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.
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.
Why can't I do the same thing with the surface of a sphere?
Take two circles, side by side, connected by an infinitesimally small overlap, and put portals all around the circles to their equivalent point in the other circle. Then fold this over and inflate it into a sphere.
If you're suspicious about the discontinuity where they overlap, I think that's a red herring - no one is claiming the sphere is isomorphic to two disjoint surfaces of any shape.
Alternatively, make it a cube. I can definitely fold a single piece of paper into a cube, or equivalently, I can put portals on my piece of paper to give it the topology of a cube. Intuitively that is flat. But the cube isn't one of the 18 forms proposed for the universe. Nor is any of the other millions of 3d shapes I can make with the same process.
Here's an easy way to test the curvature of these examples. Draw a circle (all points equidistant to a given one) centered at a point on one of the "glue" edges. The amount that the circumference of that circle is short of the expected 2πr is a measure of the curvature at that point.
In the two-discs sphere construction, a circle centered on the disc boundaries will appear half in each disc. But it will be short of the usual circumference due to the shape of the discs. All the curvature has been pushed to these boundaries.
In the cube, a circle centered at a vertex will appear on three sides; its circumference will be three-quarters of the usual value. Note that all the curvature of the cube is at the vertices, not the edges!
For the portal-torus, try drawing a circle anywhere, passing through the portals... it will have the usual circumference, zero curvature everywhere.
I love this! And even more is true -- you can read off the Euler characteristic from adding up how many fractions of a circle are lost over all the points.
For the cube, at each vertex you've lost a quarter circle, and there are 8 vertices -- hence the Euler characteristic of a cube is 2.
For the two-disk model of the sphere, a similar thing should be true, I think, but I haven't worked it out in detail -- the integral of "circles lost" over the sphere (the support of this integral is the shared boundary of the disks) should be 2 as well.
For the two-disc sphere, I can't think of an intuitive way to "see" the "circles lost" integral. But here's a different intuitive way to see the total curvature.
Another way to measure the curvature is to look at how much the sum of the interior angles of an n-sided polygon exceeds the usual sum π(n - 2). It's most common to think about triangles, but we can also think about 2-gons... these are usually degenerate shapes with a sum of interior angles of zero.
But on the two disc-sphere, draw two lines, each from the center of one disc to the center of the other disc, passing straight through the glued boundary. These form a 2-gon with sum of interior angles (and also excess over the usual value) equal to twice the angle between the lines. To get the total curvature of the whole sphere, let each of the two interior angles be 2π, for a total of 4π... two circles, the Euler characteristic.
One thing that helped me is considering how to deform the edges of your glued circle examples into a flat square. First, map the two sides of the seam two the two corners of the plane. This becomes a sphere by folding it into a triangle and then gluing (or portaling) up neighboring edges. Having the portals on neighboring edges means that they change directions in weird ways.
This is somewhat different than the torus with portals example, where the portals were placed on opposite edges of the plane.
That's a good response! I had to spend some time to work out what goes wrong here. But I figured it out.
Parallel transport is broken in your model of the sphere. Take this example. Take a vector pointing north in circle 1, and send through the north portal. It should be pointing south when it gets to the other circle. Fine -- north in circle 1 corresponds to south in circle 2.
Now send the north-pointing vector east instead. It is going to point north in circle 2.
So the north vector changes direction depending on which portal it goes through. So parallel transport changes directions. Hence your model of the sphere is not flat.
Thinking about this a bit more, I think I see what you mean - the example where you fold it into a torus is "intuitively flat" because when you leave the portal travelling north, you re-enter travelling north, etc. That property doesn't hold for my cube example.
I wouldn't say the curvature implied by this property is "intuitively" 0 - why not 1, or 360°, or 180°? - but I can see how it's an interesting property that only a few shapes can have.
The cube isn’t flat. The curvature is just concentrated into the corners. You can measure this curvature by adding the interior angles of a triangle that contains a corner.
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].
I am not a physicist, astronomer, mathematician or anything like that. But as a lifelong musician who's learned a thing or two about acoustics, I am familiar with the topology of a space affecting the perception AND measurement of vibrations produced within it.
Example - take a speaker with good bass response, play something through it holding it above the floor in the center of the room, then on the floor in the center of the room, then where the floor meets a wall, and finally in a corner where 2 walls meet the floor. You will perceive (and measure) a 3dB increase of low-frequency for each of the boundaries at that position. This is a trick I've used as a DJ to fill rooms with sound from very small speakers.
Is there any way that the topology of the universe/higher dimensions could affect measurements of things like dark matter, etc.? Again probably a very naive question as I know not all modes of vibration are identical.
You may know the reason why but for those that don’t… Bernoulli’s Principle, think thumb over a garden hose. The effect is most pronounced with low frequencies but occurs at all frequencies. We perceive loudness as an increase in pressure.
I’ve not heard it described that way before; I would love to hear more of what you mean by this. The way I learned it in my upper level acoustics courses in college was image sources: hard surfaces are effectively mirrors for sound, so just like you see a copy of an object in an actual mirror, you hear a copy of the speaker coming as if from inside the wall. With three surfaces meeting in the corner, you effectively get 8 sound sources instead of just one. For low frequencies these virtual sources constructively interfere and make louder sound. Of course, this is just a more intuitive way of thinking of the effects of boundary conditions to a partial differential equation.
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.
Yes, it does. Gödel 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.)
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.
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.
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)?"
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)
An error in this type of thinking is that it assumes that the observer isn’t also made of light.
In our physical reality all scientific instruments are made of matter, the behaviour of which is almost entirely determined by electromagnetic interactions.
Any effect that changes the light clock somehow also changes co-moving observers identically!
It’s like asking an a 2D character drawn on a rubber sheet if they think that the things they see on the rubber sheet are changing as that sheet is uniformly stretched.
Relativity was formulated in a time before we were even sure that atoms exist.
I don't think you can have computational objects on lightlike (null) trajectories? If light is interacting with light (to have computation), it's effectively moving slower than the vacuum speed of light and the information is not on a null trajectory.
There's no error, it doesn't matter what the observer is made of. Special relativity is meant to be consistent across all frames of reference.
As another reply mentioned, from the perspective of a photon which is not interacting, time stands still. This is because we see the limit of the system where a moving body has zero mass, where things become infinite. But it still holds that a non comoving observer sees the opposite limit; the photon moving at the maximum possible speed through the medium of space. (Though the photon can gain mass depending on the reference frame)
It does. This is a foundational philosophy-of-science kind of thing that is often not taught much, which is why you may be personally unfamiliar with it.
For example, a hypothetical creature living on the surface of a neutron star made up of quark matter dominated by strong forces instead of EM forces might not agree about the postulates of SR! Such a being might talk about how moving EM charges are distorted, as-if they were hyperbolically rotated, causing EM-based condensed matter instruments to see a distorted view of the universe. Not because the universe is undergoing Lorentz transformations, but because the instruments changed shaped from spherically symmetric to ovoid. Such a creature might even claim that there is a preferred rest frame, and that its apparent absence is something only EM-based matter experiments can show!
The lenses in the optics change shape, not the gridlines of the rest of the universe.
> Special relativity is meant to be consistent across all frames of reference.
Special relativity was invented by Einstein to resolve inconsistencies in electromagnetic interactions. All experiments done to this date to verify SR have been either electromagnetic in nature, or involved matter as both the subject and instruments, both of which are dominated by EM effects. Sure, GR includes gravity fields as well, but SR does not.
All of physics is a point of view from "inside" the universe where we can't disambiguate alternatives because we're affected by the same rules we're trying to observe.
Some examples:
- There's no difference between "spacetime expanding" and "matter shrinking".
- There are mathematical representations of GR that do not require curved space time.
- We can't measure the one-way speed of light, only the two-way speed of light.
> This is a foundational philosophy-of-science kind of thing that is often not taught much
Why is that?
> The lenses in the optics change shape, not the gridlines of the rest of the universe.
And so the observer can correct for distorted measurements, if they can measure the distortion factor. But a distortion factor in measurement has no bearing on what is actually being observed.
> There's no difference between "spacetime expanding" and "matter shrinking".
At first glance, but can you satisfactory explain redshift/Hubble's law and the fact that objects at sufficient distance from the observer have an apparent size which is larger than expected? As well as a host of consistency issues in other known physical constants that would arise, such as a corresponding increase in the strength of the electromagnetic force?
In general, we would have to throw away special relativity, as the only realistic explanation would be that the speed of light is slowing down. Until we can experimentally verify this, the idea that matter could be shrinking is purely theoretical, other problems aside.
I'm happy to check out any credible literature on the subject, if you can provide it.
> An error in this type of thinking is that it assumes that the observer isn’t also made of light.
Well, yes. Light has an entirely different model of the universe than we have. To a photon, clocks don't tick at all, so a photon says "What is this 'time' you speak of?" And since photons are "always" exactly "where" they intend to be without time passing, they also say "What is this 'space' you speak of?"
Our view of the universe must seem very weird to a photon.
I read the news today, oh boy
Four thousand holes in Blackburn, Lancashire
And though the holes were rather small
They had to count them all
Now they know how many holes it takes to fill the Albert Hall
- The Beatles – A Day in the Life
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.
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 a patch of the paper.
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.
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.
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.
IIRC PacMan was not a torus. Asteroids was, but not PacMan.
Edit: Just looked it up. We're both right, sort of. There were warp tunnels on the sides, but not the top and bottom. PacMan was a semi-torus. IOW a cylinder.
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.
> 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.
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