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8/21/04 |
Curved Spaces |
It is very difficult to visualize curved four-dimensional spacetime. It is even difficult to visualize curved three-dimensional space, because one must be outside the space to actually see the curvature. Fortunately, visualization is not essential to the study of curved space. In fact, the assumption of a higher dimensionality is not even required.
Only those spaces that exhibit extrinsic curvature require one to get outside of their space in order to measure their curvature. Spaces with intrinsic curvature can be measured without ever leaving the space itself. Because physics concerns itself only with what can be observed from within our own universe, general relativity concerns itself only with spaces having intrinsic curvature.
Although visualization of space curvature is not essential for its study, visualization does aid its understanding. Therefore, we will use two-dimensional examples to illustrate the various properties of curved spaces. For example, a spherical shell illustrates a space with positive curvature, a flat plane illustrates zero curvature, and a saddle-shaped surface illustrates negative curvature. From these examples we can see how parallel lines will converge in a positively curved space, remain equidistant in a flat space, and diverge in a negatively curved space.
If these lines represent worldlines in spacetime, then we can see that positive curvature will cause objects to come together, zero curvature will cause them to stay the same distance apart, and negative curvature will cause them to drift apart. Since mutual gravity pulls masses together, empty space leaves them unaffected, and tidal forces pulls things apart, then we conclude that the phenomenon of gravity exhibits all three types of curvature.
Questions:
Q1. Which are true? (A) Einstein
used the correspondence principle to guide him in the development of general
relativity. (B) The principle of equivalence asserts that it is impossible to
distinguish experimentally between a spaceship at rest in a gravitational field
and one floating freely in free space (C) The principle of equivalence requires
that the curved spacetime of general relativity must be locally Minkowski. (D)
Two of these. (E) Three of these.
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