Failure of the Galilean Transformation

The speed of light was known by the Greeks to be high - much higher than the speed of sound. You can see a bolt of lightning seconds before you hear the thunderclap. There was no, no reason for scientists to believe that light did no obey the Galilean Transformation.
Anyone carrying a light source would emit light that traveled faster in their direction of travel, and slower to a stationary observer behind the.
Repeated experiments showed however, that the speed of light was a fixed number
m/s, whatever speed of source and observer. Light, and in fact all electromagnetic radiation did not obey the Galilean transformation.
In addition there was the puzzling form of the Lorentz Force
\[\mathbf{F}=q \mathbf{E}+q \mathbf{v} \times \mathbf{B}\]
The Lorentz law implied a force that depended on speed. Because the speed changes under the Galilean Transformation, so should the force, then using
\[\mathbf{F}=m \mathbf{a}\]
, so should the acceleration. This directly contradicts the Galilean transformation, which implies forces and accelerations are unchanged by the Galilean Transformation.

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