The symmetry principle states that the absolute positions of these arrows can never be determined. Allowing such variations introduces a spurious current unless we add an extra ingredient to exactly compensate for it. This extra mathematical ingredient is the electromagnetic field. The presence of this field explains the physical properties we associate with electromagnetism. In the 1920s Maxwell's unification of electricity and magnetism, together with Einstein's new general theory of relativity, inspired Theodor Kaluza and Oskar Klein to suggest that it might be possible to unify electromagnetism and gravity in an overarching geometrical scheme involving extra dimensions.
General relativity is a wonderful example of a geometrical theory.
So Lorentz symmetry incorporates the fact that the results from physical experiments are independent of the direction from which we view them and of our velocity. Indeed, our modern geometric picture of gauge theories makes it seem almost natural. First, it failed to explain why the strength of the electromagnetic force is quite large while the gravitational force is fantastically weak. Second, quantum mechanics, which was developing rapidly at the time of Kaluza and Klein, could be incorporated into the theory of electromagnetism rather neatly, but not into the theory of gravity. Quantum gravity seemed to be plagued by infinities, which rendered calculations of physical processes useless.