What about the physics of Flatland?

[rcgldr]

Since it's abstract concept, why shouldn't gravity from a point source remain relative to 1/r^2, and from a line source to 1/r?

In the case of the single atom wide universe between two sheets, assume the sheets are kept free of any curve in the 3rd dimension.

 

[Farsight]

I don't know how you can have two-dimensional gravity. Can you even have a two dimensional mass?

I can understand it if we're talking about the bowling ball stretching a rubber sheet like Dave says. The flatlanders are part of the rubber, and are stretched along with it. So they don't see the stretch, or any distortion or curvature. They can't even see the bowling ball. But if they try to travel from one side of the rubber sheet to the other, they notice something. The rubber is more stretched closer to the bowling ball, so they go slower on that side. So they find they travel in an arc rather than a straight line. It's like something is pulling them off to one side. They can't see the distortion in the higher dimension, but they can feel it.

And do you actually need a bowling ball and gravity for this sort of thing? Can't you have something stretching the rubber sheet within its plane, giving you something akin to those tesselated Escher drawings? Or graph paper where a massive flatland object makes the surrounding squares bigger?

Edit: Or is that smaller? See this "smaller and smaller" drawing by Escher.

http://classes.yale.edu/fractals/Labs/FracPerimLab/FPerimBackground.html

 

[DaveC426913]

You seem to be asking "how can a 2D object in a 3D universe have mass?"

Instead, ask the question "how can a 2D object in a 2D universe have mass?"

Why does mass have to depend on a 3D universe? If it does, what happens if we extrapolate to a 4D universe? Does the mass of any 4D object become ... infinite?



Another way of looking at it:

The Higgs bosons in SliceLand are 2-dimensional. They imbue the 2D particles with mass in the same way 3D Higgs bosons imbue 3D matter with mass.

 

[-Job-]

Good point. It's true that 2D objects don't have a z-axis component, which we might call "thickness" or "height", but that doesn't mean they can't exist. By that same reasoning, if we were living in a 4D world, thinking about 3D, we'd ask, how can 3D objects exist if they don't have a fourth-axis component?

 

[Farsight]

Dave: I was thinking topologically, like you can't have a two-dimensional knot. I've never felt very enthusiastic about the Higgs boson.

Job: I just put a fold into a piece of paper. It's a 2D "object".

 

[WhyIsItSo]

If FLATLAND has no up or down, what does gravity matter??

Agreed. In their world, gravity is a 3rd dimension - one that doesn't exist.

Then again, I watched a show years ago about, I think, a teseract?. Anyway, it was a 4-d object, and they described it by showing what its 3-d shadow would look like. In the same way, they illustrated how an orange would appear as it passed through Flatland.

My point is that just because Flatland is 2-d, does not mean 3-d does not exist, only that it is beyond the comprehension of Flatlanders. Any up or down movement of a Flatlander would make them seem to disappear in the eyes of their fellows.

So, a "hole" in their world would be a terrible thing. Anyone venturing into it would vanish forever :(

 

[yenchin]

I found a honours thesis of my senior in 1985/86 that deals with Gravity in Flatland. I have not read through the details, but the conclusion page says that

"Newtonian theory predicts the existence of planetary orbits about a central point in the universe of Flatland whereas Einsteinian theory contradicts the very existence of such orbits (no gravity is possible in three-dimensional spacetime [i.e. Flatland with time] since curvature in such spacetime is zero). Gravity is so influential in the Newtonian case that once any planet enters the gravitational field of another body in the universe of Flatland, it is forced to go into orbit about that body and it would never be able to leave this orbit. Gravity does not exist in the Einsteinian case and so no planets would ever orbit another body in this universe".

Perhaps I should spend sometime to read through the thesis in details.

 

[DaveC426913]

I would VERY much like to read your paper. Would that be possible?

 

[yenchin]

I would VERY much like to read your paper. Would that be possible?

Well, it's not *my* paper. Since it's a thesis written by someone in the 80's, it's type-written using a ... well... typewriter. Hence no soft copy. Copyright issue in the university will prevent scanning and distributing the pages, so I am afraid I cannot do about it. The next best thing I can do is perhaps to finish reading it and then post about the main points/arguments here. It may take sometime due to my busy schedule though. I hope you don't mind. (Well, at least that gives me some pressure to go read it )

 

[DaveC426913]

Drat. Becasue that is EXACTLY what I want.

And if anyone could
- direct me to sites that deal with this stuff, or
- suggest some good physics keywords that might help me find stuff
that'd be great.

 

[Virtua]

Please let me know if I'm talking s**t at any point. Only been reading about the 4th spatial dimenion in the last week. What an amazing concept.

Often an analogy is used of a 3d object passing through a 2d plane. As it passes through it creates a 2d slice. These 2d slices can be used to recreate the 3d image, like one of those 3d puzzles. To create a 4d image we wouild need to layer 3d slices on top of each other. Although impossible to visualise, the concept seems to make sense.

However, I hit a brick wall at this point: How can you layer 2d slices on top of each other when they have no height?

There must be a straightforward answer to this small point but think I'm missing something fundamentally.

As a bonus question, why can't we see, in our 3 dimensional world, 3d shadows of 4d objects?

No misinformation / personal opinions please!

 

[DaveC426913]

However, I hit a brick wall at this point: How can you layer 2d slices on top of each other when they have no height?

You can't - at least not literally. But stacking them gives you a grid, which allows you to see the model by "sampling". Think of all those computer-graphicy images you've seen in movies where a seemingly 3D shape is formed out of a grid of lines.

 

[Virtua]

Thanks, that makes sense. So to create a 4D image you would need to stack 3D cross sections connected via the w axis. Is that right? Obviously we couldn't see it without 3D retinas but have I got the principal correct?

Nosebleed...

 

[Mk]

However, I hit a brick wall at this point: How can you layer 2d slices on top of each other when they have no height?

Calculus!