How did Maxwell's theory of electrodynamics contradict the Galilean principle of relativity? (Pre-special relativity)
My question is (1) how Maxwell's equations contradicted Galilean principle of relativity.
Maxwell's equations have wave solutions that propagate with speed $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$.
Since velocity is relative (speed c with respect to what?), it was initially thought that the what is an luminiferous aether in which electromagnetic waves propagated and which singled out a family of coordinate systems at rest with respect to the aether.
If so, then light should obey the Galilean velocity addition law. That is, a lab with a non-zero speed relative to the luminiferous aether should find a directionally dependent speed of light.
However, the Michelson–Morley experiment (original and follow-ups) failed to detect such a directional dependence. Some implications are
(1) there is no aether and electromagnetic waves propagate at an invariant speed. This conflicts with Galilean relativity for which two observers in relative uniform motion will measure different speeds for the same electromagnetic wave. This path leads to special relativity theory.
(2) there is an aether but it is undetectable. This path leads to Lorentz aether theory.
A Galilean set of frames are an obvious/common sense way of viewing motion if we assume the validity of 3 also apparently obvious postulates.
All clocks measure time at the same rate, independent of their velocity.
Objects have no limit on their potential velocity.
Rulers have the same length (difference in position between the lengths at a common time), independent of their velocity.
When Maxwell formulated/compiled his equatons, implying that light speed was invariant in every frame, Einstein was forced to consider the implications of this for Galilean transformations and their "obvious" underlying assumptions.
If light speed is invariant in all frames, then something has to give to preserve that invariance, and the 3 assumptions above needed to be abandoned to preserve Maxwell's laws.
How the Maxwell equations retained the same form in all inertial frames by obeying Lorentz transformation?
By the development of the Faraday tensor $F_{\mu v}$ based on a vector potential $\vec A $ and a scalar potential $\Phi $ .
The difference between Galilean and special relativity is the details of how spacetime coordinates change between reference frames. The Galilean transformation $t'=t,\,\mathbf{x}' =\mathbf{x}-\mathbf{v}t$ relates reference frames of relative velocity $\mathbf{v}$. This implies that, if $A$ has velocity $\mathbf{u}$ relative to $B$ and $B$ has velocity $\mathbf{v}$ relative to $C$, $A$ has velocity $\mathbf{u}+\mathbf{v}$ relative to $C$. This implies no speed can be invariant across reference systems. For example, if I shine a torch while on a train that's going past you, you and I should disagree on the speed of the torch's light.
However, Maxwell's theory contains waves of speed $c:=1/\sqrt{\mu_0\varepsilon_0}$, so cannot apply in all reference frames if they are related as per Galileo's formulae. In a region with no electric charges or currents, Maxwell's equations imply the wave equations $$\nabla^2\mathbf{B}=c^{-2}\partial_t^2\mathbf{B},\,\nabla^2\mathbf{E}=c^{-2}\partial_t^2\mathbf{E}.$$
Special relativity still claims physical laws are the same in all reference frames, but they relate their coordinates differently, viz. $$t'=\frac{t-\mathbf{v}\cdot\mathbf{x}/c^2}{\sqrt{1-v^2/c^2}},\,\mathbf{x}' =\frac{\mathbf{x}-\mathbf{v}t}{\sqrt{1-v^2/c^2}}.$$One can show that a change in reference frames preserves the above speed-$c$ wave equations.