the Schwarzschild coordinates; I'm afraid you'll have to go completely remedial to explain it to me, because I don't know the equations for the Schwarzschild coordinates or the reasoning. All I know is that time slows down as you get near the surface, and this causes light to turn toward the planet and get a higher frequency. When you're talking about the "line" at the horizon which is actually just a point, or the three-surface which is really just a two-surface, I'm not sure precisely what is meant.