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# Uniform motion of objects launched from point 0. At the end of time interval ∆t, the objects are located on the surface of a four-dimensional hemisphere with radius r = c ∆t. It is a hemisphere due to the fact that increments of coordinate cτ cannot be negative. Motion of objects in plane x-cτ is plotted here. Speed of an object can also be expressed by an angle α: sin

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## Contexts in source publication

**Context 1**

... to the 4D speed invariance postulate all objects travel the same distance during time interval ∆t. Various cases for objects moving from point 0 at different speeds v are displayed in Figure 3. ...

**Context 2**

... the 4D speed invariance post- ulate we determine that equal transit time ∆t means that the 4D paths of all balls must be equal as well. In the moment of reflection all balls are located on the surface of the four-dimensional hemisphere with radius 2 t r c ∆ = (see Figure 3). ...

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## Citations

... The problem of mutual observation of time dilation does not lie in the deformation of the time dimension but in the change of direction interpreted as the space-dimension of the observer, which causes the observed path (time) of the body in motion in E4 to seem shorter than in the observer's system -this is shown in Fig. 7 and formulas (7,8). As long as the bodies are moving in rectilinear paths, this effect is symmetrical, and both observers observe an identical dilation of time in the system of the other body. ...

... The relativistic equations for such particles, such as the equation determining the time dilation (7,8), will have a different shape than the known relativistic relationships. Is the special case the norm then? ...

The paper presents a new approach to space-time problems that is completely different from the approach used for over 100 years. The essence of the changes are two new ideas that can be treated as a complement to the theory of relativity. The first is the description of reality as a four-dimensional Euclidean space. What we observe as space-time dimensions are directions in objective (Euclidean) space, and these directions are not constant but depend on the pair of bodies-the observer and the observed body. Depending on the choice of body pair, the same direction in objective Euclidean space can be interpreted as a temporal or spatial dimension of the observer's coordinate system. The new model allows the body to be described directly as a wave and allows for a connection of the ideas of absolute space and the relativity of motion. The second idea binds the transmission of signals (quanta) to the systems of sending and receiving particles. As a result, the motion of the quantum is always constant in the system of the sending and receiving particle. This justifies the constancy of the speed of light regardless of the relative velocity of the bodies. Quanta are no longer independent particles but are disturbances of particles, which are treated as waves. The proposed changes simplify the description of relativistic phenomena, eliminate the need to apply Einstein's postulates by introducing mechanisms describing the relative motion and propagation of quantum, bind the description of relativistic and quantum phenomena by describing bodies directly as waves, extend the range of phenomena described within one model and solve many problems impossible to solve within the theory of relativity. The paper compares the descriptions of particular problems in the Theory of Relativity with the descriptions of the same issues in the new model. In most cases, the predictions of both models are similar, but the differences in the construction of the models give different conclusions in some cases, which is the basis for proposing specific experiments allowing verification of the proposed approach. Some of the proposed experiments can be carried out with the use of existing experimental devices.

... Problem wzajemnej obserwacji dylatacji czasu nie polega tu na deformacji wymiaru czasowego ale na zmianie kierunku interpretowanego jako wymiar przestrzenny obserwatora co powoduje, że obserwowana droga (czas) ciała w ruchu w E4 wydaje się krótsza niż w układzie obserwatorapokazuje to rys. 7 i wzory (7,8). Tak długo jak ciała poruszają się po torach prostoliniowych, efekt ten jest symetryczny i obaj obserwatorzy obserwują identyczne skrócenie czasu w układzie drugiego z ciał. ...

1 Streszczenie W pracy przedstawiono nowe podejście do zagadnień czasoprzestrzennych całkowicie odmienne od podejścia stosowanego od ponad 100 lat. Istotą zmian są dwie nowe idee, które można traktować jako uzupełnienie Teorii Względności. Pierwszą z niech jest opis rzeczywistości jako czterowymiarowej przestrzeni Euklidesowej. To, co obserwujemy jako wymiary czasoprzestrzenne są to kierunki, w obiektywnej (Euklidesowej przestrzeni) i te kierunki nie są stałe ale zależą od pary ciał-obserwatora i ciała obserwowanego. W zależności od wyboru pary ciał, ten sam kierunek w obiektywnej przestrzeni euklidesowej może być interpretowany jako wymiar czasowy lub przestrzenny układu współrzędnych obserwatora. Nowy model pozwala na opis ciała wprost jako fali, oraz pozwala na połączenie idei absolutnej przestrzeni i względności ruchu. Druga idea wiąże przekazywanie sygnałów (kwantów) z układami cząstek wysyłającej i obierającej. Dzięki temu ruch kwantu jest zawsze stały w układzie cząstki wysyłającej i odbierającej. Uzasadnia to stałość prędkości światła niezależnie od prędkości względnej ciał. Kwanty nie są już samodzielnymi cząstkami ale są zaburzeniami cząstek, które to cząstki są traktowane jako fale. Proponowane zmiany upraszczają opis zjawisk relatywistycznych, likwidują konieczność stosowania postulatów Einsteina przez wprowadzenie mechanizmów opisujących ruch względny i propagację kwantów, łączą opis zjawisk relatywistycznych i kwantowych poprzez opis ciał wprost jako fal, poszerzają zakres zjawisk opisywanych w ramach jednego modelu i rozwiązują wiele problemów niemożliwych do rozwiązania w ramach Teorii Względności. W pracy porównywane są opisy poszczególnych zagadnień w Teorii Względności z opisem w tych samych zagadnień w nowym modelu. W większości przypadków przewidywania obu modeli są podobne, jednak różnice w konstrukcji modeli dają w pewnych przypadkach odmienne wnioski co jest podstawą do zaproponowania konkretnych eksperymentów pozwalających na weryfikację proponowanego podejścia. Część z zaproponowanych eksperymentów jest możliwa do przeprowadzenia z wykorzystaniem już istniejących układów doświadczalnych. 2 Wstęp Jednym z podstawowych problemów w rozwoju modeli fizycznych, który już był podejmowany w przeszłości, jest pytanie, aktualne praktycznie na każdym etapie rozwoju nauk, o relację i między układem współrzędnych, którego używamy do opisu rzeczywistości a faktycznym kształtem tej rzeczywistości. Tworząc nowy model rzeczywistości początkowo stosujemy do opisu rzeczywistości układ współrzędnych jaki stosujemy na co dzień do opisu naszego bezpośredniego otoczenia. Może się jednak zdarzyć, że dotychczas stosowane przez nas układy współrzędnych nie są odpowiednim narzędziem do opisu wszystkich zjawisk i poprawnie opisują jedynie pewną klasę zjawisk występujących w naszym bezpośrednim otoczeniu. Zastosowanie niewłaściwego układu

... But none of them identifies the issue in coordi-77 nate time, and they all run into geometric paradoxes (see Sect. 4) because they don't pro-78 ject ES to an observer's reality. Only Machotka added a "boundedness postulate" to avoid 79 paradoxes [14], but this postulate sounds rather contrived. 80 It is instructive to compare ER with Newton's physics and Einstein's physics. ...

The primary concept of time in special and general relativity (SR, GR) and quantum mechanics (QM) is coordinate time t. Here I show: SR and GR are mathematically correct, but physically t has an issue. It takes an observer as the center of time, just as the geocentric model takes Earth as the center of space. In Euclidean relativity (ER), the roles of t and proper time τ have switched. Time dilation is interpreted differently: In ER, an observed clock is slow with respect to an observer in his proper flow of time (not in its proper flow of time as in SR/GR). All energy is moving through 4D Euclidean space (ES) at the speed c. All four dimensions are distance, and “cosmic time” t is the total distance covered in ES divided by c. Unlike in previous ER models, an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. The Lorentz factor and gravitational time dilation are recovered in ER. So, ER predicts the same relativistic effects as SR and GR. Yet ER outperforms SR in solving time’s arrow and the c2 in mc2. ER also outperforms a GR-based cosmology in explaining the data from high-redshift supernovae while declaring cosmic inflation, expansion of space, dark energy, and quantum gravity redundant. ER even improves our understanding of QM: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant. I conclude: The true pillars of physics are ER and QM.

... And they all run into geometric paradoxes discussed in Sect. 4 98 because they don't project ES to an observer's reality. Only Machotka added a "bounded-99 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 100 paradoxes by limiting our second postulate to an observer's reality. ...

... 766 With that said, conflicts of mankind become all so small. 767 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-768 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 769 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 770 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 771 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. In Euclidean relativity (ER), we replace Einstein time (coordinate time of an observer) with Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light. For each observed object, Euclidean time flows in a 4D direction equal to its direction of motion. The projection of the object’s 4D vector “flow of time” to an observer’s direction of motion yields its motion in his Einstein time. Because of the projection, Einstein time provides less information than Euclidean time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR. Yet ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant. We conclude: Physics based on Euclidean time penetrates to a deeper level and makes less assumptions.

... And they all run into geometric paradoxes discussed in Sect. 4 98 because they don't project ES to an observer's reality. Only Machotka added a "bounded-99 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 100 paradoxes by limiting our second postulate to an observer's reality. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (coordinate time of an observer) with universal Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light c. Euclidean time is distance covered in ES, divided by c. For each object, Euclidean time flows in a unique 4D direction. Clocks project this 4D flow to a 1D flow of time. Unlike other ER models, we claim that an observer’s reality is only a projection from ES. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant. We conclude: Physics based on Euclidean time penetrates to a deeper level and makes less assumptions.

... And they all run into geometric paradoxes discussed in Sect. 4 97 because they don't project ES to an observer's reality. Only Machotka added a "bounded-98 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 99 paradoxes by limiting reality in our second postulate. ...

... 766 With that said, conflicts of mankind become all so small. 767 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-768 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 769 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 770 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 771 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (coordinate time of an observer) with universal Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light c. Euclidean time is distance covered in ES, divided by c. For each object, Euclidean time flows in a unique 4D direction equal to its current direction of motion. Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

... And they all run into geometric paradoxes discussed in Sect. 4 97 because they don't project ES to an observer's reality. Only Machotka added a "bounded-98 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 99 paradoxes by limiting reality in our second postulate. ...

... 765 With that said, conflicts of mankind become all so small. 766 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-767 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 768 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 769 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 770 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (coordinate time of an observer) with universal Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light c. Euclidean time is distance covered in ES, divided by c. For each object, Euclidean time flows in a unique 4D direction equal to its current direction of motion. Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

... And they all run into geometric paradoxes discussed in Sect. 4 97 because they don't project ES to an observer's reality. Only Machotka added a "bounded-98 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 99 paradoxes by limiting reality in our second postulate. ...

... 765 With that said, conflicts of mankind become all so small. 766 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-767 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 768 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 769 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 770 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (coordinate time of an observer) with universal Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light c. Euclidean time is distance covered in ES, divided by c. For each object, Euclidean time flows in a unique 4D direction equal to its current direction of motion. Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

... And they all run into geometric paradoxes discussed in Sect. 4 91 because they don't project ES to an observer's reality. Only Machotka added a "bounded-92 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 93 paradoxes by limiting reality in our second postulate. ...

... With that said, conflicts of mankind become all so small. 808 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-809 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 810 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 811 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 812 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (proper time of one observer) with universal Euclidean time (proper time of all objects/observers). It originates from an absolute point O (Big Bang). In Euclidean spacetime (ES), all energy is moving radially away from O at the speed c. For each object, time flows in a unique 4D direction related to its position. Einstein time makes us believe that time would flow in one direction for all objects in the universe. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

... And they all run into geometric paradoxes discussed in Sect. 4 91 because they don't project ES to an observer's reality. Only Machotka added a "bounded-92 ness postulate" to avoid paradoxes [14], but it sounds rather contrived. We overcome such 93 paradoxes by limiting reality in our second postulate. ...

... With that said, conflicts of mankind become all so small. 808 ER solves 15 mysteries at once: (1) time, (2) time's arrow, (3) 2 , (4) relativistic ef-809 fects, (5) gravitational time dilation, (6) CMB, (7) Hubble's law, (8) flat universe, (9) cosmic 810 inflation, (10) competing Hubble constants, (11) dark energy, (12) wave-particle duality, 811 (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 812 are 15 confirmations of ER. ...

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (proper time of one observer) with universal Euclidean time (proper time of all objects/observers). It originates from an absolute point O (Big Bang). In Euclidean spacetime (ES), all energy is moving radially away from O at the speed c. For each object, time flows in a unique 4D direction related to its position. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc2. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.