Why I keep talking about it
- How necessary is general
- Infinity in a bubble
- In cosmic time
- Space curvature
- Expansion and redshift
- Adding in gravity... and taking it out
- Milne's legacy
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.
How necessary is general relativity?
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!
Infinity in a bubble
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
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
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.
In cosmic time
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
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
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 firstname.lastname@example.org, 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.)
Expansion and redshift
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
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
To what extent can we generalize this notion? To answer that
question, we'll have to move to a more realistic cosmology...
Adding in gravity... and taking it out again
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
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
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
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