I know it is impossible to visualise a 3 space and 1 time dimensions. But what about 2 space and 1 time dimensions? Will it be like similar to 3 space dimensions or different? If different will it be something like a 3 dimensions except that the past are remained? Something like if a circle move along the plane, adding the time dimension will make it look like a slanted tube?
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It's not clear what you're asking - of course you can draw a 3D spacetime as a 3D geometric object, with the understanding that the Euclidean distance in it will not correspond to the Lorentzian distance. What exactly is the question about that? – ACuriousMind Apr 21 '16 at 14:17
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Spacetimes with two spatial and one time dimension are known as 2+1D spacetimes, and there are lots of questions on this site about them. They are popular in quantum gravity studies because quantum gravity is a lot simpler in 2+1D.
However the physics of 2+1D spacetimes is very different to our 3+1D spacetime. For example stable planetary orbits are only possible with three spatial dimensions. For some additional reading you might be interested in Is 3+1 spacetime as privileged as is claimed?.
However we can often use symmetry to make spacetime easier to visualise. For example where the system is spherically symmetric we may be able to use 2D diagrams with just time and one spatial dimension. This is how spacetime diagrams are often drawn.
John Rennie
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Depends on your view on "spacetime". In the classical view on space and time, time is completely independent from space, expressed e.g. in the Galilean transformations which transform space and time independently.
https://en.wikipedia.org/wiki/Galilean_transformation
Thus you can just visualize it as $\mathbb{R}^3$, which is not hard to sketch. In the modern physical approach however, spacetime has minkowskian structure and space and time are linked, which can be easily seen in the corresponding transformation laws of special relativity, the Lorentz transformations.
https://en.wikipedia.org/wiki/Lorentz_transformation
There the structure would be $\mathbb{R}^{1,2}$, which is a bit harder to visualize. If you are not familiar with it, I recommend you to read a bit about minkowski diagrams, this may answer your question.
Moe
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