The Grand Design

by Stephen Hawking

It is easy to see how the billiard table is warped in this example because it is curving into an outside third dimension, which we can see. Since we can’t step outside our own space-time to view its warpage, the space-time warpage in our universe is harder to imagine. But curvature can be detected even if you cannot step out and view it from the perspective of a larger space. It can be detected from within the space itself. Imagine a micro-ant confined to the surface of the table. Even without the ability to leave the table, the ant could detect the warpage by carefully charting distances. For example, the distance around a circle in flat space is always a bit more than three times the distance across its diameter (the actual multiple is π). But if the ant cut across a circle encompassing the well in the table pictured above, it would find that the distance across is greater than expected, greater than one-third the distance around it. In fact, if the well were deep enough, the ant would find that the distance around the circle is shorter than the distance across it. The same is true of warpage in our universe—it stretches or compresses the distance between points of space, changing its geometry, or shape, in a way that is measurable from within the universe. Warpage of time stretches or compresses time intervals in an analogous manner.

Armed with these ideas, let’s return to the issue of the beginning of the universe. We can speak separately of space and time, as we have in this discussion, in situations involving low speeds and weak gravity. In general, however, time and space can become intertwined, and so their stretching and compressing also involve a certain amount of mixing. This mixing is important in the early universe and the key to understanding the beginning of time.

Although Einstein’s general theory of relativity unified time and space as space-time and involved a certain mixing of space and time, time was still different from space, and either had a beginning and an end or else went on forever. However, once we add the effects of quantum theory to the theory of relativity, in extreme cases warpage can occur to such a great extent that time behaves like another dimension of space.

The realization that time can behave like another direction of space means one can get rid of the problem of time having a beginning, in a similar way in which we got rid of the edge of the world. Suppose the beginning of the universe was like the South Pole of the earth, with degrees of latitude playing the role of time. As one moves north, the circles of constant latitude, representing the size of the universe, would expand. The universe would start as a point at the South Pole, but the South Pole is much like any other point. To ask what happened before the beginning of the universe would become a meaningless question, because there is nothing south of the South Pole. In this picture space-time has no boundary—the same laws of nature hold at the South Pole as in other places. In an analogous manner, when one combines the general theory of relativity with quantum theory, the question of what happened before the beginning of the universe is rendered meaningless. This idea that histories should be closed surfaces without boundary is called the no-boundary condition.

Over the centures many, including Aristotle, believed that the universe must have always existed in order to avoid the issue of how it was set up. Others believed the universe had a beginning, and used it as an argument for the existence of God. The realization that time behaves like space presents a new alternative. It removes the age-old objection to the universe having a beginning, but also means that the beginning of the universe was governed by the laws of science and doesn’t need to be set in motion by some god.