Galilei transformation

One may ask why the expression for the non-relativistic current density is so very much more complicated than the corresponding relativistic expression. The answer is that it reflects the attempt to merge equations (Schrodinger and Maxwell) with incompatible transformation properties. When Poincar6 studied the transformation properties of Maxwell s equations, he found that they did not transform according to the Galilei transformation as the equations of the Newtonian physics, and they were in fact termed non-relativistic until Einstein with the introduction of the theory of special relativity showed that they indeed possessed the physically correct transformation properties ... 

Further, the constants a,P,y,5, will be determinated, applying CPl-4 principles, in order to define the classical (also-called Galilei) transformations. 

The necessity of reconsideration of Galilei transformations was above exposed, with the opportunity the non-invariance time derivation of these transformations was highlighted. In addition, it was introduced the idea that light propagates at finite speed (by Maxwell - in theoretical way, in 1873,... 

Another formal deficiency of Newtonian mechanics and therefore of the Galilei transformations is the existence of Maxwell s equations (2.94)-(2.97), which contain the speed of light c as a parameter. Maxwell s equations directly yield the wave equation (2.121), which states that the propagation of electromagnetic fields in vacuum occurs with speed c. Either Maxwell s equations are not valid in all inertial frames of reference or the Galilei transformations cannot be the correct coordinate transformations between inertial frames. Experimentally one finds that the Maxwell equations are valid for any inertial system. 

The reader should compare this formulation with its nonrelativistic counterpart given in section 2.1.2. As a direct consequence of the covariance, i.e., invariance in form, of Maxwell s equations we obtain the desired result that the speed of light c is the same in all inertial frames of reference, cf. Eq (3.1). This immediately implies that the coordinate transformations relating the description of events in different inertial frames can no longer be the Galilei transformations but have to be replaced by more suitable coordinate transformations. 

However, in Maxwell s days everyone assumed that there had to be a mechanical underpinning for the theory of EM. Many researchers worked on very detailed hidden variable theories for the EM field, in an attempt to prove that the laws of EM were in fact a theorem in NM, just like Kepler s laws are a theorem in NM. No one noticed that it was impossible to do this, since Maxwell s equations are not Galilei invariant and Newton s laws are. That includes Lorentz who discovered around 1900 that the Maxwell equations are invariant under another transformation that now bears his name. 

In principle, the basis Mi can be chosen arbitrarily. However, if we want the order parameters x, (r, r ) to have well-defined transformation properties under the Galilei group we are led to the Balian-Werthamer matrices (Balian and Werthamer 1963) ... 

We now exploit the relativity (in the sense of Galilei) of Newton s laws for the determination of the most general Galilean transformations. Newtonian physics crucially relies on the concept of absolute time the time difference df between two events is the same in all inertial frames. The time shown by a clock is in particular independent of the state of motion of the clock. As a consequence the most general relation between the time f in IS and the time f in IS is given by... 

IS is the only parameter, cf. Figure 2.1. This transformation is the Galilei boost in x-direction and given by... 

Quantities without any indices such as the mass m, which are not only covariant but invariant under Galilean transformations, are called Galilei scalars or zero rank tensors. They have exactly the same value in all inertial frames of reference. 

For V = 0 Eq. (3.67) for a Lorentz boost in x-direction reduces to the Galilei boost as given by Eq. (2.17). But for nonvanishing velocities v the relativistic transformation is more involved and mixes space- and time-coordinates. 

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