The Ascent of Gravity
Page 16
The anomalous motion of Mercury
On Christmas Eve 1907, just after he had completed his review of special relativity, Einstein had written to his Zurich friend Conrad Habicht: I hope to explain the still unexplained secular changes in the perihelion distance of Mercury.’27 Back then, he had failed. Nevertheless, the letter reveals Einstein’s prescience in recognising that such a tiny effect was a subtle symptom of a fundamental failure of Newton’s theory of gravity.
Mercury is the Sun’s closest companion. Being so near the most massive body of all, the planet is forced to negotiate the most grossly warped space-time in the Solar System. It makes it the planet on which the effects of warped space-time leave their biggest mark.
In 1905, Einstein had discovered that all forms of energy have an effective mass. It follows that all forms of energy must exert gravity. And one of those forms of energy is gravitational energy – the energy of warped space-time itself. Remarkably, this means that warped space-time is not only gravity but a source of more gravity. Gravity creates more gravity!
Close to the Sun, therefore, gravity is stronger than predicted by Newton. It departs from a force described by an inverse-square law.
Newton’s great triumph was of course his demonstration that a body experiencing a centrally directed inverse-square law of force travels in an ellipse. It obviously follows that, if a body does not experience an inverse-square law of force, it does not travel in an ellipse. The trajectory is instead an ellipse that ‘pre-cesses’, continually changing its orientation in space, and tracing out a rosette-like pattern.
Einstein calculated Mercury’s orbit. His theory predicted that the effect of warped space-time near the Sun should indeed cause an anomalous precession of Mercury’s orbit. The amount was 43 arc seconds per century.
It was exactly the anomalous precession that had mystified astronomers for half a century. It was exactly the precession that had prompted Urbain Le Verrier to postulate the existence of the planet Vulcan.
There was, of course, no Vulcan. The anomalous motion of Mercury was not telling astronomers of the existence of an unknown planet skirting the fires of the Sun. It was signalling something far more fundamental and astonishing – something that nobody before had ever suspected: that Isaac Newton was wrong.
‘The theory agrees completely with the observations,’ Einstein concluded on presenting his Mercury result to the Prussian Academy. He had overturned 200 years of physics and shown that the greatest scientist ever to have lived had been wrong but he had managed to disguise what he was actually feeling. Inside he was in utter turmoil. He was beside himself with excitement.28 In fact, he was having palpitations.29
Physicists scrawl arcane mathematical equations across blackboards but it is an enormous leap of faith to believe that nature really obeys those equations. It invariably comes as a massive shock when it turns out that nature really does.
After an eight-year struggle, Einstein had finally reached the summit of a towering mountain. The fog that had enshrouded him every step of the climb had cleared. And stretching out below him, lit by dazzling sunlight, was a landscape no other human being had ever seen. ‘The years of searching in the dark for the truth that one feels but cannot express,’ said Einstein, ‘the intense desire and alternations of confidence and misgiving until one breaks through to clarity are known only to him who has experienced them.’30
Actually, Einstein was not the only one to suggest that the anomalous motion of Mercury could be explained if gravity close to the Sun was slightly greater than predicted by Newton’s law of gravity. In the late nineteenth century, the American astronomer Simon Newcomb had pointed out that the planet’s motion made sense if gravity weakened not according to the inverse-square of the distance between masses – that is, according to a power of 2 – but according to a power of 2.0000001612.31,32
Substituting 2.0000001612 for 2 marred the simplicity of Newton’s law of gravity. But, if nature chose ugly over beautiful, there was no choice but to accept it. What scuppered Newcomb’s idea was the recognition that, although such a messy ‘power law’ of gravity might be able to explain the motion of Mercury, it could do it only at the expense of not being able to explain the motion of the Moon.
Einstein’s explanation predicted the observations of both Mercury and the Moon. Close to the enormous mass of the Sun, space-time was warped enough to cause a noticeable anomaly in the motion of the innermost planet. Close to the far punier Earth, space-time was far less warped, so there was no noticeable anomaly in the motion of the Moon.
It was history repeating itself. Hendrik Lorentz and George FitzGerald had proposed that the length of a body contracts at speeds approaching the speed of light but had provided no fundamental explanation for it. Einstein did. Here, Newcomb had proposed that gravity close to the Sun was slightly stronger than Newton predicted but provided no fundamental – or, in this case, correct – explanation for it. Einstein did.
Einstein’s field equations
The pressure from Hilbert breathing down Einstein’s neck had had the desired effect. In the week before his fourth and final lecture, after eight years of struggle, and in the absolute nick of time, Einstein reached his goal. On 25 November 1915, his coat buttoned up against the cold, he made his way along Unter den Linden Strasse and addressed his audience at the Prussian Academy. On the blackboard he simply wrote:
Guv = 8πGTuv/c4
It is the law of gravity everyone experiences, no matter what their state of motion. It is the general theory of relativity in a nutshell. American science writer Dennis Overbye describes it as ‘the equation that rules the Universe’.33
Einstein’s equation uses super-compacted notation so, like Dr Who’s ‘Tardis’, it is bigger on the inside than on the outside. The left-hand side of the equation is in fact a 4 × 4 table of numbers known as the ‘Einstein curvature tensor’, which summarises the curvature of space-time. The right-hand side is another 4 × 4 table of numbers known as the ‘stress-energy tensor’, which summarises the ‘sources of gravity’.34
That Einstein’s equations contain 4x4 tables of numbers means that there are actually sixteen equations. In fact, Einstein was able to use ‘symmetry arguments’ to reduce the number of equations down to just ten. Nevertheless, the fact remains that he still substituted ten equations for the single equation of Newton’s theory of gravity.
Einstein’s ‘field equations of gravity’ dictate the warped spacetimes that are generated by any distribution of mass-energy. They are the mathematical embodiment of John Wheeler’s phrase: ‘Matter tells space-time how to warp. And warped space-time tells matter how to move.’ Finding space-times that satisfy the ten gravitational field equations is extremely hard. In fact, it is so hard that anyone who finds one often has the space-time named after them.
Einstein’s field equations are ‘generally covariant’, which means they do not depend on your point of view (technically, they retain their form no matter what system of coordinates they are expressed in). This is their beauty. And this is something Einstein had shed blood and tears to achieve.
But Einstein’s theory was not quite the one he originally set out to find in 1907. His aim had been to generalise his special theory of relativity by finding what must be done to the measurements of space and time of people varying their speed, or ‘accelerating’, relative to each other so that they would agree on the same laws of physics. In fact, Einstein replaced Newton’s law of gravity with a new and improved theory of gravity rather than finding a theory about accelerated observers. Such is the serendipity of science.
The bending of light by gravity
The scene, as Einstein’s chalk squeaked across a blackboard in Berlin, could not have contrasted more starkly with the outside world, where the slaughter of young men on an industrial scale had been gaining momentum. Already in 1915, gas attacks had poisoned, burnt and suffocated soldiers on all sides; Zeppelins had rained down death on British civilians; and a U-boat had torpedo
ed the ocean liner Lusitania off the coast of Ireland, with the loss of 1,198 lives.
But despite the mounting horrors, contact, incredibly, was maintained between the scientists of the warring nations. Within weeks of the publication of the general theory of relativity, copies were smuggled out of Germany to Holland, and then to England. And though the ‘war to end all wars’ would leave 10 million dead and as many again with their health wrecked for ever, within a year of the Armistice on 11 November 1918 it was an Englishman who confirmed a key prediction of Einstein’s, catapulting the German-born physicist into the scientific firmament.35
Arthur Stanley Eddington had received his smuggled-out copy of Einstein’s theory from the Dutch astronomer Willem de Sitter in Leiden. An accomplished populariser of science, the Cambridge scientist became the chief communicator of Einstein’s ideas to the English-speaking world. Asked by a journalist in 1919: ‘Is it true that only three people in the world understand the theory of general relativity?’ he replied (perhaps not as modestly as he might have done): ‘Oh. And who is the third?’
Eddington zeroed in on Einstein’s prediction of light bending by the gravity of the Sun. Einstein had realised the effect in 1907 when he had written his review article on the special theory of relativity, and had begun to think for the first time about developing a theory of gravity that, unlike Newton’s, was compatible with the new view of space, time, matter and energy.
The special theory of relativity had revealed that all energy -including light energy — has an effective mass.36 Consequently, a massive body like the Sun must attract light as surely as it attracts matter. Observing this effect would provide strong evidence for Einstein’s theory of gravity.
However, by the time Einstein had developed his full-blown theory of gravity, he realised that the bending of light by gravity was actually a more subtle effect than he had guessed in 1907.
Let’s return to the astronaut in the blacked-out cabin of a rocket accelerating at 1g, far away from the gravity of any planet. Because the astronaut’s feet are pinned to the floor and all objects fall at the same rate, irrespective of their mass, there is no way he can tell that he is not on the surface of the Earth.
Well, actually, that is not entirely true. There is one way he can tell.
The Earth is round. Consequently, all bodies fall towards the centre of the Earth. In the most extreme case, when objects are dropped on opposite sides of the globe – say, in England and New Zealand – they fall in opposite directions. Actually, wherever two objects are dropped their paths inevitably converge as they head towards the centre of the Earth.
But this is not what the astronaut in the rocket sees. If he observes two falling objects with a precise enough measuring device, he finds that their paths do not converge but stay parallel. So he is able to guess that he is not on the surface of the Earth.
Remarkably this is not a killer blow for Einstein’s theory of gravity. The Principle of Equivalence on which the great edifice of the general theory of relativity is built actually requires only that gravity and acceleration are indistinguishable locally – that is, in an arbitrarily small region of space.
But the fact that objects fall along converging paths in the vicinity of a real body such as the Earth or Sun has implications for the path of a light beam. In the neighbourhood of such bodies – as opposed to inside the cabin of the astronaut’s rocket – the beam bends by twice as much as naively expected.
The body in our vicinity with the greatest light-bending power is of course the Sun, which contains 99.8 per cent of the mass of the Solar System. The best way to see the effect, Einstein realised, is to observe a distant star whose light, on its way to the Earth, passes close to the solar disc, where the valley of spacetime is at its steepest. The path of the light will be bent as surely as the path of a hiker negotiating a hilly landscape. If the star is observed from Earth, it will be shifted from its expected position in the sky.
A tale of two eclipses
Obviously, stars close to the Sun are lost in its glare, as impossible to see as fireflies next to a car headlight. But there is one instance in which such stars can be seen: when the luminous disc of the Sun is blotted out by the disc of the Moon. In such a ‘total eclipse’, the world is plunged into darkness and for a few minutes, the stars come out in daytime.
A total eclipse occurs somewhere in the world every few years. But the necessary alignment between the Sun, the Moon and the Earth occurs only along a narrow ‘track’ on the Earth’s surface. Consequently, the chance of seeing a total eclipse at a given spot in a given year is very small – there is one only every 350 years on average.
As luck would have it, a total eclipse was visible from Russia’s Crimea peninsula – not too far from Germany – on 24 August 1914. A German expedition was mounted, led by Erwin Freundlich, an astronomer who was greatly impressed by Einstein’s ideas. On 19 July, Freundlich left Berlin with two companions and four telescopes equipped with cameras. It was a bad time to be going to Russia.
Freundlich had probably heard of the shooting three weeks earlier in Sarajevo of Austrian Archduke Franz Ferdinand by a Serbian nationalist. But, in common with everyone else in Europe, he had no inkling of the disastrous chain of events Gavrilo Prin-cip had set in motion. On 1 August, three days before Britain did the same, Russia declared war on Germany.
Overnight, Freundlich and his companions were transformed from guests of the Russians into enemy aliens. Their equipment was impounded and they were imprisoned. Consequently, they missed the total eclipse – which, anyhow, was obscured by clouds over the Crimean peninsula. But their misery was short-lived. In one of the first prisoner exchanges of the First World War, they were swapped for Russian officers, and they limped back to Berlin by the end of September.
For Einstein this was actually a piece of good fortune – and not only because Freundlich was a friend and supporter. Had the astronomer succeeded in measuring the deflection of starlight by the Sun, it would not have matched Einstein’s prediction. The reason was that, in 1914, Einstein still believed that the deflection would be 0.87 arc seconds – the figure he had obtained in 1911 -rather than the correct figure of 1.7 arc seconds he obtained from his full theory in 1915.37
The First World War finished and, on 29 May 1919, there was another total eclipse of the Sun. Eddington and an assistant set off for Principe, a small volcanic island in the Gulf of Guinea off the coast of West Africa. Weather conditions were not brilliant on 29 May. In fact, the morning began with a tropical downpour. But, although the rain eased off by the time of the eclipse in the early afternoon, Eddington and his assistant were dismayed to see the clouds thicken and clear repeatedly as the Sun was blotted out by the black disc of the Moon. There was nothing for it but to carry on regardless taking photographs and hope for the best.
Of the sixteen exposures obtained by Eddington only six were made in cloud-free conditions. Four of them could not be developed in the tropical heat of Principe but had to be packed away for transport back to England. Of the remaining two exposures, only one had captured a clear enough star-filled sky for Eddington to carry out the necessary measurements.
But one was all he needed.
On 3 June, Eddington compared the star positions recorded during the total eclipse with their positions recorded on a photograph taken back at Greenwich in England. It was a difficult measurement to make. A single arc second on the sky corresponded to a mere 1/16 millimetre on Eddington’s photographic plate. But the English astronomer rose to the challenge. He made his painstaking measurement. He checked it and rechecked it.
There was no doubt about it. The stars near the Sun were shifted in position by 1.61 arc seconds, plus or minus 0.3 arc seconds. It was within a whisker of Einstein’s prediction.
Eddington would look back on this magical moment as the single most important incident of his life. He had confirmed the general theory of relativity. Newton was wrong. A forty-year-old German had assumed his mantle. Eddington penned
the following ditty:
One thing at least is certain, light has weight
Light rays, when near the Sun, do not go straight.
Bizarrely, an eclipse expedition in 1914 which had failed because of a man called Princip had been followed by one in 1919 which succeeded on the island of Principe.
Einstein was sick and in bed when a telegram reached him from his friend Hendrik Lorentz. It did not quite say that general relativity had been confirmed. But it probably relayed the charming but brief wording of the telegram Eddington had sent from Principe back to England:
Through cloud. Hopeful.
Eddington.
It was enough. ‘I knew I was right!’ exclaimed Einstein.38
And Einstein did know he was right. It was not simply that he was cocksure – though he was certianly that – but he had a great belief that nature’s fundamental laws must be elegant and beautiful. And the equations of general relativity were certainly that. Later, he was asked by a doctoral student: ‘What if general relativity had not been confirmed by Eddington?’
‘Then I would feel sorry for the dear Lord,’ replied Einstein.39
On 7 November 1919, on page 12 of The Times of London, there appeared an article under a triple-headline:
REVOLUTION IN SCIENCE
New Theory of the Universe
Newtonian Ideas Overthrown
It was the report of a joint meeting of the Royal Society and the Royal Astronomical Society which had been held the day before. Overnight, Einstein became a superstar. He was destined to achieve the global fame of Charlie Chaplin. In fact, he would stay with Chaplin and his wife when visiting Los Angeles.40 Such would be his fame that when Edith Piaf made her first visit to America in 1947 and was asked at a press conference who she would most like to meet she did not hesitate: ‘Einstein. And I’m counting on you to get me his phone number.’41