Why is Spacetime described as flat even though we live in 3 dimensions of space?

Question

I’ve always heard and seen diagrams that show spacetime as being “flat” or in 2 dimensions with curvature. How does this correspond to the 3 spacial dimensions that we perceive to exist in?

Answer 1

"Flat space" means that on large scales, Euclidean geometry holds. All the angles in any triangle drawn in space add up to 180°; the total distance between points separated by $\Delta x$, $\Delta y$, and $\Delta z$ is $d=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}$; et cetera.

Note that this is not the case on the 2D surface of the Earth, because it is curved. If you have a globe or a basketball to play with, you can easily see that it's possible to draw a triangle with more than 180°. You can even draw one with three 90° angles for a total of 270°. On a sphere, the distance between two nearby points is $ds = \sqrt{dr^2+r^2d\theta^2+r^2sin^2\theta d\phi^2}$ where $(\theta,\phi)$ are the longitude and latitude. In general, Euclidian geometry does not apply.

The statement that in our universe, space is flat, means that on the largest scales (disregarding curvature "wrinkles" caused by stars, galaxies, black holes, etc) our 3D universe is measured to be flat such that Euclidean geometry applies. This also implies that it could be infinite in extent. If it were positively curved like a (3D) spherical surface, it might wrap in on itself and be finite in extent. The picture is somewhat analogous a smooth metallic surface. Although under a microscope you can see the surface roughness and all kinds of asperities, hills, and valleys, zooming out to a macroscopic view the overall structure is smooth and flat.

Answer 2

The meaning of "flat" here is not the colloquial meaning, and it certainly does not mean "in 2 dimensions with curvature".

Instead, the meaning of "flat" here is the mathematical meaning: there is no restriction at all on dimension, so it could be 2 or 3 or 4 or more dimensions; but the curvature in the sense of Lorentzian geometry is restricted to be $0$. So to make formal, mathematical sense out of this, you need to know something about Lorentzian geometry, a branch of the mathematical subject of differential geometry which is specialized for application to relativity.

Answer 3

Space being flat essentially means that the laws of Euclidean geometry hold. The interior angles of a triangle sum to 180 degrees. Parallel geodesics (straight lines) never intersect, etc.

Consider, in contrast, the 2D surface of the earth, which is curved. You can start on the equator facing north, walk due north to the pole, turn 90 degrees, walk due south to the equator, turn 90 degrees, walk due west to your starting position, and turn 90 degrees to face north again. So that triangle in a curved space has 270 degrees, not just 180. And two nearby lines going north from the equator start out parallel but wind up intersecting at the pole. Such non-Euclidean things are not possible in flat space.

These same concepts can be generalize to spacetime. And in a theory of gravity where the equivalence principle holds you can represent tidal effects as curved spacetime.