Expansion and Space Travel
Daryl McCullough
forever. In that case, distant galaxies will be getting farther and
farther apart, according to the Hubble law. For anyone who knows the
answers, here are some questions that I have:
1. Can a galaxy be so distant that it is impossible to travel
to it? Intuitively, can the speed of a distant galaxy be greater
than the speed of light? I know that the concept of "speed"
of distant objects is not well-defined in GR, but we can
operationalize this question by asking whether there is a
galaxy so far away that a light signal that we send today will
never reach it.
2. Is there a "point of no return" for an immortal space traveller
who travels between distant galaxies? Is it possible that a
traveller could go to a distant galaxy and be forever stuck
there because it would be impossible to return to Earth (due
to expansion)?
These questions have practical importance to me, since I'm considering
taking a vacation to a distant destination. 8^)
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
Chris Hillman
On 20 Apr 2000, Daryl McCullough wrote:
> Let's assume that the universe is open, so that it will keep expanding
> forever. In that case, distant galaxies will be getting farther and
> farther apart, according to the Hubble law. For anyone who knows the
> answers, here are some questions that I have:
>
> 1. Can a galaxy be so distant that it is impossible to travel
> to it?
Yes, this is the notion of "particle horizon" in cosmology; see for
example the discussion of the de Sitter vacuum in Hawking and Ellis or in
the textbook by D'Inverno. And indeed, our best guess model is now a FRW
model with positive cosmological constant, which implies that eventually
every other galaxy will disappear behind our particle horizon. Indeed,
eventually the universe will consist of causally isolated elementary
particles. So extreme pessimism concerning our hopes for the future is
warranted :-)
> Intuitively, can the speed of a distant galaxy be greater
> than the speed of light?
Akhkh, no, that's the wrong intuition.
> I know that the concept of "speed"
> of distant objects is not well-defined in GR,
Right.
> but we can
> operationalize this question by asking whether there is a
> galaxy so far away that a light signal that we send today will
> never reach it.
Right, and that is the idea of a particle horizon.
> 2. Is there a "point of no return" for an immortal space traveller
> who travels between distant galaxies? Is it possible that a
> traveller could go to a distant galaxy and be forever stuck
> there because it would be impossible to return to Earth (due
> to expansion)?
Yes; indeed this very point is discussed in D'Inverno's textbook.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/personal.html
Frank Wappler
> Daryl McCullough wrote:
> > I know that the concept of "speed"
> > of distant objects is not well-defined in GR
> Right.
Does this extent to the notions of "distance, x" and
"interval, t", in GR?
If so, then what do you (both) mean by
"distant object", in GR?
And if not, then what's not well-defined about
d/dt( x( t ) )?
Thanks, Frank W ~@) R
Phillip Helbig
<Pine.OSF.4.21.00042...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> writes:
> Yes, this is the notion of "particle horizon" in cosmology; see for
> example the discussion of the de Sitter vacuum in Hawking and Ellis or in
> the textbook by D'Inverno.
I usually recommend Harrison (I hear a second edition has just come out;
has anyone seen this?), in particular chapter 19; can you give a brief
description of D'Inverno? Harrison is certainly more at the level of
the poster than Hawking and Ellis.
@BOOK {EHarrison81a,
AUTHOR = "Edward R. Harrison",
TITLE = "Cosmology, the Science of the Universe",
PUBLISHER = "Cambridge University Press",
YEAR = "1981",
ADDRESS = "Cambridge"
}
Phillip Helbig
--
Kapteyn Instituut Email .............. hel...@astro.rug.nl
Rijksuniversiteit Groningen Tel. ................... +31 50 363 4067
Postbus 800 Fax .................... +31 50 363 6100
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******************************** formerly at ********************************
this will still work for a while -----> Email ......... p.he...@jb.man.ac.uk
University of Manchester Tel. ... +44 1477 571 321 (ext. 2635)
Jodrell Bank Observatory Fax ................ +44 1477 571 618
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UK-Cheshire SK11 9DL Web ... http://www.jb.man.ac.uk/~pjh/
My opinions are not necessarily those of either of the above institutes.
<A HREF=" http://gladia.astro.rug.nl:8000/helbig/hire/hire.html ">HIRE ME!</A>
t...@rosencrantz.stcloudstate.edu
Daryl McCullough <da...@cogentex.com> wrote:
> 1. Can a galaxy be so distant that it is impossible to travel
> to it?
If the expansion is accelerating (as data these days seem to suggest),
then yes; otherwise, no.
> 2. Is there a "point of no return" for an immortal space traveller
> who travels between distant galaxies? Is it possible that a
> traveller could go to a distant galaxy and be forever stuck
> there because it would be impossible to return to Earth (due
> to expansion)?
Ditto. This is possible if and only if the expansion is accelerating
(and if it keeps on accelerating forever).
In fact, if the expansion is accelerating, than *any* galaxy will do!
(Well, not the local group -- you've got to go far enough to be in the
Hubble flow.) In an accelerating Universe, you can go to any other
galaxy and sit there and wait. Eventually, we'll slip out of your
horizon and you won't be able to get home.
-Ted
squ...@my-deja.com
da...@cogentex.com (Daryl McCullough) wrote:
> Let's assume that the universe is open, so that it will keep expanding
> forever. In that case, distant galaxies will be getting farther and
> farther apart, according to the Hubble law. For anyone who knows the
> answers, here are some questions that I have:
>
> than the speed of light? I know that the concept of "speed"
> of distant objects is not well-defined in GR, but we can
> operationalize this question by asking whether there is a
> galaxy so far away that a light signal that we send today will
> never reach it.
This is always true for an open universe, assuming the rate of
expansion is the same everywhere, a quite good approximation for
realitsic universes, as far as I know.
> 2. Is there a "point of no return" for an immortal space
> traveller who travels between distant galaxies? Is it possible
> that a traveller could go to a distant galaxy and be forever
> stuck there because it would be impossible to return to Earth
> (due to expansion)?
It is possible in principle, though I am not sure what about realistic
universes.
> These questions have practical importance to me, since I'm considering
> taking a vacation to a distant destination. 8^)
Well, I recommend you to construct a "warp bubble" engine before, it'll
be a very long vacation otherwise. :-)
Regards, squark.
Sent via Deja.com http://www.deja.com/
Before you buy.
squ...@my-deja.com
100...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
> ...a FRW model with positive cosmological constant, which implies
> that eventually every other galaxy will disappear behind our particle
> horizon. Indeed, eventually the universe will consist of causally
> isolated elementary particles. So extreme pessimism concerning our
> hopes for the future is warranted :-)
I think this conclusion is only true if a one negelcts the non-
gravitational attractive forces between the particle. Or, are you
saying the rate of expansion is ever growing?
> > Intuitively, can the speed of a distant galaxy be greater
> > than the speed of light?
>
> Akhkh, no, that's the wrong intuition.
I just want to add a remark, that the speed measured as d(distance)/d
(time) where the time is the proper time of the galaxies, and the
distance is the length of the geodesic curves on the equi-temporal
hypersurfaces linking the two galaxies - the time is chosen to be the
proper time of matter in every-point, assuming matter to be in rest, in
an appropriate sense, everywhere (this is an approxiamtion, but a quite
realistic one) - might, in principle be greater than c.
> > 2. Is there a "point of no return" for an immortal space
> > traveller who travels between distant galaxies? Is it possible
> > that a traveller could go to a distant galaxy and be forever
> > stuck there because it would be impossible to return to Earth
> > (due to expansion)?
>
> Yes; indeed this very point is discussed in D'Inverno's textbook.
Actually, this follows from the above observation of galaxies leaving
each other's horizon - this might be viewed as this kind of "joureny".
Best regards, squark.
Phillip Helbig
> > 1. Can a galaxy be so distant that it is impossible to travel
> > to it? Intuitively, can the speed of a distant galaxy be greater
> > than the speed of light? I know that the concept of "speed"
> > of distant objects is not well-defined in GR, but we can
> > operationalize this question by asking whether there is a
> > galaxy so far away that a light signal that we send today will
> > never reach it.
>
> This is always true for an open universe, assuming the rate of
> expansion is the same everywhere, a quite good approximation for
> realitsic universes, as far as I know.
What do you mean by "open universe"? One that expands forever, or one
which is spatially infinite (for the experts---I'm assuming a trivial
topology here)? However, in either case, I don't think your assumption
is correct. The Einstein-de Sitter universe expands forever and is
spatially infinite, but the rate of expansion goes to 0 with time.
Thus, after a long enough time, one can journey anywhere. With a
negative cosmological constant, we can have spatially infinite universes
which contract in the future (if the cosmological constant is negative,
no matter how small, such universes will ALWAYS recollapse at some
point), with the distance between any two points becoming arbitrarily
small. If we imagine the "opposite", a finite universe which expands
forever, also an "open universe in some sense (or indeed any universe
which expands forever, except the Einstein-de Sitter model or the
"reverse-Eddington model" which starts with a big bang and then expands
to the Einstein static model after ab infinitely long time), then the
assertion is true.
squ...@my-deja.com
hel...@astro.rug.nl wrote:
> In article <8dpbpn$uk5$1...@nnrp1.deja.com>, squ...@my-deja.com writes:
>
> > > 1. Can a galaxy be so distant that it is impossible to travel
> > > to it? Intuitively, can the speed of a distant galaxy be
> > > greater than the speed of light? I know that the concept
> > > of "speed" of distant objects is not well-defined in GR, but
> > > we can operationalize this question by asking whether there
> > > is a galaxy so far away that a light signal that we send
> > > today will never reach it.
> >
> > This is always true for an open universe, assuming the rate of
> > expansion is the same everywhere, a quite good approximation for
> > realitsic universes, as far as I know.
>
> What do you mean by "open universe"? One that expands forever, or one
> which is spatially infinite (for the experts---I'm assuming a trivial
> topology here)?
I meant a spatially infinite, i.e. a non-compact one.
> However, in either case, I don't think your assumption is correct.
> The Einstein-de Sitter universe expands forever and is spatially
> infinite, but the rate of expansion goes to 0 with time.
The de Sitter universe is spatially infinte?! The de Sitter universe
can be described as a hyperboloid in 5-dimensional space, therefore,
its slices are 3-spheres which are definitely spatially finite.
> Thus, after a long enough time, one can journey anywhere. With a
> negative cosmological constant, we can have spatially infinite
> universes which contract in the future (if the cosmological constant
> is negative, no matter how small, such universes will ALWAYS
> recollapse at some point), with the distance between any two points
> becoming arbitrarily small.
I was referring to the case of a zero cosmological constant.
Regards, squark.
Matt McIrvin
>In article <8e7nqb$878$1...@info.service.rug.nl>,
> hel...@astro.rug.nl wrote:
>
>> However, in either case, I don't think your assumption is correct.
>> The Einstein-de Sitter universe expands forever and is spatially
>> infinite, but the rate of expansion goes to 0 with time.
>
>The de Sitter universe is spatially infinte?! The de Sitter universe
>can be described as a hyperboloid in 5-dimensional space, therefore,
>its slices are 3-spheres which are definitely spatially finite.
You're talking about different things.
The "de Sitter universe" is an empty manifold with a cosmological constant.
The "Einstein-de Sitter universe" is just the FRW metric (*no* cosmological
constant) with a precisely critical matter density.
For that matter, the "Einstein universe" is yet another completely different
thing: it's a hyperspherical universe with matter, made static (though, as
it turns out, unstably so) by a cosmological constant.
--
Matt McIrvin http://world.std.com/~mmcirvin/
t...@rosencrantz.stcloudstate.edu
>In article <8e7nqb$878$1...@info.service.rug.nl>,
> hel...@astro.rug.nl wrote:
>> However, in either case, I don't think your assumption is correct.
>> The Einstein-de Sitter universe expands forever and is spatially
>> infinite, but the rate of expansion goes to 0 with time.
>The de Sitter universe is spatially infinte?!
Careful! "Einstein-de Sitter" and "de Sitter" are two different
spacetimes.
Einstein-de Sitter is a spatially flat, matter-dominated FRW model.
That is, it's what used to be the "standard model" predicted by
inflation before evidence that there wasn't enough matter for it
became convincing.
de Sitter space is a vacuum solution with a cosmological constant.
It expands exponentially and has flat spacelike slices.
Both are spatially infinite.
-Ted
squ...@my-deja.com
hel...@astro.rug.nl wrote:
> In article <8egj7s$hcs$1...@nnrp1.deja.com>, squ...@my-deja.com writes:
>
> > > However, in either case, I don't think your assumption is correct.
> > > The Einstein-de Sitter universe expands forever and is spatially
> > > infinite, but the rate of expansion goes to 0 with time.
> >
> > can be described as a hyperboloid in 5-dimensional space, therefore,
> > its slices are 3-spheres which are definitely spatially finite.
>
> (lambda=0, Omega=1) and the de Sitter universe (lambda=1, Omega=0).
> In any case, they are both infinite (again assuming the trivial
> topology) and both spatially flat. Perhaps a mathematician (I think
> there's one around here somewhere) can comment on the hyperboloid
> stuff. :-)
I don't know about the Einstein-de Sitter universe, but the de Sitter
one is certainly finite. Can you please explain what is the
Einstein-de Sitter universe? Also, what do you denote by lambda and
Omega? Oh, maybe you were referring to the anti-de Sitter universe, but
of course, I don't have in mind such "pathological" models (i.e. with
closed timelike curves).
> > I was referring to the case of a zero cosmological constant.
>
> Be sure to point this out, especially since such cosmological models
> look rather inviable now.
Right. I was uncareful, sorry. However, isn't the current estimation
for the c.c. is positive?
Best regards, squark.
t...@rosencrantz.stcloudstate.edu
>I don't know about the Einstein-de Sitter universe, but the de Sitter
>one is certainly finite.
"Is not!" "Is too!"
Enlightening though this exchange is, we may want a way to move beyond
this sticking point. I suggest a game of dueling references. I'll go
first.
Misner, Thorne, and Wheeler, p. 745, eq. (27.76), give a metric for de
Sitter space that clearly indicates flat (and hence infinite if you
assume nontrivial topology) spacelike slices at constant t. To
be specific, the line element is
ds^2 = -dt^2 + A exp (Bt) (dx^2 + dy^2 +dz^2)
for some constants A and B. (MTW's equation is not in precisely
this form, but it's equivalent.)
Your turn!
>Can you please explain what is the
>Einstein-de Sitter universe?
It's a homogeneous, isotropic spacetime filled with pressureless
matter of density equal to the critical density. No cosmological
constant. It's described by a flat Friedmann-Robertson-Walker
metric with line element
ds^2 = -dt^2 + a^2(t) (dx^2 + dy^2 + dz^2).
If you solve Einstein's equation for this metric, assuming critical
density of pressureless matter, you'll find that the scale factor a(t)
is proportional to t^(2/3).
>Also, what do you denote by lambda and
>Omega?
There are unfortunately a number of slightly different
notational choices out there in the literature. Here's my
preferred notation, which may differ from Phillip's or
anyone else's.
Omega is the density of matter in units of the critical density.
Lambda (capital) is the cosmological constant (a.k.a. vacuum energy
density). It's often useful to express the vacuum energy density in
units of the critical density. This is sometimes called lambda
(lower-case), although I prefer Omega_lambda. The Universe is
spatially flat if the total density equals the critical density:
Omega + Omega_lambda = 1 (spatially flat).
The recent CMB data, interpreted in the standard way, suggest
that the Universe is close to flat: Omega + Omega_lambda is close
to one.
With this notation, Einstein-de Sitter means Omega = 1, Omega_lambda =
0; and de Sitter means Omega = 0, Omega_lambda = 1.
Some people use Omega to stand for what I call Omega + Omega_lambda
and Omega_m to stand for what I call Omega.
-Ted
Phillip Helbig
(Matt McIrvin) writes:
> In article <8egj7s$hcs$1...@nnrp1.deja.com>, squ...@my-deja.com wrote:
>
> >In article <8e7nqb$878$1...@info.service.rug.nl>,
> > hel...@astro.rug.nl wrote:
> >
> >> However, in either case, I don't think your assumption is correct.
> >> The Einstein-de Sitter universe expands forever and is spatially
> >> infinite, but the rate of expansion goes to 0 with time.
> >
> >The de Sitter universe is spatially infinte?! The de Sitter universe
> >can be described as a hyperboloid in 5-dimensional space, therefore,
> >its slices are 3-spheres which are definitely spatially finite.
>
> You're talking about different things.
Right. But they are both infinite.
> The "de Sitter universe" is an empty manifold with a cosmological constant.
> The "Einstein-de Sitter universe" is just the FRW metric (*no* cosmological
> constant) with a precisely critical matter density.
>
> For that matter, the "Einstein universe" is yet another completely different
> thing: it's a hyperspherical universe with matter, made static (though, as
> it turns out, unstably so) by a cosmological constant.
This, however, is finite.
Note that some pundits have suggested that the Boomerang data support a
closed universe and Einstein's aesthetic arguments (wrinkle your
forehead and mumble "boundary conditions at infinity" in the presence of
John Baez) are returning to current discussion in cosmology.
Phillip Helbig
> In article <8ejl1j$4nh$3...@info.service.rug.nl>,
> hel...@astro.rug.nl wrote:
> > In article <8egj7s$hcs$1...@nnrp1.deja.com>, squ...@my-deja.com writes:
> >
> > > > However, in either case, I don't think your assumption is correct.
> > > > The Einstein-de Sitter universe expands forever and is spatially
> > > > infinite, but the rate of expansion goes to 0 with time.
> > >
> > > The de Sitter universe is spatially infinte?! The de Sitter universe
> > > can be described as a hyperboloid in 5-dimensional space, therefore,
> > > its slices are 3-spheres which are definitely spatially finite.
> >
> > There is a difference between the Einstein-de Sitter universe
> > (lambda=0, Omega=1) and the de Sitter universe (lambda=1, Omega=0).
> > In any case, they are both infinite (again assuming the trivial
> > topology) and both spatially flat. Perhaps a mathematician (I think
> > there's one around here somewhere) can comment on the hyperboloid
> > stuff. :-)
>
> I don't know about the Einstein-de Sitter universe, but the de Sitter
> one is certainly finite.
No. See also Ted's followup.
> Can you please explain what is the
> Einstein-de Sitter universe?
Omega = 1.0, lambda = 0.0. See section a.i. in
http://gladia.astro.rug.nl:8000/helbig/research/publications
/gzps/angsiz_guide.ps-gz
especially the table and figure.
> Also, what do you denote by lambda and
> Omega?
The standard stuff. See Sect. 2 of
http://gladia.astro.rug.nl:8000/helbig/research/publications
/gzps/angsiz.ps-gz
> Oh, maybe you were referring to the anti-de Sitter universe, but
> of course, I don't have in mind such "pathological" models (i.e. with
> closed timelike curves).
No, no closed time-like curves here.
> > > I was referring to the case of a zero cosmological constant.
> >
> > Be sure to point this out, especially since such cosmological models
> > look rather inviable now.
>
> Right. I was uncareful, sorry. However, isn't the current estimation
> for the c.c. is positive?
Yes. Which is why you should point out that you assume something which
is non-standard. :-| The "however" seems a non-sequitur.
squ...@my-deja.com
t...@rosencrantz.stcloudstate.edu wrote:
> Careful! "Einstein-de Sitter" and "de Sitter" are two different
> spacetimes.
Okey, I got this point by now.
> Einstein-de Sitter is a spatially flat, matter-dominated FRW model.
> That is, it's what used to be the "standard model" predicted by
> inflation before evidence that there wasn't enough matter for it
> became convincing.
>
> de Sitter space is a vacuum solution with a cosmological constant.
> It expands exponentially and has flat spacelike slices.
The last I'm almost sure to be a mistake. The de Sitter universe has 3-
sphere slices and I doubt its expansion can be described
as "expotential". If I am not mistaken, it may be described as a
hyperboloid in a 5-dimensional space with (+ - - - -) signature.
> Both are spatially infinite.
Hmm... I'm not sure! I'll have to check on this issues in the library.
Phillip Helbig
t...@rosencrantz.stcloudstate.edu writes:
> >Also, what do you denote by lambda and
> >Omega?
>
> There are unfortunately a number of slightly different
> notational choices out there in the literature. Here's my
> preferred notation, which may differ from Phillip's or
> anyone else's.
>
> Omega is the density of matter in units of the critical density.
> Lambda (capital) is the cosmological constant (a.k.a. vacuum energy
> density). It's often useful to express the vacuum energy density in
> units of the critical density. This is sometimes called lambda
> (lower-case), although I prefer Omega_lambda. The Universe is
> spatially flat if the total density equals the critical density:
>
> Omega + Omega_lambda = 1 (spatially flat).
>
> The recent CMB data, interpreted in the standard way, suggest
> that the Universe is close to flat: Omega + Omega_lambda is close
> to one.
>
> With this notation, Einstein-de Sitter means Omega = 1, Omega_lambda =
> 0; and de Sitter means Omega = 0, Omega_lambda = 1.
>
> Some people use Omega to stand for what I call Omega + Omega_lambda
> and Omega_m to stand for what I call Omega.
Like Ted, my Omega is Omega_matter or Omega_m. My lambda is
Omega_lambda. I prefer the lower case lambda, since a) I have two
letters for two more-or-less equally important quantities, 2) I avoid
double indices (one often has the index "0" to denote today's values,
since in general the values are time-dependent), 3) one often has
indices on Omega to denote different types of matter (cdm, neutrinos,
baryons,...) so again I don't want more than one index and 4) by using
lambda explicitly, there is less chance of confusion that by Omega I
mean Omega + lambda. Lambda, upper case, is the "constant" cosmological
constant; lambda = Lambda/3H^2. Since in general H is not constant in
time (it is called the Hubble constant since, at a given time, it is
constant in space), lambda can be time-dependent while Lambda is not.
(Okay, some folks consider a time-dependent Lambda, but that is
something different still.)
Aaron Bergman
>>
>> de Sitter space is a vacuum solution with a cosmological constant.
>> It expands exponentially and has flat spacelike slices.
>
>The last I'm almost sure to be a mistake. The de Sitter universe has 3-
>sphere slices and I doubt its expansion can be described
>as "expotential". If I am not mistaken, it may be described as a
>hyperboloid in a 5-dimensional space with (+ - - - -) signature.
According to Hawking and Ellis p. 124, the de Sitter spacetime is
topologically R x S^3
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
Matt McIrvin
aber...@princeton.edu (Aaron Bergman) wrote:
These "finite or infinite?" discussions can be tricky because you have to
define carefully what you mean. Both sides may be right in some sense.
On p. 745 of MTW, they give the spatially-infinite-looking metric, then
follow it immediately with the "hyperboloid in five dimensions" picture!
I dimly recall reading somewhere that the hyperboloid is the *maximal
extension*, and that the conventional metric is a *piece* of that which is
topologically open and looks spatially infinite with the standard
definition of spacelike surfaces. (To understand how this could possibly
be, consider the always-instructive Milne cosmology. The region inhabited
by test particles is confined to the interior of a light-cone, yet with
the conventional FRW-style spacelike surfaces, it's spatially infinite--
the surfaces are infinite bowl-shaped hyperboloids stacked inside the
cone, the same shape as mass shells. But I could be misremembering
something in the de Sitter case.)
It may have been in Harrison's _Cosmology_, but my copy is 500 miles from
here.
Phillip Helbig
> In article <8enkdj$f...@math.ucr.edu>,
> t...@rosencrantz.stcloudstate.edu wrote:
> > Careful! "Einstein-de Sitter" and "de Sitter" are two different
> > spacetimes.
>
> Okey, I got this point by now.
I was just looking over some of the classic papers from the 1910s and
1920s by Einstein, de Sitter, Robertson, Friedmann, Eddington, Lemaitre
(wow! 6 different nationalities!) etc. Back then, these guys often used
quite different coordinate conventions than we do today, and the de
Sitter universe is often referred to as "spherical" or by some phrase
concerning the word "spherical". Maybe this is the source of the
confusion. To clarify things, my claim that the de Sitter universe is
infinite uses the "modern" notion (like it is used in the modern
literature) and means something like "if the average density corresponds
to such-and-such a number density of particles, then there are an
infinite number of these particles" (of course, this is somewhat
problematic in the actual case of the de Sitter universe, as it is
empty, but think of it as a limit). Of course, I'm talking of space,
not space-time, in all posts in this thread talking about infinity,
curvature etc.
Note that the static Einstein universe is often described as a
"cylindrical universe", the idea here being that the z-axis of the
cylinder is time, and the radius is constant in time since it is static.
Of course, it's three-dimensional shape is spherical, i.e. it
corresponds to the surface of a conventional sphere. (One suppresses
one spatial dimension in the cylinder picture, much as one suppresses
two spatial dimensions in Feynman diagrams.)
Of course, one can always use coordinates to map something in [0,\infty]
onto [0,1], but that's just playing games.
--
Phillip Helbig Email .............. hel...@astro.rug.nl
Kapteyn Instituut Email ................. hel...@man.ac.uk
Rijksuniversiteit Groningen Tel. ................... +31 50 363 4067
Postbus 800 Fax .................... +31 50 363 6100
NL-9700 AV Groningen Web ... http://www.astro.rug.nl/~helbig/
My opinions are not necessarily those of my employer.
John Baez
Norbert Dragon <dra...@itp.uni-hannover.de> wrote:
>I find it interesting to add that de Sitter space has also noncompact,
>maximally symmetric spacelike slices,
>
>the intersection of
>
>t^2 - w^2 - x^2 - y^2 - z^2 = - 1
>
>with t+w = const > 0.
>
>As the example shows the topology of maximally symmetric slices
>is not determined by the topology of spacetime.
Oh, okay! This explains the conflict between Bunn/Helbig and squark.
They're both right! Presumably Bunn/Helbig are using the slicing
that's relevant in cosmology, even though it's less obvious from
the "hyperboloid" description of de Sitter spacetime.
Thanks also go to Johan Braennlund and Matt McIrvin for clearing this up.
So there are 2 very different ways to slice de Sitter spacetime with
spacelike hypersurfaces. In one way, space looks like a perfectly
round 3-sphere that first shrinks and then grows as time passes, with
radius r = cosh(T), where T is the proper time as measured by anyone
sitting at rest at one point of this 3-sphere. In other way, space
looks like a perfectly flat Euclidean 3-space which expands exponentially
with the passage of proper time as measured by anyone sitting at rest
at one point of this 3-space. The latter slicing only covers part of
the whole spacetime. Of course, this part may also be considered as a
spacetime in its own right.
Is this correct?
Ned Wright
>
> The last I'm almost sure to be a mistake. The de Sitter universe has 3-
> sphere slices and I doubt its expansion can be described
> as "expotential".
When de Sitter first wrote the metric for this space, little was
understood about what cosmological metrics should look like. But
one can change variables to write the de Sitter metric in FRW form,
with flat spatial sections and exponential growth.
I had a lot of trouble with authors of papers to the ApJ who
didn't understand this, but I was ultimately able to convince
them of the error of their ways.
--
Edward L. (Ned) Wright, UCLA Astronomy, Los Angeles CA 90095-1562
(310)825-5755, FAX (310)206-2096 wri...@astro.ucla.edu
http://www.astro.ucla.edu/~wright/intro.html
[Moderator's note: Anyone with basic questions about the expanding
universe should check out Ned Wright's Web site. -MM]
t...@rosencrantz.stcloudstate.edu
>> Misner, Thorne, and Wheeler, p. 745, eq. (27.76), give a metric for de
>> Sitter space that clearly indicates flat (and hence infinite if you
>> assume nontrivial topology) spacelike slices at constant t.
>
>Ah, but this topology assumption is the root of the disagreement, I
>think. The coordinates MTW use do not cover all of the de Sitter
>hyperboloid, which I take as my definition of de Sitter space.
You're right. This is the key point. As it turns out, everyone's
right, according to his or her own definition of "de Sitter space"
and "spatially finite."
When cosmologists talk about de Sitter space, we slice it up into
constant-time hypersurfaces that are spatially flat and infinite.
I hadn't realized before that that only gives part of the
whole (maximally extended) spacetime manifold. It does include
the entire region causally connected to an observer at the origin,
so for many practical purposes it doesn't matter -- that's
probably why I'd never realized it before -- but it is as you
say the source of the present confusion.
So what we've got here is the following rather counterintuitive
situation: "my" de Sitter space is spatially infinite, but it's a
proper subset of "yours" which is spatially finite! It just goes to
show that the way you can slice things up into "space" and "time"
really matters.
As Matt McIrvin points out elsewhere in this thread, much the same
thing happens in the Milne model. Anyone who's interested in this
stuff should read Matt's post and think hard about the Milne model.
The basic idea is that you can impose coordinates on a patch of
Minkowski space (specifically, the interior of the forward light cone
of the origin) to make it into a negatively-curved expanding Friedmann
spacetime. With those Friedmann coordinates, the spacelike slices are
infinite, even though the patch is "really" finite at any particular
(Minkowski) time.
Back in the mid-'90's, there was a brief fad among cosmologists for
what were known as one-bubble open inflationary models. Pretty much
the same thing happens in them. These are models in which a single
bubble of "true" vacuum is nucleated in an exponentially expanding
false-vacuum Universe. The most natural coordinate system to impose
on that bubble turns out to be one in which it's an open (negatively
curved) Friedmann spacetime. In these models, we think of ourselves
as living in an infinite, open Universe (the bubble), even though the
whole thing is embedded in a "larger" spacetime, which may itself be
spatially finite. I remember being terribly bothered by this at the
time.
Incidentally, when I talked about assuming nontrivial topology in the
quoted material above, I was talking about something still different
from this. I actually said the exact opposite of what I meant -- I
meant trivial topology! If you take any flat Friedmann spacetime,
including the patch of de Sitter spacetime described by the
coordinates in MTW, you can give spacelike slices the topology of a
3-torus instead of of R^3. (Just identify points that are a distance
L_x apart in the x direction, L_y apart in the y direction, or L_z
apart in the z direction. In other words, mod out by a rectangular
lattice.) If you do that, you'll get something whose constant-time
slices are finite even in the Friedmann coordinates. It's locally
isometric to de Sitter space, but of course it's globally a completely
different manifold. I just wanted to exclude that possibility
when I said "assuming nontrivial [sic] topology."
-Ted