How Space and Time Could Be a Quantum Error-Correcting Code(quantamagazine.org)
quantamagazine.org
How Space and Time Could Be a Quantum Error-Correcting Code
https://www.quantamagazine.org/how-space-and-time-could-be-a-quantum-error-correcting-code-20190103/
135 comments
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.
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.
> 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?
> 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?
> 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
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
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.
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.
> And we've done exactly that using our cosmological models of the Big Bang
What's the result- flat or not?
What's the result- flat or not?
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.
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.
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)
(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)
The balloon-surface model works if the universe is actually positively curved. It bends around in on itself and comes back. So you would be able to see yourself (eventually) after light made its way around the surface and back to you.
As far as anyone can tell, though, the universe is flat. So photons traveling outwards from you will never come back around- they'll just keep going.
I'm partial to the raisin bread model of the universe:
https://www.loc.gov/rr/scitech/mysteries/universe.html
It's a tad misleading: the raisin bread is, probably, infinite in extent. As a practical matter, we can only see a finite portion of the raisin bread. As we gaze more deeply into the sky, we're looking further into the raisin bread. Because light only travels at a finite speed, the further something is away, the older the image we see of it is. At a certain point, all we can see is the goopy bread batter that the raisin bread used to be: that's (sort of, kinda) the Cosmic Microwave Background: it's the oldest light in the universe, and we can't see anything older than it.
So the universe looks like a bunch of galaxies, with us at the center, and a sphere of microwave radiation at the edge. But every single observer sees a similar sphere around their exact location, so it's a kind of illusion.
As far as anyone can tell, though, the universe is flat. So photons traveling outwards from you will never come back around- they'll just keep going.
I'm partial to the raisin bread model of the universe:
https://www.loc.gov/rr/scitech/mysteries/universe.html
It's a tad misleading: the raisin bread is, probably, infinite in extent. As a practical matter, we can only see a finite portion of the raisin bread. As we gaze more deeply into the sky, we're looking further into the raisin bread. Because light only travels at a finite speed, the further something is away, the older the image we see of it is. At a certain point, all we can see is the goopy bread batter that the raisin bread used to be: that's (sort of, kinda) the Cosmic Microwave Background: it's the oldest light in the universe, and we can't see anything older than it.
So the universe looks like a bunch of galaxies, with us at the center, and a sphere of microwave radiation at the edge. But every single observer sees a similar sphere around their exact location, so it's a kind of illusion.
Does the expansion of the universe mean that galaxies that are too far away from us are seemingly moving faster than the speed of light, and this is why we can't see any galaxies past a given point (because that's where they "accelerate" faster than light)?
A lot of the visible universe actually appears to be receding faster than the speed of light:
https://phys.org/news/2015-10-galaxies-faster.html
It's because the space inbetween is getting bigger. But light keeps moving towards us from those galaxies, so we see those photons eventually anyway. You can still hear things that are going above the speed of sound, as long as they're headed away from you- if they're headed toward you, you'll not hear them (probably?). But the waves they throw off will expand behind them just fine.
https://phys.org/news/2015-10-galaxies-faster.html
It's because the space inbetween is getting bigger. But light keeps moving towards us from those galaxies, so we see those photons eventually anyway. You can still hear things that are going above the speed of sound, as long as they're headed away from you- if they're headed toward you, you'll not hear them (probably?). But the waves they throw off will expand behind them just fine.
> if they're headed toward you, you'll not hear them (probably?)
Actually, you would not have much time to hear, but what you would hear (among the other effects) would be the sound “in reverse”(!)
https://physics.stackexchange.com/questions/14880/hearing-a-...
Actually, you would not have much time to hear, but what you would hear (among the other effects) would be the sound “in reverse”(!)
https://physics.stackexchange.com/questions/14880/hearing-a-...
If the universe originated with the big bang and then expanded, then unless it expanded infinitely fast it must be bounded. But if it's bounded, how can it be flat?
The math indicates that the singularity was infinitely dense and infinitely hot... if you're already allowing that something infinitely dense could be real, then infinitely fast expansion doesn't seem less crazy. Going from an infinite temperature to a finite temperature is an infinite amount of cooling, which also seems impossible; likewise, an infinite density to a finite one.
You'd need infinite expansion to do that- if it was merely finite, you'd still end up with an infinite density. Expanding an infinitely dense point to a bowling ball still gets you an infinite density; you need to expand it out "infinitely far" (whatever that means) to spread stuff out enough to get a merely finite density.
https://en.wikipedia.org/wiki/Initial_singularity
It seems more likely that the singularity at t=0 in general relativity isn't physical- what does an infinite density even mean?
You'd need infinite expansion to do that- if it was merely finite, you'd still end up with an infinite density. Expanding an infinitely dense point to a bowling ball still gets you an infinite density; you need to expand it out "infinitely far" (whatever that means) to spread stuff out enough to get a merely finite density.
https://en.wikipedia.org/wiki/Initial_singularity
It seems more likely that the singularity at t=0 in general relativity isn't physical- what does an infinite density even mean?
Now I'll take a not-very-rigorous stab at your exact question:
> what does an infinite density even mean?
In the previous reply I asked what does t=0 mean and under time-reversal marched some late time test objects towards a big-bang style singularity. Infinite density means that you can put any amount of matter down as initial values on a spacelike hypersurface at a time far from the singularity, evolve that surface in time towards the singularity, and eventually up at the singularity. The problem of the singularity is that a spacelike hypersurface there doesn't admit a sensible set of initial values, so you can't predict much about what comes out of the singularity when marching forward towards late times.
I also talked briefly about the pressure components of the stress-energy tensor.
Let's treat the stress-energy tensor as a 4x4 matrix describing the flux of i-momentum in the i-direction, and slice up spacetime into time-indexed 3-spaces. If we make a little 3d "cell" (being a little loose with terminology) then T^{00} encodes the amount of momentum sourced within the cell at t_{now} and staying within the cell at t_{immediately adjacent to now}.
Let's look at only one spatial dimension.
T^{11} encodes the amount of momentum originating to our left or right entering and leaving the cell to the left or right. Lets send rightward-going momentum into the cell from the left and mostly bounce out again to the left as leftward-going momentum, with some of the sent-in momentum sticking behind in the cell as energy (T^{00}) and some leaking out as rightward-going momentum to the next cell to the right. This is how pressure works, and shows how the stress-energy tensor can evolve as you squash things together. Momentum that is sent into a region (a cell being a sort of region) can stick around as energy. The question is, "how does it stick around"? Perhaps by increasing the frequency of the particle(s) occupying the cell.
Since T^00 usually dominates the stress-energy tensor T, and that in turn usually determines the Einstein tensor G, as more momentum-energy enters the cell without leaving, local curvature around the cell also increases. That in turn drives the split of momentum reflected back out of the cell, momentum retained within the cell, and momentum passed in another direction. This is essentially the root of the nonlinearity of the Einstein Field Equations.
The split of what energy-momentum reflects back out, flows right through, or remains within a cell depends essentially on paths of least resistance, and those depend on how the contents of the cell behave locally and also on the geometrical background. Grossly, momentum will tend to exit a cell in a downwards direction if it can, and downwards is in the direction of cells with greater energy (T^00).
At ever higher density around a cell, practically any kind of momentum entering the cell stays in the cell. As we move towards infinite density, all the "higher" (in a gravitational potential sense) cells eventually lose all of their energy (and any newly arriving momentum) in the direction of the "lowest" cell, even as we take the sizes of all cells to zero.
(For black holes rather than collapsing whole universes (or time-reversed expanding universes) only the cells inside the horizon are guaranteed to leak their entire energy and future momentum-energy towards the lowest cell.)
Unanswered questions: what quantum numbers are found in the lowest cell, especially as we shrink cell sizes? If you have a lowest infinitesimal cell holding all the sources of stress-energy, do quantum effects transfer momentum-energy from it to neighbouring higher infinitesimal cells? What quantum numbers escape the singularity? Can this happen hierarchically so as to form jets or other structures?
> what does an infinite density even mean?
In the previous reply I asked what does t=0 mean and under time-reversal marched some late time test objects towards a big-bang style singularity. Infinite density means that you can put any amount of matter down as initial values on a spacelike hypersurface at a time far from the singularity, evolve that surface in time towards the singularity, and eventually up at the singularity. The problem of the singularity is that a spacelike hypersurface there doesn't admit a sensible set of initial values, so you can't predict much about what comes out of the singularity when marching forward towards late times.
I also talked briefly about the pressure components of the stress-energy tensor.
Let's treat the stress-energy tensor as a 4x4 matrix describing the flux of i-momentum in the i-direction, and slice up spacetime into time-indexed 3-spaces. If we make a little 3d "cell" (being a little loose with terminology) then T^{00} encodes the amount of momentum sourced within the cell at t_{now} and staying within the cell at t_{immediately adjacent to now}.
Let's look at only one spatial dimension.
T^{11} encodes the amount of momentum originating to our left or right entering and leaving the cell to the left or right. Lets send rightward-going momentum into the cell from the left and mostly bounce out again to the left as leftward-going momentum, with some of the sent-in momentum sticking behind in the cell as energy (T^{00}) and some leaking out as rightward-going momentum to the next cell to the right. This is how pressure works, and shows how the stress-energy tensor can evolve as you squash things together. Momentum that is sent into a region (a cell being a sort of region) can stick around as energy. The question is, "how does it stick around"? Perhaps by increasing the frequency of the particle(s) occupying the cell.
Since T^00 usually dominates the stress-energy tensor T, and that in turn usually determines the Einstein tensor G, as more momentum-energy enters the cell without leaving, local curvature around the cell also increases. That in turn drives the split of momentum reflected back out of the cell, momentum retained within the cell, and momentum passed in another direction. This is essentially the root of the nonlinearity of the Einstein Field Equations.
The split of what energy-momentum reflects back out, flows right through, or remains within a cell depends essentially on paths of least resistance, and those depend on how the contents of the cell behave locally and also on the geometrical background. Grossly, momentum will tend to exit a cell in a downwards direction if it can, and downwards is in the direction of cells with greater energy (T^00).
At ever higher density around a cell, practically any kind of momentum entering the cell stays in the cell. As we move towards infinite density, all the "higher" (in a gravitational potential sense) cells eventually lose all of their energy (and any newly arriving momentum) in the direction of the "lowest" cell, even as we take the sizes of all cells to zero.
(For black holes rather than collapsing whole universes (or time-reversed expanding universes) only the cells inside the horizon are guaranteed to leak their entire energy and future momentum-energy towards the lowest cell.)
Unanswered questions: what quantum numbers are found in the lowest cell, especially as we shrink cell sizes? If you have a lowest infinitesimal cell holding all the sources of stress-energy, do quantum effects transfer momentum-energy from it to neighbouring higher infinitesimal cells? What quantum numbers escape the singularity? Can this happen hierarchically so as to form jets or other structures?
> what does an infinite density even mean?
What does t=0 even mean? If we switch to some coordinates where t_0 \def t_now = 0 and t_past \gt 0, as with the scale factor in cosmology, we don't have a special time coordinate at t_past ~ 13.8 Gyr, so we can still think about the stress-energy (and the mechanisms that generate it) and the curvature it sources locally.
We can also think about things under time-reversal:
A comoving test observer in vacuum just free falls without end. Our test observer is a point with no charges (including "active" gravitational charge) and no internal structure, so it feels no tidal effects.
If we have two such observers starting at arbitrarily large space-like separation, they eventually they come close to each other and ultimately end up freely falling on the same trajectory, forever.
If we complicate the test observers from this no-repulsive-interaction picture, we develop pressure (represented in the stress-energy tensor, T^11 T^22 T^33). The local increase in pressure generates local curvature in response, and if the repulsive interaction is finite then eventually the two interacting test observers end up freely falling on the same trajectory, forever. The eventuality is because the background has infinite curvature, and that and the contribution from the self-gravitation of pressure wins out over any finite repulsion.
We can complicate test object interactions further by introducing: repulsive and attractive interactions; extended observers that are composites of these test observers; and even have these objects obey the statistics for bosons and fermions. Same-charged fermions in effect resist being pushed onto the same trajectory, but the resistance is almost certainly finite. Eventually some daughter product free-falls forever.
(We can see some of this in sufficiently massive objects that repulsion or exclusion is overcome, as in the cores of stars or in neutron stars, where internal pressure tends to dominate the local stress-energy.)
What the daughter products of squashed-together Standard Model particles might be is a matter of research, both in terrestrial laboratories and in astrophysical processes (including black holes). Conversely, mirroring the time-reversal thinking above, the Standard Model particles have to freeze out of something hotter and denser that exists at greater lookback time. (cf. electroweak symmetry breaking).
Worse, at a(t_0) ("now") there is a lot of dark matter, and we don't know the details of how that interacts with the Standard Model. Those details will almost certainly matter in the time-reversal picture above.
Consequently we don't know that repulsion/exclusion is finite. Perhaps some crystalline structure develops in the time-reverse picture (or in black holes), and things just accumulate in that. Or perhaps there is a sudden drop in pressure where daughter products fly away from some ultrastrong squashing-together (cf. pair-instability supernovae).
There are quite a few options. And of course the Robertson-Walker background is already just an approximation; the real metric is likely to be very different at high lookback times.
Finally, Carroll has a nice deck (and it's filled with references) at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab...
What does t=0 even mean? If we switch to some coordinates where t_0 \def t_now = 0 and t_past \gt 0, as with the scale factor in cosmology, we don't have a special time coordinate at t_past ~ 13.8 Gyr, so we can still think about the stress-energy (and the mechanisms that generate it) and the curvature it sources locally.
We can also think about things under time-reversal:
A comoving test observer in vacuum just free falls without end. Our test observer is a point with no charges (including "active" gravitational charge) and no internal structure, so it feels no tidal effects.
If we have two such observers starting at arbitrarily large space-like separation, they eventually they come close to each other and ultimately end up freely falling on the same trajectory, forever.
If we complicate the test observers from this no-repulsive-interaction picture, we develop pressure (represented in the stress-energy tensor, T^11 T^22 T^33). The local increase in pressure generates local curvature in response, and if the repulsive interaction is finite then eventually the two interacting test observers end up freely falling on the same trajectory, forever. The eventuality is because the background has infinite curvature, and that and the contribution from the self-gravitation of pressure wins out over any finite repulsion.
We can complicate test object interactions further by introducing: repulsive and attractive interactions; extended observers that are composites of these test observers; and even have these objects obey the statistics for bosons and fermions. Same-charged fermions in effect resist being pushed onto the same trajectory, but the resistance is almost certainly finite. Eventually some daughter product free-falls forever.
(We can see some of this in sufficiently massive objects that repulsion or exclusion is overcome, as in the cores of stars or in neutron stars, where internal pressure tends to dominate the local stress-energy.)
What the daughter products of squashed-together Standard Model particles might be is a matter of research, both in terrestrial laboratories and in astrophysical processes (including black holes). Conversely, mirroring the time-reversal thinking above, the Standard Model particles have to freeze out of something hotter and denser that exists at greater lookback time. (cf. electroweak symmetry breaking).
Worse, at a(t_0) ("now") there is a lot of dark matter, and we don't know the details of how that interacts with the Standard Model. Those details will almost certainly matter in the time-reversal picture above.
Consequently we don't know that repulsion/exclusion is finite. Perhaps some crystalline structure develops in the time-reverse picture (or in black holes), and things just accumulate in that. Or perhaps there is a sudden drop in pressure where daughter products fly away from some ultrastrong squashing-together (cf. pair-instability supernovae).
There are quite a few options. And of course the Robertson-Walker background is already just an approximation; the real metric is likely to be very different at high lookback times.
Finally, Carroll has a nice deck (and it's filled with references) at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab...
[...] 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.
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.
Thank you for a terrifically clear description of difficult concepts. This helped me a lot!
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.
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.
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.
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.
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.
[deleted]
What do you mean by saying that our universe is flat. Are you saying that it’s a 2-manifold?
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
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
There isn’t an exterior in this model, just a boundary you can’t get “outside of” because everything including space and time are part of the universe. What this is all based on is the observation that the information required to describe the volume of (for example) a black hole, can be encoded on its event horizon. The theory says that the universe and it’s horizon(s) can be similarly modeled. In essence that we live not in the volume of a soap bubble but in the fluctuations of the skin of the bubble. It’s only to us, at our energy level and scale that a higher dimensional volume appears to emerge.
It’s important to say that this is all entirely speculative, based on the physical possibility which allows for the resolution of some outstanding problems in physics. That doesn’t mean it is in any way real, it is just another possible model, and one without observation to support it as the way our universe actually works.
It’s important to say that this is all entirely speculative, based on the physical possibility which allows for the resolution of some outstanding problems in physics. That doesn’t mean it is in any way real, it is just another possible model, and one without observation to support it as the way our universe actually works.
Perhaps a stupid question:
If a 3d universe can be encoded in a 2d space, then I suppose a 2d universe can be in a 1d space? And could 1 dimeneion be encoded in 0? Just some bits or fluctuations?
If a 3d universe can be encoded in a 2d space, then I suppose a 2d universe can be in a 1d space? And could 1 dimeneion be encoded in 0? Just some bits or fluctuations?
With the caveat that I am neither a physicist nor a mathematician:
1. My understanding is that some kinds of 3d universes can be encoded on a 2d surface. Whether our universe is one of those is an area of research.
2. You can't always generalize from one dimension to another in mathematics. The most common example is that knots can only exist in three dimensions; they don't exist in 2 or 4.
1. My understanding is that some kinds of 3d universes can be encoded on a 2d surface. Whether our universe is one of those is an area of research.
2. You can't always generalize from one dimension to another in mathematics. The most common example is that knots can only exist in three dimensions; they don't exist in 2 or 4.
You can knot surfaces in 4D though.
Edit: not to say that 3D space isn't special though. There are plenty of special things about it.
Edit: not to say that 3D space isn't special though. There are plenty of special things about it.
You’re not wrong, but all such knots are trivial in 4D. The math is discussed here: https://math.stackexchange.com/questions/1426501/why-are-all...
That link appears to be about unknotting any embedding of the circle into R^4 . Indeed, these can all be unknotted.
I am saying that if instead of having a 1-d curve (a circle) embedded in 3-space, you instead have a 2-d surface embedded in 4-space, that such surfaces can be knotted in 4-space just as the circle can be knotted in 3-space.
Iirc, one way to construct these surfaces is to start with a knot in a particular 3D slice of 4-space, put all of it on one side of a plane in that 3D slice, stick the knot to that plane at 3 points (remove the bit of the knot between them), and then revolve what is left of the knot around the plane (so, the projection of the knot as it is revolving into the 3D slice will look like the knot is being squished into the plane, unsquished on the other side, and then doing that backwards. But this is only the projection into the 3D slice. Really, when revolving around the plane, as the direction normal to the plane within the slice gets scaled to zero, the same amount is added first in one direction normal to the 3D slice, and then as- ... blah, I am going on too long in describing this. I'm typing on my phone, so I can't effortlessly see all that I wrote, so I might be repeating myself. It rotates around the plane. Each point in the original knot traces out a circle in 4-space. )
I am saying that if instead of having a 1-d curve (a circle) embedded in 3-space, you instead have a 2-d surface embedded in 4-space, that such surfaces can be knotted in 4-space just as the circle can be knotted in 3-space.
Iirc, one way to construct these surfaces is to start with a knot in a particular 3D slice of 4-space, put all of it on one side of a plane in that 3D slice, stick the knot to that plane at 3 points (remove the bit of the knot between them), and then revolve what is left of the knot around the plane (so, the projection of the knot as it is revolving into the 3D slice will look like the knot is being squished into the plane, unsquished on the other side, and then doing that backwards. But this is only the projection into the 3D slice. Really, when revolving around the plane, as the direction normal to the plane within the slice gets scaled to zero, the same amount is added first in one direction normal to the 3D slice, and then as- ... blah, I am going on too long in describing this. I'm typing on my phone, so I can't effortlessly see all that I wrote, so I might be repeating myself. It rotates around the plane. Each point in the original knot traces out a circle in 4-space. )
At some level of approximation you can encode any higher dimensional space into a lower dimensional one. The usual approach is to use a space-filling curve, like a z-order curve or Hilbert space filling curve, then describing a point in the higher dimensional space as a distance along the curve.
It’s easy to encode the universe into a single binary bit. It’s the decoding bit that presents a challenge.
Here's a possible answer. Imagine a 2d universe - a piece of paper. Cut that paper into infinitely thin lines, and line them up.
You have converted the 2d paper into a 1d line, but in the process have made the line much longer than the 2d version. (Like the difference between Aleph 0 and Aleph 1.)
I don't see any way to go lower than that though.
You have converted the 2d paper into a 1d line, but in the process have made the line much longer than the 2d version. (Like the difference between Aleph 0 and Aleph 1.)
I don't see any way to go lower than that though.
To go lower you cut the line into infinitely many points and stack them together :-) ?
I think a good illustration could be provided by the Gauss law. https://en.wikipedia.org/wiki/Gauss%27s_law
It states that if you have a volume of space - say, a cube 1 meter side - containing some electric charges, and you calculate the total flux of the electric field across the boundary of that volume - that is, across the surface of that cube - then you'll find that total charge Q and total flux FF are proportional, FF = Q / epsilon_0 , where epsilon_0 is a fundamental constant. And that ratio doesn't in fact depend on shape or size of that volume of space.
That means Gauss law allows you to go along the boundary surface, calculate total electric flux and calculate the total charge inside the volume within that surface.
Similarly here, "holographically dual" means that you can derive important properties of matter inside some volume from properties which are observable on the boundary surface of that volume. What are those properties is another matter - but this duality principle says that there is a certain relation between them.
It states that if you have a volume of space - say, a cube 1 meter side - containing some electric charges, and you calculate the total flux of the electric field across the boundary of that volume - that is, across the surface of that cube - then you'll find that total charge Q and total flux FF are proportional, FF = Q / epsilon_0 , where epsilon_0 is a fundamental constant. And that ratio doesn't in fact depend on shape or size of that volume of space.
That means Gauss law allows you to go along the boundary surface, calculate total electric flux and calculate the total charge inside the volume within that surface.
Similarly here, "holographically dual" means that you can derive important properties of matter inside some volume from properties which are observable on the boundary surface of that volume. What are those properties is another matter - but this duality principle says that there is a certain relation between them.
That’s a bad analogy. You can’t deduce the charge distribution within a volume from the flux through its boundary. You can only deduce its magnitude.
Parent correctly referred to "total charge", not its distribution.
That wasn’t the issue with their analogy.
The issue was that the holographic principle states that the boundary completely determines what’s inside a volume, as opposed to Gauss’s law which only talks about the total amount of charge.
The issue was that the holographic principle states that the boundary completely determines what’s inside a volume, as opposed to Gauss’s law which only talks about the total amount of charge.
You're right, Gauss law only talks about magnitude, not distribution. I chose this example because it's simpler than explaining reversal of wave fronts, so the formula FF = Q / epsilon_0 is shorter. Note I mentioned "what those properties are is another matter".
The choice of analogy is less precise but hopefully easier to understand. The idea is just that there could be relationship between boundary conditions and internal conditions.
The choice of analogy is less precise but hopefully easier to understand. The idea is just that there could be relationship between boundary conditions and internal conditions.
Not sure if it's related but this lecture gave me some basic grasp on the notion of holographic representations
https://www.youtube.com/watch?v=2DIl3Hfh9tY
https://www.youtube.com/watch?v=2DIl3Hfh9tY
Quanta is great, but there's definitely a limit to what can be portrayed to a popular audience and this is well past it.
On the contrary, this is the perfect article to explain to audiophiles why to buy my cable with quantum error correction. By increasing the qubit parity, we bring your sound system to a whole new level of clarity.
Go one step further and sell a box that does quantum error correction for the entire room, like those "ambient room conditioners": https://www.lessloss.com/blackbody-p-200.html
Like grandma used to say: if you can't do something right, do something profitable.
I think anything can be portrayed to the popular audience. Other comments here explaining the terms used in the article suggest that this is possible. The only problem is that some fine information would most likely be lost with a simplified explanation.
I think there's an additional difficulty when those unfamiliar with more rigorous (as compared to pop sci anyway) physics/mathematical discussions.
There is simply a great deal of unintuitive interactions to keep in mind all, at once, when attempting to rationalize how many of these interactions happen. Physicists and mathematicians combat this by working with specific abstractions enough to become intuitively familiar with the behavior of said abstractions.
Unfortunately, this packing away of unintuitive complexity behind intuitive interactions is unavailable to the lay reader. Any specific interaction can be explained in simple enough language, eventually, if the curious reader keeps asking why. However, when they attempting to pop back up the why stack and get back to the big picture (i.e. the world people can understand intuitively), lay readers lose the nuance in the noise created by the volume.
There is simply a great deal of unintuitive interactions to keep in mind all, at once, when attempting to rationalize how many of these interactions happen. Physicists and mathematicians combat this by working with specific abstractions enough to become intuitively familiar with the behavior of said abstractions.
Unfortunately, this packing away of unintuitive complexity behind intuitive interactions is unavailable to the lay reader. Any specific interaction can be explained in simple enough language, eventually, if the curious reader keeps asking why. However, when they attempting to pop back up the why stack and get back to the big picture (i.e. the world people can understand intuitively), lay readers lose the nuance in the noise created by the volume.
Fine information on both sides, as they say.
So, let's imagine you have a function that takes a phase space (ie, the positions, momentum, etc of a bunch of particles in normal 3+1 space time) as an argument and produces an evolution of that phase space in time. It's got a bunch of rules in the function regarding gravity.
Now imagine you've new function, which also takes a phase space as an argument, but instead of operating in 3 spacial dimensions, it has five. And that it doesn't have any rules regarding gravity.
Now imagine, you have a one to one mapping between states in those functions. So you can take a state in 3d space, translate it to 4d space -- run both functions for the same amount of time, and they both should produce states which you can still map to each other.
Now imagine you've new function, which also takes a phase space as an argument, but instead of operating in 3 spacial dimensions, it has five. And that it doesn't have any rules regarding gravity.
Now imagine, you have a one to one mapping between states in those functions. So you can take a state in 3d space, translate it to 4d space -- run both functions for the same amount of time, and they both should produce states which you can still map to each other.
To my computer scientist mind this can easily be a parallel to checksum and hashing. Quantum error correction is analogous to using checksum to validate file integrity. Memory chips use similar schemes (ie parity bits) to correct errors. The same is being applied to quantum computing, ie. a sort of quantum hashing scheme based on (spacial) logic gates.
Now new research is being conducted where this quantum hashing scheme could be used to solve some of physics hardest problems. One of them could be Hawkins paradox, where "data" gets corrupted while being "processed" by a black hole. Maybe, scientists argue, error correcting data is stored at the black hole entrance so that it can be somehow applied as correcting code at the exit, ie when Hawkins radiation is released.
Or maybe the entire universe has gone through a hashing function and now there's error correcting code keeping information error-free using the "hash value". That's what the boundary stores that describes the bulk in certain theoretical universes.
Hashes have always fascinated me. The fact that a relatively short binary sequence can uniquely describe all of Shakespeare's works. What if we could completely reverse hashes, creating the most powerful compression ever? Well quantum physicists just might do that at cosmic scales!
Now new research is being conducted where this quantum hashing scheme could be used to solve some of physics hardest problems. One of them could be Hawkins paradox, where "data" gets corrupted while being "processed" by a black hole. Maybe, scientists argue, error correcting data is stored at the black hole entrance so that it can be somehow applied as correcting code at the exit, ie when Hawkins radiation is released.
Or maybe the entire universe has gone through a hashing function and now there's error correcting code keeping information error-free using the "hash value". That's what the boundary stores that describes the bulk in certain theoretical universes.
Hashes have always fascinated me. The fact that a relatively short binary sequence can uniquely describe all of Shakespeare's works. What if we could completely reverse hashes, creating the most powerful compression ever? Well quantum physicists just might do that at cosmic scales!
Hashes do not uniquely describe things. There are an infinite number of alternative texts that also match your hash that describes all of Shakespeare's works. A hash function is pretty much just a very lossy compression algorithm.
> What does "holographically dual" mean?
> What boundary are we talking about here?
"Boundary" refers to the boundary of Anti-de Sitter (AdS) space. The model that is typically studied is that of five-dimensional AdS space for which the boundary is four-dimensional. One can now formulate a quantum field theory on this boundary that is "dual" to the theory in the interior in the sense that there is precise dictionary between quantities living on the boundary and quantities in the interior ("the bulk"). "Holographically" refers to the difference in dimensions of the theory living on the boundary and the one living in the bulk.
> What boundary are we talking about here?
"Boundary" refers to the boundary of Anti-de Sitter (AdS) space. The model that is typically studied is that of five-dimensional AdS space for which the boundary is four-dimensional. One can now formulate a quantum field theory on this boundary that is "dual" to the theory in the interior in the sense that there is precise dictionary between quantities living on the boundary and quantities in the interior ("the bulk"). "Holographically" refers to the difference in dimensions of the theory living on the boundary and the one living in the bulk.
Its cod profundity which the author shouldn't have attempted to popularize. People haven't even demonstrated quantum error correction in a single qubit in an actual quantum computer, and suddenly it's the source of space and time. Chyeah, dude; whaddevah.
The model universes they're talking about, FWIIW, are all just models, with very little relationship to the world of matter we all live in.
The model universes they're talking about, FWIIW, are all just models, with very little relationship to the world of matter we all live in.
In our universe, construct an object that is perfectly spherical, and perfectly reflective. Now glue it to a table somewhere. Put another 3D object on the table. It has mass and volume, and felt real when you held it. Now look at the sphere. The 3D object is mapped onto the 2D surface of the sphere.
Now turn reality inside out. Because of various symmetries, it looks pretty much the same as it did before. Mathematically, it isn't all that important whether the signs of various things are positive or negative.
Instead of placing a 3D object on the table, draw a 2D shape on the surface of the sphere. Because you everted reality, this causes a 3D object to be reflected onto the table. The math can't really tell whether the object causes the reflection on the sphere, or the pattern on the sphere causes the object to exist.
Now turn the sphere inside out. Put the entire universe on the inside, and all of the stuff inside it (that you knew nothing about anyway, because it's perfectly reflective) on the outside. Now that your sphere encloses the entire universe, you can draw a 2D shapes on the outer boundary and reflect them as 3D shapes somewhere in the interior.
Now turn reality inside out. Because of various symmetries, it looks pretty much the same as it did before. Mathematically, it isn't all that important whether the signs of various things are positive or negative.
Instead of placing a 3D object on the table, draw a 2D shape on the surface of the sphere. Because you everted reality, this causes a 3D object to be reflected onto the table. The math can't really tell whether the object causes the reflection on the sphere, or the pattern on the sphere causes the object to exist.
Now turn the sphere inside out. Put the entire universe on the inside, and all of the stuff inside it (that you knew nothing about anyway, because it's perfectly reflective) on the outside. Now that your sphere encloses the entire universe, you can draw a 2D shapes on the outer boundary and reflect them as 3D shapes somewhere in the interior.
I don't know much about it but from what I've read imagine that we are inside a black hole. Every black hole is a universe of its own. The boundary of the black hole is the outer boundary if we are inside it. From the outside it would be just the surface of the black hole. Something like that perhaps
I really liked cosmological natural selection of Lee Smolin in that regard. For me it's just looks so simple and obviously.
https://en.wikipedia.org/wiki/Lee_Smolin#Fecund_universes
Should be verifiable on the inside with stuff popping up at the boundary, right?
When I read these quantum articles I think it's might as well be mambo jambo. Are the fundamentals of quantum physics even falsifiable?
When I read these quantum articles I think it's might as well be mambo jambo. Are the fundamentals of quantum physics even falsifiable?
What do you mean by “the fundamentals of quantum physics”? If you’re talking about phenomena like superposition and entanglement, the answer is absolutely yes.
First observations of quantum physics was that if you light through two slices of holes in a paper it makes more than two marks on the wall behind it - this proves that the light is some kind of a waveform or could act like one. As far as I know the fun starts when you try the same experiment with exactly one photon emitted. The single photon will simultaneously go through the two hole. This caused some confusion and we now call this phenomenon and related things "quantum". But thid is just my understanding based on a book, other commenters seems to be much more knowledgeable about the topic.
The single photon registers in a single spot on the screen, as a particle should, and so there's no problem with assuming that it went through one hole or the other.
The fun starts when you start sending single photons through the holes one by one - again, every photon registers in a specific spot; but when taken in aggregate, those spots form the same interference pattern as with multiple photons.
The fun starts when you start sending single photons through the holes one by one - again, every photon registers in a specific spot; but when taken in aggregate, those spots form the same interference pattern as with multiple photons.
The boundary would likely appear to be infinitely far away, and so you could never see it actually happen.
Sounds like if you imagine the universe as a sphere, like we (me?) "normally" do (3 dimensions), you use volume to describe its contents. Well... I think what they're saying is that the outside of the sphere, in this case the universe, just its surface, describes everything (all the information) inside of it... because in fact the volumetric area we know as the universe is a projection of a "flat" surface.
And... like I'm not very smart, but this is probably a bit like the non-euclidean space demo from yesterday's front page, where the geometry is doing really weird things... perhaps someone smart will come along and give a proper explanation.
And... like I'm not very smart, but this is probably a bit like the non-euclidean space demo from yesterday's front page, where the geometry is doing really weird things... perhaps someone smart will come along and give a proper explanation.
I feel I don't have enough background information to understand this article.
>...space-time in the interior of the [anti-de Sitter] universe is a projection that emerges from entangled quantum particles living on its outer boundary
A projection? How would this differ if the quantum particles on the outside were not entangled?
>holographic “emergence” of space-time works like a quantum error-correcting code.
holographic in what way? and how is this similar to error correcting code?
>quantum error correction explains how space-time achieves its “intrinsic robustness,”
what is an example of this robustness in space-time?
>...error-correcting codes can recover the information from slightly more than half of your physical qubits, even if the rest are corrupted. This fact that hinted quantum error correction might be related to the way anti-de Sitter space-time arises from quantum entanglement.
how does the effectiveness of error correcting codes, being able to recover information despite 50% corruption, explain space-time?
if anyone could point me in the right direction, I'd really appreciate it. I find this interesting and important but am not smart enough to comprehend it.
>...space-time in the interior of the [anti-de Sitter] universe is a projection that emerges from entangled quantum particles living on its outer boundary
A projection? How would this differ if the quantum particles on the outside were not entangled?
>holographic “emergence” of space-time works like a quantum error-correcting code.
holographic in what way? and how is this similar to error correcting code?
>quantum error correction explains how space-time achieves its “intrinsic robustness,”
what is an example of this robustness in space-time?
>...error-correcting codes can recover the information from slightly more than half of your physical qubits, even if the rest are corrupted. This fact that hinted quantum error correction might be related to the way anti-de Sitter space-time arises from quantum entanglement.
how does the effectiveness of error correcting codes, being able to recover information despite 50% corruption, explain space-time?
if anyone could point me in the right direction, I'd really appreciate it. I find this interesting and important but am not smart enough to comprehend it.
Here's a talk given by Patrick Hayden, one of the guys quoted in the article, about related topics. It was to an IEEE Information Theory chapter, so may use a bit more information theoretic terminology than you're used to. Even so, I think it is very accessible.
https://www.youtube.com/watch?reload=9&v=v5UbN0xx4X0
https://www.youtube.com/watch?reload=9&v=v5UbN0xx4X0
Cool talk, thanks!
In quantum physics and cosmology, "holographic" (and "projection", in this context) are related to the holographic principle, which is kind of like a Stokes' theorem for information-- it says that specifying the state of the universe on the boundary of a manifold is sufficient to completely specify the state inside.
If someone's read the full paper, maybe they can explain the other things-- this article is a bit too vague.
If someone's read the full paper, maybe they can explain the other things-- this article is a bit too vague.
" the universe on the boundary of a manifold is sufficient to completely specify the state inside " made it click for me if I understood you right. Makes me think of cellular automata: the state of the whole grid can be specified by the information contained in the first row + the laws of physics on the grid (game of life, etc) - is that kind of like the "boundary" we're talking about here?
Yes, that's a good analogy.
roywiggins said
>you have a universe that is infinite from "inside", but "from the outside", there's an outer boundary.
and you say
>specifying the state of the universe on the boundary of a manifold is sufficient to completely specify the state inside
this would be similar to me viewing a Mandelbrot set and being able to tell its infinitely deep?
>you have a universe that is infinite from "inside", but "from the outside", there's an outer boundary.
and you say
>specifying the state of the universe on the boundary of a manifold is sufficient to completely specify the state inside
this would be similar to me viewing a Mandelbrot set and being able to tell its infinitely deep?
Maybe a better analogy would be: If you know the complete state of the universe at one moment in time, you know the state at all moments in time, because you can run the laws of physics (forward or backward) and compute the state at some other time without adding information. (In physics, this arises from a property called "unitarity").
The holographic principle says you can do the same thing, but by knowing the state of the universe on a boundary that encompasses the whole system (when you parameterize spacetime in a convenient way), as opposed to by knowing the state everywhere at one moment.
The holographic principle says you can do the same thing, but by knowing the state of the universe on a boundary that encompasses the whole system (when you parameterize spacetime in a convenient way), as opposed to by knowing the state everywhere at one moment.
You still have to know the state of the whole boundary (which is huge) but the important part is that the boundary is one fewer dimension than the interior, which (apparently) makes the math easier.
Perhaps this lecture by Leonard Susskind can help you with the hologram universe concept:
https://youtu.be/2DIl3Hfh9tY
https://youtu.be/2DIl3Hfh9tY
The only reason I have any vague understanding of this article is all thanks to PBS Space Time on Youtube.
It's an amazing series. Must watch if you want to know how much we humans understand about our universe.
It's an amazing series. Must watch if you want to know how much we humans understand about our universe.
Or equally fairly, how much we dont understand.
One of the reasons I like that series is they seem intent to do some justice to competing theories and the fact that we still don't really have a working model of the universe, or know answers as to why theories work the way they do.
Anyways this article was a really hard read for me. It's like the relatively-readable paragraphs are laughing at me--written like I should be understanding them, but I don't, ie:
>The best error-correcting codes can typically recover all of the encoded information from slightly more than half of your physical qubits, even if the rest are corrupted. This fact is what hinted to Almheiri, Dong and Harlow in 2014 that quantum error correction might be related to the way anti-de Sitter space-time arises from quantum entanglement.
One of the reasons I like that series is they seem intent to do some justice to competing theories and the fact that we still don't really have a working model of the universe, or know answers as to why theories work the way they do.
Anyways this article was a really hard read for me. It's like the relatively-readable paragraphs are laughing at me--written like I should be understanding them, but I don't, ie:
>The best error-correcting codes can typically recover all of the encoded information from slightly more than half of your physical qubits, even if the rest are corrupted. This fact is what hinted to Almheiri, Dong and Harlow in 2014 that quantum error correction might be related to the way anti-de Sitter space-time arises from quantum entanglement.
I'd also like to recommend the semi-technical talks by Nima Arkani-Hamed and Leonard Susskind. They are the only reason I have the faintest hint of what is going on in theoretical physics and cosmology.
Space Time is fantastic. Upon watching the entire series (catching up to present time, to be more exact, since new episodes are still being released weekly), I went on a graph traversal spree of related YouTube videos. Another great series I loved was from Fermilabs, with Dr. Don Lincoln.
I caught a physics bug several weeks ago and finding PBS Space Time was exhilarating. I would also highly recommend it.
How I understand it: The world around us is some sort of "projection" of stuff at the very edges of our universe, and that stuff is inherently fragile and unstable. Yet our world/reality/spacetime is not fragile and unstable - quite the opposite, despite being dependent on that fragile stuff in the first place.
Thus it must somehow be "correcting" for the intrinsic instability of the stuff on the outer edges that gives rise to our reality. Therefore it is in some sense an error correcting code.
Happy to be corrected by those with a deeper understanding!
Thus it must somehow be "correcting" for the intrinsic instability of the stuff on the outer edges that gives rise to our reality. Therefore it is in some sense an error correcting code.
Happy to be corrected by those with a deeper understanding!
The world science festival has a good video about this: https://www.youtube.com/watch?v=HnETCBOlzJs - The "correction" could very well be reconciliation, the output of what plays out in the program + the program is procedurally generated based on the outputs. Another good talk is this one by John Preskill from 2015: https://www.youtube.com/watch?v=SW2rlQVfnK0
Just came to say it makes me happy to see the term "discovered" used in a what is essentially an engineering/software context, instead of the term "invented." The more that people in our industry speak clearly about that fact that many of the constraints in our problem space are universal, and thus much of our solution space is discovered, not invented, the more we shift the culture to understand e.g. the absurdity of software patents, etc.
Do they use a similar terminology in mathematics when talking about proofs? Software solutions don't seem that much different in regards to solving specific problems.
Not sure if it was rhetorical, but to answer your actual question: Mathematicians often say they "discovered" or "found" a proof. I've never heard one say they "invented" a proof, although they might talk about "building" a formalism or framework.
This question — is maths discovered or invented? — was discussed at length in a recent BBC series: https://www.bbc.co.uk/programmes/b0bn9dth
I wonder if there are languages where there is no real dichotomy between these two terms
"Invent" comes from Latin and simply means "find". If you think about it, no matter what language, there is no fine line. An invention is just a discovery, but we typically use the word to describe a very specific discovery.
The English language often has multiple synonyms with a difference in connotation. Discover and invent may be synonyms (although many would disagree), but everybody would agree at least the connotation is different.
My hypothesis above was that there may be languages where that connotation does not exist.
My hypothesis above was that there may be languages where that connotation does not exist.
Well I always thought there was a fine line: an invention must be brought into existance witj intelligent design (probably by a human), and a discovery exists naturally.
[deleted]
a YT link to ep 1: https://youtu.be/cyvDG8qjt-M
[deleted]
All matter is information, and all information is functional. Therefore perception is the lazy evaluation of the universe, and discovery is the fountain of invention.
That depends heavily on your philosophical interpretation of mathematics: Platonism vs. Formalism
Someone linked the BBC video, which is great, but there’s also a wonderful book that discusses this and many other facets of mathematical philosophy:
https://www.amazon.com/Thinking-about-Mathematics-Philosophy...
(Sorry for the link spew) I highly recommend that book to anyone remotely interested in the field.
Someone linked the BBC video, which is great, but there’s also a wonderful book that discusses this and many other facets of mathematical philosophy:
https://www.amazon.com/Thinking-about-Mathematics-Philosophy...
(Sorry for the link spew) I highly recommend that book to anyone remotely interested in the field.
Penrose also dedicates a chapter to this in The Road to Reality
"Discovered" in place of "invented" is actually cited as a common error in many English style guides.
Unconvinced, with respect. Patents and systems of intellectual property are also codes that create space and time, in this case financial. Without them, scientific and engineering discovery/invention has often (usually?) tended to collapse in on itself.
I inferred that OP was referring to the absurdity of the current software patent system and not the idea of patenting IP directly, and I agree that the current system is in dire need of an overhaul.
That's fair. I surely agree that poorly made IP codes can also inhibit discovery and invention. And the article notes that the corrective time-space codes are "a lot smarter than us."
Do you have a source to read about this? Seems an interesting topic.
The analogy just came to me, really. Maybe some basic IP theory, or economic history?
https://www.nature.com/articles/538456a
The etymological roots of "discover" and "invent" converge.
The study of Black Holes with the lens of Quantum information has been an exciting sub-field of Physics for the past couple of years. If you are curious about it, I recommend Patrick Hayden's "The Quantum Computational Universe" lectures [0][1]. The first one is an introduction to Quantum computation and the second one is more in line with what this Quantamagazine article is about.
[0]: https://www.youtube.com/watch?v=AqWuyeh0SxQ
[1]: https://www.youtube.com/watch?v=F6vGAiMz5kQ
[0]: https://www.youtube.com/watch?v=AqWuyeh0SxQ
[1]: https://www.youtube.com/watch?v=F6vGAiMz5kQ
It reminds me of old quote "Time is what keeps everything from happening at once" [1] and similar "Space is what keeps everything from happening in the same place".
[1] https://skeptics.stackexchange.com/a/30402
[1] https://skeptics.stackexchange.com/a/30402
Could someone explain to me why we should look into this, because we do not live in an anti-deSitter universe. So what is the relevance (beyond the obvious mathematical curiosity and elegance)?.
It's a toy model, but the hope is that the insights gained by playing with it will apply to ours: "On the physics side, it remains to be seen whether de Sitter universes like ours can be described holographically, in terms of qubits and codes. “The whole connection is known for a world that is manifestly not our world,” Aaronson said. In a paper last summer, Dong, who is now at the University of California, Santa Barbara, and his co-authors Eva Silverstein and Gonzalo Torroba took a step in the de Sitter direction, with an attempt at a primitive holographic description. Researchers are still studying that particular proposal, but Preskill thinks the language of quantum error correction will ultimately carry over to actual space-time."
Also, black holes at least superficially have a spatial boundary (the event horizon) so perhaps you can model what happens inside using the holographic principle.
Also, black holes at least superficially have a spatial boundary (the event horizon) so perhaps you can model what happens inside using the holographic principle.
My understanding (from Patrick Hayden) is that in the center of AdS space, things look very much like normal space. Physicists are working in AdS space because they can actually do some analysis there, even though they would very much like to work in de Sitter space.
Because string theory gave us AdS/CFT correspondence, and maybe if we try harder than we tried for the past 20 years, something similar will pop out that works for our universe.
My personal belief is that it's very much a case of looking for the keys where the light is.
My personal belief is that it's very much a case of looking for the keys where the light is.
For the same reason as why biologist spend a lot of time studying E. Coli, fruit flies and mice (besides being interested in them in themselves). It's because those organisms are similar enough to humans, but still a lot simpler to study. In this way we get important clues and ideas of how stuff might work for human cells, which is still very hard to work out and prove, but is a lot easier than just starting in the dark with human cells.
[deleted]
[deleted]
Awesome idea, but is there any way to test the hypothesis? Or will it be forever locked in the box of "extremely elegant but untestable" along with string theory?
I'm wondering how many physicists roll their eyes each time they see similar extraordinary headlines like this one
edit: I am a physicist.
edit: I am a physicist.
Well, I think a lot of folks here find the description strange and opaque - I studied a bit differential geometry for my math degree and this seems way up there.
I'd love to hear why a physicist would roll their eyes at this.
I'd love to hear why a physicist would roll their eyes at this.
>So, how do quantum error-correcting codes work? The trick to protecting information in jittery qubits is to store it not in individual qubits, but in patterns of entanglement among many.
So are quantum encoding patterns the Proof Of Work of multidimensional holgraphic blockchains?
Edit: accidentally pasted entire article
So are quantum encoding patterns the Proof Of Work of multidimensional holgraphic blockchains?
Edit: accidentally pasted entire article
> But a fundamental problem stood in the way of actually building quantum computers: the innate frailty of their physical components.
It's entirely possible that there is some sort of physically limiting factor that makes it nearly impossible to entangle more than X qbits. Something akin to Amdahl's law for quantum computing. It wouldn't be a very sexy result.
It's entirely possible that there is some sort of physically limiting factor that makes it nearly impossible to entangle more than X qbits. Something akin to Amdahl's law for quantum computing. It wouldn't be a very sexy result.
I'm thinking of the infeasibility of building a rocket to get off a large planet (compared to earth).
The intuition of a layperson is a bad guide, but I've always had the gut feeling that quantum error correction would prove to be exponentially difficult to scale. If smart people think otherwise, I can hope they're right, but I wouldn't bet money on it until I find an explanation I can grasp. It just sounds like a free lunch, or cold fusion.
The intuition of a layperson is a bad guide, but I've always had the gut feeling that quantum error correction would prove to be exponentially difficult to scale. If smart people think otherwise, I can hope they're right, but I wouldn't bet money on it until I find an explanation I can grasp. It just sounds like a free lunch, or cold fusion.
> Ahmed Almheiri, Xi Dong and Daniel Harlow did calculations ...
It's pretty cool to see a Westerner, Middle Easterner and an East Asian dude (at least based on names ...) collaborate to push the frontiers of science like this!
It's pretty cool to see a Westerner, Middle Easterner and an East Asian dude (at least based on names ...) collaborate to push the frontiers of science like this!
This happens literally all the time. This is probably more common than uncommon.
> Westerner, Middle Easterner and an East Asian [...] based on names
This international variety of names is common outside the English-speaking world, e.g. the 3 biggest internet companies in China (Alibaba, Tencent, Baidu).
This international variety of names is common outside the English-speaking world, e.g. the 3 biggest internet companies in China (Alibaba, Tencent, Baidu).
This makes error-correcting the foundation of the universe, but I would prefer to have causality be fundamental, such as with the causal sets program. This is interesting though, especially for quantum computing.
Gee, sounds like a theory seeking facts. Isn't it supposed to work the other way around?
You know, math is really great and all, and you can transform a sphere into a torus, but meat doesn't transform into flies.
You know, math is really great and all, and you can transform a sphere into a torus, but meat doesn't transform into flies.
I have a feeling I would get a brain aneurysm if I tried to read this.
Curious what are the logic gates made of? What kind of engineering does it take to build the gates?
> The bendy fabric of space-time in the interior of the universe is a projection that emerges from entangled quantum particles living on its outer boundary.
Guess we are living in a simulation, after all. A quantum one, to be precise.
Guess we are living in a simulation, after all. A quantum one, to be precise.
Nice, feels like we are getting closer.
does this mean that out of all the possibilities space and time brings about order and chaos?
> That year — 2014 — three young quantum gravity researchers came to an astonishing realization. They were working in physicists’ theoretical playground of choice: a toy universe called “anti-de Sitter space” that works like a hologram.
Wow, so we truly are in a simulation
Edit: Sorry, I thought this was a funny play on ambiguous words, but apparently HN disagrees!
Wow, so we truly are in a simulation
Edit: Sorry, I thought this was a funny play on ambiguous words, but apparently HN disagrees!
You will often get downvoted for humor on HN. On the other hand, I have at times gotten double digits worth of karma for a humorous post. I guess you have to appeal to a certain type of person.
Yeah, I've definitely noticed. Guess I can't help taking a gamble on occasion.
On this particular occasion, I truly couldn't tell if my joke was a whoosh, or just not funny! I made myself laugh, but that's not difficult.
On this particular occasion, I truly couldn't tell if my joke was a whoosh, or just not funny! I made myself laugh, but that's not difficult.
No, not that we know.
I'm not sure I really understand the article, as it is, but space-time seemed preternaturally a macrocosm of the quantum universe, a way to smooth out the infinitely possible subspace into a predictable and understandable macro space, the collapsing of all possible realities at the atomic level into a definite reality. In other words, this seems too obvious to be this obvious, so what am I missing here?
>What am I missing here?
It's not obvious how to reconcile the laws quantum mechanics with gravity and vice-versa as they seem to make incompatible descriptions of reality.
It's not obvious how to reconcile the laws quantum mechanics with gravity and vice-versa as they seem to make incompatible descriptions of reality.
Mandelbrot cosmology offers some interesting potential resolutions, but more research definitely needs to be done.
> [...] after the renowned physicist Juan Maldacena discovered that the bendy space-time fabric in its interior is “holographically dual” to a quantum theory of particles living on the lower-dimensional, gravity-free boundary.
What does "holographically dual" mean?
What boundary are we talking about here?
> The bendy fabric of space-time in the interior of the universe is a projection that emerges from entangled quantum particles living on its outer boundary
What is the "interior" of the universe? What is the "outer boundary"?