A EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM.

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1 A EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM. S. Kanagaraj Eulidean Relativity (1 August 009) Abstrat By re-interpreting the speial relativity (SR) postulates based on Eulidean 4-spae-time, a geometrial model is formulated alled the Eulidean Spae-Time (EST) diagram. The EST diagram is a veloity vetor model with a real time parameter along the 4 th dimension and onsequently has a (++++) metri. The model is governed by a irular funtion geometry and the relativisti variations are expressed in trigonometri form as a funtion of a real spaetime angle θ (or alternatively an orientation angle φ) that uniquely orresponds to veloity. Its possible usage as a onvenient alternative to Minkowski diagram to investigate the Lorentz transformation is disussed. Keywords: Speial relativity, Eulidean 4-spae-time, Lorentz transformation. 1.1 INTRODUCTION. The fundamentals of physial phenomena an generally be interpreted in terms of simple operations of geometry and has often been applied to simplify our understanding of it. In Galilean relativity, based on the premise that mehanial laws remains the same to an observer, the priniples of mehanis were formulated. With the introdution of speial relativity (SR), mehanis and optis were inorporated into the relativity framework by postulating natural laws remains the same in any inertial referene frame and that light veloity remains the same () independent of the motion status of the soure. The Lorentz transformation equations were formulated studying two () inertial frames O and O moving along the x and x -axis using position oordinates in similarity to that as done under the Galilean transformation but with the added requirement it satisfies the SR postulates. For the speial ase of veloities muh less than, the Lorentz transformation redues to Galilean transformation to the 1 st order of approximation. The wavefront of the invariant veloity of a propagating light pulse with referene to the oordinate system of frame O is x + y + z t 0 and of frame O is x + y + z t 0. The geometrial model formulated from SR based on this position oordinates is the Minkowski spae-time (ST) diagram. The omponents of the 4-vetor for displaement in 4- dimensional (4D) spae-time is ds dx + dy + dz dt, where ds is the Lorentz invariant. The 4 th dimensional-axis in the ST diagram is imaginary and thus has a non-eulidean (+++ ) metri. A moving body is studied in terms of a omplex angular rotation in spae-time with the Lorentz transformation equations expressed as a hyperboli funtion of an angular Page 1

2 veloity parameter, rapidity α. This motivates us to seek a onvenient alternative Eulidean ( ) metri model with the relativisti equations expressed as simple trigonometri funtions of an angular veloity parameter derivable from this geometry. 1. THE CONCEPTUAL SPACETIME GEOMETRY. To formulate the oneptual geometry, we put forward the 1 st postulate. Postulate 1: Relative to an observer in any inertial frame, natural laws remains the same. This postulate 1 is the same as in SR. It implies for the ase of inertial frames O & O at rest relative to eah, the spae and time intervals in the frames for both observers O & O remains the same at its proper values. When O is moving, the intervals are not proper relative to observer O with the observations being o-variant. If instead of only inertial frames O & O, if eah is made up of a group of inertial frames O 1, O, O 3, O 4, O 5, O 6 & O 1, O, O 3, O 4, O 5, O 6 loated orthogonally moving at a uniform veloity (Fig 1), then observations of the variations between any frames in different groups also remains exatly the same. O 1 O O 1 O O 6 O 6 O 3 O O 5 O 5 O 4 O O 4 O 3 Fig 1: groups of inertial frames O and O moving uniformly at veloity. We oneptualize the rest and moving groups of frames are in Eulidean spaes relative to observers within a group and are rotated in 4-spae-time with observations being o-variant. In our approah a frame is investigated not only in relation to its rate of displaement in spae (veloity in spae, s ) onventionally alled its veloity but also in relation to its rate of displaement in time ( veloity in time, t ). This suggests that an inertial frame s rate of displaement in spaetime (its veloity in spaetime ), be represented by a veloity spaetime vetor st with its omponents as the veloity spae vetor s and veloity time vetor t. For onveniene we fix the diretion of vetor s (representing the onventional veloity) along the x 1 -axis and with that the only axes of interest are the x 1 and x 4 axes. This implies the other omponent of the frame s st vetor be represented by a veloity time vetor t normal to the s Page

3 vetor. The veloity in time t of a frame is a measure of its lokrates. Fig shows this oneptual geometrial model. x 4 -axis Veloity time vetor t Veloity spaetime vetor st θ s Veloity spae vetor x 1 -axis Fig : The oneptual spaetime model. The magnitudes of the veloity vetors s, t and st (in bold) represent the speeds s (), t and st (salar quantities, not in bold) respetively. We will all the angular inlination θ of a frame s veloity in spaetime as the spaetime angle. Relative to an observer, s and t are real thus both vetors s and t are real and onsequently st and θ are also real. Expressing the veloity vetor addition in Fig, st s + t... Eqn 1 Inertial frames within a group are in the same Eulidean spae with those in other groups in Eulidean spaes that are rotated from eah other. We next proeed with the formulation of the spaetime geometrial model. 1.3 THE SPACETIME GEOMETRY THE EUCLIDEAN SPACE-TIME (EST) DIAGRAM. To formulate the geometry we put forward the next postulate, Postulate : Relative to an observer the veloity in spaetime of any inertial frame remains the same at. Applying this postulate, the magnitude of st is an invariant. Linden [1] has proposed that veloity in time is a physial property in the time dimension analogous to veloity in spatial Page 3

4 dimensions (i.e. veloity in spae) linked by a onstant veloity. Sine st st by postulate, the salar expression of the vetor addition in Eqn 1 is s + t... Eqn From Eqn, the spaetime geometry (Fig 3) is governed by the funtions of a irle. We will all it as the Eulidean Spae-Time (EST) diagram. x 4 axis Inlined-axis t st ( ) θ O s ( ) x 1 -axis Figure 3: The irular funtion EST diagram. Inertial frames always move at veloity in spaetime along a trajetory at an angle θ that uniquely orresponds to its veloity. The magnitudes of the vetor omponents of st are s s and t t. From Fig 3, s ( ) annot exeed onsistent with ausality π requirements. For the ase when s ( ) 0, st is exatly along the x 4 -axis with θ and t. Thus a rest frame has t along the x 4 -axis whih represents the proper lokrates. Sine t is proportional to lokrates, the lokrates ratio between a moving and rest frame is t of moving frame of rest frame t t... Eqn 3 Page 4

5 If the lok readings in the moving and referene rest frame is denoted as τ and t respetively, the ratio of the differene in the lok readings between the moving and rest frame is τ τ 1 t t 1 Δ τ Δ t... Eqn 4 The ratios of the moving and rest frame s lokrates and the differene in their lok readings are the same. Equating Eqn 3 and Eqn 4, t Δτ. Δ t Expressed in differential form in the limit t 0, dτ t dt... Eqn 5. For onveniene, we fixed a frame s veloity in spae along the x 1 -axis. For the ase in any dx diretion, s. Based on the idea that all forms of matter and energy move at veloity dt of light, Bradford [] has oneptualized a irular geometry with the 4 th -dimensional veloity in time axis as d τ dt and the veloity in spae axis as dx ( ) onsistent with our EST dt diagram. The veloity in spae s for any diretion in the x 1 -x -x 3 -axes of spae is s x 1 + x x3 + 3 s dx1 dx dx + + dt dt dt... Eqn 6 Substituting t and s from Eqn 5 and Eqn 6 into Eqn. 1 dx dx dτ dx dt dt dt + dt... Eqn 7 Page 5

6 Both s ( ) and t onstitute an observer s physial reality, therefore the axes in the EST diagram are real. All the terms in Eqn 7 are motion parameters with the invariant veloity in spaetime on the LHS and its 4-vetor veloity omponents on the RHS. Re-arranging Eqn 7, ( dt) (dx 1 ) + ( dx ) + (dx 3 ) + ( dτ).. Eqn 8 whih is the Eulidean form representation of the Minkowski metri. Substituting the Lorentz invariant ds idτ, into Eqn 8, (ds) (dx 1 ) + ( dx ) + (dx 3 ) ( dt), the Minkowski metri. The postulates put forward in our formulation is based upon a Eulidean interpretation of SR. We will all it as Eulidean speial relativity [3], ESR, as reently proposed. The model orresponds with the Eulidean re-formulation of relativisti dynamis by Montanus [4] and Gersten [5]. Sine s, from Eqn, t... Eqn 9 s From the EST diagram, os θ. Substituting os θ into Eqn 9, t sin θ. st Eqn redues to the trigonometri identity, 1 os θ + sin θ. Adopting a onvention where is positive (+ve ) for reeding motion and negative ( ve ) for approahing motion, as ranges from 0 to, θ ranges from π to 0 and as ranges from 0 to ( ), θ ranges from π to π. The irular funtion for the whole range is represented by the right quadrant ( π θ 0 ) and left quadrant ( π θ π ) respetively. Sine os θ, for approahing motion range, the osine hanges from +ve to ve and only the right quadrant ( Fig 3) is needed to represent both diretions. Page 6

7 1.4 DERIVING THE RELATIVISTIC TIME AND SPACE VARIATION. The lokrates ratio of a moving and a rest frame 1 is t of moving frame of rest frame( ) t 1 Clokrates are inversely proportional to time dilation (TD), thus TD ratio 1... Eqn 10 When <<, t st and the onstany of st (universal veloity) redues to the onstany of t (universal time). Substituting os θ, the lokrate and TD ratios are sin θ and ose θ respetively. Due to time dilation, the lok readings τ represents the proper time elapsed in the moving frame. When, θ 0 and the lokrate ratio, sin θ 0 and TD ratio ose θ. When <<, θ π and the ratios are sin θ 1 and ose θ 1. In the EST diagram, the referene rest (i.e. 0) frame moves along the x 4 -axis at speed. A moving (i.e. 0) frame is inlined at an angle (θ o θ) where θ o ( π/) and θ are the spaetime angles when at rest and moving respetively. If L rest and L moving are the observed lengths of a frame at rest and moving, then L moving L rest os (θ o θ). spae variation ratio L moving L rest L rest os L rest ( θ θ ) o π os (θ o - θ) os θ sin θ Substituting os θ into the trigonometri identity sin θ + os θ 1, spae variation ratio... Eqn 11 1 Sine all bodies have a universal veloity in spaetime, moving and rest hereon refers to its veloity in spae s, whih is the same as the onventional veloity. Page 7

8 Similarly, t of a moving frame is os(θ o θ) and the time variation ratio an alternatively be derived as os (θ o θ ) / sin θ 1 π. For approahing motion, θ π and sin (π θ ) sin θ showing the spae and time variation, sin θ, remains the same for both π diretions. When <<, θ θ o and sin θ sin θo 1.The variations are negligible and it an pratially be assumed that bodies move in Eulidean spae for this ase. As, θ 0 and sin θ 0, implying a body approahes a singularity as its veloity approahes. In the EST diagram, the spae and time variation are onveniently modeled as a ommon spaetime variation with a single-axis, the veloity spaetime vetor. The spaetime aberration, π θ, is the rotation away from the proper time x 4 -axis and proper length x 1 -axis. Fig 4 shows these axes as orthogonal with both rotating in the same diretion. x 4 -axis t rest x 4 -axis π/ θ t moving θ (π/ θ) Length rest x 1 -axis Length moving x 1 - axis Fig 4: The θ relationship with spae and time variations. The variation ratio is the projetion of the inlined x 1 -axis spae and x 4 -axis time onto the t Moving proper x 1 -axis spae and x 4 -axis time respetively. The time variation ratio π os - θ t Rest sin θ. Also if the observed length of a moving and rest frame is L and L, the Page 8

9 π L os - θ L' spae variation ratio sin θ. By postulating [6] objets move in time L L (4 th speed) with a onstant flow and assuming os θ, the variations are shown as sin θ onsistent as derived above. From the EST diagram os θ. Applying trigonometri identities ose θ 1 and ot θ / The 1 st and 4 th equations of the Lorentz transformation written as x 1 1 x 1 / x st Eqn x 4 / x x th Eqn are re-expressed using the angular veloity parameter, θ, in trigonometri form as x 1 (ose θ) x 1 (ot θ) x 4. 1 st x 4 (ot θ) x 1 + (ose θ) x 4 4 th Eqn Eqn as ompared to expressing it in hyperboli form presently using SR with the angular veloity parameter rapidity α as x 1 (osh α) x 1 (sinh α) x 4 1 st Eqn x 4 (sinh α) x 1 + (osh α) x th Eqn We derived the relativisti equations onsistent with the Lorentz transformation applying relativity priniples based only on the invariane of the relations between many equivalent referene frames obeying a group law. Leblond [7] has derived the Lorentz transformation based on the invariane of the relations between inertial frames, The priniple of relativity is first stated in general terms, leading to the idea of equivalent frames of referene onneted through inertial transformations obeying a group law. This is the point of view from whih I intend to Page 9

10 ritiize on the overemphasized role of the speed of light in the foundations of speial relativity. He states, The Lorentz ase is haraterized by a parameter with the dimensions of a veloity whih is a universal onstant assoiated with the very struture of spae-time. 1.5 THE EST DIAGRAM IN TERMS OF THE ORIENTATION ANGLE φ. π The aberration θ represents a body s orientation in spaetime and we will all it as the orientation angle φ. Similar to studying relativisti variations as a funtion of θ, (f(θ)), we an alternatively study it as a f(φ). x 4 axis Inlined-axis st ( ) φ φ (π/ θ) θ O s ( ) t x 1 -axis Figure 5: The EST diagram in terms of orientation angle φ. From Fig 5, the relationship between φ, s and st is sin φ s st Sine and s, sin φ st... Eqn 1 Applying trigonometri identities, from Eqn 1, se φ 1 and tan φ / Page 10

11 The Lorentz transformation equations re-expressed as a f(φ) in trigonometri form is x 1 (se φ) x 1 (tan φ) x 4 1 st Eqn x 4 (tan φ) x 1 + (se φ) x th Eqn By postulating sin φ, Majernik [8] has expressed the Lorentz transformation equations in trigonometri form as a f(φ) and alternatively as f(θ). He states, The above trigonometri form of the Lorentz transformation is interesting not only from a formal point of view, but may serve as a starting point for expressing other relativisti quantities in terms of trigonometri funtions and for the formulation of relativisti relations in the language of trigonometry. Although a physial interpretation for φ is not shown, by investigating [9] on photons reeived from a moving body, it is shown that Majernik s spae-time angle φ is an aberration angle whih orresponds with our model. This relationship sin φ / also appears in Loedel s diagram[10]. By investigating on the spherial wavefront of a light pulse [11], a 4-oordinate manifold of SR has been modeled governed by the funtions of a irle. The relativisti variations expressed as a f(φ) are :- time variation ratio lokrates ratio os φ. Eqn 13 The inverse is time dilation ratio se φ spae variation ratio os φ... Eqn 14 When <<, φ 0 and os φ 1 and se φ 1. Thus the variations are negligible for this ase. Applying Pythagoras theorem (Fig 5), the relationship between st ( ) ; s sin φ and t os φ is the identity sin φ + os φ 1. When 0, φ 0; when, φ π/ and when π π, φ π/. The φ range orresponding to is φ. In this ase sin φ and os φ (and se φ ) for both reeding and approahing motion remains the same (+ve ) onsistent with the variations to be independent of these diretions. In the EST diagram, a body moves along an inlined trajetory at an angle φ subtended between the vertial time axis of the observer and the inlined time-axis of the observed body. Moving bodies have been investigated [13] in terms of trajetories at an angle ϕ (same as φ) between the time-axes of an observer and observed body with expressions V sin ϕ and (1 V ) os ϕ (where 1) onsistent with our model. In studying appearanes at relativisti speeds [1] by investigating how photons emitted from a moving body are reeived, the observed spae Page 11

12 1 1 variation is desribed in terms of an angular rotation sin os whih orresponds with the orientation angle φ in our model. It is of interest that the work by [9], [11] and [1] studying on photons reeived from a moving body based on the onstany of light veloity produes results onsistent with the EST diagram. 1.6 THE EST DIAGRAM OFFERS AS A VIABLE ALTERNATIVE. From the postulates of ESR, a veloity vetor spaetime model, the EST diagram, was formulated. Compared to the ST diagram as formulated from SR, (a) the 1 st postulate is applied solely based on relativisti veloities thereby avoiding the bakground dependent [14] position oordinates and (b) the nd postulate uses the onstany of veloity in spaetime for all inertial frames instead of restriting to the onstany of veloity (in spae) of light. Sine ESR is based on a re-interpretation of the SR postulates applying broader relativity priniples, the derived equations are onsistent with SR. With a real 4 th - dimensional time parameter, the EST diagram has a Eulidean (++++) metri. The relativisti variations were expressed in terms of a real spaetime angle θ (or alternatively the orientation angle φ ) governed by a irular funtion. The Lorentz transformation equations were expressed in trigonometri form using the angular veloity parameter θ or φ instead of in hyperboli form using rapidity α. It is with interest we note the de Broglie waves an be expressed in terms of φ in trigonometri form[8] with the relativisti relationships orrelated graphially by a single diagram. Also based upon a wave field system[15], a diagram orresponding with the EST diagram has been presented. In onlusion, the onsisteny of the EST diagram with Lorentz transformation equations offers a new geometrial tool to investigate relativisti observations and enourages onsiderations on its viability to serve as a onvenient alternative to the ST diagram. REFERENCES : 1. Linden, RFJ: Dimensions in speial relativity, Galilean Eletrodynamis, Vol 18, no1, pg 1, 007. Bradford, Phillips V: A geometri interpretation of the beta fator in speial relativity. Available via Fontana, Giorgio: 4 spae-time model of reality, arxiv.org.physis/ , 004. Page 1

13 4. Montanus,JMC: Proper-time formulation of relativisti dynamis, Foundations of Physis, Vol 31, no 9, pg , Gersten, Alexender: Eulidean speial relativity, Foundations of physis, Vol 33, no 8, , Sirvent, Cesar: 4-th speed projetion re-visited. Available via ited 10 Jun Leblond, J-M Levy: One more derivation of the Lorentz transformation., Am J Phys 44(3), 71-77, Majernik, V: Representation of relativisti quantities by trigonometri funtions, Am J Phys 54(6), , Wilkins D and Williams D: From rapidity to vibray, Am J Phys 69(), , Loedel E, Geometri representation of the Lorentz transformation, Am J Phys 5:37, May Crabbe, Anthony: Alternative onventions and geometry for speial relativity, Annales de la Foundation Louis de Broglie, Vol 9, no. 4, pg , Signell, Peter: Appearanes at relativisti speeds for projet Physnet, Mihigan State University, ID Sheet MISN-0-44, , Nawrot, Witold: Proposal of simple desription of SRT, Galilean Eletrodynamis, vol 18, no 3, pg 43, Smolin, Lee : The ase for bakground independene, Perimeter Inst Theor Phys & Waterloo U. Ontario, arxiv:hep-th / v1, Krogh, Kris: A new view of the universe, arxiv.org/abs/physis/961010, END Kanagaraj Page 13