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Comment by mci | Hacker News Reader

Good luck living in a world without stable orbits and without a cross product. Stable orbits only exist in 3 dimensions [1]. Cross product only exists in 3 and 7 dimensions [2] thanks to quaternions and octonions.

[1]: https://en.wikipedia.org/wiki/Bertrand%27s_theorem

[2]: https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...

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That's addressed in section 2, "Relativistic Gravity in Three Dimensions":

> As is well known, general relativity in 2 + 1 dimensions does not have any local degrees of freedom; as a result of this, the spacetime outside of, e.g., a star is locally flat, and the presence of the object is only discernible globally, though the presence of a deficit angle. Clearly solar systems could not exist in such a world, and so this is commonly used as an argument against the possibility of life in two spatial dimensions. ...

> ... it seems not unreasonable to also modify the theory of gravity so as to include local degrees of freedom. The simplest way to do this is to include a gravitational scalar field, and for completeness and concreteness I will give an example of such a theory.

> ... In the previous subsections I have presented a purely scalar theory of gravity which may allow lifein 2 + 1 dimensions; this is not intended as a complete theory, but more as a proof-of-principle,and now I will briefly discuss a few other alternatives.

Good luck assuming that life needs a planet and an orbit. All you need is a way for particles to make explicit decisions and experience some sort of consciousness (but even that is debatable, since few life forms on earth seem to have that).

And I would even question that you need particles. Particles might just be what we can experience in 3 dimensions.

I always find it funny how humans are so eager to draw generalized conclusions based on their own meager 100 years of scientific existence. We should be more humble and realize that we know jack shit about anything.

I had a similar reaction to you. People seem to have very limited imaginations when thinking about the types of things that could exist. Or maybe the title is incomplete. Maybe the paper is saying something more like, "Could life exist in a plane that otherwise has the same or very similar physics and chemistry as the universe?" In which case I could understand the skepticism about life's existence.

OP was referring to an orbit in the dynamical systems sense: https://en.wikipedia.org/wiki/Orbit_(dynamics) which is far more general. It simply refers to a dynamical system where the trajectory of a particle repeats. If you have a system of particles in 2D space, governed by central forces between the particles, you won't have any particles that behave periodically in general, regardless of what those forces actually are. (There's actually some caveats to that statement. I believe the theorem OP's quoting assumes that the system has rotational symmetry, but giving up that symmetry may or may not actually give you stable orbits). Having some underlying periodicity is pretty important for more complex structures to emerge, otherwise it's just random chaotic motion.

I mean, there are obviously a huge number of caveats associated with that, to the point that it's meaningless when we ate discussing "possible universes". There is no rule that a n^-2 force cannot exist in a 2d universe... which would obviously result in stable orbits.

I mean come on, we have forces in our universe which are not n^-2. All this proves is that gravity couldn't work in the same way in a 2d universe..... so?? Heck, what about discrete universes? What about nonlocal universes?

Stable orbits are not only about planets and stars. They are about the stability of a universe with any central forces. Take a wrong number of dimensions and your life form will collapse or explode (along with its environment).

That assumes your universe is essentially Euclidean, at least at a scale where these effects matter.

Not all spaces are Euclidean - or even approximately Euclidean via a mapping and/or at specific scales.

I had a couple thoughts on how you might get around Bertrand's Theorem, but I don't know if they would actually work.

1. Give up on central forces. Maybe instead of the particle being attracted to the position of another particle, it's to where that particle will be in 10 seconds if it maintains its current velocity, or to a point halfway between the two particles. No idea if something like that could result in a stable orbit, or what conditions would be required for that to happen.

2. Give up on rotational symmetry (which I believe Betrand's theorem implicitly requires, not positive). Have some defined "north" direction, and forces behave differently depending on how a particle is moving relative to it.

3. Maybe asymptotically stable orbits are enough for life, if they are very asymptotically stable.

Stable orbits, I get it. Why is cross product such a fundamental requirement for life?

How would electromagnetism work in a world without a cross product?

In 2d electromagnetism splits light waves into TE and TM modes that don't couple to each others (until you add a dielectric). That is unusual to our 3d picture, but doesn't break electromagnetics.

And the generalization of the cross product, the wedge product works just fine in 2d. If you take the wedge of two 1-forms (vectors) in 2d you get a 2-form. In 3d that is equivalent to a (3-2)-form or in other words a (pseudo-)vector. In 2d you get a (2-2)-form or in other words a pseudo-scalar 0-form. That is of course nothing but the z component of the resulting pseudo-vector if you had done it in 3d. So EM in 2d might get flat landers started on weird group theory a bit early, but it is not fundamentally incompatible with life.

It's a bit complex for a comment, but electromagnetism defined in terms of differential forms generalizes nicely to all dimensions. There certainly is a 2+1D electromagnetism.

It would exist but not in a nice way, with electricity and magnetism inducing each other.

You could use the exterior product.

Maxwell did not use a cross product in his original publications. In fact, the development of the cross product was a response to the longer (quaternion based) formulation of electromagnetism equations at the time.

I'm confused by GP's reasoning. Is it valid to assume that since we have an equation that requires three dimensions to show how a force reacts, that if you remove a dimension that force doesn't exist?

Isn't it just as possible that there's a different equation for how that force would work in two dimensions (or to assume our current equation is a specialized version of a general equation that works in all dimensions)?

It just seems rather odd to assume that since our understanding doesn't extend to a circumstance that means something is impossible there. Or is there some aspect of this I'm missing?

Edit: Perhaps I was misinterpreting the GP contextually. Maybe they were just asking if we have an equation for it, rather than questioning how it could exist.

If we’re imagining a 2d world why not imagine all new forces and particles to go with it?

Magnetism doesn't use a cross product; it uses a wedge product[0]. It generalizes perfectly well to 2d, you just have <<xy>> instead of <<xy,xz,yz>>, just like you have <x,y> instead of <x,y,z>.

Edit: the reason 2d doesn't have a cross product is because you can no longer misinterpret a bivector <<xy,xz,yz>> as a (1-)vector <yz,xz,xy> by confusing each basis bivector with the basis vector[1] orthogonal to it.

0: https://en.wikipedia.org/wiki/Wedge_product

1: Actually one of the two (positive or negative) possible basis vectors; this is why cross products have a right-hand-rule versus left-hand-rule ambiguity.

This one doesn't use cross products, but instead creates different particles that react differently to each other:

https://youtu.be/12lyvmLdecc

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