Special Relativity and possible Lorentz violations consistently coexist in Aristotle space-time
SSpecial Relativity and possible Lorentz violationsconsistently coexist in Aristotle space-time
Bernard ChaverondierMarch 18, 2019
Submitted to Foundation of Physics on 9 may 2008CNIM, ZI BREGAILLON, BP208, 83507 la Seyne sur mer CEDEX, FranceAbstract: Some studies interpret quantum measurement as being explicitlynon local. Others assume the preferred frame hypothesis. Unfortunately,these two classes of studies con(cid:29)ict with Minkowski space-time geometry.On the contrary, in Aristotle space-time, Lorentz invariance, interpreted asa physical property applying to all phenomena actually satisfying this sym-metry (as opposed to a geometrical constraint applying to an assumed pre-existing Minkowski space-time) consistently coexists with possible Lorentzviolations. Moreover, as will be pointed out, the geometrical frameworkprovided by Aristotle space-time is in fact necessary to derive the Lorentztransformations from physical hypotheses.Keywords: Special Relativity, preferred frame, Aristotle space-time, quantummeasurement. Quantum Non-locality and quantum preferred frame
Percival proved realistic interpretations of quantum collapse to violate Lorentzinvariance in Bell-type experiments [4]. Henceforth, as suggested by Bell, non-locally correlated quantum events can be interpreted as faster-than-light in-teractions [5, 3] complying with the causality principle provided it rests on theabsolute chronological order associated with a quantum preferred frame [6]. Sim-ilarly, EPR experiments performed in Geneva by the Group of Applied Physics[7, 8] have been analyzed according to the not Lorentz invariant preferred framehypothesis. Realistic interpretations [1] assume quantum collapse to be an objective (i.e. observerindependent) physical process. Contrary to the Everett Many Worlds Interpretation, severelycriticized by Neumaier [2] and Bell [3], they con(cid:29)ict with Lorentz invariance. Hence incompatible with Minkowski space-time. a r X i v : . [ phy s i c s . g e n - ph ] M a y Other reasons suggesting a preferred frame
Selleri argues that some superluminal e(cid:27)ects strongly suggest the need for a pre-ferred frame and its associated preferred chronology [9]. Moreover, the scalartheory of gravitation of Arminjon [10, 11], investigates a preferred frame gravi-tation approach as a possible way to make quantum and gravitation theories (cid:28)ttogether. The preferred frame, formalized as a (cid:28)eld of time-like unit vectors, isalso used in the context of preferred frame theories of gravity by authors suchas Will and Nordtvedt [12] Eling and Jacobson [13]. Moreover, some attentionwas devoted to the preferred frame hypothesis by Kostelecky as a consequenceof possible Lorentz violations in High Energy Physics [14, 15]. Now, Aristotlespace-time will provide us with a geometrical framework authorizing the peace-full coexistence of the preferred frame hypothesis with the ubiquitous Lorentzinvariance. Let us now de(cid:28)ne Aristotle space-time’s symmetry group. The PoincarØ, Galilei and Aristotle groups
Thanks to Noether’s theorem, energy and linear momentum, as well as angularmomentum conservation laws, arise from the invariance of the Lagrangian ofdynamical systems respectively with regard to the group of space-time transla-tions and the group SO (3) of spatial rotations. The semi-direct product group,arising from these two groups, is the so-called restricted Aristotle group [16]. Itwill be denoted SA (4) .This seven parameters group is also the direct product group of the Special Eu-clidean group SE (1) (i.e. the temporal translations of the 1D a(cid:30)ne Euclideanspace E ) and the Special Euclidean group SE (3) (i.e. the direct spatial isome-tries of the 3D a(cid:30)ne Euclidean space E ). So, SA (4) is the intersection of therestricted Galilei and PoincarØ groups. Hence, neither does it contain Galileanboosts, nor Lorentzian ones. Its relevance is the following: • the invariance requirement of physical laws with regard to Galilean boostscon(cid:29)icts with interactions propagating at a (cid:28)nite speed independent of thespeed of their source, hence in particular with electromagnetism • the invariance requirement of physical laws with regard to Lorentzianboosts con(cid:29)icts with interactions propagating at in(cid:28)nite speed.On the contrary, Aristotle group of symmetry complies with interactions prop-agating at the speed of light as well as possible faster-than-light interactions . Or possible instantaneous actions at a distance caused by quantum measurements [17, 18,6]. Aristotle space-time4.1 De(cid:28)nition and foliation of Aristotle space-time
Figure 1: Aristotle space-time A Aristotle space-time, arising from the Aristotle group, embodies only the Prin-ciple of Relativity with regard to space-time localization and spatial orientation.It is de(cid:28)ned as a set denoted A equipped with a bijection f from R to A pro-viding it with an action Φ of the numerical restricted Aristotle group SA (4) de(cid:28)ned as: Φ : (cid:40) SA (4) × A → A ( a, Z ) (cid:55)→ Φ a ( Z ) = f ◦ a ◦ f − ( Z ) (1)From now on, we will identify SA (4) with its representation acting on A . So,for the sake of simplicity, Φ a , the image of a by action Φ , will be identi(cid:28)ed with a and Φ a ( Z ) will be referred to as an action a of group SA (4) on Z .Aristotle space-time is endowed with 2 preferred foliations • a 1D foliation of which the 1D leaves of absolute rest are the orbits of thetime translation group, the invariant subgroup SE (1) of SA (4) . • a 3D foliation of which the 3D leaves of universal simultaneity are theorbits of the direct isometries group, i.e. the invariant subgroup SE (3) of SA (4) .As each one of these two foliations is a complete set of orbits of an Aristotleinvariant subgroup, this foliated structure is preserved under Aristotle groupactions. This foliation may be interpreted as the preferred inertial frame ofBell’s realistic interpretation of quantum collapse and may also be helpful toaccount for possible Lorentz violations [14, 15]. The quotient manifold of A by its 1D foliation is di(cid:27)eomorphic with the 3Dleaves of universal simultaneity. This 3D manifold can be equipped with anaction of SE (3) (the invariant subgroup of spatial isometries). This provides i.e. considered as a subgroup of Gl ( R ) . The chosen manifold structure of A is that induced by f , i.e. di(cid:27)eomorphisms of A arebijections F from A to A such that f − ◦ F ◦ f are di(cid:27)eomorphisms of R .
3t with the metric structure of a 3D a(cid:30)ne Euclidean space E . Similarly, thequotient manifold of A by its 3D foliation is di(cid:27)eomorphic with the 1D leavesof absolute rest and can be provided with the metric structure of a 1D a(cid:30)neEuclidean space E . Hence, Aristotle space-time can be identi(cid:28)ed as the Carte-sian product A = E × E . It is naturally equipped with two Euclidean metricswhich are invariant with regard to the Aristotle group actions: • a rank 1 temporal metric, which will be denoted dT • a rank 3 spatial metric, which will be denoted dL . The principle of relativity of motion, embodied in Minkowski space-time geom-etry, forbids granting a privileged status to a preferred inertial frame. Now, thechronological order between space-like separated events depends on the rest iner-tial frame of the observer. This prevents Minkowski space-time complying withthe existence of causal links spanning out of the light cone. On the contrary,Aristotle space-time foliation into 3D leaves of universal simultaneity enables usto de(cid:28)ne an objective chronology between any pair of events. This gives riseto a causal structure where possible faster-than-light interactions, comply withthe principle of causality prevailing in this space-time. Aristotle charts and Aristotle bases
Aristotle space-time is associated with a family of preferred coordinate systems,called Aristotle charts, preserving its foliated geometry and its metrics.
In Aristotle space-time A = E × E , any event Z reads: Z = ( T, R ) • T ∈ E denotes the moment when event Z occurs. • R ∈ E denotes the localization where event Z occurs.Events Z are localized in so-called Aristotle charts denoted A such that Z = A ( z ) , where z = ( t, r ) = ( t, x, y, z ) ∈ R are the so-called coordinates of Z inAristotle chart A . By de(cid:28)nition, Aristotle charts are such that: • they preserve the foliation of Aristotle space-time into 1D lines of absoluterest and 3D leaves of universal simultaneity. In particular, two eventsbelonging to a same simultaneity leave (i.e. occuring at the same time T ) have the same chronological coordinate t , i.e. ∃T : R → E and ∃R : R → E such that A ( t, r ) = ( T ( t ) , R ( r )) That is to say independent of the motion of inertial observers. the temporal metric be normalized, i.e. dT = dt • the spatial metric be Orthonormalized, i.e. dL = dx + dy + dz .Besides, O = E × {R (0) } will denote the motionless observer resting at thespatial origin of Aristotle chart A and {T (0) } × E is the 3D leaf of universalsimultaneity passing through origin event E = A (0) of chart A . Any Aristotle chart A is associated with a space-time basis V = ( (cid:126)t, (cid:126)x, (cid:126)y, (cid:126)z ) ofthe vector space V ⊕ V . Indeed, let us de(cid:28)ne • E = A (0) the so-called origin event of chart A• Events E t = A (1 , ; E x = A (0 , , , ; E y = A (0 , , , ; E z = A (0 , , , • Unit vectors (cid:126)t = −−→ EE t ; (cid:126)x = −−→ EE x ; (cid:126)y = −−→ EE y ; (cid:126)z = −−→ EE z (cid:126)t is a normalized vector of V and B = ( (cid:126)x, (cid:126)y, (cid:126)z ) an Orthonormalized basis of V . Let A be an Aristotle chart. Let Φ denote the action (1) of the restrictedAristotle group SA (4) on A . Any action Φ a = Φ( a ) (denoted a for the sake ofsimplicity), of a ∈ SA (4) on A entails an Aristotle chart change ( a, A ) → A a = a ◦ A = A ◦ ϕ a (2)This de(cid:28)nition ensures coordinates’ covariance, i.e. the same system will belocated by the same coordinates whenever observer and observed system bothundergo the same chart change A → A a . Coordinates z of the (cid:19)new event(cid:20) Z = a ( Z ) in the (cid:19)new chart(cid:20) A a are the same as coordinates of the (cid:19)oldevent(cid:20) Z in the (cid:19)old chart(cid:20) A . ϕ a is the numerical expression of action a inchart A . Figure 2: Change of Aristotle chart From a di(cid:27)erential geometry point of view, V = d A [16] V ⊕ V is the tangent space to A = E × E Boosts and inertial charts in Aristotle space-time
So as to express Lorentz invariance of the phenomena that actually satisfy thissymmetry, we have to de(cid:28)ne Lorentzian boosts, inertial charts and then to deriveLorentz transformations in Aristotle space-time framework. A having the following physical prop-erties
1/ When applied to motionless observers, boosts set them in motion with thesame velocity (cid:126)v called velocity of the boost.2/ The modi(cid:28)cation of durations and distances caused by the application of aboost in the vicinity of a boosted event (cid:19)is the same(cid:20) whatever this event.3/ Freely moving observers keep on freely moving after the action of a boost.4/ The covariance of boosts’ observed e(cid:27)ect is satis(cid:28)ed under any change ofAristotle chart. Loosely speaking, a boost has the same e(cid:27)ect whatever theAristotle chart where it is applied . Actually, Aristotle covariance of boostswill be assumed to hold when, more generally, a is any action of the completeAristotle group A (4) .Figure 3: Covariance of boosts’ e(cid:27)ect under any Aristotle group action a
5/ Symmetry of point of view between motionless and moving observers: looselyspeaking, we cancel the e(cid:27)ect of a boost of velocity (cid:126)v , applied to Aristotle space-time A , by applying a boost of velocity − (cid:126)v . They will be translated mathematically in sub-section 6.3 A freely moving observer is a line D with a direction D (cid:126)v = { t(cid:126)t + t(cid:126)v/t ∈ R } , where || (cid:126)v || < c (the speed of light). (cid:126)v ∈ V is called the velocity of this observer so that the rest lines ofAristotle space-time are observers freely moving with a zero velocity. This requirement expresses the homogeneity, the stationarity and the isotropy of Aristotlespace-time physical properties with regard to boosts. So, physical observers at rest in a boosted Aristotle chart (by a boost of velocity (cid:126)v ),
6/ The maximal propagation speed measured by a motionless observer is thesame as that measured by an observer at rest in a boosted Aristotle chart.6.1.2 Requirements applying speci(cid:28)cally to pure boostsSo as to de(cid:28)ne the so-called pure boosts, i.e. boosts that are not combined withAristotle group actions, we ask for the following additional properties:7/ Any pure boost is endowed with at least one so-called origin event E , invariantunder the pure boost action. Thus, a pure boost, combined with a spatialtranslation perpendicular to its velocity, is not anymore a pure boost .8/ Any pure boost is completely determined given its velocity and an originevent. Together with property 4/ (the Aristotle covariance of boosts) this makesit possible to eliminate pure boosts combined with rotations.9/ If { B Eλ(cid:126)v /λ ∈ R } is a family of pure boosts having a same origin event E andvelocities proportional to (cid:126)v , B Eλ(cid:126)v → i A when λ → . Together with the otherproperties, in particular the symmetry of point of view, this will enable us toeliminate pure boosts combined with P or T symmetries. With any pure boost B (cid:126)v of velocity (cid:126)v , of origin event E , and with any Aristotlechart A of same origin event E , we associate a so-called inertial chart A (cid:126)v mov-ing with the velocity (cid:126)v . The chart A (cid:126)v is de(cid:28)ned so as to ensure coordinates’covariance with regard to boosts, i.e. if the (cid:19)old(cid:20) event Z is localized by co-ordinates z = ( t, r ) in the (cid:19)old(cid:20) chart A , then the (cid:19)new(cid:20) event Z = B(cid:126)v ( Z ) isalso localized by these same coordinates in the (cid:19)new(cid:20) chart A (cid:126)v . So:Figure 4: De(cid:28)nition of an inertial chart A (cid:126)v Z = A ( z ) ⇒ B (cid:126)v ( Z ) = Z = A (cid:126)v ( z ) . So that A (cid:126)v = B (cid:126)v ◦ A (3)Moreover A (cid:126)v (0) = B (cid:126)v ( A (0)) = B (cid:126)v ( E ) = E , so that A (cid:126)v has same space-timeorigin E as A . observing phenomena occurring in an Aristotle chart A , will observe the same e(cid:27)ects asmotionless observers (hence at rest in A ) observing the same phenomena occurring in anAristotle chart boosted with the velocity − (cid:126)v . This assumption expresses the impossibilityfacing a steadily translating observer if he tries to detect his absolute motion when usingmeasurements and phenomena that are Lorentz-covariant. Under a space-time translation, of which the translation vector is included in the ( (cid:126)t, (cid:126)v ) plane, the origin event E of a pure boost shifts but the new boost is still a pure boost. b v of boost B (cid:126)v in Aristotle chart A will be de(cid:28)ned as follows:Figure 5: Expression of a boost B (cid:126)v in an Aristotle chart A z = ( t , r ) = b v ( z ) are the coordinates of event Z = B (cid:126)v ( Z ) in the (cid:19)oldchart(cid:20) A wheras z are its coordinates in the (cid:19)new chart(cid:20) A (cid:126)v . The covarianceof coordinates z with regard to boost B (cid:126)v means that the passive transformation A → A (cid:126)v (causing the change of coordinates z → z ) caused by boost B (cid:126)v whenapplied to the observer only (i.e. to the observation frame A and not to theobserved system) cancels the active transformation Z → Z = B (cid:126)v ( Z ) (causingthe change of coordinates z → z ), i.e. the action of this same boost B (cid:126)v whenapplied to the observed system only. Besides, we notice that, thanks to thechoice of a chart A that has same origin event E as boost B E(cid:126)v : b v (0) = A − ◦ B E(cid:126)v ◦ A (0) = A − ◦ B E(cid:126)v ( E ) = A − ( E ) = 0 Let us now translate mathematically the physical requirements of sub-section 6.1.11/ Motionless observers are set in motion with the velocity (cid:126)v of the boost.If M = E × { M } (where M ∈ E ) is a motionless observer and B (cid:126)v is a boostof velocity (cid:126)v , then B (cid:126)v ( M ) is a line D of direction D (cid:126)v = { t(cid:126)t + t(cid:126)v/t ∈ R }
2/ The e(cid:27)ect of boost B in the vicinity of event B ( Z ) does not depend on event Z . That is to say, ∀ Z , Z (cid:48) , Z , Z (cid:48) such that −−−→ Z Z (cid:48) = −−−→ Z Z (cid:48) and for any boost B : −−−−−−−−→ B ( Z ) B ( Z (cid:48) ) = −−−−−−−−→ B ( Z ) B ( Z (cid:48) ) Figure 6: E(cid:27)ect of a boost on a translation • Let T = T −−→ ∆ Z be the translation of vector −−→ ∆ Z = −−−→ Z Z (cid:48) so that Z (cid:48) = T ( Z ) and Z (cid:48) = T ( Z ) • Let T (cid:48) = T −−→ ∆ Z (cid:48) be the translation of vector −−→ ∆ Z (cid:48) = −−−−−−−−→ B ( Z ) B ( Z (cid:48) ) so that B ( Z (cid:48) ) = T (cid:48) ( B ( Z )) and B ( Z (cid:48) ) = T (cid:48) ( B ( Z )) Z ∈ A : B ( T ( Z )) = T (cid:48) ( B ( Z )) , i.e. B ◦ T = T (cid:48) ◦ B and it is easy to establishthat: dB is constant (i.e. B is a(cid:30)ne) and B ◦ T −−→ ∆ Z ◦ B − = T dB ( −−→ ∆ Z ) It is worth noticing that the above equation may also be written: B passive ◦ B active ( T −−→ ∆ Z ) = T −−→ ∆ Z (4) • the active transformation, B active ( T −−→ ∆ Z ) = T dB ( −−→ ∆ Z ) , of a space-time trans-lation T −−→ ∆ Z is a change of the observed space-time translation e(cid:27)ect (ap-plied to a given system) when the space-time translation vector −−→ ∆ Z aswell as the observed system, both undergo the same boost B . • The passive transformation, B passive ( T (cid:48) ) = B − ◦ T (cid:48) ◦ B , of a space-timetranslation T (cid:48) is a change of the observed translation e(cid:27)ect when only theobserver (i.e. the observation chart) undergoes boost B .So, requirement 2/ amounts to the covariance of space-time translations ob-served e(cid:27)ects under any boost B (ie the invariance of these observed e(cid:27)ectswhen the applied space-time translation, the observed system as well as theobserver all undergo the same boost B ). This proves requirement 2/ to expressthe principle of relativity of motion with regard to translation observed e(cid:27)ects.As seen above, this causes boosts to be a(cid:30)ne transformations.Moreover, the expression b of a pure boost B , in a chart A that has the sameorigin event E as boost B , satis(cid:28)es b (0) = 0 . Hence, b is linear.3/ Any freely moving observer keeps freely moving when boosted: as boosts area(cid:30)ne transformations, they transform a(cid:30)ne lines into a(cid:30)ne lines of Aristotlespace-time so that physical requirement 3/ of sub-section 6.1.1 is satis(cid:28)ed. Ac-tually, there is an equivalence between the requirement 2/ and the requirement3/ that lines of A be transformed into lines of A (we may have preferred toderive 2/ from 3/ instead of the other way around).4/ Covariance of boosts observed e(cid:27)ects under any change of Aristotle chart: • Let us de(cid:28)ne the active transformation of a pure boost B = B E(cid:126)v of velocity (cid:126)v and origin event E under an action a ∈ SA (4) as a pure boost of originevent a ( E ) and velocity da ( (cid:126)v ) , i.e. a active ( B E(cid:126)v ) = B a ( E ) da ( (cid:126)v ) (5)Physically, this transformation represents an action a both on the appliedboost and on the observed system . • Let us now de(cid:28)ne a passive transformation of any boost B (cid:48) as: a passive ( B (cid:48) ) = a − ◦ B (cid:48) ◦ a (6)Physically, this transformation represents an action a on the observer only,i.e. a change A → A a of Aristotle chart of observation. As concluded in sub-section 6.2. But not on the observer. a reads : a passive ◦ a active ( B ) = B (7)i.e. a − ◦ a active ( B ) ◦ a = B , so that a ◦ B E(cid:126)v ◦ a − = B a ( E ) da ( (cid:126)v ) (8)Now, hypothesis 4/ of sub-section 6.1.1 requires that the above condition musthold for any action a of the complete Aristotle group A (4) . Derivation of Lorentz transformations
A rigorous derivation of Lorentz transformations from physical hypotheses (cfsub-section 6.1.1) needs using Aristotle space-time and its symmetries (cf sub-section 7.1, 7.2) with regard to boosts e(cid:27)ects.Let us consider a boost, denoted B (cid:126)v , of velocity (cid:126)v and origin event E .Let us consider an Aristotle chart A of origin event A (0) = E and spatial origin O , having its vector (cid:126)x in the same direction as velocity (cid:126)v (i.e. (cid:126)v = v(cid:126)x ). ◦ rotation around O(cid:126)x
As a 180 ◦ rotation R π(cid:126)x around O(cid:126)x neither changes E nor changes (cid:126)v , dR π(cid:126)x ( (cid:126)v ) = (cid:126)v and R π(cid:126)x ◦ B (cid:126)v ◦ R π(cid:126)x = B (cid:126)v (9)So that, in R : r πx · b v · r πx = b v . Now, in chart A , matrix r πx of R π(cid:126)x reads: r πx = − − (10)The right multiplication of matrix b v (of boost B (cid:126)v ) by matrix r πx reverses thesigns of columns y and z of b v . The left multiplication of matrix b v by matrix r πx reverses the signs of lines y and z of b v . Consequently, any o(cid:27)-diagonal y and z term of matrix b v vanishes except the yz terms. To exemplify the physical meaning of Lorenzian boosts’ covariance property (with regardto any action a of the Aristotle group), let us consider the special case of the covariance withregard to spatial rotations. So, let us consider, for instance, a strain tensor (cid:28)eld induced in anisotropic 3D medium submitted to an homogeneous (but anisotropic) stress tensor (cid:28)eld. If werotate both the observer and the applied stress tensor (cid:28)eld, then, the passive transformation(the rotation of the observer) cancels the active transformation (the strain (cid:28)eld modi(cid:28)cationinduced by the rotation of the applied stress tensor (cid:28)eld). Because of this 3D medium isotropy,the rotated observer will observe the same e(cid:27)ect as if neither himself, nor the stress tensor(cid:28)eld had been rotated. It wouldn’t be the case if this medium were anisotropic. Similarly,we demand the invariance of space-time deformations under any Lorentzian boost, when theboost undergoes the combination of an active and a passive action of any Aristotle groupaction a . This amounts to require space-time behaving as an homogeneous, isotropic andstationary medium. .2 Covariance under a 90 ◦ rotation around O(cid:126)x axis and underthe Π (cid:126)y = P R π(cid:126)y plane symmetry
Similarly, we get byy = bzz and byz = bzy = 0 , so that we have: ∃ a, a (cid:48) , b (cid:48) , b ” and e ∈ R such that: t = at + a (cid:48) xx = b (cid:48) t + b (cid:48)(cid:48) xy = ey z = ez (11) • Applying boost B − (cid:126)v erases boost B (cid:126)v , i.e. B − (cid:126)v = B − (cid:126)v , • The maximum propagation speed is covariant with regard to any Aristotlegroup action, hence it is isotropic. Moreover, as far as Lorentz invarianceis satis(cid:28)ed, it has the same norm c in A as in A (cid:126)v .For convenience, let us introduce speed c in the previously stated equations: ∃ a, a (cid:48) , b, b (cid:48) and e ∈ R such that: ct = act + bxx = b (cid:48) ( ct ) + a (cid:48) xy = ey z = ez (12)As b − v = b − v and b − v = π x · b v · π x (where π x denotes the sign reversal of x ): e − = e and [1 / ( aa (cid:48) − bb (cid:48) )] (cid:26) a (cid:48) − b − b (cid:48) a (cid:27) = (cid:26) a − b − b (cid:48) a (cid:48) (cid:27) (13)Consequently aa (cid:48) − bb (cid:48) = 1 , a = a (cid:48) and e = 1 so that e = ± . Actually e = 1 . Indeed, according to requirement 9/ of sub-section 6.1.2, pure boost b v is assumed to tend to the identity of R when (cid:126)v tends to (cid:126) . Now, as the originof A (cid:126)v (located at x = y = z = 0 ) moves with the velocity (cid:126)v = v(cid:126)x we have x = vt . As x = y = z = 0 we have: ct = a ( ct ) and x = b (cid:48) ( ct ) . Hence a ( vt ) = ax = ab (cid:48) ( ct ) = b (cid:48) ( ct ) , so that b (cid:48) = av/c (14)Now, let us express the covariance of the relative speed c of light: x = ct ⇒ x = ct so that x = b (cid:48) ( ct ) + a (cid:48) x = ct = a ( ct ) + bx ⇒ b (cid:48) + a (cid:48) = a + b (15)11s a = a (cid:48) we get b = b (cid:48) . Hence aa (cid:48) − bb (cid:48) = 1 becomes a − b (cid:48) = 1 As b (cid:48) = av/c , this provides a − ( av/c ) = 1 so that a = ± / (1 − v /c ) / Now, we have excluded time reversal. Indeed, b v is assumed to tend to theidentity of R when v tends to so that a = 1 / (1 − v /c ) / (16)Finally, we get the Lorentz transformations: ct = ( ct + vx/c ) / (1 − v /c ) / x = ( vt + x ) / (1 − v /c ) / y = y z = z (17) Conclusion
The present article exhibits Aristotle spacetime foliated structure, its causalstructure and the peaceful coexistence, in this arena, of the phenomena actuallysatisfying Lorentz invariance with possible Lorentz violations. It provides ageometrical framework where realistic, hence explicitly non local interpretationsof quantum collapse, comply with the principle of causality and suggests thepossibility of interpreting Lorentz invariance as a thermodynamical statisticalemergence. Last but not least, Aristotle spacetime geometry modelizes theenergy, linear and angular momentum conservation laws. This (cid:28)rst step is infact needed to derive rigorously the Lorentz transformations from the observedrelativity of motion.