Implications of a 2-Dimensional Universe

[kaikalii]

I wanted to make a thread to discuss the possible physical implications of a 2D universe as opposed to our 3D one. Note this is meant to be purely to be a thought experiment rather than a true analysis of something real. Also note that when I say 2D and 3D, I am referring to the number of spatial dimensions. Both are assumed to include time. I'll start off, but please add anything else that you don't think I've considered.

I'll mostly be discussing 2-Dimensional implications on classical physics, as it it what I know best, but feel free to discuss any field you want.

First Off, I'm pretty sure almost all of Newtonian Mechanics would be unaffected. Collisions and other interactions between macroscopic objects, to my knowledge, work just as well in 2D as 3D.

Gravity, at least in the classical sense, would still function in the same way, as would the other 3 fundamental forces I believe.

Rather than pulling itself into spheres, matter would pull itself into circles. This means planets and stars that are circular rather than spherical. Atoms and their constituents would also be circular rather than spherical. Instead of atoms having a 3D electron cloud, they would simply have a 2D one.

Interestingly, any quantities that were related to volume would have to use area instead, and some uses of area would have to just use length.
For example:
Density units: kg/m² (Instead of kg/m³)
Pressure: P=F/x (Instead of P=F/A)

One of the most prominent problems one might encounter when dealing with physics in a 2D universe would be the fact that as there is no third spatial dimension, there would be no cross products. This would mean no torque (or at least no 3D torque) and also, very sadly, no magnetism or magnetic fields

Many compounds would not be able to exist, or at least not exist as we know them. Compounds that were already flat, such as water or anything with only 2 atoms, would translate perfectly into just 2 dimensions. Other compounds, like ammonia, would have to be flattened, but would still be structurally viable I think. However, most remotely complex compounds such as glucose, propane, and perchloric acid, which pretty much need 3 dimensions to be constructed correctly, would probably not work, and I only named a few of the more simple ones.

Life, as we know it, would probably be impossible. As mentioned above, complex chemicals would not work quite the same way. Because of this, proteins, DNA, and other such molecules needed for life could probably not exist. However, I'm sure that some simplified version of life, using flat variations of different compounds could be conceived of.

I do not know enough about Quantum Mechanics to make any assumptions relating to it, so I will skip it entirely. From what I do know of Special Relativity, it would be unaffected, and I believe that GR would be as well.

That's all I can think of for now. Please tell me what you think of this and add any other ideas that you can think of.

 

[mfb]

Gravity would have a logarithmic potential instead of 1/r (as Gauß' law now integrates over a circle, not a sphere), structure formation in this universe would look really different.

Instead of atoms having a 3D electron cloud, they would simply have a 2D one.

I am not sure how the solutions to the Schroedinger equation look in 2 dimensions and with a logarithmic potential. In addition, electrons and other particles would not have to be fermions or bosons any more, they could be anyons.
Based on that, don't expect that chemistry has any similarity to our 3D-chemistry.

I think you could still have something like magnetic fields, but the formalism would have to change. Electromagnetic waves as we know them would not be possible.

 

[kaikalii]

I see. It would seem that I do not know enough of the math behind things to make some of the assumptions I have made.

 

[WannabeNewton]

Well you can derive what mfb said regarding gravity rather easily for the 2 dimensional space. Consider a localized point charge in this 2 dimensional space; Poisson's equation in the vacuum region of the point mass will just be ##\nabla^{2}\varphi = 0##. The isotropy of this system in the 2 dimensional space implies that ##\varphi = \varphi(r)##, where we are using polar coordinates. Evaluating the Laplacian in polar coordinates, we then find that ##\frac{\mathrm{d} }{\mathrm{d} r}(r\frac{\mathrm{d} \varphi}{\mathrm{d} r}) = 0## i.e. ##r\frac{\mathrm{d} \varphi}{\mathrm{d} r} = \alpha = \text{const.}## so ##\frac{\mathrm{d} \varphi}{\mathrm{d} r} = \frac{\alpha}{r}## implying ##\varphi = \alpha \ln r + \beta##. The additive constant is arbitrary and can be set to zero. The multiplicative constant can be found using Gauss's Law and comes out to ##\alpha = GM## giving us ##\varphi = GM \ln r ##. Hence the gravitational field of the point charge in the 2 dimensional space is ##g = -\nabla\varphi = -\frac{GM}{r}\hat{r}##, which differs by a power from the usual 3 dimensional space.

You can go ahead and try similar things for the electromagnetic field. You have to be careful before making such grand claims about stellar development and biological development however; making claims without developing the mathematical models will hardly prove useful.

 

[InfinitySucks]

Will I be wrong to say,
1D, 2D universes cannot exist?

The minimum dimension of existence is 3D?

 

[DEvens]

Angular momentum has some wacky in 2-D. Don't forget it's a pseudo vector. So if you have an object rotating in 2-D then the angular momentum vector has to be perpendicular to the rotation. But that's outside the space.
And so you get some wacky when you have interactions between momentum and angular momentum. I've never played carefully with it so you might be able to deal with it.

Angular momentum gets especially wacky in quantum mechanics in 2-D.

If your gravity theory is general relativity (GR) then you have some wacky in less than 3-D. You find that the GR field equations in empty space will identically give you the flat metric. So gravity goes away. If your gravity theory is not GR then you'd need to know what happens to it in less than 3-D.

There are some interesting situations when you have regular 3-D space plus time, but one dimension is somehow suppressed or rendered unimportant somehow. An example is early universe cosmology. There is some reason to think that when the universe consists of hot dense plasma then time dependence will be "washed out." So you get effectively no time coordinate. So then people work with three space and no time as an embedding in standard 1 time plus 3 space geometry. Then they turn one space dimension into a time dimension and work with 2+1 space-time. The embedding can do interesting stuff such as topological mass. And the embedding also can get you away from the identically flat. These notions then motivate work on embedding of regular 3+1 space-time in some higher dimension in order to try to find interesting new things.
Dan

 

[robphy]

Here is a classic paper by Paul Ehrenfest
http://www.dwc.knaw.nl/DL/publications/PU00012213.pdf
P. Ehrenfest, In what way does it become manifest in the fundamental laws of physics that space has three dimensions?
Proceedings Koninklijke Akademie van Wetenschappen 20: 200-209 (1918)
[more from Ehrenfest: http://www.lorentz.leidenuniv.nl/IL-publications/Ehrenfest.html ]

http://space.mit.edu/home/tegmark/dimensions.pdf
Max Tegmark 1997 Class. Quantum Grav. 14 L69
On the dimensionality of spacetime