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Why Maxwell equations are not invariant under Galilean transformations, but invariant under Lorentz transformations? What is the deep physical meaning behind it?

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Are you looking for an explicit demonstration of these properties, or what....I mean, that set of equation simple has those mathematical properties. It's sort of like asking why a square ninety degree angles and not sixty degree one. The deep physical meaning is that physics is Einsteinian and not Galilean. –  dmckee Sep 11 '12 at 0:52

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The simplest answer is that Galilean transformations do not preserve the invariance of light's speed but Lorentz transformations do. Are you looking for something deeper?

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It tells you that the phenomena of electromagnetism are inherently relativistic and unlike mechanics there is no "Newtonian" low-velocity non-relativistic limit. Hence the equations have Lorentz structure built into them.

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You can make a quasistatic approximation when the timescale of the sources $T$ and the size of the system $L$ are such that $L/T \ll c$, or $L/c \ll T$. This means that the field propagates through the entire system much faster than the sources vary. This is a non-relativistic approximation since we get action at a distance. Of course, we can't capture all EM phenomena in this approximation. In particular, radiation is usually studied in the opposite regime $r/c \gg T$ where $r$ is the distance to the source. –  Robin Ekman Jul 6 at 18:05

This is because electromagnetic radiation is a manifestation of some properties of the spacetime, consequently it has to be invariant under those transformations which preserves the spacetime interval invariant.

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