So maybe you’re a non-physicist, who wonders how physicists think. Maybe you aren’t really sure how those crazy physicists come up with all of these equations and theories seemingly from thin air. Maybe you’re getting a bit bored of F=ma posts1. Well, I’m going to try my best to give you a bit of an inside look at some of the conceptual tools commonly used by physicists in a little series called: The Physicist’s Toolbox.
This week: Symmetry
Physicists are obsessed with symmetry, perhaps even more than people with an OCD. It’s human nature to look at something symmetric and call it beautiful. Even physicists call their theories and equations beautiful if they contain some kind of symmetry. The aesthetic attraction humans have towards symmetric things is not a surprise to me; the laws of the universe are founded on symmetry. It’s only natural that the things we see every day in our world reflect this symmetric undertone of the universe and we call these things beautiful.
…so what is symmetry, you ask?
You probably have a good idea of what symmetry is by just looking around your everyday life. You probably look in the mirror every morning and notice that our bodies are symmetric. If one draws a vertical line down the center of the human body, the right side is approximately the mirror reflection of the left side. This is called reflectional symmetry but it is one of many types of symmetry.
You can make the idea of symmetry more general by roughly saying that something is symmetric if you do something to it and it stays the same. The “doing something” part, physicists like to call an operation. This could mean anything; reflection, rotation, translation, magnification, etc. So to be more specific, physicists say that some “thing” is symmetric under a specified operation.
Look at that picture of the square up above, for example. Ignore the colorful dots and just concentrate on the shape. If I rotate the square by any multiple of 90 degrees, it will look exactly the same as if I hadn’t done anything. So you could say that the square has rotational symmetry. The operation here is a rotation by an angle which is a multiple of 90 degrees. So an even better thing to say is that the square is symmetric under rotations by angles of multiples of 90 degrees.
…okay, but what does this have to do with physics, you ask?
This is the cool part. In physics, instead of looking at shapes of things and studying how the shape of something is symmetric under an operation, we go a bit further. We study how the rules of nature stay the same under a certain operation. Let’s look at a concrete example. Let’s say I do a little experiment: drop a ball from some height and time how long it takes to hit the ground. Then I take myself, the ball and the entire earth and move it ten feet to the left (in other words, I preform a translation on the system). Now I redo the same experiment. The ball should take exactly the same time to hit the ground if dropped from the same height. So, the rules of nature are symmetric under translation in space.
Since so many people have heard about Einstein’s Special Relativity, let’s use that as an example too. Einstein postulated that the speed of light is constant for all “inertial” observers. This is a statement of symmetry. It’s saying that if you preform an operation on an observer, the speed of light should stay the same for that observer. The profound insight was that this was even true for operations that changed the speed of the observer. These operations are called Boosts. (Note: I don’t mean acceleration. It’s not the same thing as a boost. I mean this as a mathematical concept.) So, if you consider two different observers, with different (constant) velocities, the speed of a light ray will be the same for each of them. Noticing that the constancy of the speed of light is symmetric under boosts leads to crazy results which you probably already know about (time dilation, length contraction, etc).
In fact, when physicists notice (or impose) types of symmetry in their theories, different laws of physics just fall out of the equations. Some of these are:
- Symmetry under spatial translation gives conservation of momentum
- Symmetry under time translation gives conservation of energy
- Symmetry under boosts gauge transformations gives conservation of charge (I plead temporary insanity)
- … and much more
Symmetry is has become such a useful tool that physicists have come to assume that physics theories should abide by some standard symmetries. This is partly the reason you hear crazy ideas like the world having ten dimensions. String theory starts with the assumption that there are “strings”, it then imposes symmetry arguments and what falls out is: the world has ten dimensions.
But what happens when symmetry breaks and the laws of nature become a bit lopsided? Hey it happens! Nature isn’t a perfectly spherical cow.
That’s when things get even more interesting…
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Update: An interesting TED talk on Symmetry.
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1. I shouldn’t even bother with F=ma posts anymore. Rhett at Dot Physics is totally owning everyone with his super interesting classical mechanics posts.
