Essential Relativity pp 105-125 | Cite as

# Basic Ideas of General Relativity

## Abstract

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.

### Keywords

Manifold Radar Sine Dition Dock## Preview

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