Introduction
Hello! In the past I have given some presentations about the mathematical origins of various physical laws, so it seemed like a fun idea to go over these concepts in a few blog posts!
My love for physics always stemmed from a burning curiosity for understanding how the world works (which probably is the case for most physics students), therefore \(i\) have spent quite some time researching the ‘fundamental origins’ of the physics that we get taught in class. Of course, after every ‘why’ is another ‘why’, but trimming the number of questions down is still very satisfying.
In this short series I will try to give mathematical origins for the following concepts:
- Newton’s three laws
- Conservation of Energy and (angular) momentum
- The formula for Kinetic Energy
- Gravity
- The 4 fundamental forces
- Entropy & Temperature
- Schrödinger’s equation
Because this is quite a list, we will require some knowledge of undergraduate physics & mathematics, but I will try to keep it not too difficult.
Special Relativity
Unfortunately, we do need to assume special relativity in order to be able to derive all these other results. Over the years, I have never encountered good intuition for why special relativity must be true, except for the fact that it follows from the speed of light being constant. I do have some ideas related to positive definite Hamiltonians, but those are for another time.
Einstein’s main thought experiment was that if the speed of light is constant for all observers, then the notions of time and space cannot be absolute, but instead they depend on the observer. These distortions at high speeds are called Lorentz transformations, named after the (Dutch!) physicists Hendrik Lorentz. The formula for those is not relevant, but they do imply a very important mathematical fact \[ \Delta s^{2}= c^{2}\Delta t^{2} - \Delta \mathbf{x}^{2} \text{ is constant for all observers}\] Here \(\Delta t\) is the difference is time between two events and \(\Delta\mathbf{x}\) is the displacement vector between two events. Note that the \(c^{2}\) factor is there to make the units work out and we will ignore it from now on (work in units where \(c\) = 1), but it also gives a physical interpretation: \[ \text{If } \left| \frac{\Delta \mathbf{x}}{\Delta t} \right| = c \text{ then } \Delta s = 0\] This has a fundamental implication: observers will always agree on the speed of light, which precisely is the fundamental axiom that Einstein started out with.
On a final remark, note that \(\Delta s\) is imaginary whenever \(\Delta s^{2}\) is negative, which happens if the displacement between two events is so large that even light has not enough time to travel between them. These events are causally disconnected. The fact that \(\Delta s^{2}\) is constant for all observers hence implies that there is no amount of distortion by Lorentz transformations which can make causally disconnected events connected, or vice versa.
Another important axiom of special relativity is that of translation invariance in time and space. This means that experiments always should give the same result, regardless of where and when they are performed, as long as you do them within the same surrounding conditions. In other words, if you are in a box without windows, then there is no experiment that you can do to discover where or when in the universe you are.
Why is this important? Well, if special relativity says that \(\Delta s^{2}\) is the same for all observers and that all experiments are invariant under translations, then that imposes strict requirements on all mathematical quantities representing measurable properties of real world things. These requirements will in fact be so strong, that they will give us many of the aforementioned laws!
Next Time
Next time I will talk a bit about group theory and about how special relativity gives us the Poincaré group, which will be a key player in many of these posts. Until then!