Although Friedmann found only one, there are in fact three different kinds of models that obey Friedmann’s two fundamental assumptions. In the first kind (which Friedmann found) the universe is expanding sufficiently slowly that the gravitational attraction between the different galaxies causes the expansion to slow down and eventually to stop. The galaxies then start to move toward each other and the universe contracts. Fig. 3.2 shows how the distance between two neighboring galaxies changes as time increases. It starts at zero, increases to a maximum, and then decreases to zero again. In the second kind of solution, the universe is expanding so rapidly that the gravitational attraction can never stop it, though it does slow it down a bit. Fig. 3.3 Shows the Separation between neighboring galaxies in this model. It starts at zero and eventually the galaxies are moving apart at a steady speed.
Finally, there is a third kind of solution, in which the universe is expanding only just fast enough to avoid recollapse. In this case the separation, shown in Fig. 3.4, also starts at zero and increases forever. However, the speed at which the galaxies are moving apart gets smaller and smaller, although it never quite reaches zero.
A remarkable feature of the first kind of Friedmann model is that in it the universe is not infinite in space, but neither does space have any boundary. Gravity is so strong that space is bent round onto itself, making it rather like the surface of the earth. If one keeps traveling in a certain direction on the surface of the earth, one never comes up against an impassable barrier or falls over the edge, but eventually comes back to where one started.
In the first kind of Friedmann model, space is just like this, but with three dimensions instead of two for the earth’s surface. The fourth dimension, time, is also finite in extent, but it is like a line with two ends or boundaries, a beginning and an end. We shall see later that when one combines general relativity with the uncertainty principle of quantum mechanics, it is possible for both space and time to be finite without any edges or boundaries.
Recollapse 再坍塌
雖然弗利德曼只找到一個模型，其實滿足他的兩個基本假設的共有三種模型。在第一種模型（即弗利德曼找到的）中，宇宙膨脹得足夠慢，以至於在不同星系之間的引力使膨脹變慢下來，並最終使之停止。然後星系開始相互靠近，宇宙開始收縮。圖3.2表示隨時間增加兩個鄰近的星系的距離的變化。剛開始時距離為零，接著它增長到最大值，然後又減小到零；在第二類解中，宇宙膨脹得如此之快，以至於引力雖然能使之緩慢一些，卻永遠不能使之停止。圖3.3表示此模型中的鄰近星系的距離隨時間的變化。剛開始時距離為零，最後星系以穩恆的速度相互離開；最後，還有第三類解，宇宙的膨脹快到足以剛好避免坍縮。正如圖3.4所示，星系的距離從零開始，然後永遠增大。然而，雖然星系分開的速度永遠不會變為零，這速度卻越變越小。
第一類弗利德曼模型的異特點是，宇宙在空間上不是無限的，並且是沒有邊界的。引力是如此之強，以至於空間被折彎而又繞回到自身，使之相當像地球的表面。如果一個人在地球的表面上沿著一定的方向不停地旅行，他將永遠不會遇到一個不可超越的障礙或從邊緣掉下去，而是最終走到他出發的那一點。
第一類弗利德曼模型中的空間正與此非常相像，只不過地球表面是二維的，而它是三維的罷了。第四維時間的範圍也是有限的，然而它像一根有兩個端點或邊界即開端和終端的線。以後我們會看到，當人們將廣義相對論和量子力學的測不準原理結合在一起時，就可能使空間和時間都成為有限的、但卻沒有任何邊緣或邊界。
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