Euclidean Model of Space and Time

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1 Jornal of Modern Physis, 018, 9, ISSN Online: X ISSN Print: Elidean Model of Spae and Time Radovan Mahotka Brno University of Tehnology, Brno, Czeh Repbli How to ite this paper: Mahotka, R. (018) Elidean Model of Spae and Time. Jornal of Modern Physis, 9, Reeived: Marh 8, 018 Aepted: May 13, 018 Pblished: May 16, 018 Copyright 018 by athor and Sientifi Researh Pblishing In. This work is liensed nder the Creative Commons Attribtion International Liense (CC BY 4.0). Open Aess Abstrat The aim of this work is to show that the rrently widely aepted geometrial model of spae and time based on the works of Einstein and Minkowski is not niqe. The work presents an alternative geometrial model of spae and time, a model whih, nlike the rrent one, is based solely on Elidean geometry. In the new model, the psedo-elidean spaetime is replaed with a speifi sbset of for-dimensional Elidean spae. The work shows that for-dimensional Elidean spae allows explanation of known relativisti effets that are now explained in psedo-elidean spaetime by Einstein s Speial Theory of Relativity (STR). It also shows simple geometri-kinematial natre of known relativisti phenomena and among others explains why we annot travel bakward in time. The new soltion is named the Elidean Model of Spae and Time (EMST). Keywords Speial Theory of Relativity, Elidean Spae, For-Dimensional Spae, Time Dilation, Length Contration 1. Introdtion Albert Einstein introded his onept of a mtal relationship between spae and time known as Speial Theory of Relativity (STR) in his work Zr Elektrodynamik bewegter Körper (On the Eletrodynamis of Moving Bodies) [1]. This work was fosed on solving then existing disrepanies between theories desribing eletromagneti phenomena on one side and lassial mehanis on the other. By its natre the work is physio-mathematial and the isse of geometry is addressed only marginally. This drawbak of the original theory was eliminated a few years later by Hermann Minkowski s work Die Grndgleihngen für die elektromagnetishen Vorgänge in bewegten Körpern (The Fndamental Eqations for Eletromagneti Proesses in Moving Bodies) [] whih was fol- DOI: /jmp May 16, Jornal of Modern Physis

2 R. Mahotka lowed by his letre from 1908 Ram nd Zeit (Spae and Time) [3]. In his work, Minkowski onneted spae and time into one for-dimensional ontinm, later alled spaetime or Minkowski spae, and he defined its key featres. He introded a speifi psedo-elidean metri for spaetime whih is alled Minkowski metri after the athor. It is sally written in the form s = t x y z (1) where x, y, z and t are oordinate inrements and s is the so alled spaetime interval. This interval plays the same role in Minkowski spae as distane plays in ordinary Elidean spae. Minkowski based his thoghts on Maxwell s eqations of eletrodynami field and their intrinsi symmetry, whih reveals itself partilarly when the eqations are written with time taken as an imaginary qantity. Here the main reason for the se of imaginary nmbers in Minkowski theory and sbseqent introdtion of the Minkowski metri an be seen. Using this metri Minkowski showed that the Lorentz transformation an be nderstood as a rotation of for-dimensional spaetime by an imaginary angle []. If a well-hosen oordinate system is sed, orientation of two spatial oordinates does not hange dring the Lorentz transformation (they are invariant) and the transformation affets only one spae-like and one time-like oordinate. E.g. if oordinate system S' moves with veloity with respet to oordinate system S in the diretion of the axis x, the transformation (rotation) affets oordinates x and t, while y and z remain invariant 1. In this ase the transformation an be written as t x x t x =, y = y, z = z, t = () where x, y, z, t are oordinates of an arbitrary point in oordinate system S and x', y', z' and t' are oordinates of the same point in oordinate system S'. Minkowski s geometrial interpretation of Einstein s STR was great sess. It was qikly adopted as an integral part of the theory and as sh, it was not qestioned for years. Moreover, Minkowski metri in generalized form known as psedo-riemannian metri beome fndamental part of General Theory of Relativity. Some dobts regarding geometry of spaetime emerged with qantm mehanis arrival in nineteen thirties, bt no real alternative was fond. The problems with Minkowski onept of spaetime amlated over years mainly in onnetion with attempts of qantm gravity theory. The nsatisfatory sitation holds till now and it is refleted in many works of ontemporary sientists [4] [5] [6]. In reent years alterations of original Minkowski onept sing Finsler or Cartan geometry are proposed [7] [8], bt with no remarkable sess. For the whole time, Elidean geometry was overlooked and relegated to ax- 1 Symbol will be only sed for mtal veloity of oordinate systems, the veloity of other objets will be marked as v. DOI: /jmp Jornal of Modern Physis

3 R. Mahotka iliary roles, mainly in pedagogial tasks as a visalization tool [9] [10]. The athor is onvined it was great falt. The aim of this work is to show that Einstein-Minkowski soltion known as STR is not niqe. There exists at least one other soltion whih orresponds to or observations, whih leads to the same mathematial formlas, bt whih geometry is qite different. In reality, its geometry is Elidean!. Assmptions and Methods Let s assme that physial spae is Elidean, i.e. the axioms of Elidean geometry hold in it. Appliation of sh an assmption on the whole niverse may be diffilt, bt for or objetives it is sffiient to assme elidiity of spae in some loal sale, that is in some restrited part of the niverse sffiiently distant from mass bodies and their gravitational fields. In sh a part of the niverse, we an imagine existene of two inertial oordinate systems with their origins niformly moving eah other. We will denote one of these systems as S and its axes x, y, z, the other as S' with axes x', y', z'. The oordinate time of the first system will be denoted as t, the oordinate time of the other as t'. For the sake of simpliity, we shall assme that the orresponding axes of both systems are parallel and the origins O and O' of the systems oinide eah other at time t = t' = 0. Motion of system S' in respet to S holds in the diretion of the positive semi axis x with the speed..1. Stationary Coordinate System In frther explanation we will assme existene of one otstanding oordinate system alled the stationary oordinate system. In this oordinate system the light propagates with the same speed in all diretions. Sh a system will be denoted as S, speed of light as. We will assme that in other oordinate systems that are moving with respet to S, the above laim is not flfilled... Time Measrement In orrespondene with Einstein, we will assme that time is measred by ideal loks and all sh loks give exatly the same reslts, if they are not moving relative eah other. Sh ideal loks old be designed also as so alled light loks whih measre time on the basis of motion of light plse between two mirrors. Sh a lok learly demonstrates the slowing-down of time flow as a onseqene of speed growth. The faster the motion of the lok with respet to S the longer the light plse trajetory in one yle and the dration of the yle is ths longer (see Figre 1)..3. Distane Measrement Distanes will be determined by means of transit times of light signals. This method assmes the light signal is emitted from point A at time t A 0, refleted in the DOI: /jmp Jornal of Modern Physis

4 R. Mahotka Figre 1. Light lok. Time of flight of light plse between two mirrors depends on the state of motion of the lok. On the left lok in rest, on the right lok moving with veloity v to the right. end point of measred distane (point B) and reeived at the initial point at time A t. The transit time is measred by an ideal lok whih is plaed at point A. The relevant formla is AB A A t t0 = (3).4. Clok Synhronization In regard to oordinate time, we will assme, in orrespondene with Einstein, that in given oordinate system the time is measred by a set of mtally synhronized loks. A method desribed in Einstein s earlier ited work will be sed for the loks synhronization. The method ses light signals emitted off A B lok A at time t 0, refleted at lok B at time t 1 and reeived at lok A at A time t. The method assmes that the speed of light is the same in both diretions and ths the refletion of the signal at B ors in the middle of the time interval bonded by the signal emission and reeption at A. The midpoint of the time interval is given by the formla A A A t t0 t1 = (4) Mtal synhronization of loks will be done by setting the lok B in sh a way that t = t (5) B A 1 1 The left part of Figre shows synhronization of loks that are at rest with respet to S, the right part shows synhronization of loks loated on a moving objet. The reslts of both synhronizations are different. In terms of oordinate time t, i.e. in terms of a stationary lok, the reslt of synhronization in motion (the right part of Figre ) is inorret. On time vales pper index regards to plae, lower to event (instant of time). DOI: /jmp Jornal of Modern Physis

5 R. Mahotka Figre. Light synhronization of loks A and B, at the left: lok at rest, at the right: both loks in niform motion to the right at speed. Light signal speed with respet to the loks in motion is variable (motion to the right with speed, motion to the left with speed + ). At the end of synhronization in motion, the loks A and B will be synhronos in terms of the oordinate system rigidly opled with the objet arrying AB B A both loks (system S') ( lok_shift = lok_reading lok_reading = 0 ) AB B A and time differene between events A 1 and B 1 is eqal to zero ( t = t 1 t 1 = 0 ). The events are onrrent. In terms of stationary system S the time differene of the events A 1 and B 1 is non-zero, it eqals t = t t = AB B A 1 1 l i.e. event B 1 will or after event A 1. As regards to reading of both loks, for stationary observer the reading of lok B will be always smaller than reading of lok A by amont AB B A l lok_shift = lok_reading lok_reading = (6b) Clok shift is in absolte vale smaller than time differene of synhronizing events. It is reslt of different loks rate. Cloks A and B moving with system S' are slower than referene lok in stationary system S (see time dilation in hapter 4.1). In the Formlas (6a) and (6b) the length l' is the distane between loks A and B measred in system S', i.e. in the system to whih both loks are at rest. Violation of lok synhronization proess is ased by different speeds of light on its way to B and bak. In onseqene the relevant transit times are not eqal. As an be seen the desribed method of lok synhronization an be sed even in a moving system where the light propagates by different speeds in dif- (6a) DOI: /jmp Jornal of Modern Physis

6 R. Mahotka ferent diretions, bt the reslt of sh synhronization differ from the reslt of synhronization in a stationary system..5. Assmptions and Methods Smmary All assmptions and methods stated above orrespond with assmptions and methods of Einstein s STR as he introded them in his work Zr Elektrodynamik bewegter Körper [1]. Einstein even ses the idea of stationary system in the work, he jst defines it differently throgh the validity of Newtonian eqations. In onseqene both definitions of a stationary system are eqivalent. The only signifiant differene lies in the method of distane measrement. In his work Einstein assmes sage of rigid gages rods. As will be shown later (Chapter 4.6) both measring methods, the one stated above and the Einstein s, are eqivalent and hoie of the method has no inflene on measrement reslts. 3. Elidean Soltion D Spae Elidiity Postlate The basis for following onsiderations as same as for the whole Elidean Model of Spae and Time (EMST) is a formla belonging to the Einstein-Minkowski soltion [3] τ = t x y z (7) It is a variation of (1) that is valid in the ase t x + y + z (8) Flfilling the ineqality (8) is demanded in order for τ to be a real. Eqation (7) an be interpreted as a relation between an inrements of oordinate time t and proper time τ of a body that has moved niformly between two points in time interval t, whereas oordinate differenes of both points are x, y, z. The given formla is valid for any objet regardless if it is in motion or at rest with respet to hosen oordinate system. The qantity τ is an invariant of the Lorentz transformation as well as spae-time interval s ( s = τ ). The Elidean formla eqivalent of (7) an be aqired by a simple rearrangement t = x + y + z + τ (9) Formlas (7) and (9) are idential from a mathematial point of view bt their geometri interpretation is different. While the Formla (7) defines the Minkowski metri s = τ in for-dimensional spaetime with three spatial axes x, y, z and one time-like axis t, Formla (9) defines the Elidean metri t in for-dimensional spae with spatial axes x, y, z and τ. 3 In the Elidean onept the qantity t is not one of spae dimensions bt a measre of remoteness 3 Alternate notation w of the forth axis will be also sed in this artile to highlight its spae-like natre. In this notation the formla (9) reads t = x + y + z + w. DOI: /jmp Jornal of Modern Physis

7 R. Mahotka of two points of spae, i.e. the distane between them. As will be shown frther, for-dimensional Elidean spae (E 4 ) an be sed as a basis of an alternative theory of spae and time. On that aont this spae will be one of the ornerstones of EMST, its first postlate. 4D spae elidiity postlate sonds: Spae is for-dimensional and Elidean. To reate a realisti theory whih wold omply with the mathematial aspet of Einstein s STR it is neessary to aept two more assmptions D Speed Invariane Postlate It an be seen from Formla (9) that no objet in E 4 an be stationary. Every objet travels a distane t dring time interval t, i.e. almost 300,000 km in a seond. This also holds for objets that are seemingly stationary or that move with distintly sb-light veloities. Motion of sh objets takes plae ompletely, or in the vast majority, in the forth dimension, that is along the axis w τ. Or (three-dimensional) senses, as well as or (three-dimensional) measring eqipment, annot detet motion in the diretion of this axis. The only way we an measre it is sing a lok onneted to the objet. Among others, the Formla (9) expresses a known relativisti fat that the larger the hange of spatial oordinates x, y and z in time interval t (or more ommonly said the faster the objet is moving) the slower the flow of its proper time τ. Denotations 4D motion, 4D speed 4 et. will be sed in the following text to distingish motion in E 4 from motion in ordinary three-dimensional spae (E 3 ). A new postlate as a replaement for Einstein s speed of light invariane postlate an be formlated with the se of 4D speed. This postlate is diretly derived from Formla (9). 4D speed invariane postlate states: In a stationary oordinate system all objets move with the same 4D speed that is eqal to the speed of light. It shold be noted that the postlate refers to the stationary system only, i.e. to the system in whih the speed of light is invariable and eqal to. This postlate is an enhaned version of Einstein s speed of light invariane postlate that states: Every ray of light moves in the stationary system with the same speed, the speed being independent of the ondition whether this ray of light is emitted by a body at rest or in motion [1]. Aording to the 4D speed invariane postlate all objets travel the same distane dring time interval t. Varios ases for objets moving from point 0 at different speeds v are displayed in Figre Forth Spatial Dimension Bondedness Postlate The seond assmption neessary for aeptane of E 4 spae as a basis for the new model of spae and time is the assmption of its limited width, or more 4 Relation between ordinary and 4D speed will be desribed in detail in hapter 7.1. DOI: /jmp Jornal of Modern Physis

8 R. Mahotka Figre 3. Uniform motion of objets lanhed from point 0. At the end of time interval t, the objets are loated on the srfae of a for-dimensional hemisphere with radis r = t. It is a hemisphere de to the fat that inrements of oordinate τ annot be negative. Motion of objets in plane x τ is plotted here. Speed of an objet an also be x v expressed by an angle α: sinα = =. t arately limited sable width in the diretion of the forth spatial axis w τ. Firstly I will larify the reasons. Experiene teahes s that two objet are in ollision if these opy the same point of spae at the same time. In ordinary three-dimensional spae it orresponds to an eqality of spatial oordinates x, y, z and time vale t. In a for-dimensional spae it wold be natral to expet eqality of all for spatial oordinates with oordinate w τ among them. It an be shown, thogh, that in real world the ollisions or entirely independently of the vale of w τ. Let s have objets A and B that are both stationary in the system S. From objet A light signal L was emitted towards objet B, it was refleted there and has retrned bak to A. On its way to B and bak the light signal traveled a distane AB of AB in time taa =. Dring this time the objet A hasn t hanged its three-dimensional position, ths x = y = z = 0 and aording to A A A Formla (9) holds that taa = τ A or τ A = taa. An inrement of the forth spatial oordinate τ A of objet A is eqal to time t AA mltiplied by speed of light. On the ontrary the light signal L travels the distane t AA in time t AA and from Formla (9) it is lear that the inrement of its forth spatial oordinate eqals zero ( τ L = 0). If the light signal was emitted from objet A at a point with oordinates x A, y A, z A, τ A then the instant of retrn had oordinates x A, y A, z A, ( τ τ ) A +. On the other hand, for the light signal itself, its proper A time has not flown ( τ L = 0), its oordinate w τ had not hanged and after its retrn it had its initial oordinates x A, y A, z A, τ A. In terms of for-dimensional spae, the light signal has retrned to the initial point, bt objet A is no longer present here. Ths the ollision, i.e. the intereption of the light signal by the objet A, shold not take plae. Physial reality, thogh, is different. Another example is shown in Figre 4. It shows two objets that went forward from points A and B one against eah other at the same time t A = t B. The two objets move with different speeds. In ase we plot their trajetories in plane DOI: /jmp Jornal of Modern Physis

9 R. Mahotka Figre 4. Plot of ollision of two objets, at the top: in plane x τ, at the bottom: in ase of existene of barriers. x τ we will see that they interset at point C. However this point will not be the plae of their ollision bease they will not be present there simltaneosly. 4D distanes AC = t AC and BC = t BC are different whih means that also the times t AC and t BC are different. In fat the two objets ollide at point D whih orresponds to two separate points D 1 and D in plane x τ. 4D distanes AD = t AD and BD 1 = t BD are eqal in this ase, whih means that both objets will opy point D at the same time. In order to explain sh a strange featre of spae, following assmption has to be adopted, forth spatial dimension bondedness postlate: Material objets are in the forth spatial dimension limited to narrow allotted region. This region is ommon for all material objets and does not provide enogh room for their mtal passing. This means, that the spae and (elementary) partiles of matter are for-dimensional bt some fores or barriers keep them in a narrow strip of the spae. The dediated strip is very narrow, so the partiles are nable to pass eah other in the forth dimension withot mtal interation. Only three remaining spatial dimensions are appliable. For the sake of ompleteness we shold add that, de to the onentration of all matter partiles in a narrow strip of for-dimensional spae, all omposite objets have one dimension signifiantly smaller than the others. Ths, even thogh omposed of for-dimensional partiles, these objets exhibit only three-dimensional properties. There ertainly exists some physial explanation for the above mentioned behavior of partiles bt it is nknown to the athor at this time. For frther explanation, sh behavior of partiles in 4D spae will be thoght as basi fat, and as sh it is not to be disssed or investigated any frther. Instead, we will fos on finding of appropriate geometrial model of partiles motion Geometri Interpretation of 4D Motion Let s reate geometrial model of motion of matter partiles in spae E 4. Basis for or onsiderations will be three aforementioned postlates omplemented by five assmptions that an be onsidered natral: DOI: /jmp Jornal of Modern Physis

10 R. Mahotka 1) It holds diret proportion between partile s 4D path inrement and oordinate time t inrement expressed by Formla (9). ) 4D speed of any partile is onstant in time and is eqal to. 3) Allotted region of spae E 4, where all partiles are sitated, is too narrow in the diretion of forth spatial dimension w τ to observe this dimension in marosopi experiments. 4) 4D trajetory of any partile is ontinos, i.e. withot gaps or jmps. 5) Partiles obey laws of onservation of energy, momentm and mass. Diret onseqene of assmption 5 is: 6) Diretion of a partile s 4D motion is hanging only when it is in the interation with fore field, with other partile or with bondary of E 4 spae region. In other ases, the trajetory is straight. Let s desribe 4D motion of a partile in niform sblminal motion. For sh motion holds: 7) Projetion of the partile s 4D speed onto or ordinary three dimensional spae (E 3 ) is ordinary (3D) speed v whih is onstant in time and less than. 8) Projetion of the partile s 4D trajetory onto E 3 is a straight line. Using statements 1, and 7 it an be stated: 9) 4D veloity omponent in diretion of w τ is non-zero and its absolte vale is onstant in time. In ombination with statements 3 and 4 we aqire: 10) De the fat that allotted region is narrow in diretion of w τ, the partile is experiening osillating motion. The sign of veloity omponent in this diretion is hanging in time. And finally from statements 6 and 10: 11) Diretion of 4D motion is hanging only on the bondaries of E 4, the hange is abrpt a it affets only sign of the 4D veloity omponent in the diretion of w τ. The above findings mean that 4D trajetory of niformly moving partile is restrited to the allotted region of E 4, is sitated in a plane given by the diretion of partile s 3D motion and parallel with axis w τ, is omposed of straight, mtally onneted lines, its verties rest on the barriers bonding allotted region of E 4, its straight lines form eqal angle α with the axis w τ, the angle is given v by speed v of the partile ( sinα = ). Sh form of motion is shown on the lower part of Figre 4. Definition: Let s have a spae E 4 ontaining two distint three-dimensional hyperplanes perpendilar to the axis w τ. These hyperplanes will be alled barriers, their distane labeled w ( w= wmax wmin ). Sbset of E 4 bonded by both barriers will be alled spae E 4 -B (letter B for bonded ). Ones again, let s gather geometrial model of motion of a partile in niform DOI: /jmp Jornal of Modern Physis

11 R. Mahotka motion. We have spae E 4 -B in whih partiles of matter are moving by speed of light. Trajetory of eah partile is ontinos and is omposed of straight lines. Eah line forms an angle α with axis w τ proportional to the speed v of the partile. Motion of the partiles in the diretion of three spatial oordinates (x, y, z) is onsidered as nrestrited, while in the forth dimension w τ is limited by existene of two barriers perpendilar to the forth spatial axis w τ. All partiles of matter are loated between these barriers that onstitte fixed bondaries of their motion in the forth dimension. The distane w of the barriers is onstant, independent of the type of the partiles, time and position in spae. As an inevitable onseqene of laws of momentm, energy and mass onservation, ollisions of partiles with the barriers are ideally elasti, partiles maintain their kineti energy (i.e. both speed and mass), 3D diretion of motion and angles of inidene to the barriers are eqal to angles of refletion. Motion of partiles in the forth dimension has osillatory natre. This osillatory motion is performed by individal elementary partiles, not by objets as a whole. Desribed geometrial model old also hold partiles in arbitrary aelerated motion. It is sffiient to drop some demands on partile s trajetory as a reslt, the 3D motion of partile is now no more niform. 4D trajetory will now be omposed of lines whih are no neessary straight, lay in one plane nor form onstant angle with w τ Forth Spatial Coordinate In frther explanation I shall distingish between the vale of oordinate w τ and the atal position of an elementary partile within E 4 -B. The later will be marked as w B. While oordinate w B is hanging in a narrow range between w min and w max in a yli manner, oordinate w τ grows with every yle by vale of (w max w min ) (Figre 5). We have no means of measring oordinate w B diretly bt by sing a lok we an measre inrements of w τ. Both variants of forth spatial oordinate w B and w τ have their theoretial importane. On one side oordinate w B gives the position of elementary partiles in the spae between barriers, on the other side oordinate w τ onnets motion of partiles with time. In terms of frther explanation, there is an important fat that qantity w τ is not yli and ths it is sitable for plotting of diagrams of objet s motion. Sh a diagram does not orrespond to the real motion of partiles in 4D spae bt it is mh more desriptive than a plot of a trajetory with nmberless refletions from the barriers. A omparison of the real motion between the barriers with a plot of fititios motion in plane x τ an be seen on Figre 4 and Figre 5. A diagram with axes x and w τ is an analogy of the ommonly known Minkowski diagram. The main differene between them is that the oordinate time t is not one of the oordinates in the x τ diagram bt a length of a trajetory. The x τ diagram also allows the display of the proper time τ whih is not possible in the Minkowski diagram. DOI: /jmp Jornal of Modern Physis

12 R. Mahotka Figre 5. Two variants of plots of a partile motion in 4D spae bold: real motion between the barriers, dashed: fititios motion in plane x τ Geometri Interpretation of Time We have desribed important featres of E 4 -B spae as well as natre of the motion in it. Now, let s trn or attention to the proper time. Proper time an be defined as an inrement in oordinate w τ divided by speed. On the other hand, the proper time does not have to be nderstood solely as a distane divided by speed; there also exists an alternate view. As a natral measre of time flow, the nmber of refletions of seleted elementary partile from the barriers an be hosen. The faster the partile is refleting, the faster the time is passing. Matter itself measres its time eah partile of matter is a tiking lok. There is a diret proportion between the nmber of refletions of a partile n and a length of the orresponding time interval τ. The proportion is given by formla ( ) τ = n w w = nw (10) 4. Geometri Basis of Relativisti Phenomena max In the previos hapter we have introded E 4 -B spae as a replaement for psedo-elidean spaetime sed in Einstein s STR. In this hapter we shall demonstrate the geometri basis of the Lorentz transformation and known relativisti phenomena of time dilation and length ontration Time Dilation min As stated in the previos hapter the partiles osillate in a narrow strip of E 4 spae onfined by a ople of parallel barriers. Aording to the 4D speed invariane postlate the partiles move with 4D speed in a stationary system S. Sh motion of partiles is in priniple idential with the motion of light in a so alled light lok. We have two parallel refletive srfaes and a partile moving with the speed of light between them. The rate of the flow of time is given by nmber of refletions of the partile from the refletive srfaes. The faster the motion of sh a lok in S is, the longer the path of the partile between refle- DOI: /jmp Jornal of Modern Physis

13 R. Mahotka tions and the slower the osillation. The 4D path t of a partile drifting together with sh a lok at speed v is given by t = v t + τ (Figre 6). Hene τ = t v (11) The formla shows that the time interval τ measred by a partile lok in motion will be smaller than the orresponding τ interval measred by a referene partile lok at rest. The above effet affets all partiles of a moving objet i.e. every partile of the objet behaves as a light lok. There is no differene between behavior of the matter and light in this respet. Time behaves the same as a partile lok or light lok. This reslts from the diret proportion between τ and the nmber of osillations of a partile (see the end of previos hapter). If motion of an objet is slowing down osillations of all its elementary partiles, it signifies time itself is slowing down. Ths all loks on a moving objet are slowed down in their operation, no matter their onstrtion. 4.. Transformation of Light Wavefront Now we shall look at the transformation of spae. A geometri model of the refletion of light from an inner srfae of an ellipsoid or sphere will be sed for this prpose. Sh a pair of bodies was hosen bease a moving ellipsoid and a stationary sphere are eqivalent bodies in terms of STR they differ only de the length ontration in the diretion of the body s motion. Let s have a flattened rotational ellipsoid e' moving in stationary system S with veloity (Figre 7). Semi axis in the diretion of motion shall be denoted as d, semi axis in the transverse diretion as b. The two semi axes satisfy the ondition d = b. The shortening of semi axis d is hosen in sh a way that it orresponds to the Lorentz-FitzGerald ontration. It an be verified sing Figre 6. Partile lok. Left: stationary lok, right: lok moving with veloity v. DOI: /jmp Jornal of Modern Physis

14 R. Mahotka Figre 7. A view of light refletion in a moving flattened ellipsoid e' as seen in a stationary system S. Light emitted simltaneosly from point F 1 sessively reflets from the ellipsoid: the first refletion ors at point A, then the points of refletion gradally move to the right towards point C and frther to point B. The refleted light reahes point F 1 from all diretions simltaneosly. The geometri set of all points of refletion is a prolonged ellipsoid e with foi F 1 and F 1. the Lorentz transformation that in the view of system S' onneted with the ellipsoid, it appears as a stationary sphere k with radis b. Let s assme, that in the enter of a moving ellipsoid (point F 1 ) there is a flash of light at an arbitrary time t. The light spreads in all diretions, strikes the inner mirror-like srfae of the ellipsoid and reflets. The light propagates with speed in respet to S and the ellipsoid itself is moving as well. The qestion is whih points of S are points of light refletion and where the refleted light is heading. It an be proved that points of light refletion from e' form a prolonged rotational ellipsoid e with semi axes a (in the diretion of motion) and b (in transverse diretion). The semi axis b is idential to that of a flattened ellipsoid e'. The following formlas hold for the semi axis a and eentriity e: b b a =, e= a b = Point F 1 is one of the foi of the ellipsoid e. Aording to a known property of oni setions the refleted light will be direted to the other fos F 1 while its trajetory will be eqal to a regardless of the plae of refletion. Assming the speed of light doesn t hange de to the refletion (whih orresponds with the 4D speed invariane postlate) all of the refleted light will reah fos F 1 siml- DOI: /jmp Jornal of Modern Physis

15 R. Mahotka taneosly. The transit time will be b a e t = = = (1) The formla shows that the transit time is also the time in whih enter of the flattened ellipsoid e' reloates from fos F 1 to fos F 1 (distane of the foi is e). That has onseqenes of great importane. If an observer in system S sees the light being emitted and reeived in different points of spae (points F 1 and F 1 ) and if he is aware that refletions of light off ellipsoid e' take plae sessively, an observer, loated in enter of e' and moving with it, sees the sitation differently. He witnessed the light being sent in all diretions simltaneosly and retrning from all diretions to the initial point! From his point of view, it holds F 1 F1. Frthermore, if he assmes the speed of light is independent of the diretion, i.e. = onst., he inevitably reahes a onlsion that the refletion of light took plae on a spherial srfae and point F 1 F1 is in its enter. The radis of the sphere an be determined by the observer from the transit time. It holds r = t where t' is time interval measred in moving system S'. This is measred sing a lok moving with the system. Aording to (11) it applies: t = τ = t (13) If t from Formla (1) is sbstitted we obtain b b t = = (14) b and ths r = or r = b (Figre 8). The geometrial model desribed here leads to the same reslts as the Lorentz transformation (all to mind the hoie of parameters of the flattened ellipsoid above). The moving flattened ellipsoid e' has transformed into a stationary sphere k de to hange of referene systems. The model is all-elidean (ontrary to the Einstein-Minkowski soltion), i.e. all sed formlas were derived sing the property of Elidean spae Geometri Basis of Lorentz Transformation Using this model, let s try to nderstand a geometri basis of the Lorentz transformation. Firstly we have to notie that only one dimension of a body is hanging dring the transformation. The hanged dimension is the one in the diretion of the x axis, i.e. in the diretion of motion. The remaining spatial oordinates y and z, as DOI: /jmp Jornal of Modern Physis

16 R. Mahotka Figre 8. Refletion of light in a moving flattened ellipsoid e as seen from system S moving with the ellipsoid. The observer is not aware that in his system the speed of light is not onstant and ths interprets synhronos retrn of light to initial point F 1 F1 as proof that the refletive srfae has shape of sphere k with its enter at point F 1 F. 1 well as proper time τ, is not hanging. These oordinates perpendilar to the diretion of the motion are invariants of the transformation. Therefore, the Lorentz transformation an be written in a modified form x = x t, y = y, z = z, τ = τ (15) The disssion whether the formla for oordinate time t x t = (16) is a linear ombination of formlas stated above or a separate fifth eqation shall be left for later (hapter 6.3). It remains to explain how it is possible that the length of a moving objet in Elidean spae is hanging. The answer is omposed of two parts. One reason is the different view on the simltaneity of non-oinidental events. Events simltaneos in terms of system S are not neessarily simltaneos in terms of another system (see the method of synhronization of non-oinidental loks, hapter.4). It is easy to show that a flattened ellipsoid hanges its appearane in onjntion with a hange of simltaneity definition (Figre 9). 1) In the first ase, we will determine simltaneity by means of stationary loks. The proedre old be as follows in a pre-seleted time t we mark the position of the leading (B) and trailing (A) point of the ellipsoid on the x axis of DOI: /jmp Jornal of Modern Physis

17 R. Mahotka Figre 9. Determination of length of moving ellipsoid. At the instant of oinidene of points A A the other terminal point of the ellipsoid an be either in point B' if simltaneity is defined sing oordinate time of stationary system S or in point B if simltaneity is defined sing oordinate time of moving system S'. stationary system S. The differene in x oordinates gives s its length in the diretion of motion. We an see that the ellipsoid is flattened, i.e. its length is smaller than the transverse dimension. ) In the seond ase, we se refletions of light from the ellipsoidal srfae for simltaneity definition. The refletions are simltaneos in terms of system S' (refletions of light an be sed for synhronization of a non-oinidental lok in S'). In ase we mark position of the beginning and the end of the ellipsoid on the x axis in the moment of the light refletion, we an see that the ellipsoid is prolonged, i.e. its length is larger than the transverse dimension. The reason for sh ontraditory reslts is the fat that in terms of S the light reflets on the trailing point (A) of the ellipsoid sooner than on the leading one (B). It has to be pointed ot that both methods of ellipsoid length determination are orret. In both ases, we have marked positions of both terminal points of the ellipsoid simltaneosly. In the first ase the simltaneity was in respet to system S, in the seond in respet to system S'. It is explained in the above text that, de to the method of synhronization sed, the flattened ellipsoid in motion looks as if it were a prolonged one. Bt the text does not explain, however, why the ellipsoid appears to be a sphere to an observer in S'. Ths we get to the seond part of the answer. This is related to the speed of light in a moving system. As stated in ) the refletions from the srfae of a prolonged ellipsoid are simltaneos in terms of S'. This an only be explained as the front of the light wave in S' propagates in the diretion of axis x faster than in other diretions. To be more arate in a moving system the front of the DOI: /jmp Jornal of Modern Physis

18 R. Mahotka light wave has the form of a prolonged ellipsoid with a larger semi axis oriented in the diretion of the system motion. The enter of sh ellipsoid is moving along with the system S'. The speed of light in the diretion of motion an be determined by ombination of Formlas (13) and (1) a t x = = = (17) t t as well as the speed of light in the perpendilar diretion y b = = (18) t The ratio of both veloities is 1:. Notie that x is larger than the 1 speed of light in a stationary system. This is ased by lok deeleration in a moving system. Ths the light in a moving system behaves differently than in a stationary system. Let s assme that an observer in system S' is not aware of the dependeny of speed of light on the diretion of its propagation. De to the assmption of invariable vale of he will onsider the simltaneity of refletions from the body to be a proof of its spherial shape. Sh a laim will be spported by the fat that the light reflets bak to the point of its emission. The seond part of the mystery of transformation of a flattened ellipsoid into a sphere is ths onneted to the non-onstant speed of light in a moving system. If we look at the parameters of ellipsoid e, we will see that the ratio of its longer semi axis to the semi axis in perpendilar diretion is 1: whih is eqal to the ratio of the speed of light in diretion of the longer semi axis and the speed in a perpendilar diretion. Both effets flly ompensate eah other and they are ths ndetetable by any measrement in S'. So nothing prevents an observer in S from aepting the assmption that the speed of light is onstant in all diretions. From a mathematial point of view the transition between the prolonged ellipsoid and a sphere is solved by hanging the length sale of axis x. The nit of 1 length on axis x' will be hosen times larger than on axis x. This new nit of axis x' assres that the speed of light in S' is diretion independent and the prolonged ellipsoid e transforms in sphere k. From a geometrial point of view spae ndergoes an affine transformation. Note: The statement given above that in a moving system the light doesn t propagate with the same veloity in all diretions is related to the sage of time t from the moving system and simltaneos sage of length sale of axis from a stationary system. In the ase of sage of orresponding qantities (times and sales) the effet disappears. So it is ndetetable in physial experiments. DOI: /jmp Jornal of Modern Physis

19 R. Mahotka 4.4. Transformation of Motion of Mass Objets So far we have demonstrated that sing the light an observer in S' annot detet that the lengths and the entire spae are affine deformed in his oordinate system. In other words, he annot detet that the body from whih the light reflets is not a sphere bt an ellipsoid. It remains to be shown that affine deformation of a moving system annot be deteted by other types of measrements as well. Let s investigate length measrements based on measrements of transit times of mass objets moving with speeds smaller than that of light. These objets will behave the same way as light (with an exeption of the speed of propagation) their motion will be niform and their refletions ideally elasti. We an imagine them as idealized tennis balls. Or rrent model of sphere/ellipsoid has to be expanded by adding a forth dimension w τ. The reason is simple: so far we have modeled propagation of light. Light moves at speed and proper time doesn t flow for it. Now we will explore slower motions and the τ oordinate will aqire non-zero vales. Three-dimensional bodies a sphere and ellipsoid will be replaed by their for-dimensional variants. However, these will differ relatively little from their three-dimensional relatives. E.g. a for-dimensional flattened ellipsoid has for semi axes from whih the one oriented in the diretion of motion is the shortest; the remaining three are eqal eah other. Ths to desribe the ellipsoid, it is enogh to know the length of its two semi axes the semi axis d in the diretion of motion and the semi axis b in perpendilar diretion. These qantities are the same as those sed to desribe the three-dimensional variant of this ellipsoid. We retain nhanged labeling also for the qantities of a prolonged ellipsoid (semi axes a and b) and a sphere (radis r). We annot imagine given bodies as a whole. Regarding frther explanation, it is not a major drawbak thogh. It sffies to neglet one of the dimensions of the for-dimensional ellipsoid and it beomes an ordinary three-dimensional ellipsoid; if we omit two dimensions, we get a two-dimensional setion of the ellipsoid. Sh a setion will sffie for frther explanation, bease we will stdy niform motions only. Applied to moving bodies sh as a sphere or ellipsoid, the plane of the setion will always be hosen in sh a way that it inldes the enter of the body and the diretion of its motion the axis x. As a reslt of rotational symmetry all setions of this type are similar i.e. the shape of the setion is independent of the hoie of its seond dimension; it old be either y, z, τ or any other diretion perpendilar to the diretion of motion. Let s ondt an experiment: Tennis balls were lanhed from a point that does not move with respet to S. Eah one of them moves in a different diretion and different speed. In the diretion of motion of eah ball a solid refletive panel is positioned perpendilarly to this diretion, so the ball is refleted bak to the lanh point. The distanes of refletive panels from the lanh point are set so that all balls DOI: /jmp Jornal of Modern Physis

20 R. Mahotka retrn to the lanh point at the same time. Let s sppose that in free spae the balls move niformly and refletions do not alter their kineti energy and ths their speed. Now it is ertain that refletions of all balls happen simltaneosly, i.e. times of flight there and bak are eqal. Using the 4D speed invariane postlate we determine that eqal transit time t means that the 4D paths of all balls mst be eqal as well. In the moment of refletion all balls are loated on the srfae of the for-dimensional hemisphere with radis r = t (see Figre 3). Note: Mentioned hemisphere is of orse a mere fition as well as oordinate w τ. As a onseqene of refletions from the barriers the motion of balls won t be straight and the hemisphere will transform into a narrow objet sqeezed between the barriers limiting motion in the forth dimension. Inrements of oordinates x, y, z and w τ as well as the 4D paths of any objets will not be affeted by this transformation thogh. Now let s plae an nhanged devie with the experiment so that it is in rest in moving system S'. In terms of S it will be shortened in the diretion of its motion (in orrespondene with Lorentz transformations). The ratio of the new and original length will be 1 :1. If we repeat the experiment desribed above, an observer in S' will see it identially as an observer in S had before. All refletions will be simltaneos in respet to S' as well as retrn of balls to the lanh point. The observer will be able to verify simltaneity of refletions by means of loks plaed at the refletive panels and synhronized sing the light. He will not detet any deviation. The observer in S, thogh, will see the orse of the experiment differently. From his point of view, the points of lanh and reeption of balls will not be the same. Also times of refletions of balls will differ. On the other hand, he will not qestion the fat that all balls were lanhed simltaneosly and they also retrn simltaneosly. Using the 4D speed invariane postlate he will determine that all balls travel the same 4D distane. Bease they were lanhed from one point and after refletion retrned to another one, he will easily determine that all points of refletion mst be loated on the srfae of a prolonged for-dimensional ellipsoid. The points of lanh and reeption of balls are the foi of this ellipsoid (or semi ellipsoid, if inrements of oordinate w τ on the way to the refletive panels will be regarded as positive and on the way bak as negative). Parameters of the ellipsoid an be determined from the balls travel time. For a longer semi axis applies a = t, for eentriity e= t. Given vales an be ompared to parameters of the prolonged ellipsoid e reated by refletions of light see Formla (1). They are idential! 4.5. Analogy between Motion of Mass Objets and Motion of Light We have fond that reslts of experiments modeling motion of light and balls do not differ. What is the impliation? DOI: /jmp Jornal of Modern Physis

21 R. Mahotka Jst the same way we have desribed motion of light in the sphere/ellipsoid model by means of three interhangeable bodies flattened ellipsoid, prolonged ellipsoid and sphere, we an desribe motion of balls. The only differene is sage of a for-dimensional variant of the bodies. We have a moving flattened ellipsoid with semi axes d = b and b representing a moving devie with the experiment, sphere of radis r = b moving with S' representing a set of refletion points of balls as they appear to an observer in S', and a prolonged ellipb soid with semi axes a = and b representing a set of refletion points as they appear to an observer in S. Correspondene of all parameters of given bodies has a simple explanation, it is reslt of eqivalene of both types of motion. Whether we se the light moving with speed or tennis balls moving onsiderably slower, the 4D path is always the same d4 = a = t. The only thing it hanges is its projetion to the D three-dimensional spae. If we plot motion of a light beam and a ball in orresponding ross-setions of a for-dimensional ellipsoid we get two variants of the same pitre (Figre 10). The only differene is one axis label. Figre 10. Trajetory of beams of light inside an ellipsoid (at the top) and trajetory of a ball lanhed along the x-axis (at the bottom). The plots differ only in the plane in whih the motion takes plae. Plot of a trajetory of a ball lanhed in random diretion and speed wold differ only in the plane of motion. DOI: /jmp Jornal of Modern Physis

22 R. Mahotka Partial onlsion: Reslts of length measrements based on measrements of moving objets (photons, balls,...) transit times transforms in ompliane with Lorentz transformation. They do not depend on the speed of the objets sed for length measrement. If the simltaneity of light reeption in the sphere/ellipsoid model led s to the hange of the axis x' length sale, reslts of all other length measrements based on objets motion will lead s the same way Contration of Rigid Rods Only one hope remains for determination of real ndeformed dimensions of a moving body. That is measrement by means of rigid measring rods. With their help, is it possible to reveal that what seems to be a sphere, is a flatten ellipsoid in reality? We mst reply: no. The shape and dimensions of rigid bodies are given by fores ating between individal partiles of mass. These partiles are not in diret ontat and ths ation at a distane is responsible. Strong and weak nlear interations at in the atomi nlei, eletromagneti fore between nlei and eletrons shells as well as between adjaent atoms, and all is omplemented by gravitational fore. All these interations are, aording to modern theories, mediated by exhange of mass partiles (mesons, leptons, photons, gravitons). These partiles are in permanent motion and their interations keep the dimensions of rigid bodies nhanged. How wold the distane of two adjoining ions in a rystal lattie be affeted by motion of the whole body? Will it remain nhanged? On the basis of the tennis balls experiment we mst say no. The partiles mediating interations are only tennis balls, in their natre, i.e. objets moving there and bak with any speed less or eqal to the speed of light. Ths dring the motion of objets in S the distanes between adjoining ions (atoms, moleles, nleons,...) are shortened. Sh shortening will have the same ratio in whih the sphere moving in S had to be shortened, so that it still appeared to be a sphere in view of S'. Shortening of distanes between ions in a rystal lattie will inevitably lead to the shortening of the whole body. This shortening ors only in the diretion of the motion and the shortening is real. A similar explanation of length ontration was sggested by Lorentz in his work Eletromagneti phenomena in a system moving with any veloity smaller than that of light [11]. It remains to add the obvios the same reason will ase that all bodies, not jst measring rods, will be deformed the same way. The phenomenon also affets liqids and gases whih will hange their volme. 5. Eqivalene of Coordinate Systems Let s review onlsions onerning deformation of bodies and oordinate systems: 1) By means of observations in a moving system we annot detet shortening DOI: /jmp Jornal of Modern Physis