Is the universe finite but unbounded? - Page 4

sol invictus · 13th June 2012, 03:04 PM

Originally Posted by TubbaBlubba

I still don't quite grasp what a "flat" universe would be like.

Would it be analogous to some sort of finite unbounded plane?

A flat infinite universe is very simple - it's just an infinitely large, 3D volume. That's the only case that's both flat and isotropic.

If you want something that's flat, finite, without boundary, but not isotropic, a 3-torus is a good example. That's a 3D space that you can think of as like the interior of a cube or parallelepiped, with facing sides identified with each other.

Tim Thompson · 13th June 2012, 03:51 PM

Originally Posted by ynot

The most "credible" option I currently have is an endless Universe that has always existed in some form.

Originally Posted by dafydd

You disregard all the evidence of the Big Bang? It was the discovery of the cosmic background microwave radiation that put paid to Hoyle's Steady State infinite endless universe theory.

Originally Posted by sol invictus

All we know is what the currently known laws of physics predict (or rather postdict) for the early universe - and that's a big bang.

Sol is on the money, but we need to keep in mind just what the bang of big bang fame really is supposed to be.

Originally Posted by Brian Greene, "The Fabric of the Cosmos", page 272

A common misconception is that the big bang provides a theory of cosmic origins. It doesn't. The big bang is a theory, partly described in the last two chapters, that delineates cosmic evolution from a split second after whatever happened to bring the universe into existence, but it says nothing at all about time zero itself. And since, according to the big bang theory, the bang is what is supposed to have happened at the beginning, the big bang leaves out the bang. It tells us nothing about what banged, why it banged, how it banged, or, frankly, whether it ever really banged at all.

The quote above is the first part of the first paragraph of chapter 10, "Deconstructing the Bang", in Brian Greene's popular book Fabric of the Cosmos (Vintage Books, 2004). While it's generally a good idea to be wary of "popular" level writing, I think this does a pretty good job of making the correct point (a wordier version of what Sol said). As long as we accept general relativity as the theory of space-time, we can get perhaps arbitrarily close to the initial state of the universe (i.e., "time zero"), but we can never access that initial state. So, if one is going to hold that the universe (in the sense of literally everything that does exist or ever can exist or ever has existed) has a finite age and a true beginning, that has to be a matter of interpretation; there is no direct evidence for a true beginning, simply because neither theory nor observation (at least for now) can access the initial conditions of the universe. That's why there is such a concerted effort to discover a quantum theory of gravity, a quantum theory of space-time, which might reveal a theory for the initial conditions of the universe. Colliding brane hypotheses in string theory, or "bouncing universe" models in loop quantum cosmology are two ways to establish a theory of initial conditions for the universe, which allow for our "universe" to be a subset of an eternal universe around it.

So, we can have both a big bang beginning for our universe, and still retain an eternal universe as well. At this point in our history, theory and observation really both fall short of the goal of definitively establishing one or the other, a universe with a true beginning or an eternal universe, or even both at once. And I point out that this is very much different from the now discredited steady state cosmology of Hoyle, at al., or its more modern offspring of a "quasi steady state cosmology", advanced by Jayant Narlikar.

Perpetual Student · 13th June 2012, 04:16 PM

Originally Posted by sol invictus

We don't know any of those things. All we know is what the currently known laws of physics predict (or rather postdict) for the early universe - and that's a big bang.

Specifically, how do the EFE post-dict the big bang? I have recently spent some time studying the GR tensor equation and have looked at some specific solutions, so I am somewhat prepared for a mathematical demonstration of the big bang -- assuming there is one.

TubbaBlubba · 13th June 2012, 04:45 PM

Originally Posted by sol invictus

A flat infinite universe is very simple - it's just an infinitely large, 3D volume. That's the only case that's both flat and isotropic.

If you want something that's flat, finite, without boundary, but not isotropic, a 3-torus is a good example. That's a 3D space that you can think of as like the interior of a cube or parallelepiped, with facing sides identified with each other.

Okay, so when people talk about a "flat" as opposed to a "curved" universe, they mean something like a 3-torus as opposed to something like a 3-sphere?

Perpetual Student · 13th June 2012, 07:48 PM

Originally Posted by Perpetual Student

Specifically, how do the EFE post-dict the big bang? I have recently spent some time studying the GR tensor equation and have looked at some specific solutions, so I am somewhat prepared for a mathematical demonstration of the big bang -- assuming there is one.

I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review.

sol invictus · 13th June 2012, 08:22 PM

Originally Posted by TubbaBlubba

Okay, so when people talk about a "flat" as opposed to a "curved" universe, they mean something like a 3-torus as opposed to something like a 3-sphere?

Yes, they're usually talking about the curvature of the 3D slices of constant cosmological time. There's occasionally some confusion because (just about) all cosmologies have spacetime curvature, but the "flat" ones have no space curvature.

Originally Posted by Perpetual Student

I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review.

If you've made some progress with the math of GR, doing so should answer your previous question. FRW assumes homogeneity and isotropy, which means the metric is characterized by a single function of time (the scale factor a(t)), and Einstein's equations reduce to an ordinary differential equation for a.

Perpetual Student · 14th June 2012, 08:55 AM

Originally Posted by sol invictus

If you've made some progress with the math of GR, doing so should answer your previous question. FRW assumes homogeneity and isotropy, which means the metric is characterized by a single function of time (the scale factor a(t)), and Einstein's equations reduce to an ordinary differential equation for a.

I've made a cursory review of the Wikipedia article HERE. I intend to go over it and other sources as thoroughly as I can over the next few days (weeks?). In the meantime, I see that the key aspect of that metric is that the a(t) factor scales all the space dimensions. That explains why I have heard it said so many times that it is not the matter in the universe that is flying apart, but space itself is expanding -- over time. (

I love this stuff!)
Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift. So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space?

H'ethetheth · 14th June 2012, 09:33 AM

Originally Posted by Perpetual Student

Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift. So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space?

I think the history is the key here. I think Einstein found that his equations predicted an expanding universe, which he didn't think was right so he added the cosmological constant. He changed his mind after Hubble's observations.

Perpetual Student · 14th June 2012, 09:39 AM

Originally Posted by H'ethetheth

I think the history is the key here. I think Einstein found that his equations predicted an expanding universe, which he didn't think was right so he added the cosmological constant. He changed his mind after Hubble's observations.

Actually, I believe Einstein thought his equations predicted a contracting universe, which required the cosmological constant.

H'ethetheth · 14th June 2012, 09:54 AM

Originally Posted by Perpetual Student

Actually, I believe Einstein thought his equations predicted a contracting universe, which required the cosmological constant.

Could be. Not stationary at least.

ServiceSoon · 14th June 2012, 10:39 AM

Originally Posted by Brian-M

Okay, if you're going to refuse to accept reduced dimension analogies, how about a computer simulation?

Let's say we have a thousand underground chambers in a computer game. From any chamber you're free to move in three dimensions:

First Dimension: North/South
Second Dimension: East/West
Third Dimension: Up/Down.

If you ever found a chamber where you couldn't move in any of these directions, you'd have encountered a boundary.

But playing the game you notice that you never come to a boundary. In every chamber you're free to move in all directions. You also discover that if you move in any one direction, you end up passing the same chambers over and over again. Move ten places in any direction, and you're back where you started from.

The 3D space in this game is finite (limited 1000 cubic units), but without any boundaries on this space.

For space to be finite but have no boundaries, it must loop around. If you could simply move in a straight line and never end up in the same place twice, it would be infinite.

ETA:

The OP was asking for an explanation of what it would mean for the universe to be finite with no boundary. Not asking for evidence that it is.

If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes?

Vorpal · 14th June 2012, 06:20 PM

Originally Posted by Perpetual Student

Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift.

No, actually: many solutions in that family were discovered years before Hubble's observations.

Originally Posted by Perpetual Student

So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space?

It's not possible to completely disconnect the behavior of space from the behavior of the galaxies because of the Einstein field equation--the galaxies gravitate, so both their presence and motion affects the behavior of space. In particular, look up the Friedmann equations to see how it works out for the FLRW family. But the basic problem is that the assumptions of homogeneity and isotropy makes it impossible to fiddle with the behavior of the galaxies on the large scale: FLRW is as general as it gets. There is actually a particular case in FLRW for which the redshift is entirely due to motion of the galaxies, but it has zero density and no cosmic background radiation, so it's physically problematic.

Brian-M · 16th June 2012, 01:01 AM

Originally Posted by ServiceSoon

If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes?

Yes. But you'll find no boundaries, you just end up going past the same places over and over again while traveling in a straight line.

I that's what I assume a finite universe with no boundaries is (sort-of) like.

ynot · 16th June 2012, 01:24 PM

Originally Posted by Brian-M

Yes. But you'll find no boundaries, you just end up going past the same places over and over again while traveling in a straight line.

I that's what I assume a finite universe with no boundaries is (sort-of) like.

Anyone who believes a computer game is in any way a valid representation of the reality of the Universe must also believe it’s a valid representation of reincarnation given game characters can often come back to life after being “killed”. If “anything is possible” computer games and impossible less than 3D scenarios are the best “examples” of “finite and unbounded” then it‘s about as pathetic as believing in a god in my opinion. In the reality of a 3D Universe being bounded to a line or surface isn’t being unbounded.

Perpetual Student · 16th June 2012, 05:24 PM

Originally Posted by ynot

Anyone who believes a computer game is in any way a valid representation of the reality of the Universe must also believe it’s a valid representation of reincarnation given game characters can often come back to life after being “killed”. If “anything is possible” computer games and impossible less than 3D scenarios are the best “examples” of “finite and unbounded” then it‘s about as pathetic as believing in a god in my opinion. In the reality of a 3D Universe being bounded to a line or surface isn't being unbounded.

I think the computer game was intended as an analogy, not a "valid representation."
As I said earlier. I've come to see that my problem was one of semantics; it's possible you're having the same difficulty. Sol invictus made the point that "unbounded" and "without boundary" -- which I had been treating as more or less synonymous -- can have slightly different meanings.
The surface of a sphere is without boundary, in the sense that a point can be moved continuously anywhere on that surface indefinitely with no impediment. There is no boundary regarding the movement of that point. That's what is being said when the universe is described as finite but without boundary. This desciption may or may not truely represent the universe, but it is plausable.

theprestige · 16th June 2012, 06:50 PM

Originally Posted by ServiceSoon

If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes?

Surely it depends what you're simulating? If you're simulating a 2-d plane, then no. If you're simulating a 2-d constraint in a 3-d volume, then yes (obviously).

I mean, it's not like mathematicians are retards who never noticed that planes don't extend into the third dimension, and the only reason they talk about unbounded planes is because they're too stupid to understand the limits of planes.

ynot · 16th June 2012, 09:57 PM

Originally Posted by Perpetual Student

I think the computer game was intended as an analogy, not a "valid representation."
As I said earlier. I've come to see that my problem was one of semantics; it's possible you're having the same difficulty. Sol invictus made the point that "unbounded" and "without boundary" -- which I had been treating as more or less synonymous -- can have slightly different meanings.
The surface of a sphere is without boundary, in the sense that a point can be moved continuously anywhere on that surface indefinitely with no impediment. There is no boundary regarding the movement of that point. That's what is being said when the universe is described as finite but without boundary. This desciption may or may not truely represent the universe, but it is plausable.

Sorry I should have said valid analogy.

Vorpal · 16th June 2012, 10:40 PM

Even with the amendment, I'm failing to comprehend your post, ynot. It just seems to be a complete non-sequitur.

When saying that the space of a toroidal universe is like an Asteroids game, for example, it's just an illustration based on a particular feature of one video game many people are familiar with--the way the going to the edge of the screen gets on the opposite edge, so it's not a boundary in the game.

There is absolutely no implication that any other feature of this, of this or other games, have anything to do with reality. It's just an illustration that I thought might help some people visualize what living (er, floating) in that particular "finite but unbounded" universe might be like.

ynot · 16th June 2012, 11:55 PM

Originally Posted by Vorpal

Even with the amendment, I'm failing to comprehend your post, ynot. It just seems to be a complete non-sequitur.
When saying that the space of a toroidal universe is like an Asteroids game, for example, it's just an illustration based on a particular feature of one video game many people are familiar with--the way the going to the edge of the screen gets on the opposite edge, so it's not a boundary in the game.
There is absolutely no implication that any other feature of this, of this or other games, have anything to do with reality. It's just an illustration that I thought might help some people visualize what living (er, floating) in that particular "finite but unbounded" universe might be like.

Not sure if you meant to say “this feature or any other feature of this, or any other game”, or you're claiming that this feature does have anything to do with reality. If it's the latter I don't agree.

If I understand it correctly the theory is that the Universe is “curved” and essentially forms a shape that loops back on itself. It is therefore finite but unbounded and if you travel in a “straight” line in any direction you will eventually end up back where you started. I have a problem understanding the actual nature of the reality of this “curvature” and abstract 2D surfaces and fantasy computer games are of no help.

So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point. If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line. Also different areas of the Universe would have different curvatures. Seems to me it would have to be a curvature that was somehow equally in all directions concurrently and the same degree of curvature throughout the Universe. I can't see any possible way such a scenario could ever be possible in our 3D Universe.

HghrSymmetry · 17th June 2012, 12:04 AM

Originally Posted by ynot

So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point. If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line. Also different areas of the Universe would have different curvatures. Seems to me it would have to be a curvature that was somehow equally in all directions concurrently and the same degree of curvature throughout the Universe. I can't see any possible way such a scenario could ever be possible in our 3D Universe.

If it's "curved" in a higher dimension then it would not be possible to compensate.
The "craft" you're traveling in would read a dead straight path the entire trip so you wouldn't be able to "cancel out" the curved path because your space is shaped that way. How would a 2D creature living on a balls surface cancel out the curvature on their trip? They couldn't.
Now add one more dimension and you have the same scenario.

ynot · 17th June 2012, 01:06 AM

Originally Posted by HghrSymmetry

If it's "curved" in a higher dimension then it would not be possible to compensate.
The "craft" you're traveling in would read a dead straight path the entire trip so you wouldn't be able to "cancel out" the curved path because your space is shaped that way. How would a 2D creature living on a balls surface cancel out the curvature on their trip? They couldn't.
Now add one more dimension and you have the same scenario.

“Higher dimension” sounds a bit like “higher conciousness” or “sixth sense” to me. I only know of and accept 3 spatial dimensions.

Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid. Regardless of whatever number of dimensions there actually are, or we pretend there are, everything surely must share the same number in any particular scenario. A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path.

dafydd · 17th June 2012, 01:33 AM

Originally Posted by ynot

“Higher dimension” sounds a bit like “higher conciousness” or “sixth sense” to me. I only know of and accept 3 spatial dimensions.

Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid. Regardless of whatever number of dimensions there actually are, or we pretend there are, everything surely must share the same number in any particular scenario. A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path.

A section of the 3D ball would be a flat disk in 2D space, not the whole ball.

HghrSymmetry · 17th June 2012, 01:44 AM

Originally Posted by ynot

“Higher dimension” sounds a bit like “higher conciousness” or “sixth sense” to me. I only know of and accept 3 spatial dimensions.

The work being done trying to unify gravity with the standard model hasn't had much success until higher dimensional theories were studied. Yes, it's theoretical/mathematical at this point. The technology to harness the power levels required for direct observation are far off indeed.

Quote:

Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid.

It's mathematically valid and it gives an analogy that one can "visualize."

Quote:

Regardless of whatever number of dimensions there actually are, or we pretend there are, everything surely must share the same number in any particular scenario. A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path.

The "universe" is a 2d surface warped by a 3rd dimension. A 2d creature can cancel out a 2d curvature, but is completely helpless if its space is warped by a 3rd dimension.

If you completely dismiss higher dimensions, seems one would have a difficult time explaining a finite yet unbounded universe.

Incidentally, some may remember Sagan's old bit from the Cosmos series.
It was before the WMPA and even the COBE data, so the local geometry wasn't as well understood.

YouTube Video This video is not hosted by the JREF. The JREF can not be held responsible for the suitability or legality of this material. By clicking the link below you agree to view content from an external website.

I AGREE

ynot · 17th June 2012, 02:09 AM

Originally Posted by dafydd

A section of the 3D ball would be a flat disk in 2D space, not the whole ball.

Shaun = “Your cars got a flat tyre.”
Paddy - “To be sure, but lucky me, it's only flat at the bottom”.

Even if what you say is true, surely a 2D creature could only exist on the 2D flat disk section.

ynot · 17th June 2012, 02:35 AM

Originally Posted by HghrSymmetry

The work being done trying to unify gravity with the standard model hasn't had much success until higher dimensional theories were studied. Yes, it's theoretical/mathematical at this point. The technology to harness the power levels required for direct observation are far off indeed.

Yet the power levels required to observe three dimensions appear to be very low.

Originally Posted by HghrSymmetry

It's mathematically valid and it gives an analogy that one can "visualize."

Don't count me as a “one”.

Originally Posted by HghrSymmetry

The "universe" is a 2d surface warped by a 3rd dimension. A 2d creature can cancel out a 2d curvature, but is completely helpless if its space is warped by a 3rd dimension.

Everything that exists is 3D and a surface only exists as the outer extent of 3D things. The Universe isn't a surface and it's only a thing in that it's everything. So this “space warping” causes a 2D surface to become 3D but the creature remains 2D?

Originally Posted by HghrSymmetry

If you completely dismiss higher dimensions, seems one would have a difficult time explaining a finite yet unbounded universe.

So the ~~end~~ need justifies the means?

Vorpal · 17th June 2012, 03:37 AM

Originally Posted by ynot

Not sure if you meant to say “[/font]this feature[font=Verdana, sans-serif] or any other feature of this, or any other game”, or you're claiming that this feature does have anything to do with reality. If it's the latter I don't agree.

Neither of those. In that sentence, I meant exactly what I said.

Originally Posted by ynot

I have a problem understanding the actual nature of the reality of this “curvature” and abstract 2D surfaces and fantasy computer games are of no help.

Then you're concentrating way too much on what was a side comment to begin with. Particularly since I described what I was talking about in fair mathematical detail before making that analogy.

Originally Posted by ynot

So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point.

Actually, the example you're taking issue with has no curvature. It's completely flat. And it's still finite and without boundary.

But yes, with the common assumptions of homogeneity and isotropy, a spatially finite universe is curved. Physically, spacetime curvature is a tidal force: you have two test particles nearby each other, you see how their paths accelerate from one another a short time later, and the amount of deviation measures a component of curvature.

This has a more exact mathematical definition applicable to both spacetime and purely space manifolds, but I'm not sure it would be fruitful to you. If we're talking about only space, geodesics are paths of minimize length compared to other 'nearby' paths (intuitively: if you slightly deform the path, you always get a longer one) and curvature still describes how those paths deviate from one another in a manner similar to the above (the length along the path takes the role of 'time').

Originally Posted by ynot

If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line.

This statement doesn't make any sense.

Originally Posted by ynot

I can't see any possible way such a scenario could ever be possible in our 3D Universe.

That's partly because you keep thinking in terms of something curved in 3D space, rather than the 3D space itself being curved.

Originally Posted by HghrSymmetry

If it's "curved" in a higher dimension then it would not be possible to compensate.

As HghrSymmetry later notes, this is a visualization tool. There is as of yet no empirical evidence of more dimension nor need to postulate them that's backed by observations.

Originally Posted by ynot

Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid.

It's an analogy. Its main purpose is to illustrate the logical consistency of a 3D space that curves, because there is absolutely nothing 'logically special' about 3D compared to 2D.

I say again: a mathematical curved space exists regardless of whether or not it is embedded in any higher-dimensional space. We have good empirical evidence to believe our universe is described by a curved spacetime, but no good evidence that would support of that's its of the finite type~the question in the thread is a hypothetical.

That people like to visualize things as it having some higher-dimensional space means precisely nothing fundamental. For one thing, curvature is defined according to how things inside the space behave, without any external reference whatsoever. Which actually makes the analogy of 2D beings very apt.

Originally Posted by ynot

A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible.

That's right, if 'flat' is replaced by '2D'.

Originally Posted by ynot

A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path.

I'm unclear about what you mean. If you mean that such a being can travel along a path that's not a geodesic in the 2D space, then that's perfectly true--it would do so by accelerating. But you have the labels reversed: it's the geodesic path that's straight, and the accelerated path that's curved.

dafydd · 17th June 2012, 03:48 AM

Originally Posted by ynot

Even if what you say is true, surely a 2D creature could only exist on the 2D flat disk section.

Of course, that is the point. I can recommend this book.

http://www.amazon.co.uk/Fourth-Dimen.../dp/0140130365

TubbaBlubba · 17th June 2012, 10:50 AM

Here's a bit of an odd question - if a 2D being on a spherical surface were to measure the ratio between a circle's diameter and its circumference, would they always measure it to be pi, or would they find it to have different values based on the relative size of the circles? Does that feature only appear when a creature of a higher geometry appear to conduct measurements on curved surfaces?

dafydd · 17th June 2012, 10:56 AM

Originally Posted by TubbaBlubba

Here's a bit of an odd question - if a 2D being on a spherical surface were to measure the ratio between a circle's diameter and its circumference, would they always measure it to be pi, or would they find it to have different values based on the relative size of the circles? Does that feature only appear when a creature of a higher geometry appear to conduct measurements on curved surfaces?

This might help

''PI DOESN'T HAVE TO BE 3.14159…

One can measure the intrinsic curvature of a surface by drawing circles and comparing their circumferences to their radii.
Positive curvature yields a smaller circumference than we would expect for a given radius.
Negative curvature yields a larger circumference than we would expect for a given radius.

Instead of trying to find the principal curvatures, the ant can draw a circle on his surface and look at the ratio of the circumference to the diameter. This ratio is often known as pi, and in flat space it is about 3.14159. We usually consider pi to be a universal constant, and it can be, but that depends on which universe we are talking about. In a Euclidean universe, pi is indeed constant. In non- Euclidean universes, however, the value of pi depends on where exactly the circle is drawn—it's not a constant at all!''

http://www.learner.org/courses/mathi...extbook/06.php

TubbaBlubba · 17th June 2012, 02:21 PM

Right, but would the non-euclidean creature (and his measuring device) also have that curving, thus "cancelling out" his appreciation of pi as non-constant?

It's an odd question, but...

Perpetual Student · 17th June 2012, 02:47 PM

Originally Posted by TubbaBlubba

Right, but would the non-euclidean creature (and his measuring device) also have that curving, thus "cancelling out" his appreciation of pi as non-constant?

It's an odd question, but...

Pi would still be important in non-Euclidean spaces because it would be approached asymptotically. For example, on a sphere the ratio of the circumference to the diameter would approach pi as the circles are made smaller.

Perpetual Student · 17th June 2012, 03:44 PM

For example, if we lived on a perfectly round and smooth earth and drew a circle of 1 meter in diameter, the ratio would be:
C = 2πRsin(r/R), where R is the radius of the earth and r is the radius of the drawn circle. R is approximately 6,371 km so r/R would be 1/6,371,000 or 1.57x10-7. At those small values of θ, the sine function can be approximated by θ itself, so Rsin(r/R) ≈ r, which gives us C ≈ 2πr. The error would be very small.

TubbaBlubba · 17th June 2012, 03:45 PM

Yeah, you'd just have to define pi as a limit - but that doesn't answer my question. Is that something that's only noticeable for us as relatively "Euclidean" creatures measuring non-euclidean spaces, or does a non-euclidean creature with non-euclidean measuring the ratio also notice it?

Perpetual Student · 17th June 2012, 03:52 PM

Originally Posted by TubbaBlubba

Yeah, you'd just have to define pi as a limit - but that doesn't answer my question. Is that something that's only noticeable for us as relatively "Euclidean" creatures measuring non-euclidean spaces, or does a non-euclidean creature with non-euclidean measuring the ratio also notice it?

Well, it's difficult to speak for hypothetical non-Euclidean critters, but if they developed geometry how could they not notice pi as a limiting ratio as circles got smaller?

TubbaBlubba · 17th June 2012, 03:54 PM

Originally Posted by Perpetual Student

Well, it's difficult to speak for hypothetical non-Euclidean critters, but if they developed geometry how could they not notice pi as a limiting ratio as circles got smaller?

It's not immediately clear to me that their measurements are directly analogous to ours.

CapelDodger · 17th June 2012, 04:43 PM

Originally Posted by Perpetual Student

I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review.

I used to make plans like that. Now I just skim across conversations on wings made of sheer wonder

.

Vorpal · 18th June 2012, 02:32 AM

Radius and circumference are measured lengths along some one-dimensional curve, which can never have any intrinsic curvature. Intuitively one could repeat the same picture: if you're crawling along a curve, then you can only describe the length along it... unless you have some external to measure how it curves. Hence, only extrinsic curvature is possible. The is no meaningful metrical difference between "Euclidean" and "non-Euclidean" 1D manifolds.

Though it's probably better to also think of how you might measure the length of a curve of you were a higher-dimensional being (more than 1D, anyway). Say you look at this curve and with your trusty ruler you try to align the ruler along the curve, and repeat where the ruler ended, etc. Because the curve might not go along the ruler exactly, in general you would get better results if you used a smaller ruler (*). But as Perpetual Student illustrated with another example, curvature become becomes less and less relevant as the scale becomes smaller.

Through such a process, everyone's length measurements are analogous to everybody else's, and actually come out to be the same in the limit. Mathematically, the length is ultimately an integral of a line element, so the "ruler" you're using is actually infinitesimal in length.

(*) Assume the problem of putting the beginning of the ruler at the exact point you left off previously is not important. You're just that good at laying down rulers.

13th June 2012, 03:51 PM	#122
Tim Thompson Muse Join Date: Dec 2008 Location: San Gabriel Valley, east of Los Angeles Posts: 963	How much bang would the big bang bang if the big bang could bang big? Originally Posted by ynot The most "credible" option I currently have is an endless Universe that has always existed in some form. Originally Posted by dafydd You disregard all the evidence of the Big Bang? It was the discovery of the cosmic background microwave radiation that put paid to Hoyle's Steady State infinite endless universe theory. Originally Posted by sol invictus All we know is what the currently known laws of physics predict (or rather postdict) for the early universe - and that's a big bang. Sol is on the money, but we need to keep in mind just what the bang of big bang fame really is supposed to be. Originally Posted by Brian Greene, "The Fabric of the Cosmos", page 272 A common misconception is that the big bang provides a theory of cosmic origins. It doesn't. The big bang is a theory, partly described in the last two chapters, that delineates cosmic evolution from a split second after whatever happened to bring the universe into existence, but it says nothing at all about time zero itself. And since, according to the big bang theory, the bang is what is supposed to have happened at the beginning, the big bang leaves out the bang. It tells us nothing about what banged, why it banged, how it banged, or, frankly, whether it ever really banged at all. The quote above is the first part of the first paragraph of chapter 10, "Deconstructing the Bang", in Brian Greene's popular book Fabric of the Cosmos (Vintage Books, 2004). While it's generally a good idea to be wary of "popular" level writing, I think this does a pretty good job of making the correct point (a wordier version of what Sol said). As long as we accept general relativity as the theory of space-time, we can get perhaps arbitrarily close to the initial state of the universe (i.e., "time zero"), but we can never access that initial state. So, if one is going to hold that the universe (in the sense of literally everything that does exist or ever can exist or ever has existed) has a finite age and a true beginning, that has to be a matter of interpretation; there is no direct evidence for a true beginning, simply because neither theory nor observation (at least for now) can access the initial conditions of the universe. That's why there is such a concerted effort to discover a quantum theory of gravity, a quantum theory of space-time, which might reveal a theory for the initial conditions of the universe. Colliding brane hypotheses in string theory, or "bouncing universe" models in loop quantum cosmology are two ways to establish a theory of initial conditions for the universe, which allow for our "universe" to be a subset of an eternal universe around it. So, we can have both a big bang beginning for our universe, and still retain an eternal universe as well. At this point in our history, theory and observation really both fall short of the goal of definitively establishing one or the other, a universe with a true beginning or an eternal universe, or even both at once. And I point out that this is very much different from the now discredited steady state cosmology of Hoyle, at al., or its more modern offspring of a "quasi steady state cosmology", advanced by Jayant Narlikar.
	__________________ The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell

13th June 2012, 04:16 PM	#123
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by sol invictus We don't know any of those things. All we know is what the currently known laws of physics predict (or rather postdict) for the early universe - and that's a big bang. Specifically, how do the EFE post-dict the big bang? I have recently spent some time studying the GR tensor equation and have looked at some specific solutions, so I am somewhat prepared for a mathematical demonstration of the big bang -- assuming there is one.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

13th June 2012, 04:45 PM	#124
TubbaBlubba Knave of the Dudes Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,394	Originally Posted by sol invictus A flat infinite universe is very simple - it's just an infinitely large, 3D volume. That's the only case that's both flat and isotropic. If you want something that's flat, finite, without boundary, but not isotropic, a 3-torus is a good example. That's a 3D space that you can think of as like the interior of a cube or parallelepiped, with facing sides identified with each other. Okay, so when people talk about a "flat" as opposed to a "curved" universe, they mean something like a 3-torus as opposed to something like a 3-sphere?
	__________________ Disagreement begets progress.

13th June 2012, 07:48 PM	#125
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by Perpetual Student Specifically, how do the EFE post-dict the big bang? I have recently spent some time studying the GR tensor equation and have looked at some specific solutions, so I am somewhat prepared for a mathematical demonstration of the big bang -- assuming there is one. I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

14th June 2012, 08:55 AM	#127
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by sol invictus If you've made some progress with the math of GR, doing so should answer your previous question. FRW assumes homogeneity and isotropy, which means the metric is characterized by a single function of time (the scale factor a(t)), and Einstein's equations reduce to an ordinary differential equation for a. I've made a cursory review of the Wikipedia article HERE. I intend to go over it and other sources as thoroughly as I can over the next few days (weeks?). In the meantime, I see that the key aspect of that metric is that the a(t) factor scales all the space dimensions. That explains why I have heard it said so many times that it is not the matter in the universe that is flying apart, but space itself is expanding -- over time. ( I love this stuff!) Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift. So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space?
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

13th June 2012, 03:04 PM	#121
sol invictus Philosopher Join Date: Oct 2007 Location: Nova Roma Posts: 8,417	Originally Posted by TubbaBlubba I still don't quite grasp what a "flat" universe would be like. Would it be analogous to some sort of finite unbounded plane? A flat infinite universe is very simple - it's just an infinitely large, 3D volume. That's the only case that's both flat and isotropic. If you want something that's flat, finite, without boundary, but not isotropic, a 3-torus is a good example. That's a 3D space that you can think of as like the interior of a cube or parallelepiped, with facing sides identified with each other.

13th June 2012, 08:22 PM	#126
sol invictus Philosopher Join Date: Oct 2007 Location: Nova Roma Posts: 8,417	Originally Posted by TubbaBlubba Okay, so when people talk about a "flat" as opposed to a "curved" universe, they mean something like a 3-torus as opposed to something like a 3-sphere? Yes, they're usually talking about the curvature of the 3D slices of constant cosmological time. There's occasionally some confusion because (just about) all cosmologies have spacetime curvature, but the "flat" ones have no space curvature. Originally Posted by Perpetual Student I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review. If you've made some progress with the math of GR, doing so should answer your previous question. FRW assumes homogeneity and isotropy, which means the metric is characterized by a single function of time (the scale factor a(t)), and Einstein's equations reduce to an ordinary differential equation for a.

14th June 2012, 09:33 AM	#128
H'ethetheth fishy rocket scientist Join Date: Aug 2004 Location: among the machines Posts: 2,340	Originally Posted by Perpetual Student Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift. So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space? I think the history is the key here. I think Einstein found that his equations predicted an expanding universe, which he didn't think was right so he added the cosmological constant. He changed his mind after Hubble's observations.

14th June 2012, 09:39 AM	#129
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by H'ethetheth I think the history is the key here. I think Einstein found that his equations predicted an expanding universe, which he didn't think was right so he added the cosmological constant. He changed his mind after Hubble's observations. Actually, I believe Einstein thought his equations predicted a contracting universe, which required the cosmological constant.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

14th June 2012, 09:54 AM	#130
H'ethetheth fishy rocket scientist Join Date: Aug 2004 Location: among the machines Posts: 2,340	Originally Posted by Perpetual Student Actually, I believe Einstein thought his equations predicted a contracting universe, which required the cosmological constant. Could be. Not stationary at least.

14th June 2012, 10:39 AM	#131
ServiceSoon Muse Join Date: Oct 2007 Posts: 995	Originally Posted by Brian-M Okay, if you're going to refuse to accept reduced dimension analogies, how about a computer simulation? Let's say we have a thousand underground chambers in a computer game. From any chamber you're free to move in three dimensions: First Dimension: North/South Second Dimension: East/West Third Dimension: Up/Down. If you ever found a chamber where you couldn't move in any of these directions, you'd have encountered a boundary. But playing the game you notice that you never come to a boundary. In every chamber you're free to move in all directions. You also discover that if you move in any one direction, you end up passing the same chambers over and over again. Move ten places in any direction, and you're back where you started from. The 3D space in this game is finite (limited 1000 cubic units), but without any boundaries on this space. For space to be finite but have no boundaries, it must loop around. If you could simply move in a straight line and never end up in the same place twice, it would be infinite. ETA: The OP was asking for an explanation of what it would mean for the universe to be finite with no boundary. Not asking for evidence that it is. If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes?
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14th June 2012, 06:20 PM	#132
Vorpal Extrapolate! Join Date: Jan 2005 Posts: 1,005	Originally Posted by Perpetual Student Without researching the history of this, one would assume this idea came about because of Hubble's discovery of the cosmological red shift. No, actually: many solutions in that family were discovered years before Hubble's observations. Originally Posted by Perpetual Student So, the big question is, why would this be the interpretation of the red shift instead of merely assuming that matter is moving apart through space, with space as previously described by GR, which would not provide for an expansion of space? It's not possible to completely disconnect the behavior of space from the behavior of the galaxies because of the Einstein field equation--the galaxies gravitate, so both their presence and motion affects the behavior of space. In particular, look up the Friedmann equations to see how it works out for the FLRW family. But the basic problem is that the assumptions of homogeneity and isotropy makes it impossible to fiddle with the behavior of the galaxies on the large scale: FLRW is as general as it gets. There is actually a particular case in FLRW for which the redshift is entirely due to motion of the galaxies, but it has zero density and no cosmic background radiation, so it's physically problematic.
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16th June 2012, 01:01 AM	#133
Brian-M Daydreamer Join Date: Jul 2008 Location: Downunder Posts: 4,237	Originally Posted by ServiceSoon If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes? Yes. But you'll find no boundaries, you just end up going past the same places over and over again while traveling in a straight line. I that's what I assume a finite universe with no boundaries is (sort-of) like.
	__________________ "That is just what you feel, that isn't reality." - hamelekim

16th June 2012, 01:24 PM	#134
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by Brian-M Yes. But you'll find no boundaries, you just end up going past the same places over and over again while traveling in a straight line. I that's what I assume a finite universe with no boundaries is (sort-of) like. Anyone who believes a computer game is in any way a valid representation of the reality of the Universe must also believe it’s a valid representation of reincarnation given game characters can often come back to life after being “killed”. If “anything is possible” computer games and impossible less than 3D scenarios are the best “examples” of “finite and unbounded” then it‘s about as pathetic as believing in a god in my opinion. In the reality of a 3D Universe being bounded to a line or surface isn’t being unbounded.
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise. Last edited by ynot; 16th June 2012 at 01:28 PM.

16th June 2012, 05:24 PM	#135
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by ynot Anyone who believes a computer game is in any way a valid representation of the reality of the Universe must also believe it’s a valid representation of reincarnation given game characters can often come back to life after being “killed”. If “anything is possible” computer games and impossible less than 3D scenarios are the best “examples” of “finite and unbounded” then it‘s about as pathetic as believing in a god in my opinion. In the reality of a 3D Universe being bounded to a line or surface isn't being unbounded. I think the computer game was intended as an analogy, not a "valid representation." As I said earlier. I've come to see that my problem was one of semantics; it's possible you're having the same difficulty. Sol invictus made the point that "unbounded" and "without boundary" -- which I had been treating as more or less synonymous -- can have slightly different meanings. The surface of a sphere is without boundary, in the sense that a point can be moved continuously anywhere on that surface indefinitely with no impediment. There is no boundary regarding the movement of that point. That's what is being said when the universe is described as finite but without boundary. This desciption may or may not truely represent the universe, but it is plausable.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

16th June 2012, 06:50 PM	#136
theprestige Philosopher Join Date: Aug 2007 Posts: 8,568	Originally Posted by ServiceSoon If at any point in your simulation I am not free to travel in any one of the three dimensions, than that is a boundary. Yes? Surely it depends what you're simulating? If you're simulating a 2-d plane, then no. If you're simulating a 2-d constraint in a 3-d volume, then yes (obviously). I mean, it's not like mathematicians are retards who never noticed that planes don't extend into the third dimension, and the only reason they talk about unbounded planes is because they're too stupid to understand the limits of planes.
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16th June 2012, 09:57 PM	#137
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by Perpetual Student I think the computer game was intended as an analogy, not a "valid representation." As I said earlier. I've come to see that my problem was one of semantics; it's possible you're having the same difficulty. Sol invictus made the point that "unbounded" and "without boundary" -- which I had been treating as more or less synonymous -- can have slightly different meanings. The surface of a sphere is without boundary, in the sense that a point can be moved continuously anywhere on that surface indefinitely with no impediment. There is no boundary regarding the movement of that point. That's what is being said when the universe is described as finite but without boundary. This desciption may or may not truely represent the universe, but it is plausable. Sorry I should have said valid analogy.
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise.

16th June 2012, 10:40 PM	#138
Vorpal Extrapolate! Join Date: Jan 2005 Posts: 1,005	Even with the amendment, I'm failing to comprehend your post, ynot. It just seems to be a complete non-sequitur. When saying that the space of a toroidal universe is like an Asteroids game, for example, it's just an illustration based on a particular feature of one video game many people are familiar with--the way the going to the edge of the screen gets on the opposite edge, so it's not a boundary in the game. There is absolutely no implication that any other feature of this, of this or other games, have anything to do with reality. It's just an illustration that I thought might help some people visualize what living (er, floating) in that particular "finite but unbounded" universe might be like.
Vorpal Extrapolate! Join Date: Jan 2005 Posts: 1,005	__________________ For every philosopher, there exists an equal and opposite philosopher. They're both wrong.

16th June 2012, 11:55 PM	#139
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by Vorpal Even with the amendment, I'm failing to comprehend your post, ynot. It just seems to be a complete non-sequitur. When saying that the space of a toroidal universe is like an Asteroids game, for example, it's just an illustration based on a particular feature of one video game many people are familiar with--the way the going to the edge of the screen gets on the opposite edge, so it's not a boundary in the game. There is absolutely no implication that any other feature of this, of this or other games, have anything to do with reality. It's just an illustration that I thought might help some people visualize what living (er, floating) in that particular "finite but unbounded" universe might be like. Not sure if you meant to say “this feature or any other feature of this, or any other game”, or you're claiming that this feature does have anything to do with reality. If it's the latter I don't agree. If I understand it correctly the theory is that the Universe is “curved” and essentially forms a shape that loops back on itself. It is therefore finite but unbounded and if you travel in a “straight” line in any direction you will eventually end up back where you started. I have a problem understanding the actual nature of the reality of this “curvature” and abstract 2D surfaces and fantasy computer games are of no help. So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point. If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line. Also different areas of the Universe would have different curvatures. Seems to me it would have to be a curvature that was somehow equally in all directions concurrently and the same degree of curvature throughout the Universe. I can't see any possible way such a scenario could ever be possible in our 3D Universe.
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise. Last edited by ynot; 16th June 2012 at 11:57 PM.

17th June 2012, 12:04 AM	#140
HghrSymmetry View clearer from above. Join Date: Jul 2007 Posts: 8,356	Originally Posted by ynot So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point. If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line. Also different areas of the Universe would have different curvatures. Seems to me it would have to be a curvature that was somehow equally in all directions concurrently and the same degree of curvature throughout the Universe. I can't see any possible way such a scenario could ever be possible in our 3D Universe. If it's "curved" in a higher dimension then it would not be possible to compensate. The "craft" you're traveling in would read a dead straight path the entire trip so you wouldn't be able to "cancel out" the curved path because your space is shaped that way. How would a 2D creature living on a balls surface cancel out the curvature on their trip? They couldn't. Now add one more dimension and you have the same scenario.
	__________________ HygrSym brought up some fantastic points. He's so good, he doesn't have to use pseudo words like 'chillax'... -FSM Last edited by HghrSymmetry; 17th June 2012 at 12:06 AM.

17th June 2012, 01:06 AM	#141
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by HghrSymmetry If it's "curved" in a higher dimension then it would not be possible to compensate. The "craft" you're traveling in would read a dead straight path the entire trip so you wouldn't be able to "cancel out" the curved path because your space is shaped that way. How would a 2D creature living on a balls surface cancel out the curvature on their trip? They couldn't. Now add one more dimension and you have the same scenario. “Higher dimension” sounds a bit like “higher conciousness” or “sixth sense” to me. I only know of and accept 3 spatial dimensions. Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid. Regardless of whatever number of dimensions there actually are, or we pretend there are, everything surely must share the same number in any particular scenario. A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path.
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise.

17th June 2012, 01:33 AM	#142
dafydd Penultimate Amazing Join Date: Feb 2008 Location: Belgium (Flatland) Posts: 31,443	Originally Posted by ynot “Higher dimension” sounds a bit like “higher conciousness” or “sixth sense” to me. I only know of and accept 3 spatial dimensions. Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid. Regardless of whatever number of dimensions there actually are, or we pretend there are, everything surely must share the same number in any particular scenario. A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path. A section of the 3D ball would be a flat disk in 2D space, not the whole ball.
	__________________ Yesterday upon the stairs I met a man who wasn't there He wasn't there again today I wish that he would go away.

17th June 2012, 02:09 AM	#144
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by dafydd A section of the 3D ball would be a flat disk in 2D space, not the whole ball. Shaun = “Your cars got a flat tyre.” Paddy - “To be sure, but lucky me, it's only flat at the bottom”. Even if what you say is true, surely a 2D creature could only exist on the 2D flat disk section.
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise.

17th June 2012, 02:35 AM	#145
ynot Illuminator Join Date: Jan 2006 Location: New Zealand Posts: 4,640	Originally Posted by HghrSymmetry The work being done trying to unify gravity with the standard model hasn't had much success until higher dimensional theories were studied. Yes, it's theoretical/mathematical at this point. The technology to harness the power levels required for direct observation are far off indeed. Yet the power levels required to observe three dimensions appear to be very low. Originally Posted by HghrSymmetry It's mathematically valid and it gives an analogy that one can "visualize." Don't count me as a “one”. Originally Posted by HghrSymmetry The "universe" is a 2d surface warped by a 3rd dimension. A 2d creature can cancel out a 2d curvature, but is completely helpless if its space is warped by a 3rd dimension. Everything that exists is 3D and a surface only exists as the outer extent of 3D things. The Universe isn't a surface and it's only a thing in that it's everything. So this “space warping” causes a 2D surface to become 3D but the creature remains 2D? Originally Posted by HghrSymmetry If you completely dismiss higher dimensions, seems one would have a difficult time explaining a finite yet unbounded universe. So the ~~end~~ need justifies the means?
	__________________ Rumours of a god’s existence have been greatly exaggerated. My post are all (IMO) unless stated otherwise. Last edited by ynot; 17th June 2012 at 02:59 AM.

17th June 2012, 03:37 AM	#146
Vorpal Extrapolate! Join Date: Jan 2005 Posts: 1,005	Originally Posted by ynot Not sure if you meant to say “[/font]this feature[font=Verdana, sans-serif] or any other feature of this, or any other game”, or you're claiming that this feature does have anything to do with reality. If it's the latter I don't agree. Neither of those. In that sentence, I meant exactly what I said. Originally Posted by ynot I have a problem understanding the actual nature of the reality of this “curvature” and abstract 2D surfaces and fantasy computer games are of no help. Then you're concentrating way too much on what was a side comment to begin with. Particularly since I described what I was talking about in fair mathematical detail before making that analogy. Originally Posted by ynot So what is the actual nature of this “curvature”? I don't see that it could be a normal curvature that's in a particular direction with a discrete radius and centre-point. Actually, the example you're taking issue with has no curvature. It's completely flat. And it's still finite and without boundary. But yes, with the common assumptions of homogeneity and isotropy, a spatially finite universe is curved. Physically, spacetime curvature is a tidal force: you have two test particles nearby each other, you see how their paths accelerate from one another a short time later, and the amount of deviation measures a component of curvature. This has a more exact mathematical definition applicable to both spacetime and purely space manifolds, but I'm not sure it would be fruitful to you. If we're talking about only space, geodesics are paths of minimize length compared to other 'nearby' paths (intuitively: if you slightly deform the path, you always get a longer one) and curvature still describes how those paths deviate from one another in a manner similar to the above (the length along the path takes the role of 'time'). Originally Posted by ynot If it was it would be possible to travel at an equal and opposite curvature to cancel it out and travel in a straight line. This statement doesn't make any sense. Originally Posted by ynot I can't see any possible way such a scenario could ever be possible in our 3D Universe. That's partly because you keep thinking in terms of something curved in 3D space, rather than the 3D space itself being curved. Originally Posted by HghrSymmetry If it's "curved" in a higher dimension then it would not be possible to compensate. As HghrSymmetry later notes, this is a visualization tool. There is as of yet no empirical evidence of more dimension nor need to postulate them that's backed by observations. Originally Posted by ynot Regardless that 2D objects are pure “let's pretend” it seems ludicrous to combine a 2D creature and a 3D ball in the same scenario, so I don't think your question is valid. It's an analogy. Its main purpose is to illustrate the logical consistency of a 3D space that curves, because there is absolutely nothing 'logically special' about 3D compared to 2D. I say again: a mathematical curved space exists regardless of whether or not it is embedded in any higher-dimensional space. We have good empirical evidence to believe our universe is described by a curved spacetime, but no good evidence that would support of that's its of the finite type~the question in the thread is a hypothetical. That people like to visualize things as it having some higher-dimensional space means precisely nothing fundamental. For one thing, curvature is defined according to how things inside the space behave, without any external reference whatsoever. Which actually makes the analogy of 2D beings very apt. Originally Posted by ynot A “ball” in a 2D Universe would be a flat disc and only a 2D curvature woud be possible. That's right, if 'flat' is replaced by '2D'. Originally Posted by ynot A 2D creature travelling on a 2D surface could easily “cancel out” a 2D curved path. I'm unclear about what you mean. If you mean that such a being can travel along a path that's not a geodesic in the 2D space, then that's perfectly true--it would do so by accelerating. But you have the labels reversed: it's the geodesic path that's straight, and the accelerated path that's curved.
Vorpal Extrapolate! Join Date: Jan 2005 Posts: 1,005	__________________ For every philosopher, there exists an equal and opposite philosopher. They're both wrong.

17th June 2012, 03:48 AM	#147
dafydd Penultimate Amazing Join Date: Feb 2008 Location: Belgium (Flatland) Posts: 31,443	Originally Posted by ynot Even if what you say is true, surely a 2D creature could only exist on the 2D flat disk section. Of course, that is the point. I can recommend this book. http://www.amazon.co.uk/Fourth-Dimen.../dp/0140130365
	__________________ Yesterday upon the stairs I met a man who wasn't there He wasn't there again today I wish that he would go away.

17th June 2012, 10:50 AM	#148
TubbaBlubba Knave of the Dudes Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,394	Here's a bit of an odd question - if a 2D being on a spherical surface were to measure the ratio between a circle's diameter and its circumference, would they always measure it to be pi, or would they find it to have different values based on the relative size of the circles? Does that feature only appear when a creature of a higher geometry appear to conduct measurements on curved surfaces?
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17th June 2012, 10:56 AM	#149
dafydd Penultimate Amazing Join Date: Feb 2008 Location: Belgium (Flatland) Posts: 31,443	Originally Posted by TubbaBlubba Here's a bit of an odd question - if a 2D being on a spherical surface were to measure the ratio between a circle's diameter and its circumference, would they always measure it to be pi, or would they find it to have different values based on the relative size of the circles? Does that feature only appear when a creature of a higher geometry appear to conduct measurements on curved surfaces? This might help ''PI DOESN'T HAVE TO BE 3.14159… One can measure the intrinsic curvature of a surface by drawing circles and comparing their circumferences to their radii. Positive curvature yields a smaller circumference than we would expect for a given radius. Negative curvature yields a larger circumference than we would expect for a given radius. Instead of trying to find the principal curvatures, the ant can draw a circle on his surface and look at the ratio of the circumference to the diameter. This ratio is often known as pi, and in flat space it is about 3.14159. We usually consider pi to be a universal constant, and it can be, but that depends on which universe we are talking about. In a Euclidean universe, pi is indeed constant. In non- Euclidean universes, however, the value of pi depends on where exactly the circle is drawn—it's not a constant at all!'' http://www.learner.org/courses/mathi...extbook/06.php
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17th June 2012, 02:21 PM	#150
TubbaBlubba Knave of the Dudes Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,394	Right, but would the non-euclidean creature (and his measuring device) also have that curving, thus "cancelling out" his appreciation of pi as non-constant? It's an odd question, but...
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17th June 2012, 02:47 PM	#151
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by TubbaBlubba Right, but would the non-euclidean creature (and his measuring device) also have that curving, thus "cancelling out" his appreciation of pi as non-constant? It's an odd question, but... Pi would still be important in non-Euclidean spaces because it would be approached asymptotically. For example, on a sphere the ratio of the circumference to the diameter would approach pi as the circles are made smaller.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

17th June 2012, 03:44 PM	#152
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	For example, if we lived on a perfectly round and smooth earth and drew a circle of 1 meter in diameter, the ratio would be: C = 2πRsin(r/R), where R is the radius of the earth and r is the radius of the drawn circle. R is approximately 6,371 km so r/R would be 1/6,371,000 or 1.57x10-7. At those small values of θ, the sine function can be approximated by θ itself, so Rsin(r/R) ≈ r, which gives us C ≈ 2πr. The error would be very small.
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

17th June 2012, 03:45 PM	#153
TubbaBlubba Knave of the Dudes Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,394	Yeah, you'd just have to define pi as a limit - but that doesn't answer my question. Is that something that's only noticeable for us as relatively "Euclidean" creatures measuring non-euclidean spaces, or does a non-euclidean creature with non-euclidean measuring the ratio also notice it?
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17th June 2012, 03:52 PM	#154
Perpetual Student Illuminator Join Date: Jul 2008 Location: USA Posts: 3,703	Originally Posted by TubbaBlubba Yeah, you'd just have to define pi as a limit - but that doesn't answer my question. Is that something that's only noticeable for us as relatively "Euclidean" creatures measuring non-euclidean spaces, or does a non-euclidean creature with non-euclidean measuring the ratio also notice it? Well, it's difficult to speak for hypothetical non-Euclidean critters, but if they developed geometry how could they not notice pi as a limiting ratio as circles got smaller?
	__________________ _{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman} ξ

17th June 2012, 03:54 PM	#155
TubbaBlubba Knave of the Dudes Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,394	Originally Posted by Perpetual Student Well, it's difficult to speak for hypothetical non-Euclidean critters, but if they developed geometry how could they not notice pi as a limiting ratio as circles got smaller? It's not immediately clear to me that their measurements are directly analogous to ours.
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17th June 2012, 04:43 PM	#156
CapelDodger Penultimate Amazing Join Date: Sep 2001 Location: Cardiff, South Wales Posts: 16,740	Originally Posted by Perpetual Student I just learned of the "Friedmann–Lemaître–Robertson–Walker metric," which I plan to review. I used to make plans like that. Now I just skim across conversations on wings made of sheer wonder .
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