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I don't understand why we are able to see and measure curvature / warping of space at all.

Space as I understand it determines distances between objects, so if space were "compressed" or warped, shouldn't distances be compressed or warped the same way (like crumpling up a sheet of paper) ?

Then, however, our units of measure and frames of reference should be compressed likewise so that there should not be any visible changes to our cognition.

This would also rule out warp drives unless they form some wormhole in hyper-space (analogous to two points touching each other on the crumpled sheet of paper)...

What am I missing ?

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I don't understand why we are able to see and measure curvature / warping of space at all.

The Earth's surface is curved and this can be observed via the vast number of pictures of the Earth from space that now exist.

However, the surface curvature can also be "seen" via measurements on the surface itself.

For example, if one were start at the North Pole and travel in a "straight line" (a great circle) to the equator, then move east along the equator for a quarter of the circumference, and then move North (always along a great circle), one would eventually reach the starting point at the North Pole.

But look, one would have formed a "triangle" with the interior angles adding up to 270 degrees! This is one way that intrinsic curvature is measured.

Simply put, intrinsic curvature is mathematically characterized by the Riemann Curvature Tensor and observed via geodesic deviation.

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Seems to me that transforming a flat surface into a spheric surface as an example for space warping is misleading because the transformation actually changes the relationship of points on the surface and therefore the frames of reference, and is at best an example for extrinsic curvature. Additionally, there is no homeomorphic transformation for a rectangular surface into a sphere surface (points get merged) but space warping should be homeomorphic because space is the frame of reference for measuring distance which is an intrinsic invariant property of space as we perceive it. –  Archimedix May 12 '13 at 10:39
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Curvature affects how objects in the universe move and interact with one another, and these effects can be measured.

Take, for example, the phenomenon of gravitational lensing. Because spacetime curvature can deflect the path of light, we can potentially observe light coming from objects that are directly behind other objects. Here's a nice picture.

As another example, planets orbit around the sun and each other because of spacetime curvature, and we can certainly measure these effects. Specifically, we can measure the period of orbit of a certain planet and compare that to what general relativity tells us that the period will be given that the sun curves spacetime in a particular way.

When we make measurements of curvature, we're not literally going to some point far out in space, putting some clocks and rulers there, and "measuring" how much spacetime is warping by counting seconds and ticks on the ruler. We're making predictions about how the curvature should give rise to certain phenomena, and then we're checking those predictions.

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Well, I know about the deflection of light, however I don't get why we can see this deflection. I imagine light to be like a line on a sheet of paper, and space warping like the crumpling of that sheet which changes the shape of that line from an outside point of view but not relative to the sheet itself, assuming the sheet surface is rigid. An ant walking on the sheet would still walk the same distance if it were following the line, no matter how the sheet is crumpled (as long as there are no shortcuts through new touching points). –  Archimedix May 10 '13 at 21:44
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@Archimedix Ah, in that case it seems you have a misunderstanding about curvature in GR. What you are describing is what would be called "extrinsic curvature," which is curvature that a spacetime has as viewed from a higher-dimensional spacetime in which it is embedded. Spacetimes also have something called "intrinsic curvature," and it is this curvature that we are usually referring to. This curvature is something whose effects can be measured by someone in the spacetime regardless of whether or not spacetime is embedded in something higher dimensional. –  joshphysics May 10 '13 at 21:52
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@Archimedix To illustrate one straightforward difference, if you draw a triangle on a cylinder/curled-up-paper, the angles will add to $180$ degrees. The cylinder (intrinsically) flat. But if you draw a triangle on a sphere, they will not, and an ant crawling on its surface can tell just by measuring the angles. The sphere is curved. Imagine curvature as more warping, stretching, or contracting rather than crumpling. Although in more than $2$ dimensions, this gets more complicated. –  Stan Liou May 10 '13 at 21:58
    
I imagine light to be like a line on a sheet of paper, and space warping like the crumpling of that sheet... - The better (and more adequate) image would be an ant walking down the surface of an apple (Wheeler's image?). Apple has a recess around its stalk, and an ant would deviate towards the stalk while trying to go straight forward. Crumpling of sheet of paper (or rubber) is not what mathematicians are talking about when they discuss 'curved space'. Try to get free of the wrong image. –  firtree May 11 '13 at 6:21
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General Relativity deals with curvature of Spacetime, not just curvature of Space. You can't ignore time because clocks are affected throughout the universe. Spacetime events are what we measure and are independent of observers.

Now, let's come to point: You're asking why we're able to measure effects of Spacetime curvature with classical way when reference frames are also affected with Spacetime curvature.
Simple Answer: Any specific Spacetime curvature doesn't affect everything uniformly throughout the universe. Mars isn't affected equally as our reference frame on Earth is affected by Sun (ignoring curvature due to Earth and countless other things for simplicity).

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