Warp Speed

When yachts set sail with the tide, or people gather to witness a total eclipse of the Sun, they are trusting in Isaac Newton’s theory of gravity. For more than three hundred years his theory has proved so accurate in describing the universe that it has enabled us not only to predict tides and eclipses, but even to send spaceships to Jupiter, Saturn and beyond. One of Newton’s assumptions is that the effects of gravity are transmitted instantaneously. However, it is worth asking what ‘instantaneous’ means in this context. Since Einstein, the speed of light has been recognised as a natural limit; what, then, is the speed of gravity?

In Einstein’s universe, space and time are intimately entwined. ‘Space-time’ acts like a medium in which objects are distorted and the passage of time slowed when massive bodies such as planets and stars are present. In the jargon, space-time is ‘warped’. When galaxies collide or stars explode, the warping changes and the disturbance spreads out in ‘gravitational waves’ travelling at the speed of – what? The story of gravity, and especially of gravitational waves, has been far from clear-cut; as Daniel Kennefick points out in Travelling at the Speed of Thought, Einstein’s ‘first reaction on the completion of his theory was to conclude that gravitational waves do not exist’. The majority opinion now is that they do, even though no definitive observation of them has yet been made. Kennefick’s historical account shows that even today, as satellites carrying gravitational wave detectors are preparing for launch into space, the nature of what they are looking for is still not fully understood.

Unlike the electrical attractions and repulsions resulting from positive and negative charges within atoms, which cancel one another out, the gravitational attraction exerted by each and every particle in a large body adds up. Objects larger than about 500 km in diameter exert a powerful pull. The Sun, no bigger than a thumbnail when viewed from Earth, traps the planets in a cosmic waltz across hundreds of millions of kilometres of space. Newton posited that gravity’s pull between two bodies diminishes as the square of the distance between them increases, and that a massive body such as the Sun sends out its gravitational tentacles in all directions uniformly. His was a clockwork universe, where planets orbited permanently in regular repetitive orbits; the design seemed to accord with the perfection expected from a divine creator. But this ideal would not last.

Near the end of the 17th century, Edmond Halley examined records of medieval and ancient solar eclipses back to the time of Ptolemy. He discovered that when he used the position and trajectory of the Moon to determine retrospectively when solar eclipses should have occurred, the times calculated differed from the actual ones by up to an hour. Halley deduced that in the past the Moon must have moved across the sky from east to west more slowly than in his own time. This was a far-reaching, even heretical assertion. For the Moon to have changed its motion in such a way would imply that its course through the heavens did not repeat in periodic orbits. Such ‘secular’ changes in its orbit could eventually cause the system itself to disappear, and the Moon to fall into the Earth or escape into space. For many philosophers, to theorise that the cosmos could decay in this way was a slur on the Almighty, as it implied that God was such an unskilled craftsman as to have constructed a system of stars and planets that could fall into ruin and disorder. Nonetheless, Halley was right, as even the fundamentalists were eventually forced to concede. The question now became: what causes the secular acceleration of the Moon?

Pierre-Simon Laplace discovered in 1776 that orbits would eventually degrade if, in contrast to Newton’s theory of instantaneous action at a distance, gravitational forces took time to propagate. Laplace’s insight was that the cumulative effect of the planets on the Moon as well as on the Earth reduced the eccentricity of the Earth’s orbit bit by bit over the centuries, causing the Moon to approach the Earth and reducing its orbital period. This was consistent with Halley’s results. However, Laplace also discovered that over even greater spans of time the effect would reverse itself, causing the Moon to slow down and the orbits of the Earth and the Moon ultimately to repeat themselves. This was a tour de force, appearing to explain Halley’s observation while vindicating Newton’s theory.

All went well until the mid-19th century, when John Couch Adams showed that Laplace’s calculations were incomplete and actually accounted only for about half of the observed effect. The apparent agreement between Halley and Laplace was ruined and nationalistic passions inflamed: Laplace was French and Adams English. Parity was restored when the Frenchman Charles Delaunay showed that Laplace’s calculation did indeed account for half the effect and that tidal friction could account for the rest.

The Moon raises tides on the oceans directly below it. As the Earth rotates, it drags these tidal bulges with it, so that we see the Moon directly overhead just before high tide. This bulge in turn exerts a gravitational pull on the Moon, slowing its rotation and also the Earth’s: days are growing longer, as are the months, though the number of days per month is falling. In the far future, there will be only one day per month, the Earth always presenting the same face to the Moon, as the Moon does to us today. The empirical proofs of these theoretical calculations rely on sophisticated modern technology: atomic clocks measure the year to fractions of a microsecond – the midnight hour is periodically adjusted at New Year – and laser range-finding using a mirror placed on the Moon by Apollo astronauts confirms that the Moon is gradually moving away from us. There is no permanent clockwork cosmos.

Newton’s theory seemed to describe everything other than the bizarre motion of the innermost planet, Mercury, whose point of closest approach to the Sun (the ‘perihelion’) changes from one orbit to the next. This phenomenon could not be accounted for within Newton’s theory and was ultimately explained by Einstein’s general theory of relativity. On flat ground, the shortest distance between two points is a straight line. In Einstein’s universe, space-time behaves like an elastic solid, such as a rubber sheet. The force of gravity arises when a large mass, such as the Earth or Sun, distorts space-time; in metaphorical terms, it weighs on the sheet, warping its surface. The effects of gravity can be represented as free-fall along lines that are the shortest paths in a warped space-time.

Mercury, as the nearest planet to the Sun, experiences the strongest gravitational pull, moves the fastest and is most susceptible to the effects of relativity. The warping of space makes the distance around the Sun slightly different from its Newtonian value, so that after completing an orbit, Mercury doesn’t end up in quite the same place as it would in Newton’s picture. The result is that Mercury’s orbit differs from year to year, in agreement with Einstein’s theory. This has been confirmed, as has Einstein’s prediction that light beams curve in the presence of gravity. The one aspect of general relativity that remains to be verified is its prediction that gravitational waves should exist. If the mass that is the source of the gravitational field suddenly shifts, for example in a supernova explosion, the theory implies that gravitational waves will spread through the medium, analogous to an earthquake producing seismic waves in the solid earth. Behaviour of this kind is familiar in electromagnetism: an oscillating charge in a radio antenna emits what is known as electromagnetic ‘dipole’ radiation. At first sight, gravitational waves and electromagnetic waves appear to be analogous, and this is treated as obvious in many textbooks. In fact, the analogy isn’t so obvious; indeed it was questioned by theorists, including Einstein, for several decades before the issue was resolved.

Einstein had realised that while positive and negative electric charges, or north and south magnetic poles, are the cause of electromagnetic attraction and repulsion, there is only one ‘pole’ to the gravitational force: there is no gravitational repulsion, no ‘antigravity’. It is the duality in electromagnetism that allows for dipole radiation; its absence in gravity implies that there is no such analogue for gravitational waves. In electromagnetism there are more complicated possibilities, such as quadrupole radiation, which is emission from a system featuring motion around two separate axes of symmetry. In electromagnetism this is a secondary effect, whereas for gravitational radiation, should it prove to exist, it would be the most important.

Einstein’s first reaction on completing his theory, as recorded in a letter to Karl Schwarzschild on 19 February 1916, was that ‘there are no gravitational waves analogous to light waves.’ Kennefick takes this to imply that Einstein concluded that ‘gravitational waves do not exist.’ Einstein certainly agreed that gravitational waves are not ‘analogous’ to light waves, at least in the sense that there is no dipole radiation; whether he denied that they exist at all is a more subtle issue. Much of the confusion came from the way that gravitational waves were described by Einstein’s equations, which implied the existence of three types of gravitational waves, two of which did not transport energy. Einstein managed to show that these were spurious results, arising from the way he had defined his co-ordinates. Yet, in the following decades, mathematicians would periodically rediscover these spurious waves. Solving Einstein’s equations and correctly interpreting the results has been one of the most difficult tasks facing the physical sciences in the last hundred years.

Arthur Eddington’s The Mathematical Theory of Relativity, published in 1923, became the definitive work and influenced the field for many years. Eddington, an astrophysicist and leader of the British eclipse expedition to Africa which confirmed Einstein’s prediction that starlight is deflected by the gravitational field of the Sun, was sceptical about the analogy between gravitational and electromagnetic waves, though he had no doubt that the waves existed. But he appears to have been the first to ask at what speed they travel. It is taken as self-evident, in popular texts at least, that this must be the speed of light, though there is no obvious reason why there should be only one universal speed in nature. Before him, physicists had assumed that the consistency of physics required the speed of gravitational waves to be the same as, or ‘at any rate not greater’ than, that of light. ‘The statement that in the relativity theory gravitational waves are propagated with the speed of light’, Eddington wrote, has ‘been based entirely’ on Einstein’s presentation of the problem. ‘The result stands or falls by the choice of co-ordinates,’ he continued, and Einstein had made his choice in order to simplify the calculation. He had unwittingly assumed the speed of light waves and of gravity to be the same. If different co-ordinates had been chosen, the result might have been different.

Eddington set out to find the answer himself. He rediscovered the three types of wave, convincingly proved that only one of them transports energy, and established that its speed is indeed that of light. He noted that the other two waves were ‘merely sinuosities in the co-ordinate system, and the only speed of propagation relevant to them is “the speed of thought”’. This remark is so well known that many think Eddington claimed that all gravity waves travel at the ‘speed of thought’, and this has made many people think of him as universally sceptical. Far from it. He showed that the real waves travel at the speed of light, at least if they are of low intensity. He had made approximations that were fine for weak gravity waves but break down for waves at high intensity. Strong waves change the fabric of space-time so much that they are themselves disturbed as they travel. The waves scatter off the curves of space-time so that some parts backtrack and arrive ‘late’ as a tail to the main wave. There is still no general proof that intense gravitational waves travel at the speed of light.

The insight that prompted Einstein’s theory of general relativity was that it is impossible to tell the difference between being accelerated and being in a gravitational field: ‘Any observer in a gravitational field is always entitled to imagine herself as if she were falling freely, experiencing no weight . . . One can make gravitational fields disappear by adopting the appropriate point of view.’ There was a lot of confused debate about binary stars: as one star orbited in the gravity field of the other, it would effectively be in free fall, so would it radiate energy? This was a central question during the 1950s, when there was a resurgence of interest in the subject, with influential scientists arguing that gravitational waves would not carry away energy from binary stars nor, possibly, from other systems.

In those days the notion of energy transport in gravitational waves was not well understood. The modern belief in their reality came after the discovery of the first binary pulsar in 1974 provided the opportunity to test the theory. Over several years, precise measurements showed that its orbital period was slowing down, at a rate in line with what would be expected if it was radiating energy. Since that result the received wisdom has been that Einstein’s theory of gravitational waves is correct, although it remains the case that they haven’t been directly observed.

In 1969, Joseph Weber announced that he had detected gravitational waves, but with an intensity far greater than the theory would have expected. This persuaded other experimentalists to build detectors similar to his, but his results were never replicated and have been almost universally rejected. A project called LIGO (Laser Interferometer Gravitational Wave Observatory) plans to use satellites in an attempt to detect gravitational waves. The proponents of these experiments hope to pioneer a new field of gravitational wave astronomy, which will provide direct insight into the behaviour and properties of black holes and even detect echoes of the Big Bang.

However, to understand and interpret the results of these experiments it will be necessary to have some notion of what the theory would expect to find. This is extremely delicate, so difficult are the concepts. Today theorists use supercomputers, and numerical solutions of Einstein’s equations are being found for complex situations such as the gravitational interactions between pairs of black holes. But the controversies and uncertainties of principle that have been present over the decades have not all gone away. Computers calculate to high but not perfect accuracy and there are always residual errors in the computation, ‘so that the sums do not quite add to zero’. As Kennefick paraphrases Eddington, ‘it seems that numerical error travels with the speed of thought . . . and a supercomputer’s speed of thought is blazingly fast.’