Dec 6, 2010

Jose A. Heras

## I. INTRODUCTION

Typically, the nonrelativistic behavior of Maxwell’s equations is not extensively discussed in textbooks focused on electromagnetic theory and special relativity. This might be surprising given the emphasis these textbooks usually place on the relativistic invariance of Maxwell’s equations. The reason for this omission may be due to the subtle nature of the nonrelativistic behavior of relativistic expressions, as is the case for the Lorentz transformations that have two nonrelativistic limits.

Consider for instance, the Lorentz transformations. If we assume that the relative velocity between two inertial frames is much less than the speed of light and follow the ultra-timelike condition, then the Lorentz transformations reduce to the Galilean transformations. These transformations represent the nonrelativistic ultra-timelike limit of the Lorentz transformations. Similarly, the ultra-spacelike limit of the Lorentz transformations, known as “Carroll transformations” can also be obtained using certain conditions. However, these transformations are not always physically realizable as causal relations between events may seem to be impossible.

## II. THE NONRELATIVISTIC LIMITS

What’s fascinating about the Lorentz transformations is that there is a third limit known as the instantaneous limit. This limit is reached by letting the speed of light tend towards infinity, this changes the finite propagation speed of the electric and magnetic fields to an infinite speed. This concept of instantaneous fields is natural in a Galilei-invariant theory.

Interestingly, both the nonrelativistic ultra-timelike limit and the instantaneous limit exhibit the same form but carry different interpretations. The instantaneous limit can not usually be considered as a nonrelativistic limit due to the allowance of finite but otherwise arbitrary velocities. This concept can be extended to electromagnetic quantities leading to obtaining two nonrelativistic limits under the condition that the velocity tends towards zero while maintaining specific restrictions on the magnitude of relevant quantities.

## III. THE ELECTRIC AND MAGNETIC LIMITS OF MAXWELL'S EQUATIONS

This concept of two nonrelativistic limits for Maxwell’s equations is often expected. These limits, known as the electric and magnetic limits of Maxwell’s equations, were first obtained in SI units. It should be noted that no physical result should depend on the choice of a specific system of units. Hence, the electric and magnetic limits of Maxwell’s equations should also be obtainable in Gaussian units.

## IV. TOWARDS AN INSTANTANEOUS LIMIT

Certainly, the procedure applied to obtain the instantaneous limit of the Lorentz transformations can be applied to electromagnetic quantities. This would mean that an instantaneous limit should also be expected for Maxwell’s equations, which would allow finite but otherwise arbitrary velocities. However, the double role that the speed of light plays in Maxwell’s equations must be considered carefully. To obtain an instantaneous limit, the 'speed of light' associated with propagation must be differentiated from that associated with units.

## V. CONCLUSION

In summary, the nonrelativistic and instantaneous limits of electromagnetic quantities exhibit that Maxwell's equations have three distinct Galilean limits: the electric limit, the magnetic limit, and the instantaneous limit. Each limit carries a unique physical interpretation, and together they show the extensive scope of Maxwell's equations in illuminating our understanding of the electromagnetic world.

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