It seems maybe this has something to do with an overarching notion in Physics that things that are symmetrical look nice but I need to be educated on a deeper level about how this impacts the outcome of whether a theory is right or wrong. I have taken an undergraduate EM course and recently started reading special relativity on my own. Nowhere in the course did it mention transformations. So I'm still trying to wrap my head around the popular statement that Maxwell's equations are invariant under the Lorentz transformation. What does this mean and are there any examples in daily life that this can be applied to? More generally, why do we like transformations that don't change the physical laws? Is it a problem if a transformation does change the laws? Does the fact that the equations are invariant under Lorentz make Lorentz a "correct" transformation in the sense that makes it wrong to say that if someone is moving at 100 and I'm moving at 90 in the same direction then he's moving at 10? Does the Lorentz transformation have any other advantages over the Galilean other than the fact that it preserves the speed of light? Please do not judge the phrasing of my questions too harshly. I hope you get the gist of the matter. I just mean to know why we said that phrase in the first place and how it makes physics tick.
[Physics] the significance of Maxwell’s equations being invariant under the Lorentz transformation
lorentz-symmetrymaxwell-equationsspecial-relativity
Best Answer
It's not really a matter of symmetry groups of equations. A simple model of waves on water is $\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0$, and this is invariant under Lorentz transformations with speed of light $1$ (for the same reason $dt^2-dx^2$ is invariant under Lorentz transformations with speed of light $1$). The invariance of the simple wave equation didn't have any significance for physics.
The idea, thinking as a physicist in the year 1900 or earlier, is that either Galilean invariance is wrong or Maxwell's equations are wrong.
Possibility one: there is a privileged reference frame in which Maxwell's equations hold, possibly complicated by a "luminiferous ether" which might be "dragged along" by Earth like a fluid. The fundamental laws would obey Galilean invariance, but Maxwell's equations hold in the frame in which the "luminiferous ether" is at rest. No one ever got anywhere or did any useful physics with this approach.
Possibility two: Maxwell's equations hold in all inertial reference frames, and Galilean relativity is simply wrong. This seemed untenable to all but Einstein, because it implied no notion of simultaneity can exist, that time passes differently for different inertial observers, and all other weird special relativistic phenomena. However, it has provided countless experimentally confirmed predictions; it's how nature works.