extrinsic : the standard way known for centuries. for example, we can tell the earth is round by looking, say, at how ships leave the horizon with their masts being the last thing to see. A 3 dimensional-object (ship) (or an object traveling and having shape in 3 dimensions) telling us how the two dimensional surface of the earth is curved by traveling on it. It's extrinsic since we're trying to tell the curvature of a 2-dimensional object by looking at the behaviour of how a 3-dimensional object "travels" on it.
intrinsic : very new concept formulated by Riemann and others. We are all trapped in these dimensions. We can't look "outside" to see how, say, our 3-dimensional universe curves into higher dimensions. So how do we tell the universe we live in is curved? Well there are lots of ways. For example, the angles of a triangle won't be exactly 180 degrees if our universe is not "flat". And so on.
You don't need to travel "outside" of something to see if it's curved or not. You just need to study manifold theory:)
In fact, this was one of the big stumbling blocks that Einstein had to face when making the General Theory. How the hell do you talk about curvature without leaving the "surface"? Well luckly Einstein was born after Riemann and was clever enough to understand the quite hard mathematics (amazingly hard if you think about how abstract it was; how revolutionary it was [Euclid is wrong?!?!?]; and how the idea of using such abstract mathematics practically. We take this way of working for granted. Einstein was one of the first physicists to embrace the new maths).
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there are two ways of defining curvature.
:)
extrinsic : the standard way known for centuries. for example, we can tell the earth is round by looking, say, at how ships leave the horizon with their masts being the last thing to see. A 3 dimensional-object (ship) (or an object traveling and having shape in 3 dimensions) telling us how the two dimensional surface of the earth is curved by traveling on it. It's extrinsic since we're trying to tell the curvature of a 2-dimensional object by looking at the behaviour of how a 3-dimensional object "travels" on it.
intrinsic : very new concept formulated by Riemann and others. We are all trapped in these dimensions. We can't look "outside" to see how, say, our 3-dimensional universe curves into higher dimensions. So how do we tell the universe we live in is curved? Well there are lots of ways. For example, the angles of a triangle won't be exactly 180 degrees if our universe is not "flat". And so on.
You don't need to travel "outside" of something to see if it's curved or not. You just need to study manifold theory
In fact, this was one of the big stumbling blocks that Einstein had to face when making the General Theory. How the hell do you talk about curvature without leaving the "surface"? Well luckly Einstein was born after Riemann and was clever enough to understand the quite hard mathematics (amazingly hard if you think about how abstract it was; how revolutionary it was [Euclid is wrong?!?!?]; and how the idea of using such abstract mathematics practically. We take this way of working for granted. Einstein was one of the first physicists to embrace the new maths).
Ciao