The Twin Paradox and a Glome Shaped Universe

[jinydu]

I think it has been discussed many times that the universe might be shaped like a glome (4D sphere). But one thing I don't understand is that the twin paradox from special relativity seems to make it impossible.

In case you don't know, an explanation of the Twin Paradox can be found here: http://theory.uwinnipeg.ca/mod_tech/node141.html

The important point is that the apparent paradox can be resolved because the astronauts needed to accelerate after reaching their destination.

However, if the universe was shaped like a glome (with a finite size), it would be theoretically possible for the astronaut to travel around the entire universe, returning to Earth, without undergoing acceleration. Thus, the resolution of the paradox would fail.

Can anyone explain a way out of this?

P.S. I've been upgraded to a Linespace Citizen. Now I can move!

[mghtymoop]

firstly i hate that name for a hypersphere, 'glome' drives me nuts, mostly because the integral of a 3d sphere can be one of a number of 4d possibilities and also because it sounds stupid. adding hyper as a prefix to a 4d shape covers all integrals and is the most correct terminology.
but anyway the most common theory is that the universe is a hyperplane within the multiverse which is the hypersphere and using the fourth dimension as time removes your paradox problem
oh and i looked at that paradox site anyway, it isn't correct, the rules of special relativity always apply, problem is people confuse special relativity with general relativity which is the toned down laymens version which tells the same story but fudges the working, so if you do the calculations you get the right, or damn near the right result, but the method of getting there is wrong so it cannot be used to expand upon itself as a theory.

[pat]

firstly i hate that name for a hypersphere, 'glome' drives me nuts, mostly because the integral of a 3d sphere can be one of a number of 4d possibilities and also because it sounds stupid. adding hyper as a prefix to a 4d shape covers all integrals and is the most correct terminology.

I hate that name, too. On the other hand, I'm not following the rest of your reasoning. Firstly, I know there are infinitely many diffeomorphic structures (and thus integrals) in R4, but I was pretty sure that the 3d sphere only had one diffeomorphic structure. As such, I'm not following what you mean by "the integral of a 3d sphere can be one of a number of 4d possibilities". Further, I don't understand why "glome" implies a particular one but "hypersphere" does not.

[mghtymoop]

lets take a 2d circle and integrate it, the obvious equilateral choice is a sphere, however you can also have a cylinder, or for that matter a cone or anynumber of shapes for which the circle itself is the largest area presented in the plane around which the integration occurs.

[pat]

anynumber of shapes for which the circle itself is the largest area presented in the plane around which the integration occurs.

Ahh... your definition of integration seems odd to me. Is the differential of the cone a circle (if taken with respect to the proper variable)?

Anyhow, it still doesn't answer my question about why the term glome is vague but hypersphere is not.

[mghtymoop]

hypersphere is a simle term derived of the words hyper and sphere meaning simply the integral of a suface which includes all points a set distance from a single origin. there is of course more than one possible solution for this but the word covers all possible solutions within it's definition. i have seen the word glome used to describe many possibilities, essentailly glome should regard only the surface of the single equilateral integral of a hypersphere. I have seen the word glome directly attached to the word hypersphere as if they are the same thing, i have seen it used to describe a hyperball with is the 4d equivelant to a 3d ball whiuch incedentally is the correct term for the volume of a sphere, a glome is a 3-sphere following the formula x^2*y^2*z^2*w^2=r^2 (why alkaline uses one in the place of r^2 is a little wierd because although 1^2 is one and it works for all of his shapes given a radius of one it may seem confusing as no differentiation can be made in his forula between radius and hyperradius which is in fact a plane and not a line so this yet another insufficient explanation of the glome. i beleive that due to the extreme confusion caused by hypoer geometry the use of a term such as hypersphere with a short explanation of the integral used based upon its construction leads to a far clearer understanding.

[PWrong]

adding hyper as a prefix to a 4d shape covers all integrals and is the most correct terminology.

That's not true. Adding "hyper" to the word "cylinder" can mean three separate things (a spherinder, cubinder, or duocylinder). Alkaline explains why he prefers not to use "hyper" somewhere on the website.

I always assumed glome was a word Alkaline invented, because there are so many funny words in the glossary, and he invented his own language on one of his other websites.

By the way, your formula should be x^2+y^2+z^2+w^2=r^2, not
x^2*y^2*z^2*w^2=r^2.

What's the difference between radius and "Hyperradius"?

[mghtymoop]

my formula is correct and obviousely so, what are you smoking?
besides what i said about hyper is that it covers all three possible integrals, you are definately smoking something?
and if you've never heard the term glome outside of alkalines site you have probably never read anything else or you've just smoked so much of whatever that is you can't remember it.

[Geosphere]

Keep it civil, moop.

[PWrong]

The formula for an n-sphere basically derives from Pythagorus's theorum. It's the sum of the squares, not the product of the squares. You just used * instead of +, that's all. Simple mistake, it doesn't matter.

Your definition of integral seems odd to me, too. I've noticed it on other threads. Using your definition though, if using "hyper" as a prefix covers all integrals, then "hypersphere" should cover all integrals of a sphere. So it's no more specific than "glome". I did look that up, by the way, and you're right, it's a pretty common word. But I haven't done much actual research into 4D geometry and rotatopes, and there are a lot of made up words in Alkaline's glossary.

no differentiation can be made in his forula between radius and hyperradius which is in fact a plane and not a line

What do you mean by that? A circle's radius is a line, and a sphere's radius is a line, so why not a hypersphere?

[RQ]

Whether the universe is a hypersphere or hyperellipsoid or anything that is a hypersomething with curved edges, or it is flat doesn't matter. Either way the astronaut needs to accelarate to get over a large distance so he's not 80 years old like his twin brother. The faster you move, the faster time goes for objects that aren't at your speed at the same time as you.

So anyways, the universe can't be infinite because otherwise our night sky would be as bright as the day. True objects and matter might cast a 1billion lightyear shadow or so, but eventually they'll get heated up and will glow as bright as the object, or even brighter, that is lighting it up. Now in a universe that is infinitely wide, long and high, there will be just as much mass as there are suns. In fact if this universe was infinite, our world would be 1/infinity and wouldn't exist.

[PWrong]

In fact if this universe was infinite, our world would be 1/infinity and wouldn't exist.

There are an infinite number of things that don't exist, and only a finite number of things that do. By your logic, the number of things that do exist is therefore 1/infinity = 0.

Be careful with infinity. There is no need to divide a finite number by infinity even in an infinite universe. The size of an object is defined by the object, not by the universe.

Besides, we know the universe is finite because the edge is about 50 billion light years away. By "the edge", I mean the first light released from the big bang. It's been moving at the fastest (finite) possible speed since the beginning of time (also finite), so how can anything be beyond that? I found this information in a New Scientist article.
Obviously, it might as well be infinite, because we can never catch up to it.

[jinydu]

Here is the article that I think PWrong has in mind (sorry its not from New Scientist, but I think the content is similar):

http://www.space.com/scienceastronomy/m ... 40524.html

However, the title is somewhat misleading. In reality, the study that they're referring to determined that the universe's diameter is at least 156 billion light years. That is, 156 billion light years is a lower bound. To quote the authors of the paper:

"For a wide class of models, the non-detection rules out the possibility that we live in a universe with topology scale smaller than 24 Gpc."

Here is a link to the scientific paper itself:

http://arxiv.org/abs/astro-ph/0310233

In case anyone doesn't know, a parsec (pc) is a unit of distance in astronomy. If I remember correctly, it is about 3.26 light years. A gigaparsec (Gpc) is 1 billion parsecs. Unfortunately, I can't view the paper itself because my computer window crashes everytime I try to view a PDF.

[PWrong]

Hmm, interesting. It's not the same one though. It wasn't really an article, just a special "map of the universe" that new scientist had once. It's basically what you would see if you stared at the sky for 24 hours from some point on the equator. It's on a logarithmic scale, so everything is easy to find, and it also mentions "the very edge of space and time, now almost 50 billion light years away".

[jinydu]

50 billion light years? That contradicts the figure of 24 Gpc given in the abstract. 24Gpc is definitely larger than 50 billion light years. I'll try to post more when I get on a computer that can view PDF's.

[RQ]

1/infinity, meaning we wouldn't exist but infact be in the higher dimension, and then the higher and higher until infinity i guess, whatever that means.

By the way Hawking says that the universe expands about 5-10% per year or so. It would be cool if someday we can cross that 156 billion gap, well of course when the big crunch comes, if it does.
By the way, nice story.

[jinydu]

5-10% per year?! He couldn't have been serious. That would imply an exponential growth for the size of the Universe, and an incredibly rapid one at that, since 1 year is VERY short compared to the lifetime of the Universe. At a growth rate of 5% per year, the Universe would expand by over 131 times in a century, and over 1.54*10^21 times in a millenium!

The Universe's rate of expansion is measured by something called Hubble's Constant, H. A change of 5-10% in Hubble's Constant over 1 year would definitely have been noticed.

[jinydu]

50 billion light years? That contradicts the figure of 24 Gpc given in the abstract. 24Gpc is definitely larger than 50 billion light years. I'll try to post more when I get on a computer that can view PDF's.

Ok, my parents bought me a new computer for when I go to University later this month, and it can view PDF's without crashing! So I had a look at the paper.

There are several parts in it that are too advanced for me to understand, but I can see that the study in fact doesn't completely rule out the possibility of a Universe with a radius of less than 24 Gpc. It only does so for some topologies (shapes that the Universe might take). The authors of the paper hope that future studies can examine the remaining possible topologies, and maybe even push up that figure of 24 Gpc.

[Rkyeun]

You can do math with infinity, but you have to be very careful with it.

Infinite + Finite = Infinite
Infinite - Finite = Infinite
Finite - Infinite = -Infinite
Finite * Infinite = Infinite
0 * Infinite = 0
Finite / Infinite = Infinitessimal ... NOT ZERO
0 * Infinitessimal = 0
Finite * Infinitessimal = Infinitessimal
1 / Infinitessimal = Infinite
Finite + Infinitessimal ~ Finite.Inf (The boundlessness of the infinity must be preserved, even when it has no effect. 3 + 0.inf is not 3, because subtracting 3 does not equal 0, it equals 0.inf.)
Finite - Infinitessimal ~ Finite
Infinitessimal - Finite ~ -Finite
Infinite - Infinite = Variable. You must examine the sources of the Infinites. It may be 0, it might be a Finite, it might be another Infinite.
Infinite * Infinitessimal = Variable. You must examine the sources of the Infinites. It may be 1, Finite, Infinite, or Infinitessimal. But it cannot be 0.
Finite / 0 = No Solution
Infinite / 0 = No Solution
Infinitessimal / 0 = No Solution
Infinite ^ 0 = 1
Infinitessimal ^ 0 = 1

When using infinite math, the number 'line' is actually a circle. 0 is an unsigned number, and when divided by it will produce an equally unsigned No Solution. Positive and negative infinity reach around the circle, meeting in a discontinuity. The No Solution would be theoretically larger than infinity, and negatively larger than negative infinity, at the same point. This is because even dividing by an infinite number does not produce zero, there is still something left, something infinitely small yes, but not non-existant. The existance of 0 defines the position of the point of No Solution, which is beyond the reach of even Infinity.

Thusly the universe can indeed be infinite. A finite number of objects in it does not equal 0, because 0 times the infinite universe does not produce those finite objects again. And there's no prrof that the number of objects themselves are not infinite. Even if precisely 1/10^35 of space were filled, that would still be infinite stuff, just with a lot of empty space between.
Similarly, the sky is not infinitely bright. Assuming there are an infinite number of stars, they are not necessarily in every direction. Assuming they are, the inverse square law still applies, over how infinitely far away they probably are. The squared infinitessimal strength over their infinte distance almost certainly beats out the infinite number of stars. Except for the closest points of light, the sky is dark because of all that infinite light, it has been ruthlessly muted by the square of another infinity.

[Rkyeun]

As usual, I discover minor mistakes after posting. All those ~ Finite for approximately equal signs should be removed, and the Finites replaced with Finite.Infs.

[RQ]

I knew this blasphemous crap had to appear sometime on this forum.

First of all infinity is not a number how you treated it:

You can do math with infinity, but you have to be very careful with it.

Infinite + Finite = Infinite
Infinite - Finite = Infinite

Good

Infinite ^ 0 = 1

First of all Infinity^0 is not 1. Infinity is a set of numbers: [1,2,1,0,.....]

Put each of those numbers to the 0 power, you still have an infinite amount of elements.

Finite - Infinite = -Infinite

Bullshit. A finite element subtracted by any of infinite elements, still gives you an infinite amount of elements. No such thing as positive or negative infinity. Infinity is like 0. Infinite amount of elements, though you can say there is a negative and positive ad infinitum of numbers if that's what you mean.

Finite * Infinite = Infinite

Correct

0 * Infinite = 0

Incorrect. An infinite number of elements each multiplied by 0, does not give you 0. You still have an infinite number of elements.

Finite / Infinite = Infinitessimal ... NOT ZERO

Infinitesimal is 0, please don't make me elaborate with a proof.

0 * Infinitessimal = 0
Finite * Infinitessimal = Infinitessimal

Correct

1 / Infinitessimal = Infinite

1/0=0

Finite + Infinitessimal ~ Finite.Inf (The boundlessness of the infinity must be preserved, even when it has no effect. 3 + 0.inf is not 3, because subtracting 3 does not equal 0, it equals 0.inf.)

That's just stupid.

Finite - Infinitessimal ~ Finite
Infinitessimal - Finite ~ -Finite

Again, no such thing as infinitesimal, if you're not referring to 0.

Infinite - Infinite = Variable. You must examine the sources of the Infinites. It may be 0, it might be a Finite, it might be another Infinite.

Untrue. Subtracting an infinite amount of elements from one infinite set to another still gives you an infinite amount of elements.

Finite / 0 = No Solution

e solution.

Infinite / 0 = No Solution

Infinity is the solution.

Infinitessimal / 0 = No Solution

0 is the solution.

Infinitessimal ^ 0 = 1

0^0 does not equal 1, it's 0.

[RQ]

Also, jinydu, I'm sure I misread 5-10% per billion maybe. How can we measure anything more than 20 billion lightyears, since 20 billion years is how long it takes light to reach us from there, and the universe isn't much older than 20 billion years.

[houserichichi]

Infinity is a set of numbers: [1,2,1,0,.....]

No it's not. Infinity can be treated in many ways, and in the context of infinitesimal arithmetic (this stuff) one can treat it as the "smallest value" larger than the cardinality of the counting numbers.

A finite element subtracted by any of infinite elements, still gives you an infinite amount of elements. No such thing as positive or negative infinity.

The first part is true, but you need to consider what direction you're coming from. If you take a very small number, say 1, and subtract a very large number, say 14 trillion, you get a very "large" negative number...that is, the absolute value of that subtraction is a big number, but the sign is negative. Same goes for this...take a number, subtract an infinite value from it, you'd be heading off towards negative infinity.

If there's no such thing as positive or negative infinity, how does one differentiate the limits of the graph 1/x as x approaches zero? On the left side of the y-axis it heads straight down, and on the right it heads straight up. I'd say those are two different "values" it's approaching.

An infinite number of elements each multiplied by 0, does not give you 0.

This one's convention...they're giving serious priority to the 0 multiplication rule.

Infinitesimal is 0

This is the big difference between regular arithmetic and transfinite arithmetic...there is a huge difference between an infinitesimal and zero...so I hope you just haven't taken a course in calculus yet because that's exactly how it works.

1/0=0

Check a graph of 1/x...as x approaches zero (as I mentioned above) it tends to infinities, depending on which way you come from...it doesn't converge to zero.

That's just stupid

Prove it wrong. It's the basis of every introductory calculus class you'll ever take.

Untrue. Subtracting an infinite amount of elements from one infinite set to another still gives you an infinite amount of elements.

As a practical example, take the set of integers. Take the cardinality. Subtract that cardinality from itself and what does it give you? Infinity? No.

0 is the solution [to infinitesimal / 0]

That would mean that (I'm calling the infinitesimal "e" here) e = 0*0...but as was stated above, e is not zero.

0^0 does not equal 1, it's 0.

This one's convention as well...it makes the arithmetic work. However, 0^0 is neither one nor zero algebraically - but we're not talking algebra here, we're talking analysis. And again, infinitesimal is not the same as zero.

[Rkyeun]

For purposes of this thread, infinity refers to "The number of members in the set of real numbers."
If you instead mean it to be defined as "The set of all real numbers." then you might be better off calling it the set of all real numbers, instead of infinity. Assuming you use it to refer to any set with a infinite number of members, you would naturally have to perform the operation for each member in the series. So an infinitely long set * 0 wouldn't be 0, it would be a set containing an infinite string of zeros.

Thanks for watching my back, houserichichi. It's nice to have some support.

And on a side note RQ, keep your gods out of my math. Statements in this study are either correct or incorrect. There is no blasphemy about it, except for the vulgarity you pollute the forum with. I have done you the courtesy of not using personal insults of profane language to make my point. I expect the same in return. You can argue your results logically, you can provide counterexamples to mine, or you can concede, but I will not accept "No it's not." and "That's just stupid." as steps in your proof.

Allow me now to systematically debunk each of your arguments.

You objected to Infinite^0 being 1. Any number to the power of 0 (save for 0) is 1. When you take 3^0, you are really taking something similar to 3^1 * 3^-1. This becomes 3/3. It doesn't matter if you have an infinitely large number on both sides of the equation, they have the same source, the same value, and they cancel neatly into 1. You made a reference to sets here. If you have a set and do something like this, exponentiation applies to each member of the set individually. So all the nonzero members of your infinitely long set become 1, and all the 0s become No Solution.

You objected to Finite - Infinite being -Infinite. I am curious to know what answer you get for Y at the the highest negative value of X for the equation Y=1/X. Because I get an infinitely large number on the negative side of the Y axis.

You objected to Infinite * 0 being 0. This is part of the definition of multiplication. It doesn't matter whether you've got apples or oranges or rational numbers, or infinities. When you have zero of them, you have nothing. If anything times nothing could be something, the lack of pink elephants in your pantry would quickly reproduce to fill the entire observable universe. However you again referenced sets for some reason. If you multiply a set by 0 then yes, each member of the set becomes zero individually. The number of elements in the set does not change.

0.Inf is 0 you say? I wonder why we even have the term, then?
Bring your proof. I'll line it up right beside mine.
Given: Let X = The number of members in the set of real numbers.
Assumption: 1 / X = 0
1) X / X = 0 * X | Multiplicative Equality
2) 1 = 0 * X | Multiplicative Inverse
3) 1 = 0 | Zero Multiplication Property
4) Assumption is false. | Reflexive Property.
Infinitessimal =/= 0.

You then state that 1/0 is 0. I find this laughable. Here, I shall give you an apple. Please divide it into zero peices, all of which must have equal size.
Anything over 0 produces No Solution. This is the Zero Division Property. It was defined before you were born, will remain after you die, and it does not notice your ineffectual flailing against it.

You then state that preserving the reversability of addition is stupid. Very compelling argument. I'm totally convinced. ... No, these numbers represent the points just before and after discontinuities. While they may be ridiculously close to a given finite number, they are notable because the actual finite number doesn't exist on the graph, and they are the bounds of the equation's range or domain. If infinitessimal were equal to 0, the discontinuity would not be present because the graph would be closed.

Then you decide to just skip my line about examining the source of the infinities, and how it may indeed produce another infinity. In the case you have given, assuming you have the same infinitely long set twice and you subtract it, you will end up with a set containing an infinite number of 0s. If you have an expression like 1/X and you subtract 1/X, you get 0. Except for at X=0, where you get No Solution. If you have 1/X^2, and you subtract 1/X, you still get a positive and a negative infinity as X approaches 0 from both sides.

And then you request for me to again introduce you to the Zero Division Property.
Even if 0.inf was 0, 0/0 still isn't 0. It's No Solution.
And then I give you an infinitely large apple and tell you to divide it into zero parts, all of which must be of equal size. The answer is not to have all parts be of infinite size, as you suggest. Once again, it is No Solution, because the parts of the apple refuse to vanish for no apparent reason. If a number can be infinitely large, the capacity must also exist for that number to be infinitely small, or infinitessimal, in order to preserve the Multaplicative Inverse Property.

Then you say 0^0 is 1. And no, it is not. just like 3^0 is actually 3/3, 0^0 is 0/0. The answer is No Solution.

[houserichichi]

Just to add to the conversation in case it crops up in the 0/0 discussion on the general board...the extended reals (these things) do not form a field, so you still can't perform division by zero. Thought I'd add that in...but that's why all these infinity things are "okay" to do - because we're not dealing with regular real numbers, we're dealing with the extended reals (take the reals and add points at negative and positive infinity).

[jinydu]

Actually, I think that the system you're talking about is called the "hyperreals". Here's a link http://mathforum.org/dr.math/faq/analys ... reals.html.

Hyperreals do provide a different way to work with calculus, but not the only way. The method taught in high school and university, as far as I know, is to use the epsilon-delta definition of limits. Thus, we talk about the limit as numbers approach zero or become larger than any fixed real number, but "infinitessimal" and "infinite number" are not used.

[houserichichi]

Jinydu, you're right my friend - it is the hyperreals I had to look it up in my old textbook too to make sure, but again you have me - they taught the epsilon-delta scheme, though I never really looked at it that way until my first analysis course. However, we were given a few primers, if I recall (at least in uni) on infinitesimals through the use of Leibniz' differential notation. I always preferred his way of thinking over Newton's, so I suppose that's why my own personal interpretation of the calculus was different than that which was taught. My bad.

There IS a textbook online (I forget where I got it now) called Elementary Calculus - An Infinitesimal Approach (2nd Edition) written by Keisler who actually treats introductory calculus through the use of infinitesimals as opposed to epsilon-deltas. You may want to take a look as Robinson found ways to treat infinitesimals literally in the 50s...I found the book enlightening, at least, as it was the way I had been treating calculus concepts all along. What fun.

[houserichichi]

Found it

http://www.math.wisc.edu/~keisler/calc.html

Nothing really new, but it's a nice change in pace from the typical epsilon-deltas. Enjoy.

[jinydu]

In some sense, hyperreals seem somewhat more intuitive, but there are two things that I find unsatisfying:

1) I don't see how to write down a particular infinitessimal or infinite number. The textbook makes a lot of statements about [epsilon] and H, which are arbitrary infinitessimal and infinite numbers respectively, but never gets around to giving a particular example of either.

2) It isn't "closed" in the sense that you can apply the same process used to form hyperreals to form new numbers that are not hyperreals. For instance, I can define "super-infinitessimals" that are closer to zero than any infinitessimal

[RQ]

there is no such thing as an infinite number, because it wouldn't be a number. Thus only an infinite amount of elements can account for infinity.

This is why there is not positive or negative infinity.

I guess this would mean that infinitesimal is not 1/infinity, since 1/infinity=infinity.

infinitesimal values are 0 as you can see from:

1/9=0.111...
1/9[9]=[9]0.111...
9/9=0.9999...
1=0.999...

Thus 0.000...1 does not exist or is 0.