Accelerating the Debate on Motion

Einstein’s biggest breakthroughs came after he had made an effort to learn from his earlier mistakes

Albert Einstein: If only our bankers had his capacity for profound thought

In 1976 the composer Philip Glass wrote a surrealistic “opera” called Einstein on the Beach, recently revived at the Barbican. This curious museum piece lasts more than five hours without a break, the audience being welcome to wander in and out at will, which I did. It deals with courtrooms and lawyers, but has little to do with the beach or with Einstein, apart from a reference in one of the poems forming the text: “…green Christmas Trees. So Santa Claus has about/red. And now the Einstine [sic] trail is like in Einstein on the Beach. So this will.”

At this late point in the opera the staging shows a vision of the nuclear explosion that destroyed Hiroshima. It’s hard to remember now, but back in the 1960s and early ’70s there was huge guilt — and worry — about the nuclear bomb. Looking back at the Queen’s Silver Jubilee in 1977 one newspaper recently recalled a man saying, “We’re due a celebration . . . we’ve made it to 1977 without a nuclear war.” Do you recall CND, and those Aldermaston marches? All gone, and as for regarding Einstein as the father of the nuclear bomb, no one thinks of him this way any more. But what exactly did Einstein do?

Just after seeing the opera I received a mathematics book to review: The Universe in Zero Words by Dana Mackenzie (Princeton, £19.95). It presented, among other things, Einstein’s famous equation E=mc2, linking mass m with energy E. The large number c2 shows that a small amount of mass is equivalent to a huge amount of energy, leading to nuclear power and nuclear bombs. Einstein derived the equation in 1905. That was when he produced his first theory of relativity, showing that a moving body, particularly one travelling close to the speed of light, would appear distorted: lengths shortened, time going more slowly, and mass increased. A thought experiment about a body emitting two photons in opposite directions then led him to see that an apparent increase in mass would be matched by an apparent increase in the energy of the photons as their wavelengths were foreshortened. This led him to his famous equation.

Yet Einstein did much more than this. Before explaining what, let’s consider some other equations taken from Mackenzie’s book. One is Newton’s famous equation giving the gravitational attraction between two bodies, depending on the mass of each and the distance between them. Using his invention of the calculus, Newton then derived the equation of motion for planets orbiting the Sun, hence confirming and explaining empirical laws governing these orbits, discovered by Kepler. This stupendous achievement was the beginning of what became known as Newtonian mechanics, which reigned supreme until the late 19th century. Then came various discoveries leading to relativity theory and quantum mechanics. However, for sizeable bodies at reasonably low speeds all was well, and equations of motion could in principle be written down. Or so it seemed.

Indeed in 1954, in the early days of electronic computers, John von Neumann suggested computers would one day be able to predict the weather 30-60 days in advance. It hasn’t happened, though computer power has grown to measures unimaginable to von Neumann. One problem is that Newtonian mechanics contained surprises. The problem of two bodies was solved immediately but the three-body problem resisted all attempts.

Now you would think that given enough mathematical ingenuity equations would be found governing the motion of a mere three bodies such as Earth, Sun and Moon, affected only by gravitational forces between them. However, in the 20th century it turned out that a tiny alteration in position or velocity could lead to grotesque differences in the resulting motion. Move by a hair’s breadth and a stable motion of the three could turn unstable, with the Moon heading off to infinity. Tiny changes could lead to huge effects, and as is often said, the flutter of a butterfly’s wings in Siberia could alter the weather in London.

Chaotic behaviour became part of the story, but mostly it seemed under control. For example, in the stock market a share price goes up and down unpredictably from day to day, but there was a way to hedge against such variations by using options. This led to options trading, and Mackenzie tackles the Black-Scholes equation, which gives a means for pricing options. The trouble is that it assumes the spread of variations fits a normal distribution, and when that fails the equation delivers the wrong answer. Then all bets are off.

Now back to Einstein. After introducing his “Special” Theory of Relativity in 1905 he went further. The theory dealt with uniform motion, but what about acceleration? In the 1960s a debate raged in the pages of the Listener, where Herbert Dingle, a philosopher at University College London, insisted that Special Relativity had an inherent inconsistency. The space twins paradox brought it into focus: if one twin went off by spaceship he or she might experience a mere ten years, while the sibling on Earth might experience 20. After 40 years of earth time the twin would return a mere 20 years older. Fine, but what if you interchanged roles? Could you not argue that the travelling twin was “relatively” equivalent to the one left at home, and the 40 years and 20 years should be switched? The answer — and much hot air was expelled on this — is that the twins are not equivalent because of acceleration and deceleration for the travelling twin. But this was precisely Dingle’s point.

Years earlier, Einstein himself had thought about the problem of acceleration. He asked what the difference was between the force of gravity one feels standing in a stationary elevator, and the force one feels when the elevator sits in space and accelerates upwards? No difference, he concluded.

By 1908, Hermann Minkowski had put Einstein’s Relativity into a new form, mingling space and time in a four-dimensional space-time that made the equations of Special Relativity perfectly natural. In 1915, Einstein showed how to use Minkowski’s space-time, along with the mathematics describing curvature, in a way that incorporated gravity, and acceleration. This was the “General” Theory of Relativity, his greatest achievement, which incidentally underpins the accuracy of our modern global positioning system.

But look at how Einstein did it. He dwelt deeply on things. Would that our bankers could do the same, to say nothing of our politicians. If Special Relativity was at fault for not including acceleration, and Black-Scholes was at fault for assuming normal distributions, how do we describe the fault of those who handed out mortgages to people who couldn’t pay when house prices fell and they could no longer raise money by refinancing, or the misguided idealists who created the euro?

Einstein saw beyond his creation to correct its faults, but our politicians can’t do that. It’s no longer nuclear bombs that worry us, but the folly of human beings. An old story, I’m afraid.