expansion of the universe; infinity of "level 1"

From: geodesicallyincomplete@warpmail.net
Date: Sun Apr 20 2003 - 12:13:36 MDT

Perry Metzger writes:

> The visible universe is 42 billion light years in radius, even though
> the age of the universe is only 13.7 billion light years (to within 1%,
> according to the WMAP data). This, apparently, because of the expansion
> of the universe. I don't pretend to understand how the universe could
> have expanded such that two points separated at a rate greater than c
> -- perhaps someone with a physics background could explain about this.

There seem to be two different issues here.

Firstly, why has the visible universe expanded faster than light?

This question can be answered without referring to strange relativistic
effects. The explanation is that far away points sent out their light a
long time ago, and in the mean time these points have been receding from
us. By assuming the universe was always "matter-dominated" and taking
into account how the scale factor of the universe changes over time, you
can calculate that the present radius of the region we can see (actually,
whose past we can see) is exactly 3 times the distance light could have
travelled. (Since the assumptions made are only approximately correct, so
is the factor of 3).

It's a bit confusing to think about this, because what we see is not a
large region of space at one time, but a "slice out of spacetime" where
each point we see is at a different time. So the 42 Gly-radius visible
universe is not actually what we see through telescopes, but a
"straightened out" extrapolation; and different definitions for "visible
universe" are possible.

(I'm allowed to speak of a definite "space" and "time" in this case,
because Friedman-Robertson-Walker universes have a nice "preferred" set
of coordinates with respect to which the universe is homogeneous and

Secondly, why does the distance between different points (say, beyond the
visible universe) increase faster than light?

The answer has nothing to do with quantum mechanics or varying speed of
light. Also, while "they're *not* moving apart, it's *space itself* that
expands" is a good way to think about it intuitively, I don't think this
distinction has any absolute meaning in general relativity. All you can
say mathematically is that the difference in a certain coordinate
increases, and usually it's handy to think of this as space expanding (to
contrast with special relativity).

The short and unenlightening answer to the question is that "relative
velocity" and "frame of reference" are special-relativistic concepts
only, that special relativity is general relativity applied to Minkowski
space, and that Friedman-Robertson-Walker space (our universe, or a good
approximation) is not Minkowski space.

For a longer and more enlightening answer, I would have to show why
objects can't move apart faster than light in Minkowski space, what
coordinates in FRW space play the same roles as what coordinates in
Minkowski space, and why the reason you can't go faster than light
doesn't work. I don't know how to do this (especially not in a way that
makes intuitive sense), but I can make a few possibly useful remarks.

I said that FRW space is not Minkowski space, but this is not completely
true. A completely empty (and therefore open) FRW universe turns out to
be Minkowski space, but in weird coordinates. These coordinates are just
as valid as those used in special relativity (they're chosen to make use
of the symmetries that come from viewing the space as an expanding
universe). Special relativity still holds, and in special relativistic
coordinates no two objects can go faster than light with respect to each
other. But the "cosmic time" used when viewing it as an expanding
universe is not the same as (any of) the special relativistic time
coordinate(s), and it turns out that with respect to the "cosmic time"
things can move apart faster than light.

When there's a certain (critical) mass density, the universe becomes
flat, instead of open; now it's no longer Minkowski space in weird
coordinates, but something else entirely that looks a bit like Minkowski
space at each moment, but expands. Since space and time get mixed up in
coordinate transformations, you can no longer do everything you could do
in Minkowski space (due to the changing scale factor). For example,
measuring distances using light beams becomes messier. Apparently, this
lets distances increase faster than light without causing any
contradiction or time-loop. (Intuitively, this should be clear from the
balloon model)

Ned Wright's cosmology tutorial at


lists several different notions of distance that can be used and makes
all this handwaving of mine mathematically precise. There's also a
cosmology FAQ on that site that I recommend looking at.

Paul Hughes asks how a finite universe can become infinite in a finite
amount of time. This is a common question, and the answer is that it
can't -- infinite universes are already infinite at all times. When
people say "back then, the universe was only as large as a tennis ball",
they mean the visible universe.

You'd think the universe was only a point at the initial singularity and
therefore finite. But this is not necessarily true -- the initial
singularity is not a part of the "spacetime manifold", so in a sense, it
never existed, and it's only meaningful to speak of times after the
initial singularity. I think for this reason you can't really say whether
it's a point or something else (what do you get when you compress an
infinite space in such a way that every finite part of it gets compressed
to one point?)

It's also possible that there was no initial singularity, as in
past-eternal eternal inflation. In this case what we think of as our
universe originated in a "pre-universe" that had already existed forever.
One of the things I learned from Tegmark's paper is that apparently a
universe-bubble such as ours can be spatially finite in one set of
coordinates (from the "outside") and spatially infinite in another set of
coordinates (from the "inside").

(A semi-interesting fact I just discovered: it's apparently possible in
GR for a "naked singularity" to become a "thunderbolt singularity" or a
"wave of death", meaning that it starts expanding as a sphere at the
speed of light, "eating" everything in its path (i.e., its causal future
just stops existing). Now *that's* unFriendly.)

Hope I got all this right,

(yeah, first post, hi all)

"Doctor, why is the TARDIS bigger on the inside than it is on the
"Oh, that's because it's Dimensionally Transcendental."
"What does 'Dimensionally Transcendental' mean?"
"It means that it's bigger on the inside than it is on the outside."
  -- 'Doctor Who'
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