But the end of the lecture hinted at something interesting about symmetry. Apparently, the laws of conservation in the previous lecture relate in some way to the symmetries in this one.
1. The first part of the chapter explores several different kinds of symmetry in nature. He defines symmetry as being able to change something in some way and it looking exactly as before.
The ones he presents are:
- symmetry of translation (if you move everything over, it operates the same)
- translation of time (if you move stuff forward in time, the laws of nature operate the same)
- rotating things (if you turn everything a certain way, they operate the same)
- motion in a straight line (relative to your frame of reference, everything operates the same)
- atom replacement (you can replace one atom with another same atom and everything works the same)
- If the universe had a beginning, then the translation of time symmetry wouldn't work for the beginning of time, but usually scientists don't include that part of time in their discussion of such symmetry.
- Einstein's theory of relativity sets the speed of light as an absolute speed limit for the universe, so motion in a straight line only is symmetrical within a frame of reference, not from one frame of reference to another moving relative to it.
- Change of scale. If you make something bigger, it does not operate the same way. If you took a matchbox cathedral and increased the scale to the size of an actual cathedral, it would collapse under its own weight. This relates to one of the random discussions on Big Bang Theory on whether a giant rat was possible.
- Spinning is not symmetrical in the sense that you can tell if you are spinning (e.g., with a gyroscope). By contrast, you cannot tell if you are moving at constant velocity without looking outside your car.
- One of the most interesting apparent non-symmetries is reflection. I did a quick search to see if this non-symmetry still holds. From my brief look, it seems like it does.
But when he gets down to the atomic level, apparently all electrons spin left. You can create a positron (positively charged electron) that spins right, but it isn't (if I understand correctly) an exact mirror. So on the subatomic level, symmetry doesn't hold. You can't create a right-handed electron.
3. So we arrive at the pay off of the chapter, which is a deep connection between symmetry and laws of conservation. He lines them up this way:
- symmetry of translation relates to conservation of momentum
- symmetry in time relations to the conservation of energy
- symmetry in rotation relates to the conservation of angular momentum