Can we see the space curvature (bending) from higher dimension?

Question

Let's talk about the space without time - for simplicity, if this makes sense at all.

I want to grasp the space bending caused by massive objects with examples on 2-D space. I'm not sure if the 2-D space geometry and physics works the same way, but let's assume they are - like some kind of "projection" of a 3-D world.

Let's assume that we have a simulation of an empty 2-D surface where we can visualize the space with a grid of lines always following the same direction - and bending together with the space.

Is it eligible way to show the curvature? Is it possible to see the curvature of space if we place some massive 2-D object on our surface? Or is it relative to each object's perspective of perception and we will "see" nothing from outside?

If it still makes sense and we can "see" the curvature, then "where" is space actually curved? If we place the measuring tool (like a ruler) onto the surface it will curve with it. But we can clearly see the curvature from the "outside" (like on the screen of simulation): the grid of curved lines is different from initial straight lines - and we can measure the difference with our ruler from outside of simulation.

Does this imply that 2-D space curvature exists inside some kind of 3-D or 2.5-D space? And if not then how can we tell in the first place if space is curved or not, if there is not any kind of reference uncurved 2-D surface?

I understand to some extent that the space geometry is kind of mathematical concept that describes well the physical observations and experiments, but my questions are about visualization of this concept in some meaningful way - the simpler the better, but still almost fully relevant to our real perceived 3-D space.

Answer 1

There is no need to go to higher dimensions to measure curvature. The easiest way to measure curvature is to draw a triangle.

If the sum of the interior angles of the triangle is 180 deg then the space is flat. If it is greater than 180 deg then the space is positively curved. If it is less than 180 deg then the space is negatively curved.

This is particularly useful for a 2D space where the curvature can be defined by a single number. For higher dimensional spaces you need triangles in multiple directions.

Answer 2

Though Riemannian geometry is purely intrinsic, it is indeed possible (though not necessary) to "visualize" the curvature of space by going to higher dimensions, but I am afraid that except in the most trivial cases that won't help you with your intuition: this is asserted by the Nash embedding theorem, whose proof is very complicated, and which is so deep and counter-intuitive that nobody expected it to be true at the time. The original statement was for Riemannian manifolds, but generalizations exist for semi-Riemannian manifolds. Note that if you are only interested in local geometrical properties you could do with a local embedding around a point in Euclidean space, whose existence is easier to prove.

This would help you if you were a high-dimensional Euclidean being (the embedding dimension is quadratic in the manifold's dimension). For the few surfaces that can be isometrically embedded in $\mathbb R^3$ you are such a higher dimensional being, and I think this greatly helps intuition indeed: things like holonomy and curvature are easily understood on a 2-sphere for example and provide a paradigm for the general case.

For 3D already, and higher, you are better off thinking in terms of the intrinsic geometry. Your visualizations are through slicing, projecting, glueing, and following light rays to deduce how you would perceive things.

Answer 3

My advice is, don't get hung up on the term 'curvature'. If the word 'curvature' seems to suggest curving into another dimension, then use another word. The term 'warp' might do, as it is applied to cloth by dressmakers.

To detect such warping, one simple method is to construct a triangle, that is, a plane figure with three straight edges. The edges are straight if they are lines of least distance between their end points. There is warp if and only if the internal angles of some triangles do not add up to 180 degrees. More precisely, the angle sum exceeds 180 degrees by an amount which is proportional to the area of the triangle, for small triangles.