Basic Ideas of General Relativity
One of the most revolutionary features of general relativity is the essential use it makes of curved space (actually, of curved spacetime). Though everyone knows intuitively what a curved surface is, or rather, what it looks like, people are often puzzled how this idea can be generalized to three or even higher dimensions. This is mainly because they cannot visualize a four-space in which the three-space can look bent. So let us first of all try to understand what the curvature of a surface means intrinsically, i.e., without reference to the embedding space. Intrinsic properties of a surface are those that depend only on the measure relations in the surface; they are those that could be determined by an intelligent race of two-dimensional beings, entirely confined to the surface in their mobility and in their capacity to see and to measure. Intrinsically, for example, a flat sheet of paper and one bent almost into a cylinder or almost into a cone, are equivalent (see Figure 7.1). (If we closed up the cylinder or the cone, these surfaces would still be “locally” equivalent but not “globally.”) In the same way a helicoid (spiral staircase) is equivalent to an almost closed catenoid (a surface generated by rotating the shape of a freely hanging chain) ; and so on. One can visualize intrinsic properties as those that are preserved when a surface is bent without stretching or tearing.
KeywordsManifold Radar Sine Dition Dock
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