Neither of these examples are possible for a flat space, so even a 2D being confined to the sphere could determine that the space was not flat, without either needing or obtaining any evidence for or against a higher dimensional flat space. But as you follow each line due north the distance decreases, the angle changes, and the lines eventually intersect. Similarly, at the equator two nearby lines pointing due north are parallel. In a flat space the sum of the interior angles of a triangle is $180^$ interior angles. The way that you determine curvature of a sphere using only measurements in the 2D surface of the sphere is by finding things that violate the rules of normal flat Euclidean geometry. I learnt that a sphere has intrinsic curvature, that is a 2d creature on a 2d sphere can still find out that a sphere is curved. As a result, the circumference of the orbit of the satellite is not equal to $2\pi R$ as it would be in a flat geometry. Since the laws of physics on satellites are the same as the laws on Earth, the speed of light is the same, and consequently there must be an apparent difference in the length of the metre, when viewed from Earth. The easiest way to see that this is true is to recognise the daily fact that clocks on GPS satellites do not keep time with identical clocks on Earth. This means that maps of large regions cannot be drawn without distortion of the map. Spacetime (and space) has intrinsic curvature, but no extrinsic curvature because there is no exterior space to look at it from. A sphere has both intrinsic and extrinsic curvature, but a cylinder can be made by rolling a flat piece of paper, without distortion of geometrical shapes like triangles it is extrinsically curved and intrinsically flat. The two definitions of curvature are distinct. For example the angles of a triangle may not add to $180^\circ$. Intrinsic curvature refers to the geometrical theorems which can be proven within the space, without reference to anything outside. That's an assumption that cannot be tested.Įxtrinsic curvature refers to embedding a space in a higher number of dimensions. Unless you can observe the embedding space, then no, you cannot deduce that you exist embedded in a higher space. You have no choice but to measure things intrinsically. If you don't exist in the embedding space, then you can't use the tools of extrinsic curvature to take measurements. Just how you do the math is a bit different. Intrinsic and extrinsic curvature are connected in that they both make the same predictions. So why add something to the theory that cannot be observed? We don't assume an embedding space because we don't need to to get the right answers. Angles and distances measured are exactly what they would be if the space was curved. We call it "curvature" because it works exactly like curvature. Specifically and by definition what it means for a space to be intrinsically curved - like all these answers say - is that when you take geometric measurements they don't come out the way Euclidean geometry predicts. Just that in our space, we measure dot products of basis vectors to have some non-zero value. You don't have to envision space bending into some other space. You can consider this the definition of a curved space. If they're what you'd get in curved space, well, you're in a curved space. If those values are what you'd get with flat space, you're in a flat space. So how do you discover intrinsic geometry empirically? You measure angles, you measure dot products and you see what the values are. That is, they're in a sense empirically discovered. They're just what we see in our everyday experience. Any of these geometric elements that are postulates in Euclidean geometry aren't inherent truths about the Universe. How I think about it is this: there is really no reason that it must be true that, for example, a triangle has interior angles summing to $180^o$ or that the dot product of basis vectors is zero. Both will give you the same measurements.īut in our universe, there is not higher-dimensional embedding space we can refer to. We currently have no evidence that suggests our 4-dimensional universe is embedded in some higher dimensional space.įor a sphere embedded in a 3-dimensional space, you can elect to use intrinsic or extrinsic geometry.
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