Random questions + Geometric multidimensional patterns.

[Temporarily Blah]

I was reading through various books about black holes, time warps, and multiple dimensions, and now I am simply asking for clarification on a few things, and previous discoveries on something I noticed while looking at geometric patterns via different dimensions.


Last part first. Let me quickly make a graph showing correlation between the smallest possible objects in each dimension, and predictions for a 4-dimensional object.

http://img166.imageshack.us/img166/536/graph0ij.jpg

Notice how the number of points between each dimension goes up by one, and the number of lines for each dimension(x) goes up by x=x+1?

Anyway, Is it possible to construct the image of a solid 4-dimensional object in 3-dimensional space? Is the data I've shown above correct, and how many prisms and planes would be required for the making of a 4-dimensional object? One last note; if it is possible to render a 4-dimensional object in a 3-dimensional area, can someone construct one based on this? If you render it, can you tell me how many planes and prisms are formed?


Posting random questions at about 12:30(2 hours from now), please hold.

 

[Severian596]

Anyway, Is it possible to construct the image of a solid 4-dimensional object in 3-dimensional space?

Neat graph, I like the way it attempts to break some things down.

I can answer this question of yours with some confidence. We can not construct the image of a solid 4-D object in 3-D space unless you restrict yourself by eliminating the 4th dimension from your image. And therefore you'd have to show a sequence of these 3-D objects over time to give the overall 4-D impression of the object.

Think about it this way. If you want to show a sphere to a 2-D creature, all you can do is show him/her a series of 2-D circles changing diameter (from small to big and back to small). But that doesn't mean that the 2-D creature will have any idea what the shape SHOULD be like in its native 4-D space.

Check out this page for more: http://www.geom.uiuc.edu/docs/outreach/4-cube/

 

[pervect]

You might be interested in Euler's formula. Since you are looking at higher dimensions, a logical candidate would be to look for "Euler's formula in higher dimensions".

Google finds, for instance

http://www.math.ohio-state.edu/~fiedorow/math655/HyperEuler.html
http://www.omega-art.com/math/gauss.html

You might have to look up some webpages on what Euler's formula is - it relates the count of vertices, edges, and faces of any polyhedron.

You can visualzie a hypercube as a cube inside a larger cube, with 8 lines joining together the corresponding vertices of the inner and outer cubes. If you've ever happened to watch "Andromeda", the so-called "route of ages" is the result of a 3-d visulazation of a hypercube (translated to two dimensions for the TV).

Also note that your question really has a lot more to do with math than relativity. While we talk about curved space-time in relativity, we don't generally go to the bother of constructing an embedding diagram. Instead, we use the notion of "intrinsic curvature" aka Gaussian curvature. Depending on exactly where your interests are, you may want to consider moving this to the math forum.

It's a bit technical, but see the Wikipedia article about gaussian curvature, and skip on to the formula

$$K = \lim_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}.$$

to get some idea of how curvature can be defined from the viewpoint of someone living on a curved surface, rather than someone looking at it from outside. Since we can't step outside our universe to look at it, it is much more productive to use the "intrinsic" methods of studying curvature.

 

[Temporarily Blah]

Okay, the reason I had it here is mostly for the relativistic parts which I was going to post, but ran out of time to do so.

Thank you for the answer to the construction problem, Severian. ^_^

Pervect, I'm not much for math(10th grade does algebra, how weak), so.. I have no idea what lim r->0 means. However, the sites shown are quite interesting, and I will look into those.
(Picky alert) "visualize", not "visualzie". X.X (/picky)

Anyway, now for the random questions.

I heard of this theory: Due to the gravitational effects of a black hole, time will be slowed enough for all of the future to flash before the eyes of one falling into a black hole.

How will a black hole expand, then, if nothing hits the center? If it only all hits at the end of time, won't the black hole evaporate by then?

Next random question;

Does motion REQUIRE time? Looking at it mathematically, it seems so. However, I'm not much for the correlation between mathematics and relativity, so it seems unusual. Would "Quantum Foam" be able to move between amounts of Planck Time?

If time breaks down in a singularity, how can anything around the singularity move?

If time stops at the speed of light, how does it have a finite traveling speed?

If space and time are inadvertently connected, can you move through space without moving through time? (Relates to above question about motion w/o time)

If time loops, does space loop? (Say, time travelling, assuming that the two are connected)

Planck units - how are they determined? I understand Planck time is the amount of time light takes to move Planck Length, but how did they figure out the length? How about energy, and is there a Planck Mass?

Wormholes - would humans only be able to see the ENTRANCE of them, due to their extension beyond space? Would we ride only on the surface of wormholes, or would we be able to look at the extra dimensions contained within?

What would happen if you had 2 wormholes pointing in different directions, but starting from the same point in space? Kind of hard to explain in words, but I will draw a picture later.

Unrelated - I heard 2-D beings could not live, due to the inability to have a digestive system. What if they simply absorbed their food through their being, and then deposited it after it's used for energy?

Anyway, lunch time is over, back to class. I'll try to check up after school(1.5 hours from now), but not likely.

Cya then!

 

[Severian596]

Your curiosity is encouraging. I know that I don't know the answers to half of your questions, and I also think that some of the questions don't really make sense. But that's not your fault!

I recommend starting with the fundamentals. Although someone could run through and answer your questions, you wouldn't gain understanding, you'd gain factoids: little bits of information that someone told you and that you can recite, but you don't understand in the least. Trust me, I know what that's like.

I'm an amature at the subject. I have a lot of interest like you also have. Use the internet to learn. Explore math with great enthusiasm, and I mean that! In a lot of ways, math = physics, and physics seeks to map reality. I'm giving you this advice because I feel like I waited too long. I survived on pop-science books and topics like wormholes, black holes, etc. You won't gain understanding, though, you'll just gain little factoids. I regret not starting my (limited but enthusiastic) studies earlier, but you don't have to.

=D

Learn math, stick with it. Read some books on basic Newtonian mechanics (otherwise known as the laws of force and motion). That's a great start. You can't understand the advanced topics of your questions until you understand the rules of nature and reality as we understand it today.

 

[Temporarily Blah]

Hmm, just as curiousity...

Which questions don't make sense? (I want to see what makes sense, and what doesn't, to improve upon them)

I understand I probably didn't make the wormhole question clear, but I plan to make that picture later.

As for mathematics, I know simple (aka taught in 7th grade) science equations, mostly dealing with motion, but I'm lost on where to start in relativity.

Any help? <_>

EDIT: I'm going to spread my questions around the board, so if you see a question posted here that I posted in other areas, know that I simply made it match the place it should have gone anyway.

Any math explinations I might get, i'll learn to understand them. ^_^

 

[pervect]

Pervect, I'm not much for math(10th grade does algebra, how weak), so.. I have no idea what lim r->0 means.

Sorry, that expression is the process of taking the limit (in this case the limit as r approaches zero). It's only fully covered with calculus, though you can probably get an intuitive idea even with 10th grade algebra.

However, the sites shown are quite interesting, and I will look into those.
(Picky alert) "visualize", not "visualzie". X.X (/picky)

Sorry for the typo, but I'm afraid I do that a lot (I type very fast, and don't always use the spell checker).

Anyway, now for the random questions.

I heard of this theory: Due to the gravitational effects of a black hole, time will be slowed enough for all of the future to flash before the eyes of one falling into a black hole.

There's an easy answer to that question - it doesn't happen! At least not with a non-rotating black hole.

Take a look at the sci.phsics.faq on black holes, for instance

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html

Will you see the universe end?

If an external observer sees me slow down asymptotically as I fall, it might seem reasonable that I'd see the universe speed up asymptotically-- that I'd see the universe end in a spectacular flash as I went through the horizon. This isn't the case, though. What an external observer sees depends on what light does after I emit it. What I see, however, depends on what light does before it gets to me. And there's no way that light from future events far away can get to me. Faraway events in the arbitrarily distant future never end up on my "past light-cone," the surface made of light rays that get to me at a given time.

That, at least, is the story for an uncharged, nonrotating black hole. For charged or rotating holes, the story is different. Such holes can contain, in the idealized solutions, "timelike wormholes" which serve as gateways to otherwise disconnected regions-- effectively, different universes. Instead of hitting the singularity, I can go through the wormhole. But at the entrance to the wormhole, which acts as a kind of inner event horizon, an infinite speed-up effect actually does occur. If I fall into the wormhole I see the entire history of the universe outside play itself out to the end. Even worse, as the picture speeds up the light gets blueshifted and more energetic, so that as I pass into the wormhole an "infinite blueshift" happens which fries me with hard radiation. There is apparently good reason to believe that the infinite blueshift would imperil the wormhole itself, replacing it with a singularity no less pernicious than the one I've managed to miss. In any case it would render wormhole travel an undertaking of questionable practicality.

Note that the situation with a rotating black hole is unclear. Some simple models do predict what you describe (infinite blue shift, near the so-called "inner horizion"), but there is good reason to believe that these simple models are wrong (specifically, unstable). Working out the exact details of what happens in this case is very tricky, and I don't think there are definitive answer to the questions of what should happen when someone falls into a rotating black hole, and whether or not the inner horizon of a rotating black hole actually exists or not.

Next random question;

Does motion REQUIRE time? Looking at it mathematically, it seems so. However, I'm not much for the correlation between mathematics and relativity, so it seems unusual. Would "Quantum Foam" be able to move between amounts of Planck Time?

I don't share your distrust of math. It seems to me that if you want to define motion without time, the "ball is in your court". I can't think of anything in standard physics that suggests this.

If time breaks down in a singularity, how can anything around the singularity move?

If time stops at the speed of light, how does it have a finite traveling speed?

It's very unclear what happens in a singularity, though note that *near* a singularity there isn't any particular problem.

As far as the speed of light goes, anyone who measures the speed of light has to be going slower than 'c', so they don't have any problem defining or measuring time.

If time loops, does space loop? (Say, time travelling, assuming that the two are connected)

Usually it's thought of the other way around. If space loops (wormholes), then time can loop too - unless Hawking is right, and that wormholes that attempt to become time machines self destruct due to infinite vacuum pertubations.

Planck units - how are they determined? I understand Planck time is the amount of time light takes to move Planck Length, but how did they figure out the length? How about energy, and is there a Planck Mass?

This is getting long, so I'll refer you to the Wikipedia article on Planck units. Hopefully it will answer your question.

http://en.wikipedia.org/wiki/Planck_units

Wormholes - would humans only be able to see the ENTRANCE of them, due to their extension beyond space? Would we ride only on the surface of wormholes, or would we be able to look at the extra dimensions contained within?

The universe that we can percieve is only 4 dimensional. General Reltivity does not actually predict other spatial dimensions, they are a useful visual aid. You can use curved surfaces to visualize non-Euclidean geometry, but it is a mistake to assume that they *have to* exist just because you use them as visual aids. While you can assume they exist if you like, you will find that the "embedding" of 4-space in a higher dimensional manifold is not unique. You can also assume that they do not exist, and that the geometry of space is simply non-Euclidean.

What would happen if you had 2 wormholes pointing in different directions, but starting from the same point in space? Kind of hard to explain in words, but I will draw a picture later.

Unrelated - I heard 2-D beings could not live, due to the inability to have a digestive system. What if they simply absorbed their food through their being, and then deposited it after it's used for energy?

Anyway, lunch time is over, back to class. I'll try to check up after school(1.5 hours from now), but not likely.

Cya then!

I'm not quite sure what you're getting at with these last two questions, which seem to be drifting away from relativity again, and I'm also a bit tired from writing the long response ;-). Hope you find what I did write informative and useful.

 

[Temporarily Blah]

Hmm, I DO find what you wrote informative. ^_^


I'm going to research everything posted, and see what I come up with.

About the whole "Motion without time" thing, you see that (almost?) ANY formula describing motion requires a part related to time (Speed is m/S, for example), and without it, motion would be 0. Is this true, or am I missing something?

I'm going to make a new graph with the info I got from Euler's Formula, and try to search for more patterns from there.
Oh, and sorry about the typo thing, I'm anal about spelling.
Thanks a ton! ^_^