Einstein's hidden postulate.(Report)

I. Introduction

The Lorentz transformation (LT) [1] is the cornerstone of the special theory of relativity (SR). It leads directly to the condition of Lorentz invariance which plays an integral role in many areas of theoretical physics. It is also responsible for the concept of space-time mixing which represented a fundamental break with classical Newtonian physics and the Galilean transformation. The predictions of Fitzgerald-Lorentz length contraction and time dilation are derived in a straightforward manner from the LT. In both cases Einstein argued that measurement has a distinctly symmetric character whereby observers in relative motion must disagree as to whose clocks are running slower and whose measuring rods are shorter in length. Moreover, the amount of length contraction must vary with the orientation of the moving object to the observer according to the LT. Most surprising of all, it had to be concluded that events do not occur simultaneously for different observers as a consequence of the aforementioned space-time mixing predicted by the LT. Indeed, it is often claimed that this non-simultaneity characteristic is the central feature of Einstein's theory.

The derivation of the LT is consequently of supreme importance in fully understanding the foundations of Einstein's special theory. The first few pages of most texts dealing with relativity are consequently devoted to this topic and go to great lengths to show the inevitability of the predictions of SR that follow from the two postulates he employed to obtain his space-time transformation. There is general agreement that assumptions of homogeneity and isotropy of space and also the independence of the object's history are also implicit in his derivation [2]. However, there is another aspect that is easily overlooked in this discussion and this will be the subject of the following section.

II. Lorentz's Normalization Function

After introducing his two postulates of relativity, Einstein wrote down the following general equations to define the space-time transformation under consideration [1]:

t' = [gamma][phi](t - [vxc.sup.-2]) (1a)

x' = [gamma][phi](x - vt) (1b)

y' = [gamma][phi] (1c)

z' = [gamma][phi] (1d)

In these equations, x, y, z and t are the space-time coordinates of an object as measured by an observer who is at rest in inertial system S, whereas the corresponding primed symbols correspond to the measured values for the same object obtained by a second observer who is stationary in another inertial system S' which is moving along the common x, x' axis at constant speed v relative to S [c is the speed of light in free space and [gamma][(1 - [v.sup.2][c.sup.-2].sup.-0.5)]. The emphasis in the present discussion is on the function [phi] in the above equations.

Einstein was following Lorentz to this point in the derivation, who had published [3] the same set of equations in slightly different notation in 1899 (he used [epsilon] instead of [phi], for example [4]). Lorentz pointed out that there is a degree of freedom (normalization function) in defining the transformation that was not specified by the requirement that it leave Maxwell's equations of electricity and magnetism invariant. This is also obviously the case if no other condition needs to be satisfied than the light-speed postulate; the function [phi] merely cancels out when velocity components are formed by dividing x', y', z' by t' in eqs. (1a-d).

In order to completely specify the transformation, Einstein [1] made the following assertion (see p. 900 of ref. 1): "... and [phi] is a temporarily unknown function of v." He therefore removed from consideration the real possibility that [phi] might depend on some other variable than v in his derivation. He gives no justification for this conclusion. Indeed, he does not even declare that it is an assumption at all. He then proceeds on the basis of symmetry to show that the only possible value for [phi] consistent with this assumption is unity, and then upon substituting this value in eqs. (1a-d) he obtains the LT directly.

There are many predictions of SR that are direct applications of the LT. As discussed in the Introduction, these include Lorentz invariance, space-time mixing [see eq. (1a) with [phi] = 1], Fitzgerald-Lorentz length contraction (FLC) and its anisotropic character, time dilation and the symmetric nature of both time and length measurements, remote non-simultaneity of events for observers in relative motion, and the impossibility of v > c speeds (because that would open up the possibility that the two observers can disagree on the time-order of events). Except for time dilation, none of the above effects has ever been observed experimentally, despite the unwavering belief of most physicists in their existence. If Einstein's normalization assumption is not correct, it is clear that such faith is greatly misplaced. At the very least, there is compelling reason to examine the many other verified predictions of his theory to see if the LT is essential for any of those. To investigate this question, it is important to take a closer look at the other key transformation of Einstein's paper, the relativistic …