Reality Is Not What It Seems: The Journey to Quantum Gravity

by Carlo Rovelli

There is a way of understanding the curvature of space from within it, and without looking at it from outside, that is important for what follows. Imagine you are at the North Pole and walk southward until you reach the equator, carrying with you an arrow pointing ahead. Once you reach the equator, turn to the left without changing the direction of the arrow. The arrow still points south, which is now to your right. Advance a little toward the east along the equator and then turn again toward the north—again without changing the direction of the arrow, which will now be pointing behind you. When you reach the North Pole again, you have executed a closed circuit—a “loop,” as it is termed—and the arrow does not point in the same direction as when you started out (figure 3.14). The angle through which the arrow has turned in the course of the loop measures the curvature.