How many holes does the universe have?
scientificamerican.com
scientificamerican.com
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
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.
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.
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.
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.
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.
Edit: Found some diagrams!
https://en.wikipedia.org/wiki/Fundamental_polygon#Examples_o...
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.
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.
This is the Gauss-Bonnet theorem.
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.
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...
:stares blankly in mathematician:
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.
For sound, it should be the same with a corner of a room, it ~doubles the sound in the vicinity of the corner, hence a ~3dB increase
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.
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.
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.
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)?"
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.
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.
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.
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.
Etc...
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.
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 LifeWell, there's Bakersfield CA, that's definitely one to start with. And a goodly part of New Jersey too.
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.
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.
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.
If you were in a two-dimensional universe, you wouldn't be on the paper, you would be a patch of the paper.
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.
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.