The End of Time: The Next Revolution in Physics

The desire to express the laws of nature in progressively more general coordinates led to all Einstein’s major breakthroughs. Newton had argued that centrifugal forces proved the existence of absolute space. The laws of nature looked different in rotating systems. Einstein wanted to attack this problem head on. Could he perhaps show that, if expressed properly, the laws of nature did after all have the same form in rotating and non-rotating coordinates? The principle of equivalence suggested that what Newton had taken to be absolute inertial effects in a rotating system might be the gravitational effects of distant matter. The point is that in a rotating system the distant stars would themselves appear to be rotating. Since rotating electric charges generate electric and magnetic fields, rotating masses might generate new kinds of gravitational field. Nearly thirty years earlier, Mach had suggested that rotating matter ‘many leagues thick’ might generate measurable centrifugal forces within it. Einstein now conjectured that the gravitational field was the mechanism through which such forces could arise.

He therefore started to consider what form the laws of nature would take in a rotating system. This immediately led him to a startling conclusion: the ordinary laws of Euclidean geometry could not hold in such a system! His argument was based on the contraction of measuring rods in motion which he had proved in special relativity. First, imagine observers at rest on a surface who measure the circumference and diameter of a circle painted on it. They will find that their ratio is π. That agrees with Euclidean geometry – a recognized law of nature. Now imagine other observers on a disk above the painted circle and rotating about its centre. Their rods will undergo Fitzgerald-Lorentz contraction when laid out in the direction of motion, around the circumference. However, when laid out along the diameter, the rods will not contract. (The contraction occurs only in the direction of motion.) Therefore, the rotating observers will not find π when they measure the ratio of the circumference to the diameter. For them, Euclidean geometry will not hold.

Because Einstein wanted so passionately to generalize the relativity principle, he took this result seriously. According to the hint from the equivalence principle, novel effects in accelerated coordinate systems (as a rotating one is) could be attributed to gravitational effects. He concluded that geometry would not be Euclidean in a gravitational field. This happened during 1911/12, when he was working in Prague. Through either the suggestion of a colleague or the recollection of lectures on non-Euclidean geometry he had heard as a student, Einstein’s attention was drawn to a classic study in the 1820s by the German mathematician Carl Friedrich Gauss.

Gauss had studied the curvature of surfaces in Euclidean space. As a rule, material surfaces in space are not flat but curved. Think of the surface of the Earth or any human body. Gauss’s most important insight was that a surface in three-dimensional space is characterized by two distinct yet not entirely independent kinds of curvature. He called them intrinsic and extrinsic curvature. The intrinsic curvature depends solely on the distance relationships that hold within the surface, whereas the extrinsic curvature measures the bending of the surface in space. A surface can be flat in itself – with no intrinsic curvature – but still be bent in space and therefore have extrinsic curvature. The best illustration of this is provided by a flat piece of paper, which has no intrinsic curvature. As it lies on a desk it has no extrinsic curvature either: it is not bent in space. However, it can be rolled into a tube. It is then bent – but not stretched – and acquires extrinsic curvature.

In contrast to a sheet of paper, the surface of a sphere, like the earth, has genuine intrinsic curvature. Gauss realized that important information about it could be deduced from distance measurements made entirely within the surface. Imagine that you can pace distances very accurately, and that you walk due south from the north pole until you reach latitude 85° north. Then you turn left and walk due east all the way round the Earth at that latitude. All the time you will have remained the same distance R from the north pole. If you believed the Earth to be flat, you would expect to have to walk the distance 2πR before returning to the point of your left turn. However, you will find that you get there having walked a somewhat shorter distance. This shows you that the surface of the Earth is curved.