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.