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?
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4$\begingroup$ Note that you seem to be asking about space (3D) and not spacetime (4D). $\endgroup$– StenCommented Aug 24, 2023 at 15:47
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$\begingroup$ @Sten: That's certainly how one of the answerers addressed the question, but in writing my own answer I was not so sure about the OP's intention. $\endgroup$– Lee MosherCommented Aug 25, 2023 at 14:44
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$\begingroup$ How could any 2D diagram show anywhere as anything but 'flat' or in two dimensions? $\endgroup$– Robbie GoodwinCommented Aug 26, 2023 at 21:11
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$\begingroup$ Flat does not mean 2D. Speaking purely math, space appears flat to a hypothetical 4D observer in the same way as a 2D piece of paper appears flat to you as a 3D observer. $\endgroup$– M. WinterCommented Aug 30, 2023 at 13:02
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3 Answers
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"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.
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2$\begingroup$ How is surface of earth is 2D ? Suppose the centre of earth as origin, then we need three coordinates to specify any point of surface. $\endgroup$ Commented Aug 24, 2023 at 6:05
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8$\begingroup$ @An_Elephant you don't need three coordinates, because the radius is the same at every point on the earth's surface. $\endgroup$ Commented Aug 24, 2023 at 6:13
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14$\begingroup$ @An_Elephant you just need latitude and longitude to specify any position on earth's surface. $\endgroup$– justhalfCommented Aug 24, 2023 at 8:14
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6$\begingroup$ @An_Elephant Take the Earth's surface as given then you need just latitude and longitude. R is a given constant. It describes the curvature of the surface but latitude and longitude can be used by themselves (without R) to specify a point on the surface. $\endgroup$ Commented Aug 24, 2023 at 9:20
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7$\begingroup$ As a slight nitpick the universe being flat doesn't really imply that it is infinite in extent. It is perfectly possible to have a compact manifold which has a flat metric. The classic example would probably be the flat tourus, although an even simpler example is a circle (with the usual metric). (I don't know enough about cosmology to say whether there is any evidence about if the universe is bounded, this is just a mathematical perspective). $\endgroup$– FishbaneCommented Aug 24, 2023 at 16:10
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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.
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1$\begingroup$ Defining colloquial flatness in the first sentence, do you mean "without" rather than "with" curvature ? $\endgroup$– EvargaloCommented Aug 25, 2023 at 14:02
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2$\begingroup$ Well, it's not a question of what I mean... within those quotation marks of my first sentence, you will find an exact quote taken from the OP. $\endgroup$ Commented Aug 25, 2023 at 14:42
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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.
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2$\begingroup$ I'll add to @Dale's excellent answer. He has given an example of triangle whose angles add up to 270 degrees. It is possible to construct other triangles on the sphere whose angles add up to different values, depending on the lengths of the sides: in a flat space the sum is independent of the length. $\endgroup$ Commented Aug 25, 2023 at 3:29
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$\begingroup$ The lack of math gives this answer the widest applicability and the least redundancy. $\endgroup$– EdouardCommented Aug 29, 2023 at 20:35