why is space represented as two dimension fabric instead of three in relativity?

Question

what happened to the z-axis?

Answer 1

You are talking about general relativity.

It is impossible to demonstrate spacetime density (four dimentions) purely spatially. It is also increadibly difficult to demonstrate it in three spatial dimentions. A good demonstration is in two dimentions with the fabric. This is not the general relativity representation, just a demonstration so that it can be percieved.

Once you understand how gravity works in two dimentions, add your z-axis in your head. Then add your 'ct'-axis.

Then you'll see why they have to demonstrate it to you in 2D.

PS: I do not like this 2D fabric representation because it misleads younger students. For a start, the fabric has to curve into the third dimention, whilst spacetime is curved but in a non-euclidean sence. Secondly, this model requires gravity for its demonstration, and so doesn't demonstrate anything at all. A better description would be the diffusion of heat from a cold area to a hot area. If you imagine that a hot gas would naturally 'fall' towards a colder gas to try to achieve thermodynamic equillibrium. Now, in a gravitational sence replace the 'heat density' with a 'spacetime density'. This spacetime density causes the diffusion of massy objects through space to each other to obtain gravitational equilibrium. This can be percieved in 3D fine, but confuses the students because spacetime density is totally ficticious. Rather, quantum field theory describes that graviton emission density is the cause of gravitational attraction. This, however, would confuse students even more if they didn't understand Yukawa's theory.

Answer 2

It is not easy to represent a three dimensional object on a flat surface.
the drawings you see are a flat sheet bowed down in the center.