What about the physics of Flatland?

[DaveC426913]

Flatland deals with macroscopic issues. What about the physics of Flatland?

What is the strength of 2D gravity? Does it still fall off as the square of the distance, or would it be linear? And mass increases as the square of radius, not the diameter, right?

 

[Cyrus]

It's flatland! gravity is inconsequential! DUH!

 

[scott1]

It's flatland! gravity is inconsequential! DUH!

I don't think that would be a good answer.

I'm assuming that flatland is based off of Euclidean geometry which describes geometry of a world without matter. And according to Einstein you need matter to curve spacetime so a "flatlander" would not be effected by gravity because there is no way to for a flatlander to feel gravity.

But for some reason I still don't think this answers your question...

 

[Cyrus]

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

Have you read the book Flatland?

 

[Arian]

I disagree, flatland might be populated by flatoms. Flat-atoms.
Well, then we have the idea that these flatoms might by a strange flat force or flattey, flat-gravity, be drawn together into stars or planets. These planets would have a very large effect and if were drawn on a piece of paper, would crumple the paper downward. Things trying to escape the flearth, flat-earth, would need enough force to ovesome the crumple in the paper.

The streanght of gravity is :

2-D:2/1
3-D: 1
4-D: 1/2
5-D: 1/4
6-D: 1/8

Two times bigger in flatland.

Note: I wrote like a entire 10 page desgin of the physics of a flatland (not populated by squares of course but possible beings, like flumans. hehe

 

[scott1]

Have you read the book Flatland?

No but I have seen what 2-D looks like.(I'm currently reading it online right now).

Reread my post I ment that if a world had no depth(not up or down) there could be no gravity.

 

[scott1]

Note: I wrote like a entire 10 page desgin of the physics of a flatland (not populated by squares of course but possible beings, like flumans. hehe

Can you post it here on PF it might be useful in answearing the orginal question.

 

[Cyrus]

No but I have seen what 2-D looks like.(I'm currently reading it online right now).

Reread my post I ment that if a world had no depth(not up or down) there could be no gravity.

Up or down is exactly what makes something have depth! Flatlanders only know front, back, and side to side.

Yes, there can be gravity. But it matters not (because the flatlanderes won't have one clue about detecting it)!

 

[-Job-]

Why is depth needed in order to have gravity? And why doesn't a 2D world have depth? Is there a favored depth axis?

 

[Cyrus]

Omg... <smacks myself in the face>

 

[-Job-]

What's so funny? Are you laughing at yourself? A relativistic equivalent for gravity in a 2D universe would be that of a plane bent at a point, such as a flat plastic bag with a lead circle inside the bag. The behavior is similar. Anything inside this bag that approaches the lead circle will be accelerated towards the lead circle. So people living inside the bag will detect the influence of a force when approaching the circle, or any other body for that matter, which would be their gravity. The bag is 2D, think two flat plastic sheets close together. Bodies in the universe will be 2D objects in between the plastic sheets. They can move in 2 axis. Their depth is measured as the shortest distance to an object, just as with us in 3D.
I think what you should be laughing at is your inability to think conceptually, though it's no laughing matter. Anyway, to respond to the OP, i'd say it would still decrease quadratically with distance, rather than linearly.

 

[Cyrus]

Maybe I am missing something here...but 2D cannot have any depth to it since there is NO z coordinate.

Explain to me, how does something 'fall' when there is no z direction? I'd like to hear it...

 

[daveb]

I would think it's similar to the ball ont the rubber sheet analogy we use to picture what a 4D sphere looks like in 3D, but Flatlanders would have to conceptualize the effects of gravity as a 2D disk on a line rather than a 3D ball on a sheet. Physically, the 2D gravity would look exactly like the ball on sheet (at least to us). Just like creatures in a 5D universe would use the analogy of a 4D hypersphere on a 3d cube to demonstrate what ours looks like.

 

[Hootenanny]

Maybe I am missing something here...but 2D cannot have any depth to it since there is NO z coordinate.

Explain to me, how does something 'fall' when there is no z direction? I'd like to hear it...

How about two massive particle placed in this 2D plane. According to the laws of gravitation, would these two particle be attracted to each other?

 

[Cyrus]

Yes, but the OP said how would something 'fall' and I'm saying that entire premise is flawed!

 

[Hootenanny]

Yes, but the OP said how would something 'fall' and I'm saying that entire premise is flawed!

Where was 'falling mentioned'? I can't seem to find it

 

[Cyrus]

Ah you are right, my fault Hoot. However, this is exercise would amount to making up physical laws because we have no observation of a 2D world. It would be pure speculation, do you not agree?

 

[-Job-]

There is no preferred "up/down" axis, and no preferred depth axis. There's no "falling" in 3D either, just an acceleration due to gravity which we identify as downwards, but I'm sure you can position yourself such that it is sideways.
In 3D the gravitational interaction of two bodies can result in acceleration in any direction. The same thing would apply in 2D. The 2D disk inside the plastic sheets, under the influence of gravity, could accelerate in any direction describable by a 2D vector.
In 2D gravity causes bodies to come together, just as in 3D.

 

[Cyrus]

Would there be a gravitational attraction? I don't know... possibly. This is a 3D world, I don't live in a 2D world.

I do know that if there was no gravity, it would make no difference because they would not float away. If there was infinite gravity, they would not feel it because there would be nothing of them to be 'crushed' either. (They have no depth)

 

[Hootenanny]

There is no preferred "up/down" axis, and no preferred depth axis. There's no "falling" in 3D either, just an acceleration due to gravity which we identify as downwards, but I'm sure you can position yourself such that it is sideways.
In 3D the gravitational interaction of two bodies can result in acceleration in any direction. The same thing would apply in 2D. The 2D disk inside the plastic sheets, under the influence of gravity, would accelerate in any direction describable by a 2D vector.

While I totally agree with you there, it is possible to 'fall' towards a massive body to the left, right, forwards or backwards; normally one thinks of falling 'down', since flatlander's have only two axes they will have no concept of falling 'down' because they are only aware of left, right, forwards and backwards. Imagine yourself water skiing and the boat you are holding to begins to accelerate forwards from your veiwpoint, would you describe yourself as falling?

 

[Hootenanny]

However, this is exercise would amount to making up physical laws because we have no observation of a 2D world. It would be pure speculation, do you not agree?

I think that we could apply some of our '3D' laws here, but restrict them to two dimensions, that said it is still purely speculation since we have no 2D world to observe.

 

[Cyrus]

Even if you move side to side, mass by our definiton, takes up physical 3-d space. Now how do you even define mass when it does not have a 'thickness' to it? See what I mean when I say we are stretching things out to make them work how we 'think' they should based on our 3d assumptions.

Its like me asking you to give me the laws of a 4-d universe. How could you?

 

[Hootenanny]

Even if you move side to side, mass by our definiton, takes up physical 3-d space. Now how do you even define mass when it does not have a 'thickness' to it? See what I mean when I say we are stretching things out to make them work how we 'think' they should based on our 3d assumptions.

Its like me asking you to give me the laws of a 4-d universe. How could you?

I agree with you. Like I said in my previous post, this is only pure speculation, we would be applying our 3D laws to a universe which is not 3D, but 2D and may be governed by a totally different set of laws. I also agree that we are modifying our concepts to fit this 'restricted' universe. A place where masses could really be particles with no volume and 'thin discs' really exhist.

 

[Mk]

But what if gravity didn't come from Flatland, but from above or below flatland?

 

[DaveC426913]

Guys, thank you. You are nicely proving my point.

If there are no forces operating in any fictional 3rd dimension in Flatland, then that is not a direction I am asking about.

But there are still a whole 2 other dimensions in which forces (such as gravity) can operate.

Do I need to spell it out?

 

[Hootenanny]

But what if gravity didn't come from Flatland, but from above or below flatland?

How is this possible in a 2D universe?

 

[-Job-]

How is this possible in a 2D universe?

I think what he means is that, in the 2D model of two flat sheets close together, gravity is caused by the bending of the sheets by the lead disk. But, the disk bends the sheets (space) because it is under the influence of gravity coming from our 3D world.
This a problem in 3D as well. The gravity model of a sphere (planet) bending space and causing gravity doesn't explain why bodies would bend space, as if weighing in on the space. The idea of bodies bending space, which we use to model gravity, sounds a lot like gravity.
So it could be that gravity is coming from other dimensions.
The alternative would be to say, the lead disk bends the sheets because the sheets are moving across space at some speed, and bodies in between sheets, offering more resistance to this movement would hang back, bending the sheets, and producing gravity.

 

[DaveC426913]

Hm. I find it comforting that, judging by how much people are struggling, this is at least not a completely mundane concept.

Call it Sliceland ( (C) 2006 Dave Collins - you heard it here first).

Flatland is generally viewed as horizontal, upon which Flatlanders move (I do not know how) whereas Sliceland is vertical. (This is an entirely semantic distinction as horizontal and vertical are completely arbitrary, but this might help you "grok" it).

In Sliceland, atoms are 2D discs. Electron orbitals are 2D. All matter in Sliceland is attracted to all other matter by gravity, and will tend to clump together to form very large 2D discs called planets. Planets follow elliptical paths around 2D stars.

I just want to understand how gravity and other fundamental forces could work (as a thought experiment).



It occurs to me that, for the purpose of figuring this stuff out, it doesn't even have to be hypothetical.

With unlimited technology, we create the following conditions in deep, deep space:

We create two "force field" planes of effectively unlimited length and breadth and place them in proximity with only a *one atom gap* between them. These force field planes have no effect on matter except to constrain them to a one-atom-wide space. Matter is otherwise free to move through the other two directions as normal.

The matter will clump, but it will be acted upon by gravity in only two directions. Will the force still be calculated as the square root of the distance?

 

[Parlyne]

It is actually quite possible to meaningfully talk about what physics should be like in other numbers of dimensions. All we need is some principle to start from. As it turns out, this is easier to do for electromagnetism than gravity. In that case there is a well known symmetry principle that determines the form of the field equations, which can be examined in any number of dimensions.

In the case of gravity, if we don't want to go so far as solving the Einstein field equations of GR in arbitrary dimensions, the only thing to really guide us is that we know that there is an analogy between the Law of Universal Gravitation and Coulomb's Law. So, if we expect that this correspondance is not an accident of our living in three spatial dimensions, we can use it to tell us what 2-d gravity should look like.

Without going into undue detail, the symmetry principle I mentioned above tells us that the form of Maxwell's equations does not change with the number of dimensions (at least not as long as we write them in a notation that doesn't depend on the number of dimensions). In particular, this means that Gauss' Law will always be:

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

To find the form of Coulomb's Law, we need to solve this in 2 dimensions for a point charge. The first step is the integrate both sides over some 2-d "volume." In general, in d dimesions, we write:

$$\int_V \nabla \cdot \vec{E} d^dx = \int_V \frac{\rho}{\epsilon_0} d^dx$$

As in three dimensions, we can use the divergence theorem on the left hand side and the recognition that the right hand side will always integrate to the total charge enclosed in the volume to get:

$$\oint_S \vec{E} \cdot d\vec{a} = \frac{Q_{enc}}{\epsilon_0}$$

Here, $$d\vec{a}$$ is taken to mean an element of the infinitessimal d-1 dimensional area on the surface, S, of the volume V. Finally, we can solve for $$\vec{E}$$ by assuming that we have a point charge and taking the surface, S, to be a d-1 sphere of radius r, centered on the charge (in perfect analogy to the way this is done in 3-d). This gives an electric field:

$$\vec{E} = \frac{\Gamma \left (\frac{d}{2} \right )}{2 \pi^{d/2} \epsilon_0} \frac{q}{r^{d-1}} \hat{r}$$

In two dimensions, this is is simply:

$$\vec{E} = \frac{1}{2 \pi \epsilon_0} \frac{q}{r} \hat{r}$$

So, by the analogy above, we should expect that d-2 gravitational forces will also go as 1/r.

 

[Mk]

I think what he means is that, in the 2D model of two flat sheets close together, gravity is caused by the bending of the sheets by the lead disk. But, the disk bends the sheets (space) because it is under the influence of gravity coming from our 3D world.
This a problem in 3D as well. The gravity model of a sphere (planet) bending space and causing gravity doesn't explain why bodies would bend space, as if weighing in on the space. The idea of bodies bending space, which we use to model gravity, sounds a lot like gravity.
So it could be that gravity is coming from other dimensions.
The alternative would be to say, the lead disk bends the sheets because the sheets are moving across space at some speed, and bodies in between sheets, offering more resistance to this movement would hang back, bending the sheets, and producing gravity.

That is exactly what I meant by that sentance.

 

[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".