Milne cosmology

No, not the Hundred Acre Wood.

I mention the Milne model pretty frequently on sci.physics.research when people get confused about cosmological theories. It's gotten to the point that its official name there is now "that Milne model that Matt McIrvin likes to talk about." I shall explain why.

(Someday this should have more external links and suggestions for additional reading. For now I'll mention a major source of inspiration: Edward Harrison's classic textbook Cosmology: The Science of the Universe.)

The early and mid-20th century British astrophysicist Edward Arthur Milne considered his cosmological theory a serious threat to general relativity (which he tried to replace with something he called "kinematic relativity"). In hindsight, the Milne cosmology is no real challenger to modern general-relativistic cosmology, but it is an enlightening "toy model" of Big Bang ideas about the universe. Thinking about it seems to clarify many issues of what is meant by expanding space, the true role of gravity in cosmology, and the significance of the "cosmic time" coordinates of cosmological models. Historically it was the starting point for much work on the role of symmetry in cosmology, and it is illuminating in that regard as well.

In textbooks on cosmology, the notion of non-Euclidean space is usually introduced in the context of Einstein's general theory of relativity, in which space-time curvature arises from matter and energy, manifesting itself in the form of gravity. Space-time is no longer the rigid, static arena of special relativity; it's dynamic, and it stands to reason that the whole universe could be expanding. In standard Big Bang models, there is no "center" to the universe and no "outside" that the universe is expanding into; the entire space expands uniformly. Space-time is positively curved by the matter in it, but space is either positively curved (like a hypersphere), flat, or negatively curved (like a hyper-saddle), depending on whether the matter density is more or less than some critical value. General relativity would seem to be central to the whole system of ideas.

Indeed, it is central in modern cosmology, and it does seem to be necessary to describe today's observations and experiments, but it turns out not to be as necessary for a Big Bang model as one might think. It came as quite a shock when Milne described an ingenious model of an infinite, negatively curved, expanding universe that was based entirely on the flat space-time of special relativity, and involved no gravity whatsoever!

It works like this. The Milne universe does have an outside; the whole universe of galaxies gets created at a single point in flat space-time (interestingly, Milne tried to justify this on Christian theological grounds), and thereafter occupies the interior of a bubble that expands at the speed of light into previously empty space. We say that it fills the "future light-cone" of the Big Bang.

The galaxies are treated as non-gravitating test particles; we ignore gravity entirely. They all shoot out at different speeds, along constant velocity straight-line paths inside the bubble. The closer they are to the speed of light, the nearer they will be to the surface of the bubble. The picture shows a cross-sectional slice through the middle of the bubble.

The galaxies fill a bubble of finite size, yet there are an infinite number of them! How is this possible? Well, an effect of special relativity is that an object that moves relative to some reference frame has a reduced length, as calculated in that reference frame; it becomes squashed in the direction of motion. This is called the Lorentz contraction. The length converges to zero as the speed approaches the speed of light. Distances between moving objects are similarly contracted. So there could be a whole infinity of galaxies in that finite expanding bubble, packed in cleverly! They'd get closer and closer together, approaching infinite density, in the limit of galaxies near the surface. We can always pack in more galaxies by stuffing faster-moving and therefore flatter ones in the space remaining to the surface of the bubble.

How do things look to somebody living in one of these galaxies? Well, a fast-moving astronomer wouldn't feel squashed at all; everything's normal to him. Indeed, since the velocities of neighboring galaxies aren't that different, he wouldn't even notice anything unusual about them. They'd just be receding from him at a speed that gradually increased for more and more distant galaxies. Close in, the relation between recession speed and distance would be linear: the Hubble law.

In fact, such an astronomer would observe an expanding universe of galaxies, with recession speeds approaching the speed of light at a certain distance. In other words, he'd see something not too different from what astronomers actually do see! At the time Milne was writing, there was no way to know that we weren't actually in something like a Milne cosmology.

A second astronomer billions of light-years away from him, in one of those really fast-receding galaxies near his visible horizon, would see more or less the same thing! This is because the density and motion of galaxies has been chosen carefully to be invariant under a Lorentz transformation, the change from one special-relativistic rest frame to another. We can use a Lorentz transformation (one that keeps the Big Bang event constant) to map her local situation onto the first astronomer's; the positions of nearby galaxies might be somewhat different, but on average the universe would look just the same.

In fact, there's really no difference between the "fast-moving galaxies near the rim" and the "galaxies at rest in the center of the bubble," since we can make any galaxy and its neighborhood appear at the center by a Lorentz transformation. From what looked like an expanding bubble with a definite center and edge, we've extracted a universe of galaxies in which no particular galaxy sits contentedly at the center of the universe (or maybe they all do!) and no particular galaxies are leading the march out on the light-speed frontier.

Milne called this feature of his model the Cosmological Principle, a term still used today: he felt that good cosmological models should not single out any single galaxy for special treatment, and should especially not treat the Earth's position as special. He was, in other words, interested in highly symmetrical cosmological models-- those that were homogeneous, or the same on average everywhere-- and this is a focus of interest that has continued to the present day. Indeed, though we really have no way of knowing whether the universe is homogeneous on scales larger than we can see, it does seem to be homogeneous on the largest visible scales (allowing for the effects of light-speed delays, redshift, and so on), so this seems to have been a good guess! The Milne model is also isotropic, or more or less the same in all directions, and this also seems true of our universe, though some contrary claims have been made.

We started out with a flat space-time, thinking in terms of the space and time coordinates that seem natural in special relativity; in these terms, the Milne world is an expanding bubble of finite size, that gets bigger and bigger as our time coordinate increases. But is this really the best time coordinate to use?

Look at the expanding bubble from our original, lofty vantage point. Now imagine that each galaxy contains a stopwatch, which is created at the Big Bang with its hands at time zero, and flies along with the galaxy from then on. The galaxies that appear to us to be "near the rim" will have slowed-down clocks, because of the relativistic time dilation that occurs when things travel at close to the speed of light. The clocks near the middle of the bubble, which are moving more slowly, will run faster.

So the clocks near the rim seem to run behind the clocks in the middle. But we know that there is a Lorentz symmetry between galaxies, and it stands to reason that an astronomer in some galaxy would see that galaxy's clock running at a perfectly ordinary rate (after all, she's subject to time dilation too). If we want to choose a time coordinate that is fundamental to this universe of galaxies, that treats all galaxies the same, maybe we should pay attention to these clocks and use their hands to establish a "cosmic time" coordinate.

There's nothing that says we have to use this cosmic time coordinate, but it makes certain things easier to deal with. In particular, the surfaces of equal cosmic time are the surfaces of events that are mapped onto each other by the Lorentz transformations that are the symmetries of this cosmology. The cosmic time coordinate is the one that symmetry is telling us to use.

Now suppose I draw a space-time diagram of the Milne cosmology using our original coordinate system. We'll have to ignore one dimension of space, in order to include time as the third dimension. Because the clocks on the rim "run slower" (by our original time standard), the surfaces of equal cosmic time bend up into the future at the edges. They look like nested bowl-shaped surfaces that stack into the light cone, each one extending off to infinity as it asymptotically approaches the cone. (People familiar with special relativity may recognize this as the same shape as a "mass hyperboloid.") The picture shows one such surface. While snapshots of the universe taken according to our original time coordinate show a finite bubble with empty space outside it, the snapshots according to the new cosmic time coordinate are infinitely large and entirely filled with galaxies!

Ned Wright drew two nice pictures of the situation. The first picture below the link destination shows space-time drawn with cosmic time as the vertical coordinate, and one dimension of the equal-cosmic-time surface as the horizontal coordinate. The surfaces of equal cosmic time are horizontal in this picture, and the galaxies burst forth from the Big Bang along diagonal lines at all slopes. Light rays do not look straight in this picture; they can diverge from the Big Bang and converge again later!

In the cosmic-time view, it looks as if the speeds of the galaxies are diverging to infinity, but they really aren't going any faster than light, as you can see by looking at Wright's second picture. This second picture shows our original coordinate system: the equal-cosmic-time surfaces are hyperbolae stacked up inside the light cone. Now the light rays do look straight, and the rays that we thought were coming from the Big Bang moment actually seem to be emerging perpendicularly from the bounding light cone.

Here's an interesting thing about those pictures: In cosmic time, the universe of galaxies looks infinitely large going all the way back to the Big Bang-- even though from the initial "bubble" point of view, the whole thing got created at a single point. People often wonder how a universe that began in a Big Bang could be infinite. The cosmic-time view of the Milne cosmology shows a possible way: the density of galaxies just gets larger and larger as you look back in time.

What is the role of the space outside our original bubble? In cosmic time, that translates to an era before the Big Bang. If it weren't for the existence of this weird pre-Big-Bang era, the whole thing would look an awful lot like a modern cosmological model. Hmmm... that's curious.

These hyperboloids are not only infinitely large, they're non-Euclidean. If we pay attention not only to the galactic clocks, but to the galaxy-bound astronomers' distance calculations as well, it turns out that they find themselves in a negatively curved space (like the surface of a saddle). A very large circle in such a space will have a circumference larger than pi times the diameter. In the picture, the gray stripes on the surface are drawn so that they'd be equally wide according to an astronomer in a moving galaxy at that moment of cosmic time.

I've also included a flat disc with stripes drawn to the same width (according to a stationary observer). The number of stripes is the same on the disc and on the "bowl." So the part of the bowl that I've drawn has the same measured radius as the disc, but its circumference is larger. The curvature must be negative.

The outer stripes look stretched out because they are close to the light cone; in the Minkowskian geometry of space-time, intervals in space and time coordinates tend to cancel out close to the light cone, so that distances are shorter than they look. You might think that the stripes ought to look Lorentz-contracted instead, but that is how they'd look if our slice through the cosmology were taken at constant t, not constant cosmic time.

So we've gotten curved space out of flat space-time, by virtue of a funny embedding; this funny embedding is in turn a natural one to consider because of the temporal effects of special relativity on the particular distribution of galaxies in this model. The lesson is that space curvature and space-time curvature are not at all the same thing, and the former can change radically with a shift of coordinate systems.

To recap, this is a cosmological model that is either a finite bubble expanding from a point into an empty, flat, static universe, or an infinite, negatively curved, homogeneous expanding universe! The two descriptions sound completely different, yet all that separates them is a change of coordinates.

This sort of thing happens often in the discussion of cosmological models. Many things that you'd normally think of as objective properties of a universe end up shifting uncomfortably when the coordinates change. A while ago on sci.physics.research, there was a long argument between squark@my-deja.com, Ted Bunn, and Phillip Helbig over whether a certain cosmological model called the de Sitter universe was a finite hyperspherical universe that contracted and then expanded again, or an infinite universe that just expanded exponentially. After much confusion, it turned out in the end that both descriptions were correct! (The analogy between Milne and de Sitter universes is actually deeper than this, but that is probably a story for another time.)

In the Milne model, do the individual galaxies expand? Well, I suppose we could define them as expanding by fiat-- since we are ignoring gravity, there's no good way to model the dynamics-- but there's nothing that makes them expand. After all, in the "bubble" picture they're just flying apart through flat space-time. Lorentz contraction affects them, but a given galaxy's Lorentz contraction factor will always be the same (in a given rest frame). And, in fact, in more realistic cosmologies, the individual galaxies do not participate in the universal expansion either. Nor do the objects within them.

Tempers occasionally flare on the physics and astronomy newsgroups over the issue of whether the cosmological redshift-- the shift toward longer wavelengths of emitted light from distant galaxies and quasars-- is properly described as a Doppler shift, like the change in pitch of a car horn as it drives past you. Are the light waves being stretched out by the expansion of Space Itself, or is it just the Doppler effect of the recession velocity of the distant galaxies?

What does the Milne model say? Well, first let's take the "expanding bubble" point of view. There, space is just flat and static, so it can't do anything funny to the light waves. But the galaxies are all flying apart from each other, so the relativistic Doppler shift will come into play. The cosmological redshift is a Doppler shift.

On the other hand, if we plot the paths of light rays in cosmic time, they seem to diverge-- and that divergence corresponds nicely to the redshift. It appears that the light waves are being stretched out by the expansion of space. So maybe both descriptions are right!

To what extent can we generalize this notion? To answer that question, we'll have to move to a more realistic cosmology...

Now let's move away a bit from the pure Milne cosmology. Suppose we consider the effect of gravity on our expanding bubble of galaxies. In our original picture, the galaxies are packed in divergingly tightly near the surface of the bubble, and the individual galaxies' kinetic energy should diverge too! So the energy and momentum densities should diverge. Now, in Einstein's general theory of relativity, these things are sources for gravity-- they cause space-time curvature. When they diverge, you get singularities. So at some point the space-time is going to have to pucker off into some sort of roughly conical singularity.

Of course I am speaking vaguely here, since we had a flat space-time before, and now it isn't flat any more. Also, to do this right we have to perturb the model so that the symmetry remains intact-- even though it's no longer quite a Lorentz transformation. But the basic idea is sound: we lost the whole empty outside of the cone, and instead there's a singularity somewhere near where the edge used to be.

In cosmic time, this looks much more familiar. Remember that before, "the outside of the bubble" turned into "the time before the Big Bang." In our curved space-time with gravity, the conical singularity turns into a singularity sometime in the past. It's an infinite universe that expands from a Big Bang singularity! Now this is sounding familiar. There's no more outside, no more pre-existing space-- the whole space begins at that singularity.

In fact, the Milne cosmology is nothing more than the zero-density limit of the expanding Friedmann-Robertson-Walker (FRW) metrics with no cosmological constant-- the textbook universe models of every class on cosmology. An FRW universe with a sub-critical matter density looks, in cosmic time, like an infinite, negatively curved, expanding universe with a Big Bang singularity in the past. Galaxies in such a universe fly apart after the Big Bang and gradually slow (but never stop) in their recession from one another.

The matter density is an adjustable parameter-- a kind of "knob"-- on these models. Different settings give us different histories for the universe. If we turn the knob to a lower and lower density setting, the space gets more negatively curved, the space-time flattens out, and the recession of the galaxies becomes more and more constant with time. At zero, the space-time is flat, the galaxies (now gravityless test particles) recede linearly, and the singularity at the Big Bang disappears! All of a sudden we can extend the space-time into the region on the other side of the Big Bang... and we've got the Milne model back again.

What about the question of the redshift? In a more realistic model, is it a Doppler shift or isn't it? Well, the special relativistic Doppler formula is not going to apply any more at very long distances, because the space-time itself is not flat any more. But it turns out that in a cosmic-time picture, you can still chart the paths of light rays as diverging under the expansion of space. So the "expansion of space" explanation of the cosmological redshift is more robust than the "Doppler shift" explanation.

But for distances small compared to the curvature of space-time, the Doppler formula is approximately correct, and it's still sensible to speak of a Doppler shift in that limit. After all, a police speed trap happens in curved space-time, but the effect of the curvature is small enough that the Doppler formula applies to the cop's radar gun. The key thing is not to apply the special-relativistic Doppler formula blindly when the redshifts get large, because it won't apply any more.

Milne's kinematic relativity theory turned out to be a bust, and his cosmology is not a modern contender, but his Cosmological Principle-- the central role of symmetry in categorizing cosmological models-- lives on.

Before Milne, there were many fine cosmological models under consideration, built on the general theory of relativity: in fact, Friedmann had already written down the equations for the evolution of a spherically symmetric universe that eventually led to the FRW models. But Milne's emphasis on symmetry, on the Cosmological Principle, made more explicit a property that most of these models already had, and I believe it partly inspired the classification work of Walker that provided the scaffolding for modern cosmology.

The radical, gravityless nature of his cosmology also made much clearer what you really needed gravity for, and what you didn't need it for. Even Einstein occasionally got confused about the distinction between curvature arising from gravity, and curvature arising from a funny embedding of a space into space-time. The Milne model puts the distinction in sharp relief.

Finally, the Milne model is a terrific pedagogical tool because it exhibits so many of the properties of a realistic cosmological theory while only relying on special relativity, a theory that is much easier to understand than general relativity. It is an excellent means of demystifying the notion of an expanding universe.