From: baez@galaxy.ucr.edu (John Baez) Subject: de Sitter spacetime Date: 5 May 2000 17:02:22 GMT Newsgroups: sci.physics.research Summary: [missing] In article , Aaron Bergman wrote: >In article <8epqh0$ho2$1@nnrp1.deja.com>, squark@my-deja.com wrote: >>Ted Bunn wrote: >>> 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 "exponential". 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 You guys are confusing the heck out of me! I can never remember which is de Sitter spacetime and which is anti-de Sitter spacetime, but they're both Lorentzian 4-manifolds, and one of them is a hyperboloid in a flat 5d spacetime with signature (+ - - - -), while the other is a hyperboloid in flat 5d spacetime with signature (+ + - - -). The (+ - - - -) case is easier for me to visualize, since I can just visualize the lower-dimensional (+ - -) case and tack on two extra dimensions. So let me do the (+ - - - -) case and hope, along with squark, that this is the de Sitter universe. So, we start with 5d spacetime with metric dt^2 - dw^2 - dx^2 - dy^2 - dz^2 and then we pick out a hyperboloid using the equation t^2 - w^2 - x^2 - y^2 - z^2 = -1 Any slice of this hyperboloid with t = constant is a spacelike 3-sphere whose equation is w^2 + x^2 + y^2 + z^2 = 1 + constant^2 It follows that the topology of this spacetime is R x S^3, in agreement with what Aaron quotes Hawking and Ellis saying about de Sitter spacetime. Good, so maybe I really picked the right case! It seems like the radii of these spacelike 3-spheres grows roughly exponentially with time. Huh? Well, by "time" I don't mean the t coordinate, which is just some random coordinate function we happen to be using. I mean the proper time T as measured by someone sitting at rest with respect to spherical coordinates on these 3-spheres. If I haven't screwed up, this is related to t by this equation: t = sinh T and the radius r of the 3-sphere where t takes some constant value satisfies r^2 = 1 + t^2, so we get r = cosh T which grows roughly exponentially with T. Is this what de Sitter spacetime is like? Anyway, this spacetime is maximally symmetric and its isometry group is the 10-dimensional group SO(1,4), for obvious reasons. The other spacetime, coming from a hyperboloid in the flat 5d spacetime with metric of signature (+ + - - -), is also maximally symmetric, and its isometry group is the 10-dimensional group SO(2,3). All of this stuff generalizes to every dimension and signature in an obvious way. ============================================================================== From: ted@rosencrantz.stcloudstate.edu Subject: Re: Expansion and Space Travel Date: 3 May 2000 19:18:41 GMT Newsgroups: sci.physics.research In article <8epn42$e1b$1@nnrp1.deja.com>, wrote: >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 ============================================================================== From: Johan Braennlund Subject: Re: Expansion and Space Travel Date: Thu, 04 May 2000 12:10:06 GMT Newsgroups: sci.physics.research In article <8epu2h$nlh$1@pravda.ucr.edu>, ted@rosencrantz.stcloudstate.edu wrote: > In article <8epn42$e1b$1@nnrp1.deja.com>, wrote: > >I don't know about the Einstein-de Sitter universe, but the de Sitter > >one is certainly finite. > Misner, Thorne, and Wheeler, p. 745, eq. (27.76), give a metric for de > Sitter space that clearly indicates flat (and hence infinite if you > assume nontrivial topology) spacelike slices at constant t. Ah, but this topology assumption is the root of the disagreement, I think. The coordinates MTW use do not cover all of the de Sitter hyperboloid, which I take as my definition of de Sitter space. Using their notation and setting Lambda=3 for convenience, we get t=ln(z^0+z^4), so only t-coordinates above the plane z^0+z^4=0 are allowed (incidentally, this part of de Sitter space is the Einstein-de Sitter spacetime). With the hyperboloid definition of dS, its topology is RxS^3. Johan ============================================================================== From: ted@rosencrantz.stcloudstate.edu Subject: Re: Expansion and Space Travel Date: Sat, 6 May 2000 06:04:27 GMT Newsgroups: sci.physics.research In article <8erpal$mib$1@nnrp1.deja.com>, Johan Braennlund wrote: >In article <8epu2h$nlh$1@pravda.ucr.edu>, > ted@rosencrantz.stcloudstate.edu wrote: >> 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 ============================================================================== From: helbig@astro.rug.nl (Phillip Helbig) Subject: Re: Expansion and Space Travel Date: 4 May 2000 15:58:57 GMT Newsgroups: sci.physics.research In article <8epn42$e1b$1@nnrp1.deja.com>, squark@my-deja.com writes: > In article <8ejl1j$4nh$3@info.service.rug.nl>, > helbig@astro.rug.nl wrote: > > In article <8egj7s$hcs$1@nnrp1.deja.com>, squark@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. ============================================================================== From: Ned Wright Subject: Re: Expansion and Space Travel Date: Sat, 6 May 2000 06:03:39 GMT Newsgroups: sci.physics.research squark@my-deja.com wrote: > > 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 wright@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]