Symmetry in Physical Law - Symmetry Lectures 1
This is my write-up of the first lecture in a series that I mentioned yesterday. The first lecture was fairly brief and was more of an introduction to the style of the lectures. Which was helpful, because the second lecture was by far the most difficult of the six, so having as easier start beforehand was probably a good idea.
With that in mind, I'm going to try to present the content much as it was in the lecture first. Then I'll add some commentary afterwards.
Symmetry is a topic with multiple levels - the first time you encounter the word is probably in Primary School, where you are told that something is symmetric if it looks the same when it's reflected in a mirror. Then, when you get to Secondary School, you might enounter rotational symmetry - something has rotational symmetry if you can take it, rotate it, and it looks the same. This is more nuanced than the simple reflection idea, because we now have degrees of symmetry - we say a square has rotational symmetry of order four, an equalateral triangle of order three, a scalene triangle has order one and a circle has infinite rotational symmetry. Whatever angle you rotate a circle by it looks the same.
This is all very fascinating, sure (at least, it is if you're a maths fanatic; just think, spheres have infinite rotational symmetry around two axes, not just the one for a circle. Can we create a shape with rotational symmetry around a point that's not in the shape? Maybe an infinite cut-out pattern would do that around the holes?). But it doesn't seem relevant to physics yet. Really that's because we're looking at geometrical symmetries. We need a broader definition of symmetry, which is provided by Professor Weyl, who said "A symmetry is something you can do to an object, so that after you have done it, you get the same object back". This is an improvement, because now we can talk about any object and in particular, we want to talk about the symmetry of physical laws.
What does it mean for a law to be symmetric? Well, when we say that we mean that our representation is symmetric - if we pick a co-ordinate system, then we want our laws to be correct if we change co-ordinates. This is quite an abstract idea, so we want to use a concrete example: Newton's Law of Universal Gravitation. The law says that the gravitational force between two objects varies in proportion to the product of the two objects' masses and inversely as the square of the distance between them.
I apologise that I can't format maths in something like LaTeX here, but you could write this as follows:
F=(m_1)(m_2)/(r^2)
Here r^2 means 'r squared'. This law is not dependant upon our co-ordinate frame, because it talks only about properties of the objects. The two objects have masses which are (in the Newtonian view) real and physical. The distance between them is a fact about nature, not about our co-ordinates. This seems trivially true, but some people who know a little maths will want to work out the 'r' in that formula in terms of co-ordinates. They'll say "r^2=x^2+y^2", and now you have a difficulty, because your formula DOES include facts about your co-ordinate system, and it's not supposed to. To show why it's not supposed to, imagine two people are given a physics problem for homework - the question gives some details about an orbiting planet, and asks the students to (a) setup a coordinate system and express the positions of the planet as a function of time, and (b) solve for the orbital period. Now, both students are free to pick different coordinate systems if they choose. One might centre their's on the Sun and use polar coordinates, the other might use rectangular coordinates centred on the Earth. But should they get a different orbital period? Absolutely not! The orbital period is a real thing that you could do an experiment and go and measure - there is no way of changing the answer through sly algebra.
In the lecture, Dr. Low showed that the distance is preserved when you rotate the co-ordinate axes. The first step is to name the new coordinate x' and y', and to write these in terms of the old ones, x and y.
This gives you: x' = xcos(a) + ysin(a) ; y' = ycos(a) - xsin(a) , where a is the angle rotated counter-clockwise around the origin. (You can draw a diagram to see this, or you can make an educated guess that the answer is going to involve trig functions since we're talking about circles. The next clue is that x'=x and y'=y when a=0 , so you expect x'=xcos(a) + correction and then some playing around with rotations of 90 degrees ought to convince you that the formulas given are correct).
Now, our formula depends on distances, not absolute co-ordinates, so we want to show that the quantity: x^2+y^2, is indepenant of our choice of a (i.e. rotating doesn't affect distance). This is quite easily done by simply expanding the definitions and simplifying up with a trig identity. Newton's Laws is therefore independant of the choice of coordinates. (You also should have to show translation invariance, which is that you can move the origin around and the distances don't change. This is again easy enough to show, and is fairly obvious. Nonetheless it is a positive statement that you can use to prove facts about Nature, in particular conservation of momentum).
Why are these symmetries useful? For a start, they tell us if our equations are wrong. If our equations involve facts about th coordinate system, then they are probably not correct. To be correct we should limit ourselves to scalar quantities like mass and vectors like displacement. (But wait, you cry: those are the only two types of number we know! No, a component of a vector is not a vector or a scalar, because it changes based on the coordinate system.) But a more impressive use of symmetry involves a more subtle symmetry: uniform velocity in a straight line. The entirety of special relativity is founded upon the following symmetry:
If you work out the laws of physics, then they will be the same in two co-ordinate systems that are moving uniformly relative to each other in straight lines. More simply, there is no experiment that can tell you whether you are moving at 50mph or 50 000mph. This was long accepted before Einstein came along, in fact Galileo came up with it, and Newton actually proved the statement using his laws of motion. The controversy arose because James Clerk Maxwell later showed the existance of electromagnetic waves. And the electromagnetic waves travelled at 300,000km/s. Not relative to anything, they just travelled at 300,000km/s.
The laws of electricity were pretty new, and relativity was pretty old. It was expected that something had gone wrong. That the speed was relative to something, that the laws were wrong. Again, this was a common opinion, and the real genius of Einstein with Special Relativity was not that re came up with relativity, or even that he championed when others didn't. The master stroke was to challenge not Maxwell's laws, but Newton's. Imagine a clock composed of two mirrors. They're spaced 300,000 km apart, so that a light ray bouncing between them hits a mirror once a second. We now want to know what velcity we're moving at, and have a clever idea: if we're moving faster, then the light will no longer be bounding in straight lines between the mirror, but will be taking a diagonal path through space. The light has longer to go, we measure the difference compared to a regular clock, and because the speed of light is constant we can calculate our velocity. Right?
Well, no. Relativity stops us. We're just not allowed to measure our velocity like this. What's it relative to? The answer is seriously strange: the light clock and our pocketwatch read the same time. Whilst we know that the light clock should be slow, all our other clocks are also running slow: time itself has slowed down. Every conceivable clock must run slow, and that includes timing how long cells take to grow, how fast your heart beats, how quickly you think. That's what it means to slow time. A related phenomena is that energetic particles called muons have lifespans that are on the order of tens of seconds, or minutes (I can't remember which, sorry). But they travel much further than they should be able to in that time - the answer is also strange: the distance they travel foreshortens, letting them cover it quickly. From the muons perspective, the distance shortens, from ours it stays the same, but the muon's lifespan slows down by the same factor. These two phenomena are called time dilation and length contraction. They're given by formulas called the Lorentz transforms which sound scary but are actually derivable from our light clock simply by using Pythagoras' theorem if you want to have a go.
The final topic of the first lecture is tensors. Tensors are mathematical objects that are necessarily symmetric. Scalars are rank 0 tensors. Vectors are rank 1 tensors. A group of objects called matrices are rank 2 tensors and you can extend the idea to arbitrary ranked tensors (up to and including infinity. I don't know about fractional ranks, but I would guess their either impossible by definition or very cool). This was covered very quickly and was more about using matrices to transform vectors than about tensors in general. One of the beatiful things about tensors is the notation for them, which is indicial and follows Einstein sumamtion convention. Basically, you write the symbol for your tensor (say R) and then you give the 'index' of the component of the tensor by writing subscripts/superscripts for the index. Two tensors next two each other implies a product of those components, as always in maths, but if the same index is repeated in both, once subscript and once superscript then a summation is implied.
For instance, R_1_2 represents a single number which is the second element of the first row in a 2-dimensional matrix. R_4 would represent the fourth component of a vector. A dot product between two vectors, which would usually be written in vector algebra something like:
A . B = A_1B_1 + A_2B_2 + A_3B_3 = (Sum over i)A_iB_i
Would be written with tensors as: A . B = A_iB^i. The summation is implied. What's more, you can extend the notation to matrix multiplication effortlessly: A_i_k = A_i_kB^k^j
Tensors are important because they represent symmetrical situations - if you transform the coordinate system but you're working with tensors, then your equation is still valid. That means that you've said something of use about the universe, and not just your particular coordinate system.
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I was going to write some comments here about the lecture, but evidently writing up a one-hour lecture results in a very long blog post. This was the simplest lecture as well, in my opinion, so there probably won't be any comments on this series. What I will do, is reccommend anyone interested in this YouTube channel about Tensor Calculus:
https://www.youtube.com/watch?v=e0eJXttPRZI&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq
Further reading : Why E=mc2 by Brian Cox and Jeff Forshaw. Also pretty much half of all popular science books, but this is a good one to start with in my opinion.
