# How does mass work if reality has 2 spatial dimensions?

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.

• Do you mean 2+1D or 1+1D? Commented Apr 13, 2022 at 12:14

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.