If a bird-watching physicist falls off a cliff, he doesn’t worry about his binoculars; they fall with him.
SIR HERMANN BONDI
One thing at least is certain, light has weight … Light rays, when near the Sun, do not go straight.
ARTHUR EDDINGTON
Einstein’s theory of relativity is a recipe for predicting what must happen to space and time in order for everyone to measure the same speed for a beam of light.1 By ‘everyone’, Einstein meant people moving at constant speed relative to each other. A moment’s thought, however, reveals that this is a very special circumstance. Very few bodies move with uniform speed. A car in traffic slows down and speeds up before coming to a halt at traffic lights. A rocket rising on a column of orange flame and white smoke gets ever faster until it attains the 29,000 kilometres an hour necessary to stay in orbit above the Earth.
So all Einstein had figured out in 1905 was what the world looks like from the point of view of atypical, or ‘special’, observers, moving at constant speed with respect to each other. For his next trick, he needed to figure out what the world looks like to typical, or ‘general’, observers, who are varying their speed with time, or accelerating, with respect to each other. Somehow, he had to turn his special theory of relativity into a general theory of relativity. It was a monumental task that would take him a decade of mental struggle, but it would cement his place in history as the greatest physicist since Isaac Newton.
In attempting to generalise special relativity, Einstein faced a serious problem. Not only does special relativity describe a special situation, it is completely incompatible with one of the great cornerstones of science – Newton’s theory of gravity.
According to Newton, there is a force of attraction between any two bodies – for instance, the Sun and the Earth – that depends on their separation and on their masses. However, special relativity says that all forms of energy have an effective mass. If you heat a cup of coffee, for instance, its heat energy makes it marginally more massive than when it is cold. Consequently, all forms of energy must exert a gravitational force on each other, not just mass energy. It is energy not mass, as Newton believed, that is the source of gravity. Mass energy is simply the most familiar form.
And this is not the only incompatibility between special relativity and Newton’s law of gravity. According to Einstein, light sets the ultimate cosmic speed limit. His theory therefore predicts that, if the Sun were to vanish suddenly – an unlikely scenario but imagine that it did – the Earth would not realise right away. For the time it takes gravity, travelling at the speed of light, to go between the Sun and the Earth, the planet would continue blithely in its orbit. Only after 8½ minutes would it realise that the Sun had disappeared and fly off on a tangent towards the stars.
Contrast this with the prediction of Newton’s law of gravity. Two bodies feel an instantaneous force between them, which is synonymous with saying that the gravitational influence travels between them infinitely fast. Newton’s theory therefore predicts that, if the Sun were to vanish suddenly, the Earth would notice immediately, in violation of Einstein’s cosmic speed limit.
In concocting special relativity, Einstein had therefore inadvertently smashed one of the foundation stones of physics – Newton’s law of gravity. He must have felt like a vandal who topples a beautiful building without the slightest idea how to build a replacement. But, just when he was despairing of ever finding a way, he had a brainwave. It concerned a simple observation that had been known about for centuries but whose significance no one before had recognised.
The seventeenth-century Italian scientist Galileo Galilei is supposed to have dropped a heavy mass and a light mass from the leaning Tower of Pisa and observed them hit the ground together. The same experiment, minus the complicating effect of air resistance, was later carried out on the Moon in 1972. Apollo 15 commander Dave Scott dropped a hammer and a feather together and, from the simultaneous puffs of moon dust, demonstrated that they hit the ground at the same instant.
Think for a moment how peculiar this is. If you were to take a small mass and a big mass and push both of them with exactly the same force, it is obvious that the small mass will gain the most speed. It is common experience that a big mass such as a fridge resists being moved more than a small mass such as a stool – it is the very basis of our definition of mass. But, when the force of gravity pulls a small mass and big mass groundwards, the two masses gain speed at exactly the same rate. In other words, the force of gravity goes up perfectly in step with the mass. Somehow, it knows to be bigger for a bigger mass. But how? It was Einstein’s genius to think of a circumstance in which such an adjustment would come about perfectly naturally.
Imagine an astronaut in a rocket far from the gravity of any planet or moon. The rocket is accelerating, at 9.8 metres per second per second.2 Since this is precisely the rate at which a falling body accelerates towards the Earth – 1g, in the jargon – the astronaut’s feet are glued to the floor of his cabin just as if they were on the surface of the Earth. Now imagine that the astronaut holds a paperclip and a golf ball at the same height above the floor and lets them go simultaneously. Unsurprisingly, perhaps, they hit the floor at the same time, just as on Earth.
Now zoom out. Imagine you are floating outside the rocket with X-ray eyes that reveal to you the interior of the rocket (this not a realistic story). What do you see from your godlike point of view? The astronaut lets go of the paperclip and the golf ball and they hang motionless in space. How could they not do? The rocket, after all, is far from the gravity of any planet. But, as the two objects hang there, unmoving, the floor of the cabin accelerates upwards to meet them.
Now, recall that on Earth it was a complete mystery how gravity achieves the trick of adjusting its strength so that it pulls a big mass down at exactly the same rate as a small mass. But, in the rocket scenario, there is no mystery at all. Since it is the floor of the cabin that accelerates upwards to meet the motionless paperclip and golf ball, how could they not meet the floor at the same time?
But, wait a minute, the rocket scenario can explain gravity only if gravity is the same as acceleration. Exactly! Einstein’s genius was to realise that the two things are completely indistinguishable. If the port holes of the rocket are blacked out and the vibration of the rocket is imperceptible, the astronaut experiences exactly the same thing as he would if he were in a blacked-out room on Earth. Gravity, Einstein realised, is acceleration.
Bizarrely, then, we are accelerating and we do not realise it. And, because we do not realise it, we have invented a force to explain what we experience: gravity.
It turns out that there is a point of view from which this fact is completely obvious, just as in the case of the rocket. But to appreciate it, it is first necessary to know a little background.
The rocket thought experiment showed Einstein that gravity and acceleration are the same. If he could therefore find a theory of what the world looks like from the point of view of an accelerated person, he would automatically have a theory of gravity as well. Two theories for the price of one. But how? At this point, Einstein, still working as a Swiss patent clerk, had what he later called his greatest thought. ‘The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly, the thought struck me: If a man falls freely, he would not feel his own weight.’
Why was this a breakthrough? Well, since gravity and acceleration are the same thing, someone experiencing no weight – that is, no gravity – would not be accelerating. In other words, his situation would be described perfectly by special relativity, a theory that depicts what the world look likes to a non-accelerating observer. Not only had Einstein found that a theory of acceleration is one and the same thing as a theory of gravity, he had found the crucial bridge that connects it with special relativity, which he already had in his possession. A falling person feels no gravity and therefore his view of the world is described by special relativity.
A person accelerating with respect to him – that is, one experiencing gravity – could at each instant be assumed to be moving at constant speed. It was therefore possible to use special relativity to predict what his world looked like at one instant, then at the next instant, and so on.
Not surprisingly, since time appears to slow for someone moving with respect to you, time also appears to slow for someone accelerating with respect to you. But, since acceleration and gravity are the same, this means time flows more slowly for someone experiencing stronger gravity.
In other words, gravity slows time.
Take two people working on the ground floor and top floor of a building. The person on the ground floor is closer to the mass of the Earth, and so experiences marginally stronger gravity. Time therefore flows more slowly for them. If you want to survive a long time, live in a bungalow.
This slowing of time, or time dilation, is fantastically tiny, and you would need a super-precise atomic clock to show it. But, incredibly, in 2010, physicists at the National Institute of Standards Technology in the US were able to show that, if you were to stand one step lower than someone else on a staircase, time would flow marginally more slowly for you.3
The slowing of time is appreciable, however, when gravity is strong. And the strongest source of gravity we know of is a black hole.4 If you could hover near the edge, or horizon of a black hole, time would flow so slowly for you that you would be able to watch the entire future history of the Universe flash past your eyes like a movie in fast-forward.
Back to that question – so far unanswered – of why, if gravity is just acceleration, do we not realise we are accelerating?
Think of the rocket accelerating at 1g again. Imagine the astronaut shines a laser beam across the cabin, from one wall to the other, perfectly horizontally – at a height, say, of 1 metre above the floor. What does he see? The beam strikes the far wall of the cabin at a height of less than 1 metre.
This may seem peculiar. However, it is not unexpected. Although light is the fastest thing in the Universe, it nevertheless takes time to cross the cabin. And, during its flight, the floor of the cabin accelerates up towards it. The astronaut therefore sees the light beam curve downwards towards the floor. (For an acceleration of as little as 1g, the effect would be very tiny but it would be measurable by precision instruments.)
Two things. First, one of the defining characteristics of light is that it always takes the shortest path between any two points. The astronaut would therefore have to conclude from the trajectory of the laser beam that the shortest path in an accelerating rocket is not a straight line but a curve. Secondly – and this is the big thing – since acceleration is indistinguishable from gravity, the astronaut would have to conclude that the path of a light beam in the presence of gravity is a curve. In other words, gravity bends light.
Actually, Einstein had guessed, even before he came up with his general theory of relativity, that gravity bends the path of light. Special relativity, after all, predicts that all energy has an equivalent mass and therefore is affected by gravity (not to mention exerts gravity too). Since particles of light, or photons, possess energy, they have an effective mass and so should be bent by gravity. (‘Photons have mass?!?’ said Woody Allen. ‘I didn’t even know they were Catholic.’5)
Einstein’s theory of gravity, however, adds a new and subtle twist to this light bending. The claim is that gravity and acceleration are equivalent. But, in the case of the rocket, this is not completely true. From the point of view of the astronaut, the two objects they release ‘fall’ towards the floor along parallel trajectories. However, this is not what happens if the same two objects are dropped on Earth. The reason is that gravity is always directed towards the centre of the Earth (in the extreme case of people living on opposite sides of the Earth, gravity pulls in opposite directions).6 Because of this effect, the bending of a light beam by gravity is twice as big as naively expected.
Einstein’s prediction of the gravitational bending of light was triumphantly confirmed on 29 May 1919 during a total eclipse of the Sun. Since in a total eclipse the glare of the Sun is blotted out by the Moon, it is the only time stars can be seen very close to the disc of the Sun.7 As their light passes the enormous mass of the Sun on its way to the Earth, it should be deflected from its path by the gravity of the Sun. Sure enough, an expedition led by English astronomer Arthur Eddington to Principe, an island off the west coast of Africa, confirmed that the light bending was exactly as predicted by Einstein’s general theory of relativity – twice the value expected from special relativity.
The bending of light by gravity provides the vital clue to answering the question: if gravity is acceleration, why do we not realise we are accelerating? Light, recall, always takes the shortest path between two points. Why, then, in the presence of gravity, does it follow a curved path?
Think of a hiker taking the shortest path through a range of hills. From the point of view of a high-flying bird, it is clear that the hiker does not follow a straight-line path. Instead, because of the undulations of the landscape, he pursues a tortuous curved path. The shortest path through a curved landscape is therefore not a straight line but a curve. See the parallel? If light follows a curved path in the presence of gravity, then it implies space in the presence of gravity is curved.
In fact, this is all gravity turns out to be: warped space, or, more precisely, warped space–time.
Nobody, before Einstein, suspected this. And no wonder. Space–time is a four-dimensional thing – it extends in the directions north–south, east–west, up–down and past–future. Since we are mere three-dimensional creatures, we are incapable of experiencing a four-dimensional reality directly.
Now, finally, we can understand why we are pinned to the surface of the Earth. There is no ‘force ’ of gravity pinning us there – no invisible elastic holding us to the ground. Instead, space–time in the vicinity of the Earth is warped. We are at the bottom of a shallow valley of space–time. And we are accelerating downwards as surely as a ball heading to the bottom of a real valley. Only there is something in the way, stopping us: the ground. It is preventing us from falling. By pushing back, it is giving us the sensation of gravity.
It is no wonder that nobody before Einstein guessed that gravity was acceleration. Not only can we not see the valley of space–time we are in but the Earth’s surface is obstructing our free fall.
Take another familiar example: the Moon, orbiting the Earth, is not held in the grip of the force of gravity as if attached to the Earth by a long piece of elastic. Instead, according to Einstein, the Earth warps the space–time around it, creating a valley. And the Moon flies around the rim of the valley like a roulette ball around a roulette wheel.
We realise none of this because we cannot directly experience the warpage of space–time. It took the genius of Einstein to guess its existence.
This analogy may help. You are a passenger in a car that makes a sharp turn. You feel yourself thrown outwards. And you attribute this to a force. If you know any physics, you will call it centrifugal force.
However, from the point of view of someone standing beside the road, no such force exists. You, the passenger, are simply continuing to move in a straight line. As the car rounds the bend, it is the body of the car that comes towards you. In the rocket example, the astronaut thinks he is experiencing gravity. But, from the point of view of someone outside (admittedly with X-ray eyes), there is no such force. He is just floating motionless. It is the floor of the rocket that comes up to meet him.
So, living on the surface of the Earth, we think there is a force of gravity because we do not realise we are accelerating and have hit something unmovable – the ground. We are accelerating because, unknown to us, space–time is curved.
There is no such thing as the ‘force ’ of gravity. We are simply moving under our own inertia through curved space.
Einstein’s theory of gravity – the general theory of relativity – can actually be encapsulated in a single sentence. It is due to the American physicist John Wheeler, who coined the term ‘black hole ’. ‘Matter tells space–time how to curve,’ said Wheeler, ‘and curved space–time tells matter how to move.’ That’s all there is to it.
The devil, of course, is in the detail. General relativity is notorious for being easy to describe in words – and even in mathematical equations – while its implications in the real world are very hard to tease out.8 Not only that but spotting the hand of general relativity in the outside world is extremely difficult. This is because its predictions tend to diverge from those of Newton’s law of gravity only when gravity is strong. And gravity, on the Earth and in the Solar System, is very weak. If you do not think the Earth’s gravity is weak, hold your arm out straight from your body. The Earth has a mass of 6,000 billion billion tonnes yet the gravity of all that matter is incapable of pulling your arm downwards.
The general theory of relativity would have been found earlier had the evidence for it not been so subtle. But there was no need for it. Einstein’s motivation was simply to generalise a theory he himself had concocted. This makes the general theory very unusual in the annals of science. Perhaps uniquely, it was not motivated by an observation of the world that did not fit the prevailing theory. Instead, it was one man’s obsession.
Nevertheless, at the time Einstein was devising his general theory of relativity, there was an observation that contradicted a prediction of Newton’s law of gravity. Few knew about it, let alone considered it important. It concerned the orbit of the planet Mercury.
Newton had discovered that the force of gravity between two masses weakens in a very particular way: if their separation is doubled, the force becomes four times weaker; if they are moved three times as far apart, it becomes nine times weaker; and so on.9 Newton further showed, in a mathematical tour de force, that the path of a body, under the influence of such an inverse-square-law force, is an ellipse.10 This explained the observation of the German astronomer Johann Kepler that the orbits of the planets around the Sun are not circular, as the Greeks maintained, but elliptical.
Actually, it is not quite true that each planet moves under the influence of an inverse-square-law force directed towards the Sun. In addition to being tugged by the Sun, each planet is tugged by every other planet, most significantly Jupiter, which is about 1/1000th the mass of the Sun. As a result, its orbit is not an unchanging ellipse but one that very gradually changes its orientation in space, or precesses, tracing out a rosette-like pattern. This is as true of Mercury as it is of any other planet. However, puzzlingly, Mercury’s orbit precesses above and beyond that expected from the effect of the combined pull of all the other planets.
The mystery of Mercury’s anomalous precession – the planet traces out a rosette that repeats roughly once every 3 million years – is explained by general relativity. According to Einstein, all forms of energy have an effective mass – heat energy, light energy, sound energy, and, crucially, gravitational energy. This means that, like all mass, it gravitates. In other words, gravity creates more gravity.
The effect of this is tiny and appreciable only where gravity is relatively strong – close to the Sun. Mercury is the planet closest to the Sun. Consequently, it experiences slightly stronger gravity than Newton would have predicted. Since a planet orbits in an exact ellipse only under the influence of a Newtonian inverse-square-law force, general relativity predicts that Mercury should show an anomalous precession, over and above that caused by the pull of the other planets. Einstein calculated the effect. Mercury, he discovered, should trace out a rosette in space that repeats about once every 3 million years. Exactly what is observed.
The fact that Einstein’s theory can explain such an esoteric observation as the anomalous precession of Mercury was hardly likely to set the world on fire. What did that was the confirmation of the bending of light by gravity during the total eclipse of 1919. The observation – a confirmation by an Englishman of the prediction by a German, coming so soon after the catastrophe of the First World War – propelled Einstein into the scientific firmament. Instantly, he was hailed as the greatest physicist since Isaac Newton.
Light bending by the gravity of the Sun confirms that energy warps space–time while the anomalous precession of the orbit of Mercury confirms that all forms of energy – including gravitational energy – have gravity. But another prediction of Einstein’s theory is that gravity slows down time. Long before it could be checked on Earth with super-sensitive atomic clocks, the effect was looked for in space – in the light emitted by white dwarfs.
A white dwarf is the endpoint of the evolution of a star such as the Sun. Having exhausted its heat-generating nuclear fuel, such a star continues to shine as its stored internal warmth gradually trickles away into space. A white dwarf packs the mass of the Sun into a volume no bigger than the Earth, making each sugar-cube-sized chunk of its matter roughly the weight of a family car. Crucially, such a dense object has a surface gravity about 10,000 times stronger than that of the Sun, which means that, according to Einstein, time should flow noticeably more slowly than on Earth.
For such an effect to be observable, the surface of a white dwarf must possess a clock that is easily visible. Remarkably, it does.
An atom of a particular element, such as sodium or iron, emits light of characteristic colours. These are unique to the element and are essentially its light fingerprint. Colour is merely a measure of how fast a light wave oscillates up and down. Such a regular oscillation is exactly like the ticking of a clock. And, sure enough, when astronomers observe a white dwarf and the light coming from a particular element on the star, they find that it oscillates more sluggishly than it would on Earth. In other words, the clock on the white dwarf ticks more slowly. And that slowing is precisely that predicted by Einstein.
Red light oscillates about half as fast as blue light. Consequently, the slowing down of the vibration of light shifts the light towards the red end of the spectrum.11 This is why it is called the gravitational red shift.
And this red shift – or red shirt, as an article in the science magazine New Scientist once referred to it – is observed in other contexts too. For instance, when astronomers observe light from distant galaxies, they find that it too is oscillating more sluggishly than it would on Earth. This is the cosmological red shift. And it has the same cause as its counterpart on a white dwarf.
When we see distant galaxies, we see them as they were when the Universe was younger because their light has taken a long time to travel to us across space. When the Universe was younger, it was smaller. This is because the Universe is expanding, its constituent galaxies flying apart like bits of cosmic shrapnel in the aftermath of the big bang. Distant galaxies therefore inhabited a Universe where cosmic matter was squeezed to a higher density on average and had correspondingly stronger gravity than today. In such a Universe, time flowed more slowly than today, according to Einstein. We observe this in the sluggish oscillation of the light from distant galaxies – the cosmological red shift.
In physics, however, there is often more than one way to skin a cat. Another, entirely equivalent, way of viewing the red shift of light from distant galaxies and from white dwarfs is to say that, in climbing out of the strong gravity, the light loses energy. Since the energy of light is related to how fast it is oscillating, with high-energy light oscillating quickly, light that loses energy oscillates more sluggishly. It becomes red-shifted.
There remains one prediction of Einstein’s theory of gravity that is yet to be confirmed directly: gravitational waves. In the general theory of relativity, space–time is not merely a passive canvas against which the drama of the Universe is played out. It is an active medium that can be warped by the presence of matter. In fact, it can be jiggled up and down too, creating a wave that propagates outwards like concentric ripples on a pond. This is a gravitational wave.
But space–time is not as elastic as the skin of a drum. It is a billion billion billion times stiffer than steel. This makes jiggling space–time to create strong gravitational waves very hard indeed. In practice, it requires some of the most extreme upheavals of matter in the Universe – the merger of two super-dense neutron stars12 or black holes.13
But significant gravitational waves are generated by neutron stars and black holes long before they coalesce into one object – in fact, when they are still spiralling together. In 1974, this permitted an ingenious and elegant test of Einstein’s theory. American astronomers Russell Hulse and Joseph Taylor discovered a system that consists of two neutron stars in orbit about each other. One is a pulsar, which, as it spins, sweeps a lighthouse beam of radio waves around the sky.
By carefully observing the binary pulsar, or PSR B1913+16, Hulse and Taylor determined that the two neutron stars are spiralling together, getting closer by about 3.5 metres each year. In the jargon, they are losing orbital energy. And, crucially, this lost energy is exactly the amount Einstein’s theory predicts they should be radiating into space as gravitational waves. For this indirect proof of the existence of gravitational waves, Hulse and Taylor shared the 1993 Nobel Prize for Physics.
The race is now on to detect gravitational waves directly. In the beginning, people looked for them with huge suspended metal bars. The theory was that such a bar, when buffeted by a passing gravitational wave, would ring like a bell. But a myriad mundane terrestrial vibrations, such as waves sloshing on beaches thousands of kilometres away, can drown out such a minuscule signal.
In recent years, the technology of choice has been the laser interferometer which attempts to measure the deformation of space with rulers made out of laser light. Gravitational waves have the peculiar property that, as they pass, they simultaneously stretch matter in one direction while squeezing it in a perpendicular direction. Giant gravitational-wave detectors, each with two perpendicular arms to detect this effect, have been built in Europe and the US. The Laser Interferometer Gravitational Observatory (LIGO), for instance, consists of detectors in two different US states, and has perpendicular arms 4 kilometres long.
Physicists operating such detectors face apparently insurmountable difficulties. By the time gravitational ripples from even the most powerful astrophysical sources reach the Earth, they are enormously attenuated by distance. Experimenters are faced with detecting a deformation of space so tiny that it would change the distance between the Earth and the Sun by less than a tenth of an atomic diameter.
Likely sources of a strong enough pulse are not only a pair of neutron stars or black holes spiralling together but the birth of a black hole in the catastrophic collapse of the core of a star. The latter process is believed to occur when an extremely massive star detonates as a supernova.
Although there is an enormous amount of indirect evidence that black holes exist, there is no direct evidence because, by their very nature, they are very small and very black. However, in the formation of a black hole, the membrane, or event horizon, that surrounds it is expected to vibrate violently, generating copious gravitational waves. Crucially, just as the sound from a bell is unique to the bell, revealing its shape and size, the ripples in space–time spreading outwards from the birth of a black hole are expected to be an unmistakable signature of the event. The detection of the birth cry of a black hole will not only confirm Einstein’s theory of gravity but at long last will provide the definitive proof of the existence of black holes.
The comparison of gravitational waves with sound waves is apt. For all of human history, we have obtained our knowledge of the Universe essentially from light – our sense of sight.14 As far as the Universe is concerned, we have been stone deaf. Gravitational waves are the sound of gravity. Once we detect them, we shall at last be able to ‘hear’ the Universe.
1 Although relativity predicts that someone moving relative to you should appear to shrink in the direction of their motion, this is not exactly what you would see. There is another effect at play. Light takes longer to reach you from more distant parts of the person than from nearer parts. This causes them to appear to rotate. So, if their face is pointing towards you, you will see some of the back of their head. This peculiar effect is known as relativistic aberration, or relativistic beaming.
2 The disintegration, or decay, of muons is an unpredictable, random process. However, physicists talk of their half-life. After a period of one half-life, half the muons are left; after two half-lives, half as many again – that is, a quarter; after three half-lives, one-eighth, and so on.
3 This scenario is not difficult to imagine. Think of two fireworks that appear to go off at the same time from the point of view of someone standing midway between them. Switch to the point of view of someone else who sees one firework behind the other. The light from the most distant explosion will arrive later at their location, so they will see the two events at different times.
4 See Chapter 18, ‘The roar of things extremely small: Atoms’.
5 According to an idea proposed in 1964 by English physicist Peter Higgs and five others, the mass of fundamental particles such as the electron is not intrinsic but extrinsic. It is endowed on them by their interaction with the Higgs field, which pervades all of space. The field is like an invisible cosmic treacle that impedes the passage of subatomic particles. Resistance to motion is what we think of as mass. If you push a loaded fridge, it resists. In the Higgs picture, this is because you are pushing it through the cosmic treacle. The Higgs particle is the quantum of the Higgs field, just as the electron is the quantum of the electric field.
6 Technically, the space–time interval that is the same for all observers is √(x2 + y2 + z2 – c2t2), where x, y, z are the space interval between events.