[HN Gopher] How many holes does the universe have?
___________________________________________________________________
How many holes does the universe have?
Author : Brajeshwar
Score : 32 points
Date : 2024-05-31 16:57 UTC (6 hours ago)
(HTM) web link (www.scientificamerican.com)
(TXT) w3m dump (www.scientificamerican.com)
| perihelions wrote:
| Does the topology of the universe mix the space and time
| dimensions? Or would it be nonsense to have timelike loops on a
| cosmological scale? (Like with the torus example-one loop in a
| spatial coordinate, the other in a time coordinate).
|
| They're talking about *three*-dimensional manifolds so I assume
| it's an automatic no, but I don't have the background to
| understand why it's an obvious no.
| pavel_lishin wrote:
| I also share your assumption, and also share your ignorance! I
| have no idea.
|
| Closed time-like loops are possible (mathematically, at least),
| but I don't know if that's something that can be a consequence
| of the universe's shape.
| soulofmischief wrote:
| It's important to understand what "time" is.
|
| To construct a clock, we must have some sort of device which
| exhibits exact periodicity, and use it as a frame of reference.
| This is the only way we can measure the passing of "time", via
| referencing some periodic phase change. Modern atomic clocks
| observe the oscillation of states in a Cesium atom to define
| the Second. The quantum nature and energy potentials involved
| at this atomic scale lead to an exact, measurable periodicity.
|
| Einstein essentially describes a "clock", wherein a particle
| (or EM wave) periodically travels between two spatially
| distinct points A and B. First it travels from A -> B, and then
| back from B -> A, striking a sensor on both sides. That is one
| pulse. We would then reference all timekeeping via some
| coefficient of that pulse.
|
| Let's assume this particle is traveling at the constant "speed
| of light", the highest possible speed for a massless particle
| that we have observed. In a "stationary" frame of reference,
| the amount of "time" it takes for the particle to go from A ->
| B should be the same as B -> A, so we can say 1 pulse is 2AB,
| twice the distance of A to B. But what happens if A and B are
| both moving in the direction of A -> B at the same speed? Well,
| we could imagine that it would take the particle longer to
| travel from A -> B than before, but quicker to travel from B ->
| A by the same factor.
|
| Yet, when we conduct such an experiment from the same co-moving
| reference point (when we move alongside A and B at the same
| speed), we discover that sensors on both ends confirm that the
| "time" it takes for the particle to travel from A -> B is still
| the same exact amount of time that it takes to travel back from
| B -> A.
|
| What gives? How can this be possible, if the particle is not
| able to increase its speed while traveling from A -> B, since
| it already is traveling at the maximum observed speed of light?
|
| Well, when we instead measure the particle's travel from a
| stationary reference point outside A and B, we do in fact
| measure the results we expect. It does take longer to travel
| from A -> B than it does from B -> A. What??? How can we get
| two different measurements? Isn't there a single, objective
| reality?
|
| Well, it turns out, if we measure the distance from A to B
| while in the stationary outside frame, the distance _shrinks_
| with proportion to the speed of the moving A & B system. So
| because the distance shrinks, the "time" it takes to complete a
| round trip actually shortens. The particle doesn't gain more
| speed; it just has to travel less.
|
| But when we measure the distance while comoving with A and B,
| we find it _hasn 't_ shrunk. To make things worse, imagine A
| and B are inside a box. How would they ever know how fast they
| are _really_ moving? Maybe they seem still, but are they
| orbiting a star? A galaxy? A cluster?
|
| So we can only discern movement by observing from an outside
| frame of reference, in the context of two or more distinct
| frames/objects. We can only measure a relative speed of any
| given object. From any particular frame, the "speed of light"
| seems to hold.
|
| That was the mental experiment which led Einstein to uncover
| the principle of special relativity. And we have since
| experimentally confirmed both spatial and temporal dilation.
| Because if space is dilating, our perception of time must be
| dilating as well, seeing as how the entire measurement of
| "time" in this scenario is rooted in the spatial distance
| traveled by the particle.
|
| Does this make sense? This is why we have "spacetime". Time is
| a direct consequence of measuring the spatial difference
| between two states of the universe. It's crazy, it's weird, and
| it asks the question of "what is the speed of light, and why is
| it relative?", but it's internally consistent and
| experimentally verified.
|
| To answer your question about timelike loops on a cosmological
| scale: the energy involved in maintaining the stability of such
| a system would be astronomically insane. Even the smallest of
| theoretical wormholes are rifled with issues concerning
| temporal stability. Under a very particular, theoretical
| construction of the universe, stable closed timelike curves
| could be possible, but it's not likely.
|
| Further reading:
|
| _On the Electrodynamics of Moving Bodies_
| https://www.fourmilab.ch/etexts/einstein/specrel/www/
|
| Don't be scared to take a peek at the paper, it involves some
| light maths, but is largely conceptual and surprisingly
| digestible.
| drdeca wrote:
| I don't see what the first part of your comment brings to the
| last part of your comment, the part which addresses the
| question.
|
| I think the question asked is maybe something along the lines
| of, "If we consider the pseudo-Riemannian manifolds with
| signature (3,1), considered up to topological equivalence,
| are all the topological differences between these, determined
| by the 3-dimensional spacelike slices (independent of choice
| of foliation)?"
| soulofmischief wrote:
| I took it as OP seeking to understand how the topology of
| space and time are intertwined, and asking whether a
| spatially closed universe would also be somehow temporally
| closed.
|
| With this in mind, I felt like establishing the exact
| relationship between time and space might explain how a
| spatially closed universe would not necessarily be
| temporally closed, as the measurement of time involves the
| measurement of relative distance over multiple frames,
| which wouldn't necessarily change when measuring at the
| "boundaries" of a closed spatial loop (since these
| boundaries are themselves relatively defined)
|
| Apologies to OP if I misunderstood the question.
| antognini wrote:
| Yes, it does. Godel found a solution to the Einstein field
| equations that shows that it is possible to have closed
| timeline curves on a cosmological scale under particular
| conditions. (The universe is rotating and has a carefully
| chosen value for the cosmological constant.)
|
| https://en.wikipedia.org/wiki/G%C3%B6del_metric
| deadbabe wrote:
| If the universe is a donut, that means there could be an outer
| side and an inner side? Which one are we in?
| feoren wrote:
| No, that is a mistaken visualization that comes from embedding
| the donut in a higher space. No embedding is necessary, nor
| does anyone think the universe is embedded in anything larger.
|
| Think about PacMan, or the old Asteroids game, where going off
| one end of the screen would put you on the other side. That's a
| donut. (The 4 corners of the screen make up the single hole*.)
| Which "side" is the inside? The question doesn't make sense.
|
| *Edit: as rightly pointed out below, the location of the hole
| is an arbitrary choice that comes from trying to map the space
| to a sphere, and does not actually exist anywhere in the space
| itself.
|
| An interesting experiment is this: imagine yourself existing in
| the space, which is otherwise empty. PacMan alone in the middle
| of the screen. Throw a stretchy rope to yourself, horizontally
| or vertically, catch the other end, and tie it together. Then
| walk around the space without turning the rope at all. Notice
| that no matter how you walk around, the rope will always be the
| same length. Now imagine the same thing on the surface of a
| sphere. Walking around makes the rope larger or smaller, and
| there's always a point you can walk to where the rope will
| completely collapse to a single point.
| francoi8 wrote:
| I don't understand why the 4 corners of the screen make up
| the single hole. You could scroll the screen by 1 "square"
| which would change the corners which makes me feel there is
| nothing special about the original four corners.
| contravariant wrote:
| Yeah the corner can correspond to any one point on the
| torus. They are all the same point, but other than that
| there's nothing really interesting about the corner(s).
|
| The edges are more interesting, two of them go around the
| 'hole' of the 'donut' and the other two wrap 'around' the
| 'donut' itself (i.e. around the dough if it was an actual
| american style donut). There's no way to tell which is
| which.
|
| These edges have the interesting property that you can't
| shrink them to a point (compared to say a loop on a globe
| which you _can_ make smaller until its a single point).
| Except when the donut is not hollow in that case one of the
| loops becomes contractible, turning the space into the
| equivalent of a circle.
| feoren wrote:
| You're right that the original 4 corners are arbitrary. The
| hole isn't physically present in the actual space. In fact
| the word "hole" comes from visualizing the space embedded
| in a higher space, which we know is not necessary and
| invites misconceptions. So let's call it a discontinuity.
|
| The discontinuity shows up when you try to continuously map
| the space to the surface of a sphere. You can _almost_ do
| it, except for one point. Different nearly-continuous maps
| have a different point of discontinuity -- it 's basically
| your choice when doing the mapping. I think the 4 corners
| feels like a natural place for the discontinuity when I
| visualize that mapping -- and scrolling feels like
| selecting a different mapping -- but indeed it could be any
| point in the space if you visualize that mapping
| differently.
| Terr_ wrote:
| Another way to think about it is that in order to be on
| either "side" of the flat sheet, you've implicitly introduced
| depth, and it's not really two-dimensional anymore.
|
| If you were in a two-dimensional universe, you wouldn't be on
| the paper, you would be part of the paper.
| jbstack wrote:
| I'm struggling to understand why the analogy of folding a piece
| of paper into a torus works. I can see that I can easily roll the
| paper up into a cylinder. But if I then want to bend that
| cylinder so that the two ends meet, I'd have to stretch the outer
| part the torus. This is clearly visible on their diagram where
| what were squares on the flat paper are wider on the outer part
| of the torus than on the inner. If I then unfold it, I won't have
| a flat piece of paper anymore. In other words, it seems to me,
| you cannot in fact make a torus from a flat piece of paper and
| the surface of a torus is therefore not flat. Additionally,
| looking at the torus diagram, straight lines that run from the
| outer part to inner part get closer together as they approach the
| inner, meaning that the distance between two "parallel" straight
| lines varies depending on where you are on the line. I didn't
| think a flat surface can have such a property.
|
| What am I missing here?
|
| EDIT: here is a stackexchange answer claiming that the surface of
| a donut-shaped torus is indeed not flat:
| https://math.stackexchange.com/a/4377256
| Icy0 wrote:
| You're right that there are no (smooth) flat embeddings of a
| torus into 3-space.
|
| To understand how a torus can be flat, it's best to replace the
| idea of folding with the idea of placing portals on edges.
| Start with a square and put portals between the north and south
| edges and between the left and right edges. Intuitively this is
| flat, and this intuition does indeed capture the mathematical
| notion that a torus is flat.
| guyomes wrote:
| You can fold a paper to get a torus [1]. With those foldings,
| the distances on the torus embedded in 3D are the same as the
| distances on the flat paper.
|
| It is even theoretically possible to embed the flat paper as a
| torus in 3D with a C^1 surface, without polyhedral edges [2,3].
| However, this surface has a fractal structure.
|
| Finally, any torus surface embedded in 3D that is at least C^2
| (with a continuous second derivative) will nessecarily stretch
| some distances [4].
|
| [1]: https://www.imaginary.org/hands-on/diplotori-flat-
| polyhedral...
|
| [2]: https://aperiodical.com/2012/05/torus/
|
| [3]: https://www.pnas.org/doi/full/10.1073/pnas.1118478109
|
| [4]:
| https://math.stackexchange.com/questions/2291382/c2-isometri...
| raldi wrote:
| Can someone post a quick clickbait antidote for those of us stuck
| behind the paywall?
| moomin wrote:
| Some theoretical physicists have demonstrated there are some
| theoretical models of the universe consistent with current
| observations that aren't just regular spacetime.
|
| There's no evidence _for_ exotic spacetime. Just there isn't
| evidence that rules it out.
|
| The rest of it is an ELI5 explanation of topology concepts and
| pablum about how important this research is.
| loaph wrote:
| What about more clickbait?
|
| > Because of the many twists, the universe could contain copies
| of itself that might look different from the original, making
| them less easy to spot in maps of the cosmic microwave
| background.
___________________________________________________________________
(page generated 2024-05-31 23:00 UTC)