How does mass work if reality has 2 spatial dimensions?

Question

I am not a physicist, and neither do I have a full understanding of particle physics nor the theory of relativity.

I have been thinking about how the mass would work in a 2D reality. In my understanding, a plane is an infinitely thin surface approaching 0 on the $z$-axis, and that mass is defined by the volume and the density of an object. Supposedly I had a 1kg sphere with a 1m radius that I wanted to turn into a 2D circle by changing the radius in the $xy$ plane and the height of it approaching infinitely the 0 value of the $z$-axis while preserving the mass/energy of the sphere as it cannot be created nor destroyed. If the laws of physics didn't exist, the radius would now approach infinity, as the height decreases to 0. However, if we include physics there would be a stretch in the subatomic particles who would now need to change their shape and turn into 2D as well, and the same logic for the sphere above would apply to protons and neutrons, which makes me come to the conclusion that objects that are defined by the mass/energy relation would not be able to exist in a single 2D plane as even the tiniest of objects would stretch infinitely into that reality. I do however see a possibility of mapping a 3D object into separate 2D planes, for every elementary particle but that goes again beyond my understanding of subatomic particles.

Having arrived at that conclusion, how can a 2D mass and energy work, and where is my line of thinking flawed?

I thank in advance everybody who can give me a better understanding of the 2D model.

Answer 1

You have shown that you cannot map a three-dimensional object into a two-dimensional universe while preserving its mass, energy etc. without encountering unphysical infinities (specifically, without creating an infinite size or an infinite density). But that does not mean that an object that was created as a two-dimensional object in a two-dimensional universe (and so never extended into a third dimension) cannot exist.

Your argument can be adapted to show that a four-dimensional object cannot be mapped into a three-dimensional universe while preserving its mass, energy etc. - but nevertheless we know that three-dimensional objects can exist in a three-dimensional universe.

Answer 2

One way of making a 2D physics is to just consider particles and bodies as lines or cylinders extending infinitely in the z-direction. In a sense you are still doing 3D physics, but the forces and accelerations will only be along the x-y direction. If you do the calculation you will get a gravitational force between two objects going as $F(r)=G\lambda_1 \lambda_2 /r$, where $\lambda$ is the linear density of the objects. This replaces mass in this physics.

The real, complicated answer is to try to do general relativity and quantum field theory in 2D. Famously, gravitation in a 2+1 dimensional GR world is simpler than in our 3+1 dimensional world since there are no gravitational waves and there is no Newtonian limit: each particle acts like a cosmic string solution in our world, bending light and trajectories but not exerting a force in the usual sense. As for the quantum stuff, I don't know much, but it looks like 2+1 spacetimes are a popular subject and produce nontrivial physics.

Answer 3

Our universe exists as the way it is, we can sometimes formulate problems in two dimensions, that doesn't mean there aren't three dimension it just means the system is constrained to move in a plane (right-left up-down), of course if we lived in two dimensions things would be different, but as of right now, not the thinnest piece of paper is in two dimensions even if it looks a lot like it is, and atoms can't have only two dimensions too