**From:** *geodesicallyincomplete@warpmail.net*

**Date:** Sun Apr 20 2003 - 12:13:36 MDT

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Perry Metzger writes:

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

isotropic)

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

http://www.astro.ucla.edu/~wright/cosmo_02.htm#MD

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,

Nus

(yeah, first post, hi all)

--- "Doctor, why is the TARDIS bigger on the inside than it is on the outside?" "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' -- http://www.fastmail.fm - Send your email first class

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