Unifying the Galilei Relativity and the Special Relativity

We present two models combining some aspects of the Galilei and the Special relativities that lead to a unification of both relativities. This unification is founded on a reinterpretation of the absolute time of the Galilei relativity that is considered as a quantity in its own and not as mere reinterpretation of the time of the Special relativity in the limit of low velocity. In the first model, the Galilei relativity plays a prominent role in the sense that the basic kinematical laws of Special relativity, for example, the Lorentz transformation and the velocity law, follow from the corresponding Galilei transformations for the position and velocity. This first model also provides a new way of conceiving the nature of relativistic spacetime where the Lorentz transformation is induced by the Galilei transformation through an embedding of 3-dimensional Euclidean space into hyperplanes of 4-dimensional Euclidean space. This idea provides the starting point for the development of a second model that leads to a generalization of the Lorentz transformation, which includes, as particular cases, the standard Lorentz transformation and transformations that apply to the case of superluminal frames. 1. Introduction It is common to consider the Galilei relativity as the low velocity limit case of the Special relativity (SR). This is revealed by the behavior of the Lorentz transformation when we take ; for example, In this view, the absolute time of the Galilei relativity, henceforth denoted by , is not considered as an independent quantity but it refers to a particular situation of SR when it is possible to identify and , which allows us to take . It is in this sense that the absolute time is usually conceived. Therefore, the fact that one generally has seems to deny the possibility of having an absolute time in SR. We refer to the time of the SR as the physical time. In our work we intend to reinterpret the notion of absolute time in such a way that the Galilei relativity recovers its role as a theory in its own, correcting then a common view that treats the principles of the Galilei and the Special relativities as irreconcilable notions. One of the difficulties we encounter to carry out this goal is related to the way we currently understand the concept of time, which is somehow already shaped by ideas of the SR. Here, we focus our efforts on the development of two models, called models I and II, that are built in order to allow to introduce the concept of absolute time in a general setup that also incorporates the concept of time of SR. In model I, the fact