Physicists talk a lot about symmetry. Listen to an article about string theory and you might get the idea that symmetry is some sort of mysterious, mystical principle of beauty, inexplicable to the common man or woman.
Well, if it was inexplicable, I wouldn’t be blogging about it, now would I?
Symmetry in physics is dead simple. At the same time, it’s a bit misleading.
When you think of symmetry, you probably think of objects: symmetric faces, symmetric snowflakes, symmetric sculptures. Symmetry in physics can be about objects, but it can also be about places: symmetry is the idea that if you do an experiment from a different point of view, you should get the same results. In a way, this is what makes all of physics possible: two people in two different parts of the world can do the same experiment, but because of symmetry they can compare results and agree on how the world works.
Of course, if that was all there was to symmetry then it would hardly have the mystical reputation it does. The exciting, beautiful, and above all useful thing about symmetry is that, whenever there is a symmetry, there is a conservation law.
A conservation law is a law of physics that states that some quantity is conserved, that is, cannot be created or destroyed, but merely changed from one form to another. Energy is the classic example: you can’t create energy out of nothing, but you can turn the potential energy of gravity on top of a hill into the kinetic energy of a rolling ball, or the chemical energy of coal into the electrical energy in your power lines.
The fact that every symmetry creates a conservation law is not obvious. Proving it in general and describing how it works required a major breakthrough in mathematics. It was worked out by Emmy Noether, one of the greatest minds of her time, which given that her time included Einstein says rather a lot. Noether struggled for most of her life with the male-dominated establishment of academia, and spent many years teaching unpaid and under the names of male faculty, forbidden from being a professor because of her gender.
Noether’s proof is remarkable, but if you’re not familiar with the mathematics it won’t mean much to you. If you want to get a feel for the connection between symmetries and conservation laws, you need to go back a bit further. For the best example, we need to go all the way back to the dawn of physics.
Christiaan Huygens was a contemporary of Isaac Newton, and like Noether he was arguably as smart as if not smarter than his more famous colleague. Huygens could be described as the first theoretical physicist. Long before Newton first wrote his three laws of motion, Huygens used thought experiments to prove deep facts about physics, and he did it using symmetry.
In one of Huygens’ thought experiments, two men face each other, one standing on a boat and the other on the bank of a river. The men grab onto each other’s hands, and dangle a ball on a string from each pair of hands. In this way, it is impossible to tell which man is moving each ball.
From the man on the bank’s perspective, he moves the two balls together at the same speed, which happens to be the same speed as the river. The balls are the same size, so as far as he can see they should have the same speed afterwards as well.
On the other hand, the man in the boat thinks that he’s only moving one ball. Since the man on the bank is moving one of the balls along at the same speed as the river, from the man on the boat’s perspective that ball is just staying still, while the other ball is moving with twice the speed of the river. If the man on the bank sees the balls bounce off of each other at equal speed, then the man on the boat will see the moving ball stop, and the ball that was staying still start to move with the same speed as the original ball. From what he could see, a moving ball hit a ball at rest, and transferred its entire momentum to the new ball.
Using arguments like these, Huygens developed the idea of conservation of momentum, the idea of a number related to an object’s mass and speed that can never be created or destroyed, only transferred from one object to another. And he did it using symmetry. At heart, his arguments showed that momentum, the mysterious “quantity of motion”, was merely a natural consequence of the fact that two people can look at a situation in two different ways. And it is that fact, and the power that fact has to explain the world, that makes physicists so obsessed with symmetry.