It is only slightly overstating the case to say that physics is the study of symmetry.
PHILIP ANDERSON 1
Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?
WILLIAM BLAKE, Songs of Experience (1794)
The world is not without order. The Sun comes up every morning. People grow old not young. An effect always follows a cause. There is a regularity to the loom of the world. ‘Nature uses the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry,’ said American physicist Richard Feynman. That organisation hints that, behind the scenes, beneath the skin of reality, there are rules – laws – that orchestrate the world.
For much of human history, because of a belief that a Supreme Being was running things, people did not dig beneath the surface of reality. But Newton changed everything. Despite being a deeply religious man, he wanted to know the mind of God. He wanted to know the rules of the game by which the Supreme Being was orchestrating his creation. And, incredibly, Newton fathomed a universal law – one that applies at all places and at all times. The law of universal gravity quantifies how the force of attraction between two bodies depends on their masses and on the gulf of space between them. And it is remarkable in several ways.
First, formulating the law of gravity took a tremendous leap of the imagination, not to mention courage. At the time, the heavens were considered the domain of God. Even the Greeks believed that our planet was made of earth, air, fire and water – and that the heavens were made of an entirely different, and ethereal, ‘fifth essence ’. Newton dared to see heaven and earth as one and the same, subject to the same laws.
But how to tease out one of the laws? Newton’s genius was to realise that a force of attraction exists between all masses, and that that force not only causes an apple to fall towards the Earth but the Moon to fall as well. The Moon, of course, does not look as if it is falling. However, the trajectory of a massive body in the absence of any force is a straight line, as spelled out in Newton’s first law of motion (This not dissimilar from the law of cat inertia: ‘A cat at rest will tend to remain at rest unless acted upon by some outside force such as the opening of cat food, or a nearby scurrying mouse.’2) Something must therefore be continually bending the Moon’s path away from a straight line and towards the Earth. The something is the force of gravity between the two bodies. The Moon is falling towards the Earth as surely as an apple falls from a tree. The difference is that, as fast as the Moon falls, the Earth’s surface beneath curves away from it. So it never gets any closer. Instead, it falls in a perpetual circle.
The Moon is falling far more slowly than an apple – it takes 27 days, after all, to circle the Earth. This enabled Newton, who knew how much further away the Moon was from the centre of the Earth than a falling apple, to deduce precisely how the force of gravity weakened with distance. It turns out it obeys an inverse-square law. In other words, the force of attraction between two masses becomes four times weaker if their separation is doubled; nine times weaker if it is tripled; and so on.
The details are not crucial. The key thing is that Newton had identified a universal law of physics. Today, we know that his law of gravity not only describes the fall of an apple from a tree and the motion of the Moon circling the Earth but also stars orbiting the centre of the Milky Way and galaxies orbiting within great galaxy clusters.3
One remarkable feature of Newton’s universal law of gravity is that it is simple. Because the world around us looks bewilderingly complex, it might be expected that the laws that govern it are similarly complex – so complex that they are quite beyond the capabilities of our puny ape minds to grasp. But they are not. The Universe is so simple that, more than 350 years ago at the very dawn of science, one man was able to discern a universal law – one that operates at all times and in all places, from the beginning of time to the end of time, from one end of the Universe to the other.
But Newton not only discovered a simple and universal law, he discovered one that could be expressed mathematically. His compact formula summarises a huge range of phenomena.4 Specifically, it relates the force of gravity between two bodies to their respective masses and their separation. Scientists ever since Newton have followed the great man’s lead. And they have had ever more success in finding universal laws of nature, all of which are mathematical. This caused the Austrian physicist Eugene Wigner to remark on the ‘unreasonable effectiveness of mathematics in the natural sciences’. Or, as the English physicist Paul Dirac put it, ‘God is a mathematician of a very high order.’5
Most physicists still cannot quite believe that nature really dances to the tune of the symbols and mathematical relationships they scribble on a whiteboard or on a scrap paper. Time after time, they miss the messages in their own equations, just as Einstein missed the big bang in the equations he obtained that described the Universe. ‘Our mistake is not that we take our theories too seriously,’ said Nobel Prizewinner Steven Weinberg, ‘but that we do not take them seriously enough.’
‘The remarkable feature of physical laws is that they apply everywhere, whether or not you choose to believe in them,’ says American astronomer Neil deGrasse Tyson.6 ‘Anyone who believes that the laws of physics are mere social conventions is invited to try transgressing those conventions from the windows of my apartment,’ says American physicist Alan Sokal.7 ‘I live on the twenty-first floor.’ But where do the laws of physics come from?
A clue – or, at least, an incisive way of thinking about this question – came from a German mathematician called Emmy Noether. In 1918, she made arguably one of the most significant discoveries in the history of science. She proved that the laws of physics are consequences of deep symmetries.
Symmetries are aspects of the world that are unchanged, or are invariant, under changes, or transformations.8 ‘A thing is symmetrical if there is something you can do to it so that, after you have finished doing it, it looks the same as before,’ said the German physicist Herman Weyl. Take a starfish, which everyone knows has five arms.9 A rotation of one-fifth of a turn leaves it looking exactly the same. Mathematicians say a starfish has five-fold rotational symmetry.
On the surface this may appear to have nothing to do with physics. However, nothing could be further from the truth. Noether’s discovery is that symmetry gives rise to laws of physics.
Take the fact that, if you do an experiment today or next week, you will, all things being equal, get the same result. This time-translation symmetry spawns one of the most famous laws in physics – the law of conservation of energy, which says energy can neither be created or destroyed, merely morphed from one type into another – for instance from the chemical energy of petrol into the energy of motion of a car.
Then there is the fact that, if you do an experiment in London or New York, all things being equal, you get the same answer. This translational symmetry leads to the law of conservation of momentum, known instinctively by all snooker players. There exists a quantity known as the momentum, which is simply the mass of a body multiplied by its velocity. If a cue ball cannons into, say, a stationary red ball, the combined momentum of the cue ball and the red ball before the collision must be equal to the momentum of the cue ball and the red ball afterwards. This is the law of conservation of momentum.
Then there is a symmetry of two-dimensional space: rotational symmetry. If you do an experiment aligned in, say, a north–south direction, you get the same result as if the experiment is aligned east–west. This spawns the law of conservation of angular momentum. No need to know the details here. But it is the reason an ice skater spinning on the spot spins faster if she pulls in her arms.
But, of course, we live in a world with three dimensions of space and one of time. Actually, because of the constant speed of light, space and time are inextricably linked and we actually live in a universe with four dimensions of space–time. And rotational symmetries of the four space–time dimensions lead to the laws of Einstein’s special theory of relativity. Still other symmetries of space–time spawn his general theory of relativity.
But this is by no means the end of the connection between symmetries and the laws of physics. Not by a long way. Quantum entities such as electrons can be imagined living in totally abstract spaces with dimensions that have a mathematical rather than a real existence. For instance, two quantum particles need six space dimensions to describe them; three particles nine, and so on. And, remarkably, symmetries of these abstract spaces spawn all the laws of quantum theory. In fact, so immensely powerful is Noether’s theorem that all of physics can be seen as a consequence of deep, underlying symmetries. ‘There is no law of physics that does not lend itself to most economical derivation from a symmetry principle,’ said American physicist John Wheeler.
Noether’s discovery that symmetry lies behind the basic laws of nature is the single most powerful idea in fundamental physics. Einstein agreed. On Noether’s death in 1935, in a letter to the New York Times, he wrote, ‘Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.’
But what are we to make of all these symmetries? Well, a striking aspect of all of them is that they are symmetries of nothing. That’s right. The symmetries of normal space and time that spawn the conservation of energy and so on are also the symmetries of empty space and time – the symmetries of a void. And, it goes without saying, that the symmetries of abstract spaces that spawn the laws of quantum theory are also the symmetries of nothing. After all, they are symmetries of abstract mathematical spaces with no real existence. What all this means is that the laws of physics that orchestrate the Universe we live in are exactly the same as the laws of physics of an entirely empty universe. ‘God made everything out of nothing, but the nothingness shows through,’ said the French poet Paul Valéry.
All this is very suggestive. The ultimate cosmological question, after all, is: how did something come from nothing? The laws of physics are telling us that the Universe is closer to a state of nothing than we might have supposed. It is structured nothing. Maybe it was not as hard as we think to go from nothing to something?
As substances cool, they become less symmetric. Take water. It looks exactly the same at every location. However, when water freezes to make ice, the ice develops cracks and bubbles. It looks different at different locations.
Something like this is believed to have happened as the Universe expanded and cooled in the aftermath of the big bang. Originally, it was in a highly symmetric state. But, as the Universe cooled, more and more structure ‘froze out’, reducing the symmetry. Physicists talk of the symmetry being ‘broken’ so that today it is so well hidden that it takes the extraordinary ingenuity of physicists to uncloak it. This means that, when physicists recreate the conditions of the big-bang fireball by smashing together subatomic particles at the Large Hadron Collider, they recreate the more symmetric state of the primordial Universe, where it is easier to see the basic symmetries and deduce the fundamental laws.
Back to the water example again. At a lower temperature, the structured nothing of ice is simply more stable than the unstructured nothing of water. Could this be why we live in the Universe we do? The American physicist Victor Stenger thinks so. There is something rather than nothing, he says, because something is more stable than nothing. We, and everything around us, are simply patterns in the void.
1 Douglas Adams, The Hitch Hiker’s Guide to the Galaxy.
2 The ‘surface of last scattering’ marks the point at which the fireball of the big bang had cooled sufficiently for nuclei and electrons to combine to make the first atoms. Whereas free electrons are good at scattering, or redirecting, light, electrons in atoms are not. Consequently, before the epoch of last scattering, the Universe was a fog, impenetrable to light. After, light was able to travel unhindered in straight lines, and the Universe became transparent. Today, we see the light from this epoch as the cosmic background radiation. See Chapter 21, ‘The day without a yesterday: Cosmology’.
3 If the Universe is a place where space is turned into time, by contrast, ‘museums are places where Time is transformed into Space’ (Orhan Pamuk, The Museum of Innocence).
4 See Chapter 16, ‘The discovery of slowness: Special relativity’.
5 See Chapter 17, ‘The sound of gravity: General relativity’.
6 Charles Misner, Kip Thorne and John Wheeler, Gravitation, p. 937.
7 See Chapter 14, ‘We are all steam engines: Thermodynamics’.
8 The second law of thermodynamics, for all its subtlety, is virtually a tautology. As American physicist Larry Schulman of Clarkson University in New York, points out, it really says only: ‘More likely things are more likely to happen.’
9 The number of ways the components of a body can be rearranged and still make the body – technically, the ‘number of microstates corresponding to a macrostate ’ – is dubbed W in physics. The entropy, S, is then given by: S = k log W, where k is known as Boltzmann’s constant. This is the definitive statement of the second law of thermodynamics. It is one of the most beautiful and powerful equations in all of physics and it is inscribed on the headstone of Boltzmann’s grave in Vienna.
10 A highly ordered state synonymous with a highly unlikely state. Consequently, this has caused much unease among physicists. However, Larry Schulman says the key to why the early Universe was in a highly ordered state is the ‘epoch of last scattering’. At this time, about 379,000 years after the birth of the Universe, the big-bang fireball had cooled enough for atomic nuclei and electrons to combine into the first atoms. Crucially, free electrons interact strongly with photons whereas electrons trapped in atoms do not. Since there were about 10 billion photons of the big-bang fireball for every electron, this meant that, before the Universe was 379,000 years old, matter was blasted apart before gravity could pull it together. Afterwards, however, gravity could no longer be thwarted. And gravity is the key. The matter of the cooling fireball was spread extremely smoothly throughout space. But, although this is the most likely, disordered, state in the absence of gravity, in the presence of gravity, it is actually a very unlikely, ordered, state (the most likely state for matter in the presence of gravity is clumped, as can be seen from the galaxies and stars in today’s Universe). So, even though the distribution of matter in the Universe did not change at the epoch of last scattering, the ‘switching on’ of gravity was responsible for the Universe suddenly finding itself in an extremely unlikely, ordered, state. (See L. S. Schulman, ‘Source of the Observed Thermodynamic Arrow’, Journal of Physics: Conference Series, vol. 174 no. 1 (2009), 12,022.) A similar argument to Schulman’s has been proposed by the British physicist Roger Penrose.
11 James Hartle, ‘The Physics of Now’, American Journal of Physics, vol. 73, issue 2 (February 2005), p. 101; http://arxiv.org/abs/gr-qc/0403001.