by Marcus Chown
Generalising relativity
The special theory of relativity swept aside Newton’s concepts of absolute space and absolute time, revealing Newtonian physics to be wrong, although still a fantastically good description of the everyday world. But, despite the theory’s tremendous success in radically transforming our picture of reality, it had several problems.
The first was that it spelled out what must be done to the measurements of space and time of people moving at constant speed relative to each other so that they agree on the same laws of physics – that is, the same laws of motion and the same laws of optics, principally the law of the constancy of the speed of light. But people moving at constant speed relative to each other are not typical. In the real world, observers change their velocity. A car slows to a halt on a red traffic light, then speeds up again on green. A rocket gets ever faster until it attains the speed required to orbit the Earth.
The task facing Einstein was clear. He needed to find 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. Those laws should look the same no matter how people are moving – falling, spinning or being pressed into the seat of an accelerating car. He needed to turn his ‘special’ theory into a ‘general’ theory of relativity.35
There was nothing mysterious about Einstein’s desire. If the laws of physics are to have the status of universal laws they should be independent of our point of view. It should not matter, for instance, whether we are sitting next to a bar magnet, moving past it at constant speed or accelerating past it. We should see the same fundamental law of magnetism.
But special relativity had other problems in addition to not dealing with accelerated motion. Most seriously, it was in fundamental conflict with Newton’s theory of gravity.
All Newton’s law of gravity did was specify the strength of the gravitational force at every distance from a massive body like the Sun. This is tantamount to saying that the gravity of a massive body is felt everywhere instantaneously, and this in turn is equivalent to saying that the effect of gravity propagates at infinite speed. But, according to special relativity, nothing, not even gravity, can surpass the cosmic speed limit set by the speed of light.
The predictions of Newton’s theory of gravity and Einstein’s special theory of relativity would be most at odds if the Sun were to vanish. Obviously, this is an unlikely event! If it did happen, the Earth would notice immediately according to Newton, and fly off on a tangent towards the stars. But, according to Einstein, the planet would continue merrily in its orbit for the time it takes light to travel between the Sun and Earth. Only after an interval of 8½ minutes would it notice that the Sun had gone and head for the stars.
The way to incorporate the cosmic speed limit set by light into a theory of gravity, Einstein realised, was to use the concept of a ‘field’. This had been invented by the English scientist and electrical pioneer Michael Faraday in the early nineteenth century.36 Faraday had a strong sense, when he held a piece of iron in the vicinity of a magnet and felt it gripped by a powerful force, that there was an invisible force-field extending outwards from the magnet. In fact, when he sprinkled iron filings around the magnet he was able to make visible the ‘lines of force’.
In Faraday’s view, a magnet does not exert a force directly on a piece of iron. Instead, a magnet sets up a magnetic field of force around it, like a Star Trek tractor beam, and it is the field that exerts a force on the iron. It might seem like a subtle difference. But not only does this picture give the field a physical existence – in the case of the electromagnetic field, a vibration passing through it is a physical electromagnetic wave (light) -but it admits the possibility of the field propagating outwards at some speed.37
By analogy with electromagnetism, Einstein needed to create a theory in which a mass produces a gravitational field and the gravitational field in turn exerts a force on other masses. Crucially, it is possible for such a field to propagate at a particular speed and so to incorporate the cosmic speed limit of the speed of light.
But creating a field theory of gravity that is compatible with special relativity was only the second of Einstein’s problems. A third problem arose because the ‘source’ of gravity in Newton’s theory is mass. But Einstein had discovered that all forms of energy possess an effective mass, and so exert gravity. Consequently, the ultimate source of gravity cannot be mass. It must be energy.
Einstein had almost certainly been aware of all of these problems with special relativity since its completion in 1905. But things came to a head in October 1907. It was then that he was invited by the German physicist Johannes Stark to write a comprehensive summary of special relativity in The Yearbook of Radioactivity and Electronics.
Einstein was still working at the Swiss Federal Patent Office in Bern, though since 1 April 1906 he had the elevated rank of Technical Expert, Class II. Working after office hours, he completed the review article in two months, delivering it to Stark on 1 December 1907. The first four of its five sections set out the basic ideas of special relativity and worked through their consequences for space, time, matter and energy. The fifth section was entitled ‘The relativity principle and gravitation’.
While other physicists were still struggling with the counterintuitive ideas of special relativity, Einstein had already realised that the theory was nothing more than a beginning. In a letter to his friend Conrad Habicht at the end of December, he said he was pursuing another relativity theory. ‘But so far it does not seem to work out,’ he admitted in a postscript.38
Those were prescient words. It would take him another eight years to achieve his goal of extending the relativity principle to gravity and obtaining a ‘general’ theory of relativity. And it might have taken him even longer had it not been for a crucial insight he had while staring out of a window at the Patent Office.
Further reading
Bais, Sander, Very Special Relativity, Harvard University Press, Cambridge, MA, 2007.
Einstein, Albert, Relativity: The Special and General Theory, Folio Society, London, 2004.
Fölsing, Albrecht, Albert Einstein, Penguin, London, 1998.
Jaffe, Bernard, Michelson and the Speed of Light, Anchor Books, Garden City, NY, 1960.
Overbye, Dennis, Einstein in Love: A Scientific Romance, Viking, London, 2000.
Pais, Abraham, ‘Subtle is the Lord . . .’: The Science and the Life of Albert Einstein, Oxford University Press, Oxford, 1983.
6
Ode to a falling man
How Einstein realised that the ‘force’ of gravity is an illusion and all there really is is warped space-time
If a bird-watching physicist falls off a cliff, he doesn’t worry about his binoculars; they fall with him.
Sir Hermann Bondi1
In some sense, gravity does not exist; what moves the planets and the stars is the distortion of space and time.
Michio Kaku2
A falling person does not feel their weight. This insight, which occurred to Einstein in 1907, would become the foundation stone on which he would build the edifice of a new and revolutionary theory of gravity. But frustratingly, just as with his insight that catching up a light beam is impossible, Einstein did not reveal the precise circumstances that triggered the revelation. Instead we can only speculate. We know that, at the time, Einstein was living and working in the Swiss capital. ‘The breakthrough came suddenly one day,’ he wrote. ‘I was sitting on a chair in my patent office in Bern.’
I imagine Einstein at his desk, considering the last patent application of the day, which he has read all the way to the bitter end:
47242
Allgemeine Elektricitätgesellschaft, Berlin
Nägeli & Co., Bern
Alternating current machine
He dabs the nib of his fountain pen on the blotter before taking a fresh piece of Swiss Federal Patent Office paper from the stationery tray. He h
esitates no more than a second or two to compose his thoughts. Then, swiftly (and somewhat devastatingly), he writes: ‘Point 1: The patent claim is incorrectly, imprecisely, and unclearly prepared.’3
He does not get to ‘Point 2’.
The scream jolts him like an electric shock. Jumping to his feet, he sees the roofer skittering down the tiled roof of the building across the street. The man is flailing his arms desperately and picking up speed inexorably. But, before he flies off the edge of the roof and plummets five storeys to the busy Genfergass – at the very last instant possible – he lunges for a flagpole. It appears too flimsy to hold him. But – miracle of miracles – it bends but does not snap.
I picture Einstein watching the whole drama unfold from the Patent Office on the top floor of Bern’s new Postal and Telegraph Administration. Only when he sees the roofer being hauled back to safety by his workmates, does he sit back down, relieved, at his desk. With his heart still hammering, it is a while before he can focus again on Patent Application 47242.
Is his comment too harsh? Has he allowed himself to be influenced by his father’s bitter experience in Munich, where Elektrotechnische Fabrik J. Einstein бс Cie had competed, unsuccessfully, with a number of highly aggressive companies – among them AEG – for the contract to light the city centre? No, he is confident he is simply being honest not vindictive. But in ‘Point 2’ he is careful to spell out in more conciliatory language his specific objections to Patent Application 47242. He then blots his page, leans back in his chair and looks at his now-empty in-tray with satisfaction.
His boss and saviour, Friedrich Haller, is off in Zurich on business, and his office-mate and friend, Josef Sauter, has taken advantage of Haller’s absence to go to the Bear Pits, where he thinks he left his favourite umbrella at the weekend, and to buy his wife an anniversary present. (With a pang of guilt, Einstein realises he has never bought Mileva an anniversary present.)
The office is empty and quiet. I see Einstein leaning back in his chair to think. He recalls the dramatic events across the street and replays the alternatives in his head. The roofer skitters down the roof and grabs the flagpole which bends but holds his weight. The roofer skitters down the roof, grabs the flagpole but it bends and snaps and he sails out into space.
He imagines what it would be like to fall down to the street and his stomach lurches. He grips the desk. He catches his breath. In such circumstances, he has heard it said, subjective time slows almost to a standstill and a whole lifetime of events parades past one’s eyes. But what if it were possible to fall for ever?
He imagines falling in a place where there is no air or wind to slow him down. He is falling through time and space and stars and sky and everything in between. He is falling until he forgets he is even falling.4
And then it hits him like a bolt of lightning!
He leaps up abruptly, knocking his chair backwards. Instantly, he knows he has stumbled on the foundation stone on which he can build a new reality. In later years, he will call it the happiest thought of his life. It is so obvious that in the empty office he laughs out loud. Of course!
A falling person does not feel their weight!
Did a roofer fall down a roof and furnish Einstein with his moment of inspiration? Or did some other event, something rather less dramatic, trigger his light-bulb moment? Though it is fun to imagine, we will of course never know. All that Einstein told us is that some time in 1907, he had the seemingly innocuous thought that set him on the road to overthrowing the worldview of Newton.
But why is the realisation that a falling person does not feel their weight such a key insight? Picture the following situation.
A man is travelling in a lift when – disaster – the cable snaps.5 Instantly, he finds himself in free-fall. Say, he is standing on a set of scales on the floor of the lift. (This is not a very realistic scenario!) One moment, the scales read 70 kilograms, the next exactly zero. This, in concrete terms, is what it means to not feel your weight when you are falling.
According to Newton, it is impossible to get beyond the pull of gravity since it merely weakens with distance but never completely disappears. According to Einstein, however, it is easy to get beyond gravity. All you have to do is free-fall. Gravity vanishes and you become weightless.
The situation of a falling man who feels no gravity is indistinguishable from that of a man floating in empty space far from the gravity of any planet. This provides a bridge between a theory of gravity and special relativity since, in both situations, the laws of special relativity apply.
The weighing scales read zero when the man is in free-fall because, as fast as he plunges down towards the scales, the scales plunge downwards away from him. In other words, the man falls at exactly the same rate as the scales, despite the man having a mass of 70 kilograms and the scales considerably less.
That all things — not just a 70-kilogram man and a set of scales – fall at the same rate under gravity was first noted by Galileo in the seventeenth century. According to legend, he dropped a heavy mass and a light mass from the Leaning Tower of Pisa and saw that they hit the ground at the same instant.
On Earth, such an experiment is inevitably complicated by the effect of air resistance, which creates a greater drag on a body with a larger surface area. But, in 1972, Apollo 15 commander Dave Scott repeated Galileo’s experiment on the Moon where, of course, there is no air. He dropped a hammer and a feather from the same height. And, sure enough, two simultaneous puffs of lunar dust confirmed that the two objects struck the ground at the same time.
That all bodies fall at the same rate under gravity irrespective of their mass is actually extremely odd. Think of applying the same force to a big mass and a small mass – say, a loaded fridge and a wooden stool. Everyday experience tells us that the fridge will speed up, or ‘accelerate’, the least because bigger masses are harder to budge than smaller masses.6 They have greater reluctance to move, or ‘inertia’. In fact, this reluctance is the very basis of our concept of ‘mass’.
But the weird thing in the case of masses experiencing the force of gravity is that, even though a bigger mass is harder to budge than a smaller mass, the force of gravity appears to adjust itself so it is greater on the bigger mass – and by exactly the amount necessary for it to fall at the same rate as the smaller mass. So a body that has twice the mass of another experiences twice the gravitational force, a mass that is three times as massive, three times the force, and so on. Drop a loaded fridge and a stool from the Leaning Tower of Pisa, or, better still – not only for safety reasons (!) but to avoid air resistance – drop them on the Moon, and they will hit the moon dust just as simultaneously as Dave Scott’s hammer and feather.
Technically, the resistance of a body to being budged depends on its ‘inertial mass’, mt (as embodied in Newton’s second law, which says that, when a body is subjected to a force, F, it accelerates by an amount F/mt). And, technically, the force applied by gravity to a body depends on its ‘gravitational mass’, mg.
A body with twice the inertial mass of another has twice the resistance to being budged. It falls at exactly the same rate as a smaller mass only because it also experiences twice the force of gravity. In other words, a body’s resistance to motion, which depends on its inertial mass, goes up exactly in step with the force of gravity, which depends on its gravitational mass. This is tantamount to saying that gravitational mass, mg, and inertial mass, mg, are identical.
Everyone since Galileo believed that a body’s resistance to being budged and the force it experienced from gravity are two entirely different things. They certainly appear unconnected. It was Einstein’s genius to realise that everyone since Galileo was wrong and that they had entirely missed what was staring them in the face. That a falling person does not feel their weight – or, equivalently, that all bodies accelerate at the same rate under gravity – can mean only one thing. Gravitational mass and inertial mass are the same. In other words, gravity is acceleration.
&n
bsp; As already pointed out, in 1907 Einstein knew that he needed to generalise his theory of relativity so that it described the world from the point of view of not only people moving at constant speed relative to another but also accelerating with respect to each other. He also knew he needed a new theory of gravity since Newton’s theory of gravity is incompatible with special relativity. How remarkable, then, to discover that a generalised theory of relativity is automatically a theory of gravity. It is the ultimate ‘buy one, get one free’.
The power and simplicity of Einstein’s key insight takes a little thought to appreciate. If gravity and acceleration are identical, then there is no need for gravity to adjust itself so that all bodies, no matter what their mass, fall at the same rate. It happens entirely naturally and automatically. This is how . . .
Rocket man
Say an astronaut wakes up on a rocket far from the gravity of the Earth or of any planet. The rocket is accelerating at 1g so his feet are glued to the floor of his cabin and he can walk about inside just as if he is on the Earth’s surface.7 In fact, if the windows of the rocket are all blacked out, he may think he is in a room on the surface of the Earth. Einstein went further than this. His contention was that there is no way the astronaut can prove that he is not on the surface of the Earth. In practice, gravity is indistinguishable from acceleration.
Now say the astronaut – out of boredom or curiosity – tries to recreate the experiment of Galileo and Dave Scott. At shoulder height he holds out a hammer and a feather, before releasing them together. They appear to fall at the same rate and hit the floor of the cabin simultaneously. Of course, the astronaut, who has no idea he is in a rocket and thinks he is on the surface of the Earth, attributes this to gravity, which causes all bodies to fall at the same rate.