The Duality of Time Theory

As we have seen in section 1 above, Lorentz believed in the luminiferous aether hypothesis, so he looked for the transformations under which Maxwell’s equations are invariant when transformed from the aether to a moving frame. This led him to the ideas of local time and space contraction before Einstein. Also before Einstein, in 1905, Poincaré was the first to recognize that such transformations have the properties of a mathematical group, and he named it after Lorentz. Later in this year, Einstein postulated the two principles of Special Relativity, and derived Lorentz transformations without any reference to aether.

Any event in space-time is specified by its three Cartesian coordinates and time. The same event in any another inertial frame, moving at a constant speedwith relation to the observer, has the “primed” coordinates. Lorentz transformations relate between these two coordinates by the following relations, considering for simplicity that motion is only on theaxis, so that the other two axes are unaffected (,), while (,) and (,) are related by the following two equations, that give the mathematical calculation of space contraction and time dilation, respectively:

Here,is the ratio of the velocityover the speed of light, andis called Lorentz factor and it is given by:

When the relative speed between the observersis very small, in relation to the speed of light, as it is the case in most classical situations, the value ofwill be effectively considered zero, and the value ofwill be equal to one. Under these conditions, Lorentzian transformations reduce back to the Galilean transformations as expressed in equation 3.1 above.

Therefore, the Lorentzian transformations of Special Relativity have superseded the Galilean transformations of Newtonian physics. The Galilean transformations do not account for time dilation or length contraction, as the Lorentzian transformations do.

In other words, Lorentz transformations are nonlinear rescaling of the Galilean transformations, in order to satisfy Einstein’s two principles of Special Relativity, but they have been derived before Special Relativity. They were the result of the attempts by Lorentz, Poincaré and others to understand the symmetries of the laws of electromagnetism and to explain how the speed of light was observed to be independent of the reference frame, as all the results of Michelson-Morley experiments indicated.