## The Large Scale Structure of Space-time － The role of gravity の図に関連する部分を訳す（１）

The Large Scale Structure of Space-timeの"1.The role of gravity"を眺めているんですが、FIGURE 1 に関する部分の文章を備忘録として訳しておきます。

One can express the dragging back of light by a massive body more precisely using Penrose's idea of a closed trapped surface. Consider a sphere $$\mathcal{J}$$ surrounding the body. At some instant let $$\mathcal{J}$$ emit a flash of light. At some later time $$t$$, the ingoing and outgoing wave fronts from $$\mathcal{J}$$ will form spheres $$\mathcal{J}_{1}$$ and $$\mathcal{J}_{2}$$ respectively. In a normal situation, the area of $$\mathcal{J}_{1}$$. will be less than that of $$\mathcal{J}$$ (because it represents ingoing light) and the area of $$\mathcal{J}_{2}$$ will be greater than that of $$\mathcal{J}$$ (because it represents outgoing light; see figure 1). However if a sufficiently large amount of matter is enclosed within $$\mathcal{J}$$ , the areas of $$\mathcal{J}_{1}$$ and $$\mathcal{J}_{2}$$ will both be less than that of $$\mathcal{J}$$ . The surface $$\mathcal{J}$$ is then said to be a closed trapped surface. As $$t$$ increases, the area of $$\mathcal{J}_{2}$$ will get smaller and smaller provided that gravity remains attractive, i.e. provided that the energy density ofthe matter does not become negative.Since the matter inside $$\mathcal{J}$$ cannot travel faster than light, it will be trapped within a region whose boundary decreases to zero within a finite time. This suggests that something goes badly wrong. We shall in fact show that in such a situation a space-time singularity must occur, if certain reasonable conditions hold.

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FIGURE 1.At some instant, the sphere $$\mathcal{J}$$ emits a flash of light. At a later time, the light from a point $$p$$ forms a sphere $$\mathcal{P}$$ around $$p$$, and the envelopes $$\mathcal{J}_{1}$$ and $$\mathcal{J}_{2}$$ form the ingoing and outgoing wavefronts respectively. If the areas of both $$\mathcal{J}_{1}$$ and $$\mathcal{J}_{2}$$ are less than the area of $$\mathcal{J}$$ , then $$\mathcal{J}$$ is a closed trapped surface.

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