Unifying the Galilei Relativity and the Special Relativity

Abstract

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 𝑣/𝑐→0; for example, ^→𝑥^′=^→𝑥−1−𝛾^→𝑥⋅^→𝑣𝑣^2→𝑣−𝛾^→𝑣𝑡⟶^→𝑥^′=^→𝑥−^→𝑣𝑡𝑡,_′=𝛾𝑡−^→𝑥⋅^→𝑣𝑐²⟶𝑡^′=𝑡. (1) In this view, the absolute time of the Galilei relativity, henceforth denoted by 𝜏, is not considered as an independent quantity but it refers to a particular situation of SR when it is possible to identify 𝑡 and 𝑡^′, which allows us to take 𝜏≡𝑡=𝑡^′. It is in this sense that the absolute time is usually conceived. Therefore, the fact that one generally has 𝑡^′≠𝑡 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 that 𝑡≠𝑡^′ suggest us not to identify the absolute time with 𝑡 but with a function depending on 𝑡 and, perhaps, on the spatial location where the event occurred. Then, if for a certain instant 𝜏 we have (𝑡,^→𝑥), (𝑡^′,^→𝑥^′) representing the same event with respect to two inertial frames, we would expect the absolute character of 𝜏 to imply that 𝜏(𝑡,^→𝑥)=𝜏(𝑡^′,^→𝑥^′), with this expression being verified for any value of the ratio 𝑣/𝑐. In addition, we would also expect to obtain the Lorentz transformation by replacing 𝜏(𝑡,^→𝑥) into the Galilei law ^→𝑥^′=^→𝑥−^→𝑣𝜏. Having established this, the next step would be to obtain the velocity law of SR from the corresponding velocity law of the Galilei relativity, 𝑑^→𝑥^′/𝑑𝜏=(𝑑^→𝑥/𝑑𝜏)−^→𝑣. The fulfillment of these two laws places the Galilei relativity as a guiding principle for deriving some of the SR laws.

From a calculational perspective, the Lorentz transformation can be obtained by considering as variables the set {𝑡,^→𝑥,𝑡^′,^→𝑥^′}, together with the equation ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡². In model I, we will show that in order to combine the Galilei and the Special relativities, it is sufficient to enlarge the variables’ set of SR to {𝑡,^→𝑥,𝑡^′,^→𝑥^′,𝜏} and to consider the pair of equations ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡²,_→𝑥^′=^→𝑥−^→𝑣𝜏. (2) Here, the effect of this enlargement is to produce not only the usual Lorentz transformation, but also an extra equation relating the absolute and the physical time, which gives an operational definition for the absolute time associated to the occurrence of an event.

In what concerns the structure of spacetime, we notice that both relativities are formulated in terms of a 4-dimensional space ℝ⁴≡ℝ×ℝ³. In the Galilei view, two observers belonging to inertial frames 𝑆, 𝑆^′ describe the occurrence of an event, respectively, as a point (𝜏,^→𝑥)∈ℝ_𝑆×ℝ³_𝑆≡ℝ⁴_𝑆 and (𝜏,^→𝑥^′)∈ℝ_𝑆′×ℝ³_𝑆′≡ℝ⁴_𝑆′. Here, due to the nature of the absolute time, for a fixed 𝜏, the Galilei transformation reduces itself to a map ℝ³_𝑆→ℝ³_𝑆′.

In the SR view, the observers describe the event as a point (𝑡,^→𝑥)∈ℝ_𝑆×ℝ³_𝑆, and (𝑡^′,^→𝑥^′)∈ℝ_𝑆′×ℝ³_𝑆′ where now, due to the nature of the physical time, the Lorentz transformation is not restricted to be a map ℝ³_𝑆→ℝ³_𝑆′ as before, but it is considered as a map on the whole space, ℝ⁴_𝑆→ℝ⁴_𝑆′. In model I, the existence of the absolute time allows us to unveil an additional structure present in the SR spacetime that is already present in the Galilei view. In fact, when we express the absolute time in terms of the physical time, 𝜏(𝑡), 𝜏(𝑡^′), we will show that we can define embeddings ℝ↪³_𝑆^𝑖𝜏ℝ⁴_𝑆, ℝ↪³_𝑆′^𝑖′𝜏ℝ⁴_𝑆′ in terms of certain hyperplanes 𝜎_𝜏⊂ℝ⁴_𝑆, 𝜎^′_𝜏⊂ℝ⁴_𝑆′. The Lorentz transformation is then seen as a map 𝜎_𝜏→𝜎^′_𝜏 between these 3-dimensional hyperplanes that is induced by the Galilei transformation acting on ℝ³_𝑆→ℝ³_𝑆′, for a certain 𝜏. Therefore, we may see the spacetime as the union of these hyperplanes, for example, ℝ⁴_𝑆=∪_𝜏𝜎_𝜏. In the spacetime of Galilei relativity this splitting is self-evident and corresponds to the hyperplanes of 𝜏=constant.

Inspired by the equation that defines the embeddings ℝ↪³_𝑆^𝑖𝜏ℝ⁴_𝑆, ℝ↪³_𝑆′^𝑖′𝜏ℝ⁴_𝑆′, we develop model II by taking 𝛼𝑥⁰−^→𝛽⋅^→𝑥=𝛼𝑥^′0+^→𝛽⋅^→𝑥^′ as our fundamental equation, where 𝛼,^→𝛽 are arbitrary parameters that will allow us to define a generalization of the Lorentz transformation. In model II, the absolute time is introduced by modifying the previous equation to 𝜏≡𝛼𝑥⁰−^→𝛽⋅^→𝑥=𝛼𝑥^′0+^→𝛽⋅^→𝑥^′. (3) Here, the Galilei relativity arises assuming in addition the relation ^→𝑥^′=^→𝑥−^→𝑣𝜏. This is sufficient to reduce the generalized transformation to the particular form of the standard Lorentz transformation, showing then the consistency of both relativities. In model II, however, we will show that we are allowed to have the Lorentz transformation weather or not we consider the Galilei relativity, a situation that does not happen in model I.

Some works [1, 2] deal with the Galilei and the Special relativities but focus on opposite goals not proposing a scheme for unifying both relativities. In fact, in [1] the assumption that the Galilei law of velocity applies to the motion of bodies, signals, and forces leads to inconsistencies that are solved by the introduction of the postulate of the constancy of the speed of light, which ultimately leads to SR. In [2], a transformation relating the Galilei and the Minkowski-Einstein coordinates is established, which may signalize that relativistic effects are due to motion relative to an actual 3-space. (The author continues this development in a model that is known by Process Physics.) In our work, we follow another direction as our goal is to harmonize both relativities by a convenient treatment of the absolute time and the fundamentals laws of both relativities. We are not aware of any work devoted to this issue; therefore, we hope that our work may provide a convenient starting point for further investigation on this topic.

Our work is organized as follows. Model I is developed in Section 3. In Section 3.1 we introduce a set of assumptions settling the main properties of the physical space and time that are necessary to develop a model incorporating both the Galilei and the Special relativities. We axiomatize the existence of two times, the absolute time of the Galilei relativity and the physical time of SR, and we discuss an important issue, due to Møller, concerning the correct interpretation of the physical space where (1) is defined. In Section 3.2 we obtain the Lorentz transformation from the Galilei law ^→𝑥^′=^→𝑥−^→𝑣𝜏 and the assumptions of Section 3.1. In Section 3.4 we obtain from the law of velocity transformation of Galilei relativity the corresponding velocity law of SR. We also discuss a problem associated to the so-called Thomas precession. In Section 3.5 we give a geometric interpretation for the spacetime of SR and show how the concept of absolute time allows to define a partition of the spacetime in terms of certain 3-dimensional hyperplanes. Model II is developed in Section 4. In Section 4.1 we establish a set of assumptions that is slightly different than the assumptions of model I, namely, in what concerns the introduction of the Special and the Galilei relativities (resp., numbered as assumptions (III) and (IV)). In Section 4.2 we derive a transformation that we call Generalized Lorentz Transformation, and in Section 4.3 we analyze the velocity transformation associated to it. We then show that the Generalized Lorentz Transformation includes, as particular cases, the standard Lorentz transformation together with transformations that may be used when we have superluminal frames. We then analyze in Section 4.4 how to introduce the Galilei relativity in the framework of model II. In Section 4.5 we analyze the conservation of momentum and obtain the corresponding relativistic expression for the mass. Finally, in Section 5 we analyze how elementary considerations from the differential structure of the projective space ℝP(5) allow us to think on the concept of the absolute time as associated to one of the dimensions of Euclidean space ℝ⁴.

2. Basic Definitions

We recall some basic definitions that are necessary for the statement of our assumptions.

An event is any physical occurrence taking place on a certain location and in a certain instant. We adopt the standard definitions of observer and reference frame as stated concisely in [3]; that is, by an observer we understand any entity equipped with a standard rod and a standard clock that allows for the measurement of length and time (in fact, the physical time as we will introduce in the assumption (II)) and that is able to communicate with other observers by means of light signals. A reference frame is understood as an infinite set of observers each one at rest relative to the other and having their standard clocks synchronized. Abstractly, the infinite set of observers composing a reference frame is idealized in such a way that for any event there is an observer present on the same location where the event took place, which then establishes the space coordinate of the event. This observer also determines, by the reading of his clock, the instant of time when the event occurred. We will consider reference frames endowed with a rectangular coordinate system. By an inertial frame we understand any frame in which a body free of forces is unaccelerated.

We use the notation (𝑡_𝑆𝑃,^→𝑥_𝑆𝑃) to denote the description of an event 𝑃 relative to a frame 𝑆. We write (𝑡_𝑆𝑃,^→𝑥_𝑆𝑃)∼(𝑡_𝑆′𝑃,^→𝑥_𝑆′𝑃) to indicate the same event 𝑃 as seen by the frames 𝑆,𝑆^′. In the particular case when one frame 𝑆 analyzes the movement of another frame 𝑆^′, we will assume that the frame 𝑆^′ is entirely represented by its origin and write ^→𝑥_𝑆𝑆′ for the position of 𝑆^′ relative to 𝑆, and 𝑡_𝑆𝑆′ for the time measured by 𝑆. Finally, in our work by a Lorentz transformation we always mean a Lorentz boost, except in Section 5 where a Lorentz boost attains its precise meaning as a particular Lorentz transformation.

3. Model I

3.1. The Assumptions

In what concerns space and time, our basic assumptions are as follows. (I)Space. Each inertial frame describes space as being an euclidean 3-dimensional vector space. (II)Time. We model time as any variable that can be used to describe what we intuitively understand as the “instant when events occur.” We distinguish two kinds of choices as follows. By physical time, denoted by 𝑡, we understand a choice for the time variable that can be measured with the use of standard devices, like atomic clocks, under suitable arrangements (e.g., when they are the clocks of a reference frame and are all conveniently synchronized, etc.). By absolute time, denoted by 𝜏, we understand a choice for the time variable having the property that given an event all observers assign to it the same value for the instant when the event occurred. (Later, through the analysis of their transformation properties, we will identify the physical time as the ordinary time of the special relativity, while the absolute time will be identified with the time of the Galilei relativity.) As a principle, we assume that any variable that serves to describe time may be expressed in terms of the physical time in such way that by the measurement of the latter one can determine the value of the former. Therefore, for any frame in which one knows how to shift from 𝜏(𝑡)↔𝑡(𝜏) it is possible to describe an event writing its coordinates as (𝑡,^→𝑥) or (𝜏,^→𝑥). (III) Given two inertial frames 𝑆, 𝑆^′ moving with relative velocity ^→𝑣 and an event 𝑃 whose coordinates are (𝑡_𝑆𝑃,^→𝑥_𝑆𝑃), (𝑡_𝑆′𝑃,^→𝑥_𝑆′𝑃) relative to 𝑆,𝑆^′, we have ^→𝑥²_𝑆𝑃−𝑐²𝑡²_𝑆𝑃=^→𝑥²_𝑆^′_𝑃−𝑐²𝑡²_𝑆′𝑃. (4) When we consider speed calculated in terms of derivatives relative to the physical time, the relation (4) expresses the constancy of the speed of light. (IV)The Galilean Relativity Principle. Given two inertial frames 𝑆, 𝑆^′ moving with relative velocity ^→𝑣 and an event 𝑃 whose coordinates are (𝜏,^→𝑥_𝑆𝑃), (𝜏,^→𝑥_𝑆′𝑃) relative to 𝑆,𝑆^′, we have componentwise that ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−^→𝑣_𝑆𝑆′𝜏. (5)(V) The relation between (𝑡_𝑆𝑃,^→𝑥_𝑆𝑃) and (𝑡_𝑆′𝑃,^→𝑥_𝑆′𝑃) is linear.

Remarks. (i) In assumption (I), it is assumed that observers in inertial frames see physical space endowed with an euclidean structure. This assumption is supported by the lack of evidence signalyzing deviations of space properties from the euclidean structure (see the discussion of French in [4, pages 59–61]). The position of an event 𝑃 relative to inertial frames 𝑆, 𝑆^′ is written as ^→𝑥_𝑆𝑃≡(𝑥¹,𝑥²,𝑥³)∈ℝ³_𝑆, ^→𝑥_𝑆′𝑃≡(𝑥^′1,𝑥^′2,𝑥^′3)∈ℝ³_𝑆′ and since each frame is endowed with its own coordinate system, the spaces ℝ³_𝑆 and ℝ³_𝑆′ are conceived as distinct spaces; therefore, we must understand the relation between the vectors ^→𝑥_𝑆𝑃 and ^→𝑥_𝑆′𝑃 established in (5) not as a vector equation defined in one and the same vector space. However, it is possible to do so if we reinterpret the vectors ^→𝑥_𝑆𝑃 and ^→𝑥_𝑆′𝑃 as follows. (This reasoning is originally due to Møller [5].) Given two frames 𝑆, 𝑆^′ we assume that there is a single space ℝ³_𝑆𝑆′ where the components (𝑥¹,𝑥²,𝑥³),(𝑥^′1,𝑥^′2,𝑥^′3) of the vectors ^→𝑥_𝑆𝑃,^→𝑥_𝑆′𝑃 are seen as images of maps ℝ³_𝑆⟶ℝ³_𝑆𝑆′∶^→𝑥_𝑆𝑃⟶𝑥¹,𝑥²,𝑥³,ℝ³_𝑆′⟶ℝ³_𝑆𝑆′∶^→𝑥_𝑆′𝑃⟶𝑥^′1,𝑥^′2,𝑥^′3 (6) and such that for a fixed 𝜏 they satisfy (5); that is, we can think of the vectors ^→𝑥_𝑆𝑃, ^→𝑥_𝑆′𝑃 as vectors in ℝ³_𝑆𝑆′, which justify using the same notation for them. Equation (5) is now understood as an equation defined in ℝ³_𝑆𝑆′.

From the perspective of the Galilei relativity, it is clear that the vector ^→𝑥_𝑆′𝑃∈ℝ³_𝑆′ can be measured directly from the observers of the frame 𝑆 that at the instant 𝜏 are placed, respectively, at the positions corresponding to the origin of the frame 𝑆^′ and the event 𝑃; that is, componentwise we identify ℝ³_𝑆′∋^→𝑥_𝑆′𝑃↔^→𝜂∈ℝ³_𝑆, (7) where ^→𝜂∶=^→𝑥_𝑆𝑃−^→𝑥_𝑆𝑆′=^→𝑥_𝑆𝑃−^→𝑣_𝑆𝑆′𝜏. (8) This corresponds to the same equation (5), now referred entirely to the space ℝ³_𝑆. This procedure is equivalent to identify ℝ³_𝑆𝑆′ with ℝ³_𝑆. A similar procedure allows to identify ℝ³_𝑆𝑆′ and ℝ³_𝑆′. The assumption that spaces ℝ³_𝑆, ℝ³_𝑆′ are euclidean is then consistent with the interpretation we obtain for (5) through the identification ℝ³_𝑆≃ℝ³_𝑆𝑆′, ℝ³_𝑆′≃ℝ³_𝑆𝑆′.

A different picture emerges in SR. Now, instead of (5) we have the relation (it seems it was Møller (see [5, Section  2.4]) who first noticed the importance of giving a correct interpretation for (9) as an equation established in an abstract vector space, where it would make sense to relate to the vectors ^→𝑥_𝑆𝑃 and ^→𝑥_𝑆′𝑃 that, in principle, belong to different spaces. Møller did so thinking exclusively from the perspective of the SR. However, the same question is already manifest from the perspective of the Galilei relativity if we intend to interpret correctly (5)) ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−1−𝛾^→x_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′^→𝑣_𝑆𝑆′−𝛾^→𝑣_𝑆𝑆′𝑡_𝑆𝑃. (9) Here, the vectors ^→𝑥_𝑆𝑃∈ℝ³_𝑆, ^→𝑥_𝑆′𝑃∈ℝ³_𝑆′ are mapped into an abstract space 𝑉_𝑆𝑆′, where we understand that (9) is defined. From the perspective of SR and according to the frame 𝑆, if the event 𝑃 occurs at the instant 𝑡_𝑆𝑃, the observers on the frame 𝑆 that occupy, respectively, the positions of the origin of the frame 𝑆^′ and the event 𝑃 allow us to define a vector ^→𝜂^→𝜂∶=^→𝑥_𝑆𝑃−^→𝑣_𝑆𝑆′𝑡_𝑆𝑃, (10) which is not componentwise equal to the vector ^→𝑥_𝑆′𝑃 as measured by the frame 𝑆^′(11) In fact, using the relation 𝑡_𝑆′𝑃=𝛾𝑡_𝑆𝑃−^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑐² (12) together with (9) to write ^→𝑥_𝑆𝑃 in terms of ^→𝑥_𝑆′𝑃,𝑡_𝑆′𝑃, we obtain componentwise that ^→𝜂=^→𝑥_𝑆′𝑃+1−𝛾𝛾^→𝑥_𝑆′𝑃⋅^→𝑣_𝑆′𝑆𝑣²_𝑆′𝑆^→𝑣_𝑆′𝑆. (13) Therefore, from the perspective of SR, we cannot identify the vectors ^→𝜂 and ^→𝑥_𝑆′𝑃 as we did before. Physically, this means that measurements performed by the observers of the frame 𝑆 are not sufficient to identify (10) with (9) of 𝑉_𝑆𝑆′. We then distinguish the spaces ℝ³_𝑆𝑆′ and 𝑉_𝑆𝑆′ in the sense that we cannot identify ℝ³_𝑆 or ℝ³_𝑆′ with 𝑉_𝑆𝑆′ as we did in the case of the Galilei relativity.

(ii) In assumption (III), we consider that (4) holds not only for the events associated with the movement of a light ray, but also for every event. As we know [6], in the standard treatment of SR if we assume that (4) is verified for a light ray, then from the assumption of the linearity of the transformation involving ^→𝑥,𝑡 and ^→𝑥^′,𝑡^′ we obtain that condition (4) is also verified for any event. In our approach, since we work essentially with two equations (4), (5) and an extended set of variables {𝜏,𝑡,𝑡^′,^→𝑥,^→𝑥^′}, there is no guarantee that assuming that (4) is true for a light ray would imply its validity for all events. (For the case of the standard treatment of the SR, Landau and Lifshitz give a heuristic argument justifying it in [7, page 5], while Einstein put it axiomatically in [8, appendix one].) Therefore, in the development of our model it is necessary to assume from the beginning that (4) is verified for not only for those events associated to the movement of a light ray, but also all events. This will become evident in the derivation shown in Section 3.2.

(iii) The physical and the absolute time introduced in assumption (II) assume their specific characteristic from the conditions they have to obey in assumptions (III) and (IV). These conditions identify 𝑡 as the time of Special relativity and 𝜏 as the time of Galilei relativity.

(iv) Whenever we set a transformation between two frames we assume that at 𝜏=0, or equivalently at 𝑡=𝑡^′=0, the origins of both frames coincide and their coordinate axes are parallel.

(v) In assumption (V), by a linear relation between the pairs (𝑡,^→𝑥),(𝑡^′,^→𝑥^′) we mean that 𝑡^′ and ^→𝑥^′ depend only on the first-order power of 𝑡,^→𝑥.

3.2. Deriving the Transformation

In order to obtain the transformations we start from assumption (IV) ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−^→𝑣_𝑆𝑆′𝜏 (14) which gives ^→𝑥²_𝑆^′_𝑃=^→𝑥²_𝑆𝑃−2^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝜏+𝑣²_𝑆𝑆′𝜏². (15) From assumption (III) we obtain 𝑣²_𝑆𝑆′𝜏²−2^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝜏+𝑐²𝑡²_𝑆𝑃−𝑡²_𝑆′𝑃=0 (16) that allows us to write 𝜏=^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′±^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′²−𝑣²_𝑆𝑆′𝑐²𝑡²_𝑆𝑃−𝑡²_𝑆′𝑃𝑣²_𝑆𝑆′. (17) Now, according to assumption (V), we look for a relation between 𝑡_𝑆𝑃 and 𝑡_𝑆′𝑃 such that the corresponding expression for 𝜏 given in (17), when replaced into (14), results in a transformation involving at most terms to the first power in ^→𝑥_𝑆′𝑃, ^→𝑥_𝑆𝑃. In order to obtain that, let us consider the following particular relation between 𝑡_𝑆𝑃 and 𝑡_𝑆′𝑃, for example, 𝑡_𝑆′𝑃=𝑎𝑡_𝑆𝑃+𝑏^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′ (18) which gives 𝑡²_𝑆𝑃−𝑡²_𝑆′𝑃=(1−𝑎²)𝑡²_𝑆𝑃−2𝑎𝑏^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑡_𝑆𝑃−𝑏²(^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′)². The constants 𝑎 and 𝑏 must be chosen in such way that (^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′)²−𝑣²_𝑆𝑆′𝑐²(𝑡²_𝑆𝑃−𝑡²_𝑆′𝑃) becomes a perfect square, that is, ^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′²−𝑣²_𝑆𝑆′𝑐²𝑡²_𝑆𝑃−𝑡²_𝑆′𝑃=1+𝑣²_𝑆𝑆′𝑐²𝑏^2→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′²+2𝑣²_𝑆𝑆′𝑐²𝑎𝑏^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑡_𝑆𝑃+𝑎²−1𝑣²_𝑆𝑆′𝑐²𝑡²_𝑆𝑃≡^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′1+𝑣²_𝑆𝑆′𝑐²𝑏²−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃² (19) which gives 𝑎²=1+𝑣²_𝑆𝑆′𝑐²𝑏². (We could have written (19) in the form (^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′)1+𝑣²_𝑆𝑆′𝑐²𝑏²+√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃. As we will see, the minus sign adopted in (19) becomes necessary if we intend to obtain the standard Lorentz transformation of SR.) We have then 𝜏=^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′±^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′|𝑎|−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃𝑣²_𝑆𝑆′. (20) Here, the choice for the sign of 𝜏 is fixed as follows. Replacing 𝜏 given by (20) into (14) and considering the time transformation (18), we assume that the transformation (𝑡_𝑆𝑃,^→𝑥_𝑆𝑃)→(𝑡_𝑆′𝑃,^→𝑥_𝑆′𝑃) is invertible upon replacing (𝑡_𝑆′𝑃,^→𝑥_𝑆′𝑃,^→𝑣_𝑆′𝑆)↔(𝑡_𝑆𝑃,^→𝑥_𝑆𝑃,^→𝑣_𝑆𝑆′) with ^→𝑣_𝑆′𝑆=−^→𝑣_𝑆𝑆′. Therefore, we notice the following. (i) If ^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′|𝑎|−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃≥0, we must choose 𝜏=^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′−^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′|𝑎|−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃𝑣²_𝑆𝑆′=(1−|𝑎|)^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+√𝑎²−1𝑣_𝑆𝑆′𝑐𝑡_𝑆𝑃. (21)(ii) If ^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′|𝑎|−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃<0, we must choose 𝜏=^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′+^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′|𝑎|−√𝑎²−1𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑃𝑣²_𝑆𝑆′=(1−|𝑎|)^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+√𝑎²−1𝑣_𝑆𝑆′𝑐𝑡_𝑆𝑃. (22)In both cases, we obtain the transformation ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−(1−|𝑎|)^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′^→𝑣_𝑆𝑆′−√𝑎²−1𝑣_𝑆𝑆′𝑐𝑡^→_𝑆𝑃𝑣_𝑆𝑆′,𝑡_𝑆′𝑃=|𝑎|𝑡_𝑆𝑃−√𝑎²−1𝑣_𝑆𝑆′𝑐^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′,𝜏∶=(1−|𝑎|)^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+√𝑎²−1𝑣_𝑆𝑆′𝑐𝑡_𝑆𝑃. (23)(24) As we have expected, the transformation (23) satisfies the condition ^→𝑥²_𝑆^′_𝑃−𝑐²𝑡²_𝑆′𝑃=^→𝑥²_𝑆𝑃−𝑐²𝑡²_𝑆𝑃 and leaves 𝜏 invariant, that is, 𝜏^′=(1−|𝑎|)^→𝑥_𝑆′𝑃⋅^→𝑣_𝑆′𝑆𝑣²_𝑆′𝑆+√𝑎²−1𝑣_𝑆′𝑆𝑐𝑡_𝑆′𝑃=(1−|𝑎|)^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+√𝑎²−1𝑣_𝑆𝑆′𝑐𝑡_𝑆𝑃=𝜏 (25) since it was built in order to satisfy those assumptions. Here, we assume that the parameter 𝑎 depends on the relative speed 𝑣_𝑆𝑆′, which for convenience we denote by 𝑎_{𝑣′𝑆𝑆}. Under this assumption, we also obtain that 𝑎_{𝑣𝑆′𝑆}=𝑎_{𝑣′𝑆𝑆}.

The transformation (23) depends on the velocity ^→𝑣_𝑆𝑆′thatis defined as the derivative relative to the absolute time: ^→𝑣_𝑆𝑆′=𝑑^→𝑥_𝑆𝑆′.𝑑𝜏 (26) Now, considering the physical time 𝑡_𝑆𝑆′ we can define another velocity 𝑣̃^→_𝑆𝑆′𝑑∶=^→𝑥_𝑆𝑆′𝑑𝑡_𝑆𝑆′. (27) We obtain a relation between 𝑣̃^→_𝑆𝑆′, ^→𝑣_𝑆𝑆′ as follows. Identifying 𝑃 with the origin of the frame 𝑆^′ we have from (24) 𝜏=𝑎²_{𝑣′𝑆𝑆}−1𝑎_{𝑣′𝑆𝑆}𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑆′ (28) which allows us to write ^→𝑥_𝑆𝑆′=^→𝑣_𝑆𝑆′𝑎²_{𝑣′𝑆𝑆}−1𝑎_{𝑣′𝑆𝑆}𝑐𝑣_𝑆𝑆′𝑡_𝑆𝑆′ (29) and then 𝑣̃^→_𝑆𝑆′=^→𝑣_𝑆𝑆′𝑎²_{𝑣′𝑆𝑆}−1𝑎_{𝑣′𝑆𝑆}𝑐𝑣_𝑆𝑆′. (30) Since the parameter 𝑎_{𝑣′𝑆𝑆} depends on 𝑣_𝑆𝑆′, we obtain that 𝑣̃_𝑆𝑆′ also depends on 𝑣_𝑆𝑆′ through the expression 𝑣̃²_𝑆𝑆′𝑐²=𝑎²_{𝑣′𝑆𝑆}−1𝑎²_{𝑣′𝑆𝑆} (31) which gives 𝑣̃_𝑆𝑆′<𝑐. (32) Therefore, in our formalism every speed calculated as the derivative relative to the physical time is always less than 𝑐. Since in SR this is the type of derivative we consider, it seems that the problem of tachyons is ruled out in SR. From (31) we may express 𝑎_{𝑣′𝑆𝑆} in terms of 𝑣̃_𝑆𝑆′ as 𝑎_{𝑣′𝑆𝑆}=11−𝑣̃²_𝑆𝑆′/𝑐²≡𝛾𝑣̃_′𝑆𝑆, (33) and the transformation (23) becomes ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−1−𝛾𝑣̃_′𝑆𝑆^→𝑥_𝑆𝑃⋅^→𝑣̃_𝑆𝑆′𝑣̃²_𝑆𝑆′^→𝑣̃_𝑆𝑆′−𝛾𝑣̃_′𝑆𝑆𝑡^→_𝑆𝑃𝑣̃_𝑆𝑆′,𝑡_𝑆′𝑃=𝛾𝑣̃_′𝑆𝑆𝑡_𝑆𝑃−^→𝑥_𝑆𝑃⋅^→𝑣̃_𝑆𝑆′𝑐², (34) that is, the familiar Lorentz transformation. Hence, we have obtained a complete equivalence between the Galilei and the Lorentz transformation.

From (30) we obtain a similar expression relating 𝑣̃^→_𝑆′𝑆 and ^→𝑣_𝑆′𝑆,thatis,𝑣̃²_𝑆′𝑆/𝑐²=(𝑎²_{𝑣𝑆′𝑆}−1)/𝑎²_{𝑣𝑆′𝑆}, and since 𝑎_{𝑣𝑆′𝑆} depends on the absolute value |^→𝑣_𝑆′𝑆|=|^→𝑣_𝑆𝑆′|, we end up with 𝑣̃_𝑆′𝑆=𝑣̃_𝑆𝑆′. (35) From (30) we could also have considered 𝑣̃^→_𝑆𝑆′ as a vector in ℝ³_𝑆𝑆′. However, we avoid this interpretation as it would lead to an inconsistency associated to the so-called Thomas precession that we will analyze in Section 3.4.3.

3.3. On the Role of the Parameter 𝑎

As we have seen, the parameter 𝑎 was introduced in (18) as a free parameter in the sense that it was assumed to depend only on the relative speed 𝑣_𝑆𝑆′ between the frames. The assumptions give no prescription on how to fix this dependence. Having established the relation between the absolute and the physical time, we introduced another velocity 𝑣̃^→_𝑆𝑆′, considered as a derivative relative to the physical time, which is related to ^→𝑣_𝑆𝑆′ and the parameter 𝑎_{𝑣′𝑆𝑆} through (30), (31). Whatever the form we may take for 𝑎_{𝑣′𝑆𝑆}, once we obtain 𝑣̃_𝑆𝑆′ as a function of 𝑣_𝑆𝑆′ we may rewrite 𝑎_{𝑣′𝑆𝑆} in terms of 𝑣̃_𝑆𝑆′, which results in a_{𝑣′𝑆𝑆}=𝛾𝑣̃_′𝑆𝑆 (33). This gives a “universal” character for the Lorentz transformation with the transformation (23) becoming (34). Despite this, the constraint between ^→𝑣_𝑆𝑆′ and 𝑣̃^→_𝑆𝑆′ arising from a particular choice of 𝑎_{𝑣′𝑆𝑆} may signalize different behaviors.

In fact, let us assume 𝑣_𝑆𝑆′>𝑐 and choose 𝑎 such that 𝑎_{𝑣′𝑆𝑆}≡1𝛾∶=1−𝑐²/𝑣²_𝑆𝑆′. (36) With this choice, the transformation (23) becomes ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−1−𝛾^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′^→𝑣_𝑆𝑆′−𝛾𝑐²𝑣²_𝑆𝑆′𝑡^→_𝑆𝑃𝑣_𝑆𝑆′,𝑡_𝑆′𝑃=𝛾𝑡_𝑆𝑃−^→𝑥_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′ (37) that agrees with the expression obtained originally by Shankara (see [9, equations (16), (17)]) and also by Duffey [10] in the description of superluminal frames. From (31) we obtain 𝑣̃_𝑆𝑆′=𝑐²𝑣_𝑆𝑆′, (38) a relation that is also present in the tachyonic model of [9], whose context is of a wave propagating in a medium with the velocities being interpreted in a different way; namely, 𝑣̃^→_𝑆𝑆′ is the group velocity of the wave, while ^→𝑣_𝑆𝑆′ is the phase velocity. Here, even though the transformation (37) reduces to the form given on (34), which has the usual Lorentz form, the physical distinction between ^→𝑣_𝑆𝑆′ and 𝑣̃^→_𝑆𝑆′ may be employed to select one of the forms of the transformation as being physically relevant for the problem one is analyzing. We return to this issue in Section 4.3.2.

3.4. The Velocity Transformation

Following the program of deducing transformations directly from the assumptions given in (I)–(IV) and the results previously obtained, we now intend to obtain the law of velocity transformation and analyze some of its consequences.

3.4.1. Obtaining the Transformation from the Galilean Velocity Law

Let us assume two frames 𝑆 and 𝑆^′ moving with velocity ^→𝑣_𝑆𝑆′ and such that at 𝜏=0 (or equivalently at 𝑡=0=𝑡^′) both origins coincide. The position of a moving particle is written as ^→𝑥_𝑆𝑃(𝜏), ^→𝑥_𝑆′𝑃(𝜏), and from ^→𝑥_𝑆′𝑃=^→𝑥_𝑆𝑃−^→𝑣_𝑆𝑆′𝜏 we obtain the rule of velocity addition in Galilei relativity: 𝑑^→𝑥_𝑆′𝑃=𝑑𝑑𝜏^→𝑥_𝑆𝑃−𝑑𝜏^→𝑣_𝑆𝑆′. (39) From (24), and using the chain rule, we have 𝑑^→𝑥_𝑆′𝑃𝑑𝑡_𝑆′𝑃−1−𝑎_{𝑣′𝑆𝑆}𝑑^→𝑥_𝑆′𝑃𝑑𝑡_𝑆′𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+𝑎²_{𝑣′𝑆𝑆}−1𝑣_𝑆𝑆′𝑐⁻¹=𝑑^→𝑥_𝑆𝑃𝑑𝑡_𝑆𝑃1−𝑎_{𝑣′𝑆𝑆}𝑑^→𝑥_𝑆𝑃𝑑𝑡_𝑆𝑃⋅^→𝑣_𝑆𝑆′𝑣²_𝑆𝑆′+𝑎²_{𝑣′𝑆𝑆}−1𝑣_𝑆𝑆′𝑐⁻¹−^→𝑣_𝑆𝑆′. (40) Let us denote 𝑣̃^→_𝑆𝑃≡𝑑^→𝑥_𝑆𝑃𝑑𝑡_𝑆𝑃,^→𝑣̃_𝑆′𝑃≡𝑑^→𝑥_𝑆′𝑃𝑑𝑡_𝑆′𝑃. (41) Taking the scalar product by ^→𝑣_𝑆𝑆′ on both sides of (40) we obtain 𝑣̃^→_𝑆′𝑃⋅^→𝑣_𝑆𝑆′=𝑐𝑣_𝑆𝑆′𝑎_{𝑣′𝑆𝑆}^→𝑣̃_𝑆𝑃⋅^→𝑣_𝑆𝑆′−𝑎²_{𝑣′𝑆𝑆}−1𝑐𝑣_𝑆𝑆′−𝑎²_{𝑣′𝑆𝑆}−1^→𝑣̃_𝑆𝑃⋅^→𝑣_𝑆𝑆′+𝑎_{𝑣′𝑆𝑆}𝑐𝑣_𝑆𝑆′. (42) Using this last expression back in (40) together with (30) and (31), we obtain the final form 𝑣̃^→_𝑆′𝑃=^→𝑣̃_𝑆𝑃−𝛾𝑣̃_′𝑆𝑆^→𝑣̃_𝑆𝑆′−1−𝛾𝑣̃_′𝑆𝑆^→𝑣̃_𝑆𝑃⋅^→𝑣̃_𝑆𝑆′/𝑣̃²_𝑆𝑆′^→𝑣̃_𝑆𝑆′𝛾𝑣̃_′𝑆𝑆1−^→𝑣̃_𝑆𝑃⋅^→𝑣̃_𝑆𝑆′/𝑐². (43) Since 𝑣̃^→_𝑆′𝑃 and 𝑣̃^→_𝑆𝑃 are derivatives of the position vectors ^→𝑥_𝑆′𝑃, ^→𝑥_𝑆𝑃 with respect to the physical times 𝑡_𝑆′𝑃 and 𝑡_𝑆𝑃, we see (43) as the “physical time” counterpart of the rule of velocity addition of the Galilean relativity. It is straightforward to obtain that 𝑣̃²_𝑆′𝑃𝑐²𝑣̃=1+^2𝑆𝑃/𝑐²−1𝛾𝑣̃_′𝑆𝑆1−^→𝑣̃_𝑆𝑃⋅^→𝑣̃_𝑆𝑆′/𝑐²², (44) and we conclude that 𝑣̃_𝑆𝑃>𝑐⇒𝑣̃_𝑆′𝑃>𝑐 and 𝑣̃_𝑆𝑃<𝑐⇒𝑣̃_𝑆′𝑃<𝑐. However, for a particle moving with a constant velocity relative to the frames 𝑆, 𝑆^′, we also have similar relations as the ones given in (31): 𝑣̃_𝑆𝑃<𝑐,𝑣̃_𝑆′𝑃<𝑐. (45) Therefore, the only possibility for having 𝑣̃_𝑆𝑃>𝑐, 𝑣̃_𝑆′𝑃>𝑐 is related to a situation that does not demand relation (31), perhaps the case of an accelerated particle that would result in a relation for 𝜏 different from the one obtained in (24) that is the base for deducing (31).

3.4.2. The Light Speed as a Derivative Relative to 𝜏

Let us assume the propagation of a light ray with ^→𝑥_𝑆𝑃, ^→𝑥_𝑆′𝑃 being the position of a point 𝑃 in the wave front as seen by 𝑆 and 𝑆^′; that is, 𝑑^→𝑥_𝑆𝑃𝑑𝑡_𝑆𝑃=^→𝑐,𝑑^→𝑥_𝑆′𝑃𝑑𝑡_𝑆′𝑃=^→𝑐^′ (46) with |^→𝑐^′|=|^→𝑐|=𝑐. We then have 𝑑^→𝑥_𝑆𝑃=𝑑𝑑𝜏^→𝑥_𝑆𝑃𝑑𝑡_𝑆𝑃𝑑𝑡_𝑆𝑃𝑑𝜏⟹𝑐_𝑆=𝑐(1−|𝑎|)^→𝑐⋅^→𝑣_𝑆𝑆′/𝑣²_𝑆𝑆′+√𝑎²−1𝑐/𝑣_𝑆𝑆′,𝑑^→𝑥_𝑆′𝑃=𝑑𝑑𝜏^→𝑥_𝑆′𝑃𝑑𝑡_𝑆′𝑃𝑑𝑡_𝑆′𝑃𝑑𝜏⟹𝑐_𝑆′=𝑐−(1−|𝑎|)^→𝑐^′⋅^→𝑣_𝑆𝑆′/𝑣²_𝑆𝑆′+√𝑎²−1𝑐/𝑣_𝑆𝑆′ (47)(48) that implies different values for the speed when considered as a rate of change relative to the absolute time; that is, 𝑐_𝑆=𝑑^→𝑥_𝑆𝑃𝑑𝜏≠𝑐^′_𝑆=𝑑^→𝑥_𝑆′𝑃𝑑𝜏. (49) Here, there is one more consistency check to be performed. We expect to have the following relation: |^→𝑥_𝑆𝑃|=𝑐_𝑆Δ𝜏=𝑐Δ𝑡. In fact, consider an event 𝑃∶(𝑡,^→𝑥)∼(𝜏,^→𝑥) as described by 𝑆. At this instant, assume that the observer in the frame 𝑆that is at the same location of the event 𝑃 sends a light ray towards the observer that is at the origin of 𝑆. The arrival of the light ray at the origin of 𝑆 corresponds to another event described by 𝑆 as (𝑡₁,0)∼(𝜏₁,0). Here, in order to relate 𝑡 and 𝜏, and 𝑡₁ and 𝜏₁, we assume any auxiliary frame 𝑆^′ moving relative to 𝑆 with velocity ^→𝑣 and employ (25). For the events (𝑡,^→𝑥)∼(𝜏,^→𝑥), (𝑡₁,0)∼(𝜏₁,0), we then have 𝜏=(1−|𝑎|)^→𝑥⋅^→𝑣𝑣²+√𝑎²−1𝑣𝜏𝑐𝑡,₁=√𝑎²−1𝑣𝑐𝑡₁. (50) Denoting Δ𝑡∶=𝑡₁−𝑡 and expressing ^→𝑥=−^→𝑐Δ𝑡, we obtain Δ𝜏∶=𝜏₁−𝜏=√𝑎²𝑐−1𝑣+(1−|𝑎|)^→𝑐⋅^→𝑣𝑣²Δ𝑡. (51) Then, ^→𝑥𝑐=𝑐Δ𝑡=√𝑎²−1(𝑐/𝑣)+(1−|𝑎|)^→𝑐⋅^→𝑣/𝑣²Δ𝜏, (52) and from (47) we obtain |^→𝑥|=𝑐_𝑆Δ𝜏.

3.4.3. The Composition of Velocities

Let us assume three frames Σ,𝑆,𝑆^′ moving relative to each other with velocities ^→𝑣_Σ𝑆, ^→𝑣_Σ𝑆′,^→𝑣_𝑆𝑆′, having their axes parallel and their origins coinciding at 𝜏=0. According to Galilei relativity, we write ^→𝑥_Σ𝑆′=^→𝑥_Σ𝑆+^→𝑥_𝑆𝑆′ (53) with ^→𝑥_Σ𝑆′=^→𝑣_Σ𝑆′𝜏,^→𝑥_Σ𝑆=^→𝑣_Σ𝑆𝜏,^→𝑥_𝑆𝑆′=^→𝑣_𝑆𝑆′𝜏. (54) In particular, we have ^→𝑥_Σ𝑆′=−^→𝑥_𝑆′Σ, which results in ^→𝑣_Σ𝑆′=−^→𝑣_𝑆′Σ. (55) We now analyze the movement of 𝑆^′ relative to 𝑆 and Σ and then the movement of Σ relative to 𝑆 and 𝑆^′ using the same procedure that led to (43).

(i)   The Movement of 𝑆^′ Relative to Σ and 𝑆. For the movement of 𝑆^′ relative to Σ and 𝑆, we have (𝑡_Σ𝑆′,^→𝑥_Σ𝑆′)∼(𝑡_𝑆𝑆′,^→𝑥_𝑆𝑆′) and 𝑎𝜏=1−_𝑣Σ𝑆^→𝑥_𝑆𝑆′⋅−^→𝑣_Σ𝑆𝑣²_Σ𝑆+𝑎²_𝑣Σ𝑆−1𝑣_Σ𝑆𝑐𝑡_𝑆𝑆′1−𝑎_𝑣Σ𝑆^→𝑥_Σ𝑆′⋅^→𝑣_Σ𝑆𝑣²_Σ𝑆+𝑎²_𝑣Σ𝑆−1𝑣_Σ𝑆𝑐𝑡_Σ𝑆′. (56) From (53) we obtain 𝑑^→𝑥_Σ𝑆′𝑑𝑡_Σ𝑆′𝑑𝑡_Σ𝑆′=𝑑𝜏^→𝑣_Σ𝑆+𝑑^→𝑥_𝑆𝑆′𝑑𝑡_𝑆𝑆′𝑑𝑡_𝑆𝑆′.𝑑𝜏 (57) Denoting 𝑣̃^→_Σ𝑆′≡𝑑^→𝑥_Σ𝑆′𝑑𝑡_Σ𝑆′,^→𝑣̃_𝑆𝑆′≡𝑑^→𝑥_𝑆𝑆′𝑑𝑡_𝑆𝑆′ (58) and using (56), we obtain by the same procedure employed in obtaining (43) the following expression: 𝑣̃^→_Σ𝑆′=^→𝑣̃_𝑆𝑆′+𝛾𝑣̃_Σ𝑆^→𝑣̃_Σ𝑆−1−𝛾𝑣̃_Σ𝑆^→𝑣̃_𝑆𝑆′⋅^→𝑣̃_Σ𝑆/𝑣̃²_Σ𝑆^→𝑣̃_Σ𝑆𝛾𝑣̃_Σ𝑆1+^→𝑣̃_𝑆𝑆′⋅^→𝑣̃_Σ𝑆/𝑐². (59)

(ii)  The Movement of Σ Relative to 𝑆 and 𝑆^′. For the movement of Σ relative to 𝑆 and 𝑆^′, we have (𝑡_𝑆Σ,^→𝑥_𝑆Σ)∼(𝑡_𝑆′Σ,^→𝑥_𝑆′Σ) and 𝑎𝜏=1−_{𝑣𝑆′𝑆}^→𝑥_𝑆Σ⋅−^→𝑣_𝑆′𝑆𝑣²_𝑆′𝑆+𝑎²_{𝑣𝑆′𝑆}−1𝑣_𝑆′𝑆𝑐𝑡_𝑆Σ,1−𝑎_{𝑣𝑆′𝑆}^→𝑥_𝑆′Σ⋅^→𝑣_𝑆′𝑆𝑣²_𝑆′𝑆+𝑎²_{𝑣𝑆′𝑆}−1𝑣_𝑆′𝑆𝑐𝑡_𝑆′Σ. (60) From (53) and using that ^→𝑥_𝑆′Σ=−^→𝑥_Σ𝑆′, ^→𝑥_Σ𝑆=−^→𝑥_𝑆Σ, ^→𝑥_𝑆′𝑆=−^→𝑥_𝑆𝑆′, we now have 𝑑^→𝑥_𝑆′Σ𝑑𝑡_𝑆′Σ𝑑𝑡_𝑆′Σ=𝑑𝑑𝜏^→𝑥_𝑆Σ𝑑𝑡_𝑆Σ𝑑𝑡_𝑆Σ+𝑑𝜏^→𝑣_𝑆′𝑆. (61) Denoting 𝑣̃^→_𝑆′Σ≡𝑑^→𝑥_𝑆′Σ𝑑𝑡_𝑆′Σ,^→𝑣̃_𝑆Σ≡𝑑^→𝑥_𝑆Σ𝑑𝑡_𝑆Σ, (62) we obtain 𝑣̃^→_𝑆′Σ=^→𝑣̃_𝑆Σ+𝛾𝑣̃_′𝑆𝑆^→𝑣̃_𝑆′𝑆−1−𝛾𝑣̃_′𝑆𝑆^→𝑣̃_𝑆Σ⋅^→𝑣̃_𝑆′𝑆/𝑣̃²_𝑆′𝑆^→𝑣̃_𝑆′𝑆𝛾𝑣̃_′𝑆𝑆1+^→𝑣̃_𝑆Σ⋅^→𝑣̃_𝑆′𝑆/𝑐², (63) where we have used that 𝑣̃_𝑆𝑆′=𝑣̃_𝑆′𝑆 to identify 𝛾𝑣̃_′𝑆𝑆=𝛾𝑣̃_𝑆′𝑆. We recognize (59), (63) with the velocity transformations obtained by Møller in his analysis of the Thomas precession [5]. As we know, in this particular case, the velocities 𝑣̃^→_Σ𝑆′, 𝑣̃^→_𝑆′Σ are related by a rotation [5, 11]. We notice that each of (59), (63) involves vectors belonging to different spaces; therefore we should interpret each equation in the same way as we did for (9); that is, 𝑣̃^→_Σ𝑆′,^→𝑣̃_𝑆′Σ are vectors belonging to an abstract space 𝑉 where (59), (63) make sense. It is in this space that we have 𝑣̃^→_Σ𝑆′≠−^→𝑣̃_𝑆′Σ, (64) which accounts for the Thomas precession. However, the vectors ^→𝑣_Σ𝑆′,^→𝑣_𝑆′Σ are defined in the space ℝ³_{Σ𝑆𝑆′} where they satisfy (55), which is a consequence of the frames Σ,𝑆,𝑆^′ having their axis parallel during its translational movement. Now, the reason for not identifying the velocities 𝑣̃^→_Σ𝑆′, 𝑣̃^→_𝑆′Σ as vectors in ℝ³_{Σ𝑆𝑆′} becomes clear. Indeed, if this was the case, then using (30) we would associate each of them with the vectors ^→𝑣_Σ𝑆′, ^→𝑣_𝑆′Σ through 𝑣̃^→_Σ𝑆′=^→𝑣_Σ𝑆′𝑎²_{𝑣′Σ𝑆}−1𝑎_{𝑣′Σ𝑆}𝑐𝑣_Σ𝑆′,^→𝑣̃_𝑆′Σ=^→𝑣_𝑆′Σ𝑎²_{𝑣𝑆′Σ}−1𝑎_{𝑣𝑆′Σ}𝑐𝑣_𝑆′Σ, (65) and from (55) we would have 𝑣̃^→_Σ𝑆′=−^→𝑣̃_𝑆′Σ, which contradicts (64).

3.5. The Structure of Space and Time

Spacetime is the physical arena where events occur. Therefore, in order to describe spacetime, we must endow it with a suitable coordinate system, for example, the one provided by a reference frame, which is how we describe events. In our approach, we have an equivalence between the absolute and the physical time that allows an observer to associate coordinates with events according to (𝜏,^→𝑥) or (𝑡,^→𝑥), and from each of these coordinatizations originates a particular view for the spacetime. In what follows, we search for a description of spacetime that combines the views of the Galilei and the Special relativities.

Given two frames 𝑆, 𝑆^′, let us write the Galilei transformation as 𝜏_𝑆,^→𝑥_𝑆⟶𝜏_𝑆′,^→𝑥_𝑆′∶𝜏_𝑆=𝜏_𝑆′≡𝜏_→𝑥_𝑆′=^→𝑥_𝑆−^→𝑣𝜏. (66) Due to the absolute character of 𝜏, we have that for a fixed 𝜏≡𝜏_𝑆=𝜏_𝑆′ the Galilei transformation is essentially a map ℝ³_𝑆→ℝ³_𝑆′, where ℝ³_𝑆∋^→𝑥_𝑆,ℝ³_𝑆′∋^→𝑥_𝑆′ denote the 3-dimensional Euclidean space. When spacetime is described using (𝜏,^→𝑥) as coordinates, we refer to it as the Galilei spacetime that admits the following representation: ∪_𝜏∈ℝ({𝜏}×ℝ³)≡ℝ×ℝ³. In this sense, the Galilei spacetime decomposes in terms of 3-dimensional spaces ℝ³ that are indexed by 𝜏∈ℝ.

Now, let us analyze the transformation under the form (23). For the same fixed value of 𝜏, we rewrite (25) as 𝜏=𝑥⁰𝑛⁰+^→𝑥⋅^→𝑛=𝑥^′0𝑛^′0+^→𝑥^′⋅^→𝑛^′ (67) with 𝑛^𝜇≡𝑛⁰,^→𝑛√∶=𝑎²−1𝑣𝑎|),−(|−1^→𝑣𝑣²,𝑛^′𝜇≡𝑛^′0,^→𝑛^′√∶=𝑎²−1𝑣,(|𝑎|−1)^→𝑣𝑣² (68) and 𝑥^𝜇≡(𝑥⁰,^→𝑥)∶=(𝑐𝑡,^→𝑥),𝑥^′𝜇≡(𝑥^′0,^→𝑥^′)∶=(𝑐𝑡^′,^→𝑥^′). We identify (67) as the equation of two hyperplanes 𝜎_𝜏∶𝜏=𝑥⁰𝑛⁰+^→𝑥⋅^→𝑛 and 𝜎^′_𝜏∶𝜏=𝑥^′0𝑛^′0+^→𝑥^′⋅^→𝑛^′ defined, respectively, in the spaces ℝ⁴_𝑆 and ℝ⁴_𝑆′ (the spaces ℝ⁴_𝑆∋(𝑐𝑡,^→𝑥), ℝ⁴_𝑆′∋(𝑐𝑡^′,^→𝑥^′) serve here as the background in terms of which each observer describes spacetime). Then, for a fixed value of 𝜏, the transformation (23) is seen as a transformation 𝜎_𝜏→𝜎^′_𝜏 between these hyperplanes. Here, when spacetime is described using (𝑡,^→𝑥) as coordinates (in fact, in order to have coordinates with the dimension of length we would write (𝑐𝑡,^→𝑥) and (𝑐𝜏,^→𝑥) in place of, respectively, (𝑡,^→𝑥) and (𝜏,^→𝑥)), we refer to it as the SR spacetime and it decomposes as ∪_𝜏∈ℝ𝜎_𝜏≡ℝ⁴.

From the previous considerations, the properties of the spacetime that arise from our unified model are summarized as follows. First, we notice that the Galilei and SR spacetime refer to the same spacetime in fact, they are just different descriptions of the same structure. Second, both the Galilei and the SR spacetime are sliced by 3-dimensional spaces, with the Galilei spacetime being “sliced” by hyperplanes that are normal to the 𝜏-direction or, equivalently, by planes that correspond to the space ℝ³. For the SR spacetime, the “slicing” is made by hyperplanes 𝜎_𝜏, 𝜎^′_𝜏 that are orthogonal to the direction of the vectors 𝑛 and 𝑛^′. In particular, the relation between the Galilei transformation and the Lorentz transformation is clarified by the diagram below: (69) where for a fixed 𝜏 and relative to two frames of reference 𝑆,𝑆^′ that move with relative velocity ^→𝑣 we have defined mappings 𝑖_𝜏∶ℝ³_𝑆⟶ℝ⁴_𝑆,_→𝑥⟶𝑐𝑡𝜏,^→𝑥,^→𝑥,𝑡𝜏,^→𝑥=𝑣𝑐√𝑎²𝑎|)−1𝜏+(|−1^→𝑥⋅^→𝑣𝑣²,𝑖^′_𝜏∶ℝ³_𝑆′⟶ℝ⁴_𝑆′,_→𝑥^′⟶𝑐𝑡^′𝜏,^→𝑥^′,^→𝑥^′,𝑡^′𝜏,^→𝑥^′=𝑣𝑐√𝑎²𝑎|)−1𝜏−(|−1^→𝑥^′⋅^→𝑣𝑣². (70) We notice that endowing ℝ³_𝑆, ℝ³_𝑆′, ℝ⁴_𝑆, ℝ⁴_𝑆′ with the standard Euclidean topology we obtain 𝑖_𝜏(ℝ³_𝑆)=𝜎_𝜏, 𝑖^′_𝜏(ℝ³_𝑆′)=𝜎^′_𝜏 as embedded, respectively, in ℝ⁴_𝑆 and ℝ⁴_𝑆′. Then, given a Galilei transformation (𝜏,^→𝑥)−−→^𝜑𝐺(𝜏,^→𝑥^′), the embeddings 𝑖_𝜏,𝑖^′_𝜏 transfer 𝜑_𝐺 to the spaces ℝ⁴_𝑆, ℝ⁴_𝑆′ where it assumes the form of the Lorentz transformation (𝑐𝑡,^→𝑥)−−→^𝜑𝐿(𝑐𝑡^′,^→𝑥^′), which is shown in the scheme below (71) This view of the transformation provides an alternative way to analyze some aspects of the composition of Lorentz transformations. Indeed, let us write the Lorentz transformation as 𝜑_𝐿𝑣_𝑆𝑆′=𝑖^′_𝜏𝑣_𝑆𝑆′∘𝜑_𝐺𝑣_𝑆𝑆′∘𝑖_𝜏⁻¹𝑣_𝑆𝑆′ (72) and consider three frames 𝑆,𝑆^′, and 𝑆^′′ that move with relative velocities ^→𝑣_𝑆𝑆′, ^→𝑣_{𝑆𝑆′′}, and ^→𝑣_{𝑆′𝑆′′}. Consider then the following diagram (73) We notice that since the hyperplanes 𝜎^′_𝜏 and 𝜎_𝜏^′∗ are different, the Lorentz transformations 𝜑_𝐿(𝑣_𝑆𝑆′) and 𝜑_𝐿(𝑣_{𝑆′𝑆′′}) may be composed only through the maps 𝑖^′_𝜏(𝑣_{𝑆′𝑆′′})∘𝑖_𝜏^′−1(𝑣_𝑆𝑆′), for example, 𝜑_𝐿𝑣_{𝑆′𝑆′′}∘𝑖^′_𝜏𝑣_{𝑆′𝑆′′}∘𝑖^′_𝜏⁻¹𝑣_𝑆𝑆′∘𝜑_𝐿𝑣_𝑆𝑆′ (74) and we have 𝜑_𝐿𝑣_{𝑆′𝑆′′}∘𝑖^′_𝜏𝑣_{𝑆′𝑆′′}∘𝑖^′_𝜏⁻¹𝑣_𝑆𝑆′∘𝜑_𝐿𝑣_𝑆𝑆′=𝑖_𝜏^′′𝑣_{𝑆′𝑆′′}∘𝜑_𝐺𝑣_{𝑆′𝑆′′}∘𝜑_𝐺𝑣_𝑆𝑆′∘𝑖_𝜏⁻¹𝑣_𝑆𝑆′=𝑖_𝜏^′′𝑣_{𝑆′𝑆′′}∘𝜑_𝐺𝑣_{𝑆𝑆′′}∘𝑖_𝜏⁻¹𝑣_𝑆𝑆′≠𝜑_𝐿𝑣_{𝑆𝑆′′}. (75) Therefore, from this interpretation of the Lorentz transformation acting on hyperplanes associated to the same value of 𝜏, we reobtain the common fact that the composition of two Lorentz transformations is not a Lorentz transformation. Here, the reason is that even though the composition 𝜑_𝐺(𝑣_{𝑆′𝑆′′})∘𝜑_𝐺(𝑣_𝑆𝑆′) may be identified with 𝜑_𝐺(𝑣_{𝑆𝑆′′}), the latter induces the Lorentz transformation 𝜑_𝐿(𝑣_{𝑆𝑆′′}) only through the embeddings 𝑖_𝜏^′′(𝑣_{𝑆𝑆′′}), 𝑖_𝜏⁻¹(𝑣_{𝑆𝑆′′}).

4. Model II

4.1. Assumptions

We propose another model based on the following assumptions. (I)Space. Each inertial frame describes space as an euclidean 3-dimensional vector space. (II)Time. Time is described by a variable 𝑡 that depends on the frame in such a way that two inertial frames 𝑆 and 𝑆^′ measure time 𝑡 and 𝑡^′. (III) The speed of light is constant relative to inertial frames. (IV) Relative to two inertial frames, any event (𝑡,^→𝑥)∼(𝑡^′,^→𝑥^′) satisfies 𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′ (76) with 𝛼,^→𝛽 arbitrary parameters. (V) The relation between (𝑡,^→𝑥) and (𝑡^′,^→𝑥^′) is linear.

Remarks. (i) Postulates (I) and (II) are essentially the same postulates (I) and (II) of model I, except that in the current postulate (II) of model II there is no mention of the absolute time.

(ii) The condition 𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′ is suggested by a previous condition given in (67), but now with the constants 𝛼, ^→𝛽 being arbitrary parameters. Here, (76) is the starting point from which we will derive the transformation. In this sense, it plays the same role as the relation ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡² of the standard SR formulation. However, while in SR this last equation may be expressed in terms of the invariance of the quadratic form 𝑄(𝑥^𝜇)∶=^→𝑥²−𝑐²𝑡², which is a quantity defined uniquely in terms of the spacetime coordinates, the same cannot be said of (76) since it contains explicitly the parameters 𝛼,^→𝛽 that, in general, do not refer to any known property of the space and time. Therefore, unless we are able to identify the parameters 𝛼,^→𝛽 with any intrinsic property of the spacetime, it will not be possible to associate any mathematical structure with (76) as it was done with 𝑄(𝑥^𝜇).

4.2. The Generalized Lorentz Transformation

From (V) we write 𝑥^′0=𝐴⁰₀𝑥⁰+𝐴⁰_𝑖𝑥^𝑖,𝑥^′𝑖=𝐴^𝑖₀𝑥⁰+𝐴^𝑖_𝑗𝑥^𝑗. (77) From (IV), we obtain by replacing them into (76) that 𝐴⁰₀1=1+𝛼𝐴^𝑖₀𝛽^𝑖,𝐴⁰_𝑖=1𝛼𝛽^𝑖+𝐴^𝑗_𝑖𝛽^𝑗 (78) which momentarily gives 𝑡^′=11+𝛼𝐴^𝑖₀𝛽^𝑖1𝑡+𝛽𝛼𝑐^𝑖+𝐴^𝑗_𝑖𝛽^𝑗𝑥^𝑖,𝑥^′𝑖=𝐴^𝑖₀𝑐𝑡+𝐴^𝑖_𝑗𝑥^𝑗. (79) Here we assume 𝐴^𝑖₀=𝜆𝛽^𝑖,𝐴^𝑖_𝑗=𝛿^𝑖_𝑗+𝜉𝛽^𝑖𝛽^𝑗 (80) with 𝜆,𝜉 arbitrary quantities that will be fixed later. We require that the inverse transformation is obtained by the change of (𝑡,^→𝑥,𝛽^𝑖)↔(𝑡^′,^→𝑥^′,−𝛽^𝑖) into (79). This fixes 𝜆=𝜉𝛼. Let us consider now the expression for the velocity. Here, we have 𝑑𝑥^′𝑖𝑑𝑡^′=𝐴^𝑖₀𝑐+𝐴^𝑖_𝑗𝑑𝑥^𝑗/𝑑𝑡1+(1/𝛼)𝐴^𝑖₀𝛽^𝑖+(1/𝛼𝑐)𝛽^𝑖+𝐴^𝑗_𝑖𝛽^𝑗𝑑𝑥^𝑖./𝑑𝑡 (81) From (III) the speed of light is constant relative to any inertial frame. Then for a light wave we have 𝑑^→𝑥^′𝑑𝑡^′=^→𝑐^′,𝑑^→𝑥=𝑑𝑡^→𝑐≡𝑐𝜂with𝜂⋅𝜂=1,^→𝑐^′=^→𝑐. (82) From (81), (80), using 𝜆=𝜉𝛼, and due to the arbitrariness of the direction of the unit vector 𝜂, we obtain 2𝜉=𝛼²1−𝛽²/𝛼² (83) which then fixes all the transformation constants 𝐴⁰₀,𝐴⁰_𝑖,𝐴^𝑖₀, and 𝐴^𝑖_𝑗 in terms of 𝛼,𝛽 as follows 𝐴⁰₀=1+𝛽²/𝛼²1−𝛽²/𝛼²,𝐴⁰_𝑖=𝐴^𝑖₀=21−𝛽²/𝛼²𝛽^𝑖𝛼,𝐴^𝑖_𝑗=𝛿^𝑖_𝑗+21−𝛽²/𝛼²𝛽^𝑖𝛼𝛽^𝑗𝛼 (84) and finally leaves us with the transformations 𝑡^′=1+𝛽²/𝛼²1−𝛽²/𝛼²2𝑡+𝑐1+𝛽²/𝛼^2→𝛽𝛼⋅^→𝑥,_→𝑥^′=^→𝑥+21−𝛽²/𝛼^2→𝛽𝛼⋅^→𝑥^→𝛽𝛼+2𝑐𝑡1−𝛽²/𝛼^2→𝛽𝛼. (85) We have then obtained a class of transformations that is compatible with the invariance of the speed of light and that depends on arbitrary parameters 𝛼,^→𝛽. We call this transformation Generalized Lorentz Transformation (GLT). It is immediate to check that it satisfies ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡² (and not only for a light wave), which shows that it must include the standard Lorentz transformation as a particular case.

Remark. It is possible to work with a more general relation than the one given in (76); for example, if we assume that ^→𝛽 can be transformed, then we would write 𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽^′⋅^→𝑥^′, (86) and the effect of the invariance of ^→𝛽^′⋅^→𝑥^′ in this new condition corresponds to replace ^→𝛽→ℛ^→𝛽, ^→𝑥→ℛ^→𝑥 in (85), that is, 𝑡^′=1+𝛽²/𝛼²1−𝛽²/𝛼²2𝑡+𝑐1+𝛽²/𝛼^2→𝛽𝛼⋅^→𝑥,_→𝑥^′=ℛ^→𝑥+21−𝛽²/𝛼^2→𝛽𝛼⋅^→𝑥ℛ^→𝛽𝛼+2𝑐𝑡1−𝛽²/𝛼²ℛ^→𝛽𝛼 (87) with ℛ being an arbitrary space rotation.

In [12] Kapuscik using only the assumption of the linearity of the transformation and the constancy of the speed of light proposed a generalization of the Lorentz transformation having the form 𝑡^′=𝐴𝑡+^→𝐵⋅^→𝑥,_→𝑥^′=𝐴²−𝑐^2→𝐵²ℛ^→𝑥+𝐴−𝐴²−𝑐^2→𝐵²_→𝐵²ℛ^→𝐵^→𝐵⋅^→𝑥+𝑐²ℛ^→𝐵𝑡. (88) The comparison of the time transformation given in (87) and (88) suggests the identification 𝛽𝐴=1+²/𝛼²1−𝛽²/𝛼²,^→𝐵=2𝑐1−𝛽²/𝛼^2→𝛽𝛼 (89) that also shows the equivalence of the space transformation. In our formalism, up to the derivation of (87), there is no restriction on the parameters 𝛼,^→𝛽, while in the development of [12] they have to obey 𝐴²−𝑐^2→𝐵²>0. Since in [12] the relative velocity 𝑉^→ between the frames is identified with −𝑐²(^→𝐵/𝐴), the previous inequality leads to the restriction 𝑉<𝑐. As we will show in Sections 4.3.1 and 4.3.2, in our formalism there is no restriction on the relative speed of the frames. Having obtained the GLT given in (85) we analyze in the sequence some of its properties.

4.3. Transformation of Velocity

Associated with GLR, we obtain by replacing (84) into (81) the associated transformation of velocity as 𝑑^→𝑥^′𝑑𝑡^′=𝑑^→𝑥+𝑑𝑡2𝑐1−𝛽²/𝛼^2→𝛽𝛼+21−𝛽²/𝛼^2→𝛽𝛼⋅𝑑^→𝑥𝑑𝑡^→𝛽𝛼×1+𝛽²/𝛼²1−𝛽²/𝛼²+2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝑑^→𝑥𝑑𝑡⁻¹,𝑑^→𝑥^′𝑑𝑡^′2−𝑐²=𝑑^→𝑥𝑑𝑡²−𝑐²×1+𝛽²/𝛼²1−𝛽²/𝛼²+2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝑑^→𝑥𝑑𝑡²⁻¹. (90)(91) Therefore we obtain for the GLT that 𝑑^→𝑥𝑑𝑑𝑡<𝑐⟺^→𝑥^′𝑑𝑡^′𝑑<𝑐or^→𝑥𝑑𝑑𝑡>𝑐⟺^→𝑥^′𝑑𝑡^′>𝑐 (92) That is, relative to the two frames described by transformation (85), a particle that is described as subluminal (superluminal) particle by one frame will also be described as subluminal (superluminal) particle by the other frame.

4.3.1. A Transformation Involving Subluminal Frames: 𝑣̃<𝑐

As a particular case of transformation (85) let us choose the parameters 𝛼,^→𝛽 in such way that 1+𝛽²/𝛼²1−𝛽²/𝛼²=11−𝑣²̃/𝑐² (93) with 𝑣̃<𝑐. We obtain ^→𝛽𝛼=−𝑐1−𝑣̃1−²𝑐^2→𝑣̃𝑣²̃ (94) that replacing in (85) leads to the transformation 𝑡^′=𝛾𝑡−^→𝑥⋅^→𝑣̃𝑐²_→,𝑥^′=^→𝑥−1−𝛾^→𝑥⋅^→𝑣̃𝑣̃^2→𝑣̃−𝛾𝑡^→𝑣̃ (95) with 𝛾∶=1/1−(𝑣²̃/𝑐²) that corresponds to the Lorentz transformation.

4.3.2. A Transformation Involving Superluminal Frames: 𝑣̃>𝑐

Let us choose 𝛼,^→𝛽 satisfying 1+𝛽²/𝛼²1−𝛽²/𝛼²=11−𝑐²/𝑣²̃ (96) with 𝑣̃>𝑐. Now we obtain ^→𝛽𝛼1=−𝑐1−𝑐1−²𝑣̃^2→𝑣̃ (97) and replacing it into (85) gives the transformation 𝑡^′=𝛾𝑡−^→𝑥⋅^→𝑣̃𝑣²̃_→,𝑥^′=^→𝑥−1−𝛾^→𝑥⋅^→𝑣̃𝑣̃^2→̃𝛾𝑐𝑣−²𝑣²̃𝑡^→𝑣̃ (98) with 𝛾∶=1/1−(𝑐²/𝑣²̃). Here, we reobtained Shankara and Duffey’s transformation for tachyons [9, 10]. This transformation (98) appeared originally in [9] in the description of the propagation of a wave in a medium with one of the frames being at rest relative to the medium and the other moving with a speed greater than 𝑐 (in fact, in [9] since the medium is not the vacuum, instead of 𝑐 the light speed is denoted by 𝑤₀). The physical content of this transformation has been discussed more generally in [10], to which we refer the reader for details. Here we wish to analyze the role played by the velocity 𝑣̃^→ in the transformation. We consider two situations.

First, let us consider the movement of an object such that according to the frame 𝑆 the object is at rest relative to the frame 𝑆^′. Assume that 𝑣̃^→ is the relative velocity between the frames. Then the frame 𝑆 writes for the position of the object ^→𝑥=^→𝑣̃𝑡. Replacing it in (98), we obtain 𝑡^′=0 and ^→𝑥^′=(1/𝛾)𝑡^→𝑣̃=(1/𝛾)^→𝑥. Therefore, for an interval (0,Δ𝑡), 𝑆 associates with the movement of the object the set of events {(𝑡,^→𝑥)∶=(𝑡,^→𝑣̃𝑡)∶0≤𝑡≤Δ𝑡}, while 𝑆^′ associates with the set of events {(𝑡^′,^→𝑥^′)=(0,(1/𝛾)^→𝑥)∶^→𝑥=^→𝑣̃𝑡}. Since the whole movement of the object as described by 𝑆 reduces in 𝑆^′ to a static situation where 𝑡^′=0 and since there are many ^→𝑥^′ corresponding to the many possible values of ^→𝑥, we conclude that relative to 𝑆^′ the object seems to have a spatial extension. Therefore, from the perspective of the frame 𝑆^′ the object cannot be seen as a particle in the classical context. This is reinforced by noticing that the possibility of considering ^→𝑥=^→𝑣̃𝑡 leads to the vanishing of the denominator of (90) that shows that the velocity transformation law does not apply.

Let us consider now the movement of an object such that according to the frame 𝑆^′ the object is at rest relative to the frame 𝑆^′,thatis, ^→𝑥^′=0. Replacing it in (98) we now obtain 𝑡^′=(1/𝛾)𝑡 and ^→𝑥=(𝑐²/𝑣²̃)^→𝑣̃𝑡. Therefore we get 𝑑^→𝑥=𝑐𝑑𝑡𝑣²̃, (99) and with 𝑣̃>𝑐 we end up with |𝑑^→𝑥/𝑑𝑡|<𝑐. Here, it is natural that instead of 𝑣̃^→ we assume the relative velocity between the frames to be 𝑉^→∶=(𝑐²/𝑣²̃)^→𝑣̃, in terms of which the transformation (98) becomes 𝑡^′=𝛾𝑡−^→𝑥⋅𝑉^→𝑉²_→,𝑥^′=^→𝑥−1−𝛾^→𝑥⋅V^→𝑉^2→𝑉−𝛾𝑡𝑉^→ (100) with 𝛾=1/1−(𝑉²/𝑐²). Then, from a theoretical perspective, we conclude that a transformation like (98) may refer to different situations according to the interpretation one gives to 𝑣̃^→.

4.3.3. Another Superluminal Transformation

As another class of transformation let us choose 𝛼,^→𝛽 such that 1+𝛽²/𝛼²1−𝛽²/𝛼²=1𝑣²̃/𝑐²−1 (101) with 𝑣̃>𝑐, which gives ^→𝛽𝛼=−1−𝑣²̃/𝑐²−11+𝑣²̃/𝑐²−1^→𝑣̃𝑣̃. (102) Replacing it in (85) we obtain the transformation 𝑡^′=1𝑣²̃/𝑐²−1𝑡−𝑣̃2−²𝑐^2→𝑥⋅^→𝑣̃𝑐𝑣_→̃𝑥^′=^→𝑥+1−𝑣²̃/𝑐²−1𝑣²̃/𝑐²−1^→𝑥⋅^→𝑣̃𝑣̃^2→𝑣̃−2−𝑣²̃/𝑐²𝑣²̃/𝑐²𝑐−1𝑣̃𝑡^→𝑣̃. (103) The condition of having real transformations requires that 𝑣̃<2𝑐, which becomes an upper bound velocity for tachyonic motion.

4.4. Introducing the Galilei Relativity

Let us modify condition (76) writing it in the form 𝜏=𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′, (104) where we have introduced the absolute time 𝜏. Therefore, assuming the Galilei transformation ^→𝑥^′=^→𝑥−^→𝑣𝜏 (105) and using (85) and (104), we obtain that ^→𝑣2=−1−𝛽²/𝛼^2→𝛽𝛼². (106) From (104) we also obtain 𝑑𝑡=1𝑑𝜏𝛼𝑐+^→𝛽⋅𝑑^→𝑥./𝑑𝑡 (107) Let us define ^→𝑢𝑑∶=^→𝑥,𝑑𝜏^→𝑑𝑢∶=^→𝑥𝑑𝑡_→𝑢^′𝑑∶=^→𝑥^′,𝑑𝜏^→𝑢^′𝑑∶=^→𝑥^′.𝑑𝑡 (108) Then we obtain ^→𝑢^′=^→𝑢^′𝛼𝑐−^→𝛽⋅^→𝑢^′,^→𝑢=^→𝑢𝛼𝑐+^→𝛽⋅^→𝑢. (109) From the Galilei relativity we have ^→𝑢^′=^→𝑢−^→𝑣. (110) Therefore, assuming ^→𝑢^′=0, we get ^→𝑢=^→𝑣, ^→𝑢=^→𝑣̃, and ^→𝑣=^→𝑣̃𝛼𝑐+^→𝛽⋅^→𝑣̃. (111) Replacing this last expression into (106) we end up with 𝑣̃^→=−2𝑐1+𝛽²/𝛼^2→𝛽𝛼. (112) In particular, we notice that 𝑣̃<𝑐. (113) Then, the assumption of an absolute time and the Galilean relativity law in model II also leads to a situation where there are no superluminal frames.

The expression we obtained for 𝑣̃ in (112) also leads to the same transformation of Section 4.3.1; therefore, when we incorporate the Galilei relativity into model II, we reduce the model to the standard Lorentz situation. However, different from model I, as we have seen in Section 4.3.1, we are allowed to obtain the Lorentz case without the need to impose the Galilei relativity.

4.5. The Momentum

We now search for an expression for the momentum that is conserved in collision processes. As it is well known [5], the conservation of momentum in two frames 𝑆 and 𝑆^′ brings to the analysis the transformation between them. Then, the problem here is to find an expression for the momentum that satisfies a conservation law as seen by frames related by the GLT.

Let us perform our analysis considering a particular collision process involving two particles with the same mass and such that relative to 𝑆 we have before the collision the particles’ velocities ^→𝑢₁=𝑢𝜂, ^→𝑢₂=−𝑢𝜂, and after the collision ^→𝑢^∗₁=𝑢𝜂^∗, ^→𝑢^∗₂=−𝑢𝜂^∗. (As it is typical in the analysis of collision processes, for a more general collision we would obtain constraints on the final velocities, which are not essential in order to fix the form of the function 𝑓(^→𝑢) introduced in (116), (117).) According to 𝑆^′, we obtain from (90) ^→𝑢^′₁=2𝑐1−𝛽²/𝛼^2→𝛽𝛼+𝑢^→𝜔×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹,_→𝑢^′₂=2𝑐1−𝛽²/𝛼^2→𝛽𝛼−𝑢^→𝜔×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹,_→𝑢₁^′∗=2𝑐1−𝛽²/𝛼^2→𝛽𝛼+𝑢^→𝜔^∗×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1,_→𝑢₂^′∗=2𝑐1−𝛽²/𝛼^2→𝛽𝛼−𝑢^→𝜔^∗×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1 (114) with ^→2𝜔∶=𝜂+1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^→𝛽𝛼_→,𝜔^∗∶=𝜂^∗+21−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗→𝛽𝛼 (115) being two vectors having arbitrary directions due to their dependence on the arbitrary unit vectors 𝜂,𝜂^∗. Following [5], we write for the momentum conservation in the frame 𝑆𝑓^→𝑢₁^→𝑢₁+𝑓^→𝑢₂^→𝑢₂=𝑓^→𝑢^∗₁^→𝑢^∗₁+𝑓^→𝑢^∗₂^→𝑢^∗₂ (116) and relative to 𝑆^′ it becomes (assuming that the form of the function 𝑓 is invariant) 𝑓^→𝑢^′₁^→𝑢^′₁+𝑓^→𝑢^′₂^→𝑢^′₂=𝑓^→𝑢₁^′∗→𝑢₁^′∗+𝑓^→𝑢₂^′∗→𝑢₂^′∗ (117) that is equivalent to 𝑢^→𝜔𝑓^→𝑢^′₁×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹−𝑓^→𝑢^′₂×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹−𝑢^→𝜔^∗𝑓^→𝑢₁^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1−𝑓^→𝑢₂^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1+^→𝛽𝛼2𝑐1−𝛽²/𝛼²×𝑓^→𝑢^′₁×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹+𝑓^→𝑢^′₂×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹−𝑓^→𝑢₁^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1−𝑓^→𝑢₂^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1=0. (118) From the arbitrariness of the scattering directions 𝜂, 𝜂^∗, we must have 𝑓^→𝑢^′₁1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹=𝑓^→𝑢^′₂×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂⁻¹=𝑓^→𝑢₁^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²+𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1=𝑓^→𝑢₂^′∗×1+𝛽²/𝛼²1−𝛽²/𝛼²−𝑢2𝑐1−𝛽²/𝛼^2→𝛽𝛼⋅𝜂^∗−1. (119) In order to satisfy this condition we notice that from (91) we obtain 𝑢1−^′2𝑐²=𝑢1−²𝑐²×1+𝛽²/𝛼²1−𝛽²/𝛼²+21−𝛽²/𝛼^2→𝛽𝛼⋅^→𝑢𝑐⁻¹. (120) For both cases we analyzed in Sections 4.3.1, 4.3.2, we observe that 𝛽/𝛼<1 and for the case of a subluminal particle, that is, 𝑢<𝑐,𝑢^′<𝑐, we obtain 1+𝛽²/𝛼²1−𝛽²/𝛼²+21−𝛽²/𝛼^2→𝛽𝛼⋅^→𝑢𝑐>0 (121) which suggest us to take for 𝑓(𝑢) the following expression: 𝑓(𝑢)=𝑘1−𝑢²/𝑐². (122) In the case of a superluminal particle,thatis,𝑢>𝑐,𝑢^′>𝑐, we have in general 1+𝛽²/𝛼²1−𝛽²/𝛼²+21−𝛽²/𝛼^2→𝛽𝛼⋅^→𝑢𝑐<0 (123) which suggest us to take 𝑓(𝑢)=𝑘sgn1+𝛽²/𝛼²+2^→𝛽/𝛼⋅^→𝑢/𝑐1−𝑢²/𝑐². (124) The constant 𝑘 is identified with the rest mass of the particle in the case that one can effectively define such frame. In the case of tachyons we refer to the discussion of [13]. The term in the numerator, sgn[1+(𝛽²/𝛼²)+2(^→𝛽/𝛼)⋅(^→𝑢/𝑐)], becomes necessary in order to fulfill the conservation of momentum as expressed in condition (119). This term introduces a dependence of the tachyon mass with the quantity ^→𝛽/𝛼, which is ultimately a parameter depending on the relative velocity between the frames. A detailed discussion on the implications of this term may be found in [13].

5. An Alternative Way to Induce a Concept of Absolute Time

The concept of the absolute time was introduced in models I and II by means of certain relations involving the space and time variables, (5), (104). These relations may be seen as kinematical requirements allowing to bring the special and the Galilei relativity into the models. In this view, we could think on the physical and the absolute time without having to consider them as coordinates of a 4-dimensional space. Now, we reverse the construction and we will show that by considering a higher dimensional space, we can think on the absolute time with no reference to kinematical relations.

5.1. Setting the Lorentz Transformation in ℝ^(1,4)

We denote by ℝ^(1,4) the space ℝ⁵ endowed with the quadratic form 𝑄𝑥^𝑎=−𝑥⁰²+𝑥¹²+𝑥²²+𝑥³²+𝑥⁴². (125) Here ℝ^(1,4)∋𝑥≡(𝑥^𝑎),𝑎=0,1,2,3,4. The transformations leaving 𝑄 invariant have the form 𝑥^′𝑎∶=Λ^𝑎_𝑏𝑥^𝑏 (126) with [Λ^𝑎_𝑏]∈𝑆𝑂(1,4). We consider now two subgroups of 𝑆𝑂(1,4) as follows. The first consists of transformations defined by matrices 𝐿_ΛΛ∶=^𝜇𝜈𝟎𝟎1 (127) with Λ∈𝑆𝑂(1,3) (𝜇,𝜈=0,1,2,3). The second consists on transformations defined by matrices 𝐿_ℛ∶=1𝟎𝟎ℛ^𝑖_𝑗 (128) with ℛ∈𝑆𝑂(4) (𝑖,𝑗=1,2,3,4).

Now, we define new coordinates as follows. (i)(𝑥^𝑎/𝑥⁴)=(𝑥^𝜇,1) with 𝑥^𝜇≡𝑥^𝜇/𝑥⁴. For the subgroup {𝐿_Λ}⊂𝑆𝑂(1,4) the transformation (126) becomes 𝑥^′𝜇=Λ^𝜇_𝜈𝑥^𝜈𝑥^′4=𝑥⁴ (129) and in particular 𝑥^′𝜇=Λ^𝜇_𝜈𝑥^𝜈. (130)(ii)(𝑥^𝑎/𝑥⁰)=(1,𝑥^𝑖), with 𝑥^𝑖≡𝑥^𝑖/𝑥⁰. For the subgroup {𝐿_ℛ}⊂𝑆𝑂(1,4) the transformation (126) becomes 𝑥^′0=𝑥⁰,𝑥^′𝑖=ℛ^𝑖_𝑗𝑥^𝑗 (131) and in particular 𝑥^′𝑖=ℛ^𝑖_𝑗𝑥^𝑗. (132)We notice that these coordinates 𝑥≡(𝑥^𝜇) and 𝑥≡(𝑥^𝑖) cannot be identified with the coordinates of the hyperplanes 𝑥⁰=0, 𝑥⁴=0 of ℝ^(1,4).

5.2. Inducing Lorentz Transformation in Euclidean Space ℝ⁴

There is a natural way to reinterpret the coordinates 𝑥=(𝑥^𝜇), 𝑥=(𝑥^𝑖) as local coordinates on certain open sets of the real projective space ℝP⁴ [14]. In fact, let ℝP⁴∋[𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⁴]∶={𝜆(𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⁴)∶𝜆∈ℝ^∗}. Recall that a smooth atlas for ℝP⁴ is given by {(𝑉_𝑎,𝜑_𝑎)}_𝑎=0,…,4, where 𝑉_𝑎∶={[𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⁴]∶𝑥^𝑎≠0} and 𝜑_𝑎∶𝑉_𝑎⟶ℝ⁴[𝑥]≡𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⁴⟶𝜑_𝑎𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⁴𝑥∶=⁰𝑥^𝑎𝑥,…,^𝑎−1𝑥^𝑎,𝑥^𝑎+1𝑥^𝑎𝑥,…,⁴𝑥^𝑎 (133) which shows that 𝑥=𝜑₄([𝑥]) and 𝑥=𝜑₀([𝑥]). (Notice that the space ℝ⁴, appearing as the image of the homeomorphisms 𝜑_𝑎, is not seen as a subset of ℝ^(1,4).)

The transformation on ℝ^(1,4), 𝑥^′𝑎=Λ^𝑎_𝑏𝑥^𝑏, induces a transformation on ℝP⁴∶[𝑥]→[Λ𝑥] that has the particular cases 𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⟶^4𝐿ΛΛ⁰_𝛼𝑥^𝛼,Λ¹_𝛼𝑥^𝛼,Λ²_𝛼𝑥^𝛼,Λ³_𝛼𝑥^𝛼,𝑥⁴,𝑥⁰,𝑥¹,𝑥²,𝑥³,𝑥⟶^4𝐿ℛ𝑥⁰,ℛ¹_𝑖𝑥^𝑖,ℛ²_𝑖𝑥^𝑖,ℛ³_𝑖𝑥^𝑖,ℛ⁴_𝑖𝑥^𝑖. (134) Therefore, 𝜑₄∘𝐿_Λ∘𝜑₄⁻¹ reduces to an ordinary Lorentz transformation in 𝜑₄(𝑉₄)≃ℝ⁴∶𝑥^𝜇→𝑥^′𝜇=Λ^𝜇_𝜈𝑥^𝜈, while 𝜑₀∘𝐿_ℛ∘𝜑₀⁻¹ reduces to a 4-dimensional rotation in 𝜑₀(𝑉₀)≃ℝ⁴∶𝑥^𝑖→𝑥^′𝑖=ℛ^𝑖_𝑗𝑥^𝑗. We have then modelled Minkowski and Euclidean four-space as homeomorphic to open sets 𝜑₄(𝑉₄), 𝜑₀(𝑉₀) of projective space, where Lorentz and 4-dimensional rotations are induced by transformations (129), (131) of 𝑆𝑂(1,4) acting in ℝ⁵. In the intersection of the neighborhoods, 𝑉₀∩𝑉₄, we have 𝑥^𝑖=𝜑₀∘𝜑₄⁻¹(𝑥), that is, 𝑥¹=𝑥¹𝑥⁰,𝑥²=𝑥²𝑥⁰,𝑥³=𝑥³𝑥⁰,𝑥⁴=1𝑥⁰. (135) Therefore, in the intersection 𝑉₀∩𝑉₄ an ordinary Lorentz transformation on 𝜑₄(𝑉₄), 𝑥^𝜇→𝑥^′𝜇=Λ^𝜇_𝜈𝑥^𝜈, induces on 𝜑₀(𝑉₀) the transformation 𝑥^𝑖⟶𝑥^′𝑖=𝐿^𝑖_Λ0+𝐿^𝑖_Λ𝑗𝑥^𝑗𝐿⁰_Λ0+𝐿⁰_Λ𝑗𝑥^𝑗 (136) that also obeys a group law, for example, 𝑥⟶^Λ(1)𝑥⟶^Λ₍₁₎⁽²⁾𝑥₍₂₎𝑥↓↓↓_𝐿(1)⟶𝑥⟶^𝐿₍₁₎⁽²⁾𝑥₍₂₎≡𝑥−−−−−−→^Λ(2)Λ(1)𝑥₍₂₎𝑥↓↓_{𝐿(2)𝐿(1)}−−−−−−→𝑥₍₂₎. (137) That is, 𝑥^𝑖₍₂₎=𝐿₍₂₎𝐿₍₁₎^𝑖₀+𝐿₍₂₎𝐿₍₁₎^𝑖_𝑗𝑥^𝑗𝐿₍₂₎𝐿₍₁₎⁰₀+𝐿₍₂₎𝐿₍₁₎⁰_𝑗𝑥^𝑗. (138) Equation (136) may be seen as a nonlinear realization of the Lorentz group in 𝜑₀(𝑉₀).

5.3. Interpreting the Boost Transformation in Euclidean Space ℝ⁴

Now, let us introduce some physics in our analysis. So far, we have considered homogeneous coordinates 𝑥,𝑥 that carry no dimension. In order to give a physical meaning to them let us assume that there is a scale factor 𝜆 with dimension of length and let us define 𝑦^𝑖∶=𝜆𝑥^𝑖, 𝑦^′𝑖∶=𝜆^′𝑥^′𝑖. Here 𝜆^′ is the transformed scale factor whose form is left arbitrary for a moment. Then, transformation (136) may be seen as a transformation 𝑦^𝑖⟶𝑦^′𝑖=𝜆^′𝐿^𝑖_Λ0𝜆+𝐿^𝑖_Λ𝑗𝑦^𝑗𝐿⁰_Λ0𝜆+𝐿⁰_Λ𝑗𝑦^𝑗. (139) For a boost we have Λ⁰₀=𝛾,Λ⁰_𝑖=Λ^𝑖₀=−(𝛾/𝑐)𝑣^𝑖,Λ^𝑖_𝑗=𝛿^𝑖_𝑗−(1−𝛾)(𝑣^𝑖𝑣^𝑗/𝑣²), and we obtain 𝑦^′𝑖=𝜆^′𝑦^𝑖−1−𝛾𝑦^𝑗𝑣^𝑗/𝑣²𝑣^𝑖−𝛾/𝑐𝜆𝑣^𝑖𝛾𝜆−(1/𝑐)𝑦^𝑗𝑣^𝑗,𝑦^′4=𝜆^′𝑦⁴𝛾𝜆−(1/𝑐)𝑦^𝑗𝑣^𝑗, (140) where 𝑖=1,2,3. Now, let us assume that the scale factor writes as 𝜆^′=𝑐𝑡^′,𝜆=𝑐𝑡 with 𝑐 being the speed of light and 𝑡 and 𝑡^′ having the dimension of time. Here, denoting ^→𝑦=(𝑦^𝑖),^→𝑣=(𝑣^𝑖) (140) becomes ^→𝑦^′=𝑡^′→𝑦−1−𝛾^→𝑦⋅^→𝑣/𝑣^2→𝑣−𝛾𝑡^→𝑣𝛾𝑡−^→𝑦⋅^→𝑣/𝑐²,𝑦^′4=𝑡^′𝑦⁴𝛾𝑡−^→𝑦⋅^→𝑣/𝑐². (141) If 𝑡^′=𝛾(𝑡−((^→𝑦⋅^→𝑣)/𝑐²)), we obtain ^→𝑦^′=^→𝑦−1−𝛾^→𝑦⋅^→𝑣𝑣^2→𝑣−𝛾𝑡^→𝑣,𝑦^′4=𝑦⁴. (142) Equation (142) together with 𝑡^′=𝛾(𝑡−((^→𝑦⋅^→𝑣)/𝑐²)) constitutes the standard transformation of special relativity, and 𝑦^′4=𝑦⁴ can be considered as defining the absolute time, for example,𝜏≡𝑦^′4/𝑐=𝑦⁴/𝑐. In this way, the absolute time acquires an intrinsic characteristic as the coordinate (in fact with 𝑦⁴/𝑐) of a 4-dimensional Euclidean space ℝ⁴≃𝜑₀(𝑉₀), while the physical time of SR appears as a noninvariant scale factor (through the relation 𝜆=𝑐𝑡,𝜆^′=𝑐𝑡^′) transforming homogeneous coordinates 𝑥^𝑖 of Euclidean space ℝ⁴ into dimensional coordinates 𝑦^𝑖. If in addition we impose the kinematical relation of the Galilei relativity, ^→𝑦^′=^→𝑦−^→𝑣𝜏, we end up with 𝜏=(1−𝛾)((^→𝑦⋅^→𝑣)/𝑣²)^→𝑣+𝛾^→𝑣𝑡.

6. Conclusion

In our work we have consistently unified the Galilei and the Special relativities. In model I the main equations were ^→𝑥^′=^→𝑥−^→𝑣𝜏 and ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡², while in model II we have used ^→𝑥^′=^→𝑥−^→𝑣𝜏 and 𝜏=𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′. Here, contrarily to the intrinsic geometric meaning associated with the relation ^→𝑥^′2−𝑐²𝑡^′2=^→𝑥²−𝑐²𝑡², the role played by 𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′ is not clear. If there is a certain anisotropy in space, represented by a particular direction in space (not necessarily related to the relative velocity of the two frames), then we could encode it into ^→𝛽. In this way, the physics derived from the general transformation (85) may provide a framework to describe situations other than the one we contemplated in our work.

The assumption of the absolute time leads to a distinction between the velocities ^→𝑣=𝑑^→𝑥/𝑑𝜏 and 𝑣̃^→=𝑑^→𝑥/𝑑𝑡. As we have seen in (32), (113), the Galilei relativity imposes a relation between these velocities in such way that 𝑣̃ becomes less than 𝑐, which rules out tachyons since in SR we consider velocities only as derivatives with respect to 𝑡. However, as we have seen in model II, it is possible to have superluminal frames provided we do not impose the existence of the absolute time and the Galilei relativity. Having distinguished between the absolute and the physical time, the assumption of the constancy of the speed of light is understood only with respect to the derivative relative to the physical time. In fact, as we have seen in the discussion of Section 3.4.2, for the light speed calculated as a derivative relative to the absolute time, we obtained 𝑐_𝑆≠𝑐_𝑆′.

Finally, the implications of the existence of an absolute time in the formulation of a quantum theory are an old topic. Dirac, for example, has used it in his attempt to develop electrodynamics [15]. In the context of our work, one possibility that arises is related to the development of quantum theory following the Schwinger Quantum Action Principle (SQAP). When applied to quantum mechanics this approach uses an hamiltonian formalism that singles out the time as parameter. As we have shown in [16], a consequence of the application of a modified form of the SQAP to quantum mechanics has lead to the commutator of the Lie algebra of the Galilei group. Now, we could apply the same procedure using the absolute time as parameter together with its relation to the physical time as given by (76) and search for the corresponding modifications that would originate in the Galilei algebra. Then, we could check if it corresponds to the group of transformation leaving the quantity 𝜏=𝛼𝑥⁰+^→𝛽⋅^→𝑥=𝛼𝑥^′0−^→𝛽⋅^→𝑥^′ invariant.

Acknowledgments

The authors thank all the staff of the library of Centro Brasileiro de Pesquisas Físicas (CBPF), in particular Vera, Fátima, Edileuza, Rosa, and Ramalho. Marcelo Carvalho thanks Aurelina Carvalho, José E. Carvalho, Aureliana Cabral Raposo, and the enlightened conversations with Yohan Ricardo. Alexandre Lyra thanks Eliana N. Lyra de Oliveira for the permanent support and Maria I. Lyra (in memoriam) and dedicates this work (in memoriam) to his sister Anita Lyra de Oliveira with Down syndrome, recently deceased. Both authors thank professor A. F. da F. Teixeira of CBPF for some comments on part of work. This work was done in honour of ICXC, MPΘΥ.