The dividing line between the two is the critical density, which corresponds to zero curvature. (Credit: NASA/WMAP science team.)Ĭurvature. A universe with a high density can have a positive curvature, whereas a universe with a low density can have a negativeįIGURE 18.1 The curvature of space: positively curved, negatively curved, and flat two- dimensional surfaces. (Remember that mass and energy are equivalent via E = me 2.) In particular, it is useful to refer to the density of the Universe, where both mass and energy can contribute to the overall density. The curvature of space is determined by the overall amount of mass and energy in the Universe. Picturing the equivalent geometries for four-dimensional spacetime is not easy to do, but the mathematics of such spaces can be extrapolated from the more familiar examples that exist in fewer dimensions (Figure 18.1). A less familiar two-dimensional example of this type of space is the surface of a saddle. Conversely in a space with a negative curvature, lines that are initially parallel eventually diverge, and the angles inside a triangle add up to less than 180°. A familiar two-dimensional example of such a space is the surface of a sphere. In a space with positive curvature, initially parallel lines eventually converge, and the angles inside a triangle add up to more than 180°. However, in the curved four-dimensional spacetime near massive objects, these two rules may not apply! You have already read (in Chapter 14) about the way that matter curves space according to Einstein’s general theory of relativity, so it should come as no surprise that the geometry of space depends on the amount of matter within it. These two geometric examples correspond to what may be called flat space or space with zero curvature. On an everyday scale, most people are familiar with the idea that parallel lines remain parallel, no matter how far they are extended in any direction, and that the three angles inside a triangle add up to 180°.
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