[roywiggins]

anti-de Sitter universes are bounded by a horizon. The example given in the article is an Escher print with an infinite number of tiles bounded by a circle. They get smaller as they get closer to the edge, but there's an infinitude of them, so you have a universe that is infinite from "inside", but "from the outside", there's an outer boundary. As far as occupants of a hyperbolic universe go, they can't see the horizon directly because there's an infinite number of tiles between them and the edge.

That boundary has lower dimensionality than the universe itself (the Escher universe boundary is 1D and the interior is 2D).

Holographic duality is where you can describe the entire interior of the universe by characterizing "stuff happening" on the boundary- that the stuff happening inside the universe looks 2D, but is fundamentally one dimensional. Real-world holograms work like this- they encode a 3D scene onto a 2D substrate.

Our universe is not anti-de Sitter- it appears to be flat and does not pack away nicely into a bounded area like the Escher universe does, so it's as yet unclear how to apply the stuff they've found in their model universe to our own.

[floatrock]

> They get smaller as they get closer to the edge, but there's an infinitude of them, so you have a universe that is infinite from "inside",

> Our universe is not anti-de Sitter- it appears to be flat and does not pack away nicely into a bounded area like the Escher universe

How do you tell from the "infinite inside" whether it packs away nicely into a bounded area? Or is figuring that out the trick?

[antidesitter]

> How do you tell from the "infinite inside" whether it packs away nicely into a bounded area?

Roughly speaking, you measure how fast the volume of a ball grows with its radius [1]. If it grows faster than you’d expect in Euclidean space, you know you’re in a hyperbolic space.

You can also draw a big triangle and see whether the interior angles add up to less than 180 degrees [2].

[1] https://en.wikipedia.org/wiki/Scalar_curvature#Direct_geomet...

[2] https://en.wikipedia.org/wiki/Hyperbolic_triangle#Properties

[roywiggins]

And we've done exactly that using our cosmological models of the Big Bang:

https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.htm...

One side of the triangle is the width of Cosmic Microwave Background variations (calculated from models of the early universe); the other two sides are known from how far away the CMB appears to be, which is known from independent measures of the expansion of the universe. Some trigonometry will tell you what the interior angle that should make with the Earth, which you'll see as the angular width of the variations in the sky. You can compare those two numbers (the one by trigonometry and the one observed), and determine whether they are different enough to exclude a flat universe.

In other words, if someone holds an 1 foot ruler at a distance, you can use trigonometry to work out how far away it is by using the apparent size. But if you know how far away it is, and the apparent size disagrees with the trigonometry, then the shape of the universe must not be flat. The longer the ruler and the further away it is, the greater the deviation will appear (if there is one). The CMB is very far away indeed so it makes for a good ruler.

[hammock]

> And we've done exactly that using our cosmological models of the Big Bang

What's the result- flat or not?

[roywiggins]

Flat, as far as WMAP can tell!

https://wmap.gsfc.nasa.gov/universe/uni_shape.html

Of course the curvature could maybe be arbitrarily small, but... it's definitely very close to flat.

[jazzyjackson]

Hold up, so the flat-earthers just need to think a little bigger?

(Legit tho I've been trying to understand the 'shape' of the universe for a while now -- in the sense of, when I look at stars and galaxies in the sky above me, what direction am I looking as it relates to where things are in the universe relative to each other? As a kid I took Bill Nyes word for it: everything is on the surface of a balloon, expanding from the center. But do I ever see the far side of the balloon? Or is everywhere I look somehow constrained to the surface in every direction away from me, such that the 'other side of the balloon' is infinitely far away... but if I could see far enough, my own spot on the surface of this sphere would be visible to me, minus a few billion years... I want to understand this but I am very confused! This looks like an informative website so I'll keep reading ... thank you)

[danbruc]

[...] whether it packs away nicely into a bounded area [...]

That is nothing you should worry about, with enough deformation you can make a map of any shape you like for any space. You must not conclude from looking at a map of the Earth using for example a Mercator projection that there are somewhere four straight edges meeting at right angles and where you can just fall off Earth. You also must not naively make conclusions about the relative sizes of things as they get heavily distorted. The same holds for this map of hyperbolic space, it is really misleading - in the space there is no boundary, there is no special point in the middle, it does not pack nicely into a circle in any meaningful way, ... Those are all just artifacts of the way this map is drawn and it has nothing to do at all with the underlying space.

[spenczar5]

Thank you for a terrifically clear description of difficult concepts. This helped me a lot!

[badloginagain]

They talk about black holes being possible in AdS universe- in the Escher universe does that look like a group of the fish "missing"?

Also, near the end they get into black hole information preservation. What information are they referring to? My assumption has always been that black holes essentially zero out all information, like making all bits zero on a hard-drive.

[hammock]

Maybe a black hole looks similar to the boundary, but instead centered on a single point- the fish closest to the point getting smaller and smaller.

[nabla9]

How do black holes fit into this picture? Are they anti-de Sitter locally?

I mean if our universe maps to 2d boundary, you could find it out if you try to pack 3d volume of space full of information/matter. The volume must become full earlier than if the the information is limited by the surrounding area and not by the volume.

[0db532a0]

What do you mean by saying that our universe is flat. Are you saying that it’s a 2-manifold?

[pixl97]

https://en.m.wikipedia.org/wiki/Shape_of_the_universe

[0db532a0]

Makes sense. Thanks.

[raattgift]

Flatness is a property of the Robertson-Walker metric, as the k parameter at https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr... where you'll find a reasonably rigorous discussion.

More figuratively, if you take any circular planar slice of the spatial part of an expanding spacetime such that all points in the plane have the same coordinate time and then evolve the slice forward in time or examine it over cosmological distances (meaning more than megaparsecs) the plane is always flat like a disc rather than curving like a bowl.

With the constraint that there are cosmological observers -- they just float with the metric expansion of space and always observe the largest-scale distribution of matter as isotropic and homogeneous: mostly meaning similar-looking galaxy clusters (and cosmic microwave radiation) along every line of sight -- then a Robertson-Walker metric has identical constant curvature everywhere, so it doesn't matter what planar slice you choose: you'll always see the same circular disc, spherical cap, or their equivalent on a negatively-curved plane. Moreover, it does not matter when we take the planar slice; the spatial curvature is constant in the whole spacetime. [1]

Thus, we have excellent evidence for the spatial flatness of the universe from the small variations in the blackbody temperature of the Cosmic Microwave Background, and the evidence improves as we compare those variations with galaxy clusters at different redshifts (and other measures of distance).

In his last paragraph (which provoked your question) roywiggins means that our universe is expanding and at cosmological scales matches the Robertson-Walker metric extremely well. If we look into the far future, the matter dilutes away enough that it will look very similar to de Sitter space (an empty expanding spacetime) rather than anti-de Sitter (an empty collapsing spacetime).

- --

[1] Just to clarify: we are talking about spatial curvature here, where we have sliced up the 4d spacetime into 3d spaces indexed by time. In an expanding universe, whatever the spatial curvature (even if it's exactly zero), spacetime curvature is large: freely-falling objects diverge with the expansion. The spatial curvature mostly deals with relative distortions of the shapes of similar galaxies (e.g. barred spirals, or ellipticals) at different distances. With zero spatial curvature, faraway barred spirals have the same shape as nearer barred spirals.

With positive spatial curvature, further away barred spirals would be less arm-y and more core-y than nearer ones, while with negative spatial curvature they would be more arm-y and less core-y. The light from the outer parts of the arms and the core are all emitted essentially as parallel rays. With positive spatial curvature, the rays converge, so the arm-rays get squashed towards the core-rays as they travel cosmological distances. With negative spatial curvature, the initially parallel rays diverge.

We don't see that kind of distortion in images like this, with a large number of faraway galaxies looking very similar to foreground galaxies like Andromeda (M31) https://en.wikipedia.org/wiki/Hubble_Ultra-Deep_Field