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Only Euclidean Relativity Provides a Holistic View of Nature

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09 April 2024

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10 April 2024

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Abstract
Special and general relativity (SR/GR) describe nature “subjectively”, that is, from the perspective of just one observer at a time (one group of observers, to be exact). Mathematically, SR/GR are correct. I show: (1) Physically, SR/GR have an issue. Despite the covariance of SR/GR, there is always just one active perspective. Because of this constraint, there is no holistic view of nature. The issue shows itself in unsolved mysteries. Still, the Lorentz factor and gravitational time dilation are correct. This is why the concepts of spacetime in SR/GR work well except for cosmology and quantum mechanics. (2) Euclidean relativity (ER) describes nature “objectively”, that is, from the perspectives of all objects at once. An object’s (!) proper space d1, d2, d3 and proper time τ span 4D Euclidean spacetime (ES), where d1, d2, d3 and d4 = cτ are treated the same. All energy moves through ES at the speed c. An observer’s reality is created by projecting ES orthogonally to his proper space and to his proper time. In SR, these concepts are considered coordinate space and coordinate time. Neither their reassembly to a non-Euclidean spacetime nor the parameterization in SR/GR provides a holistic view. The scalar τ, in particular, cannot factor in an object’s 4D vector “flow of proper time” τ. The SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts. A master reality (described by ER) is beyond each observer’s reality (described by SR/GR). ER solves 15 mysteries at once, such as time’s arrow, the Hubble tension, dark energy, and non-locality.
Keywords: 
Subject: Physical Sciences  -   Theoretical Physics
This paper is not about a minor issue. It is about a reformation of physics. There are two approaches to describing nature: “subjectively” (from the perspective of just one observer or one group of observers at a time) or “objectively” (from the perspectives of all objects at once). Special and general relativity (SR/GR) take the first approach (Einstein 1905b; Einstein 1916). SR/GR are mathematically correct, but they lack a holistic view of nature. Euclidean relativity (ER) takes the second approach. ER is mathematically and physically correct because it provides a holistic view. My theory was rejected by several top journals in physics. I was told that manuscripts are not considered if they challenge SR/GR. Many attempts to challenge SR/GR failed, but we must not reject all attempts. Scientific theories must always be falsifiable (Popper 1935). This is why I now submit to a journal in philosophy. May the cradle of physics give physicists a hand! Subjectively, we live in a curved, non-Euclidean spacetime. Objectively, we live in a flat, Euclidean spacetime.
Six pieces of advice: (1) Do not take SR/GR as the ultimate truth. Correct predictions do not prove SR/GR. ER predicts the same relativistic effects as SR/GR. Some reviewers made a systematic error when evaluating ER with concepts of SR/GR. ER is different. In ER, all energy moves at the speed c . (2) Be patient and fair. One paper cannot address all of physics. SR/GR have been tested for 100+ years. ER deserves the same chance. (3) Do not reject ER on a knee-jerk reaction. Why should we not strive for an objective description of nature? (4) Do not be prejudiced against a theory that solves many mysteries. New concepts often do so. (5) Appreciate illustrations. Geometric derivations are as powerful as equations but far more descriptive. (6) Consider that you may be biased. Some concepts of today’s physics are obsolete in ER. If you are an expert in such a concept, you may feel offended.
To sum it all up: Predictions made by SR/GR are correct, but ER penetrates to a deeper level. I apologize for my many preprints, but I received almost no support. The final version is all that is needed. The earlier versions document how I got there. It was quite tricky to figure out why the concepts of spacetime in SR/GR work so well despite an issue. Sect. 2 is about this issue. Sect. 3 describes the physics of ER. Sect. 4 recovers the Lorentz factor and gravitational time dilation. In Sect. 5, ER solves 15 mysteries at once.

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. In SR, he merges them into a flat spacetime described by an indefinite distance function. SR is often presented in Minkowski space time because it illustrates the invariance of the spacetime interval very well (Minkowski 1910). Predicting the lifetime of muons (Rossi and Hall 1941) is an example that supports SR. In GR, curved spacetime is described by a pseudo-Riemannian metric. Predicting the deflection of starlight (Dyson et al. 1920) and the high accuracy of GPS (Ashby 2003) are examples that support GR. Quantum field theory (Ryder 1985) unifies classical field theory, SR, and quantum mechanics (QM) but not GR.
Two postulates of ER: (1) All energy moves through 4D Euclidean spacetime (ES) at the speed of light c . (2) The laws of physics have the same form in each “observer’s reality”, which is created by projecting ES orthogonally to his proper space and to his proper time. To improve readability, I refer to each observer as “he”. To make up for it, I refer to nature as “she”. My first postulate is stronger than the second SR postulate: c is absolute and universal. My second postulate refers to realities rather than to inertial frames. I also introduce a generalized concept of energy: All energy is “wavematter”, which may appear as a wave packet or as a particle depending on the perspective (see Sect. 5.12).
Newburgh and Phipps (1969) pioneered ER. Montanus (1991) described an absolute Euclidean spacetime with a “preferred frame of reference” (a pure time interval is a pure time interval for all observers). Montanus (2023) claims: Without the preferred frame, we would face the twin paradox, non-contact collisions, and a “character paradox” (confusion of photons, particles, and antiparticles). I will show that the preferred frame is obsolete. Whatever is time for me, it may be space for you. There is no twin paradox. There are no non-contact collisions. The character paradox is reasonable. Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time. Montanus (2023) even tried to formulate Maxwell’s equations in ER but wondered about a wrong sign. He overlooked that the SO(4) symmetry of ES is incompatible with waves.
Almeida (2001) investigated geodesics in ES. Gersten (2003) showed that the Lorentz transformation is an SO(4) rotation in a “mixed space” (see Section 3). van Linden (2023) runs a website about various ER models. Physicists are still opposed to ER because dark energy and non-locality make cosmology and QM work, waves are excluded, and paradoxes may turn up if ER is not interpreted correctly. This paper marks a turning point: I disclose an issue in SR/GR. I justify the exclusion of waves. I avoid paradoxes by projecting ES.
It is instructive to contrast Newton’s physics, Einstein’s physics, and ER. In Newton’s physics, all energy moves through 3D Euclidean space as a function of independent time. The speed of matter is v 3 D ≪ c . In Einstein’s physics, all energy moves through 4D non-Euclidean spacetime. The speed of matter is v 3 D < c . In ER, all energy moves through ES. The 4D speed of all energy is u 4 D = c . Newton’s physics (Newton 1687) influenced Kant’s philosophy (Kant 1781). Will ER reform both physics and philosophy?

2. Disclosing an Issue in Special and General Relativity

In SR (Einstein 1905b), there are two concepts of time: coordinate time t and proper time τ . The fourth coordinate in SR is t . In § 1 of SR, Einstein provides an instruction on how to synchronize two clocks at P and Q. At “P time” t P , a light pulse is sent from P to Q. At “Q time” t Q , it is reflected. At “P time” t P * , it is back at P. The clocks synchronize if
t Q − t P   =   t P * − t Q .
In § 3 of SR, Einstein derives the Lorentz transformation. The coordinates x 1 ,   x 2 ,   x 3 ,   t of an event in a system K are transformed to the coordinates x 1 ′ ,   x 2 ′ ,   x 3 ′ ,   t ′ in K’ by
x 1 ′   =   γ   ( x 1 − v 3 D   t )   ,   x 2 ′   =   x 2   ,   x 3 ′   =   x 3 ,
t ′   =   γ   ( t − v 3 D   x 1 / c 2 ) ,
where K’ moves relative to K in x 1 at the constant speed v 3 D and γ = ( 1 − v 3 D 2 / c 2 ) − 0.5 is the Lorentz factor. Mathematically, Eqs. (1) and (2a–b) are correct for an observer R in K. There are covariant equations for an observer B in K’. Physically, SR and also GR have an issue. They describe nature from the perspective of just one observer at a time (one group of observers, to be exact). In SR, a group consists of observers who do not move relative to each other. In GR, a group consists of observers who share the same gravitational field. The physical issue lies in the fact that there is always just one active perspective. Because of this constraint, there is no holistic view of nature. In particular, observers do not always agree on what is past and what is future. Physics paid a very high price for surrendering simultaneity as a general concept: By replacing SR/GR with ER, we will solve not less than 15 fundamental mysteries of physics. Thus, the issue is real.
The issue in SR/GR is very similar to the issue in the geocentric model: In either case, there is no holistic view but just one active perspective. In the old days, it was natural to believe that all celestial bodies would revolve around Earth. Only the astronomers wondered about the retrograde loops of planets and claimed: Earth revolves around the sun. In modern times, engineers have improved the precision of rulers and clocks. Eventually, it was natural to believe that it would be fine to describe nature as accurately as possible but from just one active perspective. The human brain is very powerful, but unfortunately it often deems itself the center/measure of everything in the universe.
The analogy is strong: (1) It holds despite the covariance of SR/GR. After a transformation (or else after replacing the center Earth), there is again just one active perspective. (2) SR/GR miss the big picture just like the geocentric model. Retrograde loops are obsolete but only in the holistic view of the heliocentric model. Dark energy and non-locality are obsolete but only in the holistic view of ER. (3) In the old days, alternatives to the geocentric model were not taken seriously. Today, alternatives to SR/GR are not taken seriously. Have physicists not learned from history? Does history repeat itself?

3. The Physics of Euclidean Relativity

The indefinite distance function in SR (Einstein 1905b) is usually written as
c 2   d τ 2   =   c 2   d t 2 − d x 1 2 − d x 2 2 − d x 3 2 ,
where d τ is an infinitesimal distance in proper time τ , while d t and d x i ( i = 1 ,   2 ,   3 ) are infinitesimal distances in coordinate spacetime x 1 ,   x 2 ,   x 3 ,   t . This spacetime is construed because coordinate space x 1 ,   x 2 ,   x 3 and coordinate time t are subjective concepts: They are not immanent in rulers and clocks but construed by an observer. Rulers measure proper distance. Clocks measure proper time. I introduce ER by defining its metric
c 2   d t 2 = d d 1 2 + d d 2 2 + d d 3 2 + d d 4 2 ,
where d d i = d x i ( i = 1 , 2 , 3 ) and d d 4 = c d τ are infinitesimal distances in 4D Euclidean spacetime (ES). In ER, the roles of t and τ are switched: The fourth coordinate is an object’s proper time τ . The invariant line element is cosmic time t . The metric tensor is the identity matrix. Because of the identity matrix, the math of ER is much simpler than the one of GR. I retain the symbol t because it is the preferred symbol for cosmic time. I prefer the indices 1–4 over 0–3 to stress the symmetry. An object’s (!) proper space d 1 , d 2 , d 3 and proper time τ span ES, where d 1 , d 2 , d 3 and d 4 = c τ are treated the same. This spacetime is natural because all d μ ( μ = 1 , 2 , 3 , 4 ) are objective concepts: They are immanent in rulers and clocks. Do not confuse Eq. (4) with a Wick rotation (Wick 1954), where time is imaginary.
For each object, we are free to label the four axes of ES. We always take d 4 as the axis in which it currently moves at the speed c . The d 4 axis is not observable by the object but experienced as proper time. Another object may move in d 4 ′ at the speed c . It experiences d 4 ′ as proper time. Since proper time τ may flow in different 4D directions, there is a 4D vector “flow of proper time” τ . This vector τ is missing in SR/GR! An “object’s reality” is created by projecting ES orthogonally to its proper space and to its proper time.
τ   =   d 4 / c   ,   τ ′   =   d 4 ′ / c ,
τ   =   d 4   u / c 2   ,   τ '   =   d 4 ′   u ' / c 2 ,
where u is the 4D velocity of an object in ES. For all objects, there is u μ = d d μ / d t , where t is cosmic time. Thus, Eq. (4) is equivalent to my first postulate.
u 1 2 + u 2 2 + u 3 2 + u 4 2   =   c 2 .
My second postulate revises the principle of relativity and defines an “observer’s reality”, which is created by projecting ES orthogonally to his proper space and to his proper time. In SR, these concepts are considered coordinate space and coordinate time. Neither their reassembly to a non-Euclidean spacetime nor the parameterization in SR/GR provides a holistic view. The scalar τ , in particular, cannot factor in an object’s 4D vector τ . Since replacing coordinate time with cosmic time is a discontinuous operation, there is no continuous transition between SR and ER. In SR, an object is described by the four coordinates x 1 ( τ ) ,   x 2 ( τ ) ,   x 3 ( τ ) , t ( τ ) , where τ is the parameter and t is coordinate time. In ER, an object is described by the four coordinates d 1 ( t ) ,   d 2 ( t ) ,   d 3 ( t ) ,   d 4 ( t ) , where cosmic time t is the parameter and d 4 relates to τ according to Eq. (5).
It is instructive to contrast three concepts of time. Coordinate time t is a subjective measure of time: It is equal to τ = | τ | for the observer only. Proper time τ is an objective measure of time: It is independent of observers. Cosmic time t is the total distance covered in ES (length of a geodesic) divided by c . By taking cosmic time as the parameter, all observers agree on what is past and what is future. Since cosmic time is absolute, there is no twin paradox in ER. Twins share the same age in cosmic time! However, ER also seems to have an issue (I will explain in Sect. 6 why it is not an issue): Only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. This is why I introduced ER in Eq. (4) by defining its metric in proper coordinates. Be aware that ER is not a physical problem that we could solve using a Lagrangian or Hamiltonian. ER is an innovative description of nature that is based on a Euclidean metric. I like to call ES the “master reality” because it is “beyond” (prior to) the projections.
Let us compare SR with ER. We consider two identical clocks “r” (red clock) and “b” (blue clock). In SR, “r” shall be “at rest”: It moves only in the c t axis at x 1 = 0 . Clock “b” starts at x 1 = 0 , but it moves in the x 1 axis at a constant speed of v 3 D = 0.6   c . Figure 1 left shows the instant when either clock moved 1.0 s in the coordinate time of “r”. Clock “b” moved 0.6 Ls (light seconds) in x 1 and 0.8 Ls in c t ′ . Thus, “b” displays “0.8”. In ER, no clock is at rest: Figure 1 right shows the instant when either clock moved 1.0 s in cosmic time. Both clocks display “1.0”. Clock “b” moved 0.6 Ls in d 1 and 0.8 Ls in d 4 .
Let “r” (or “b”) be the clock of an observer R (or else B). In the red frame of Figure 1 left, “b” displays t ′ = 0.8   s at the instant when “r” displays t = 1.0   s . In SR, time dilation with respect to “r” occurs in t ′ of B. In the red frame of Figure 1 right, “b” is at d 4 = 0.8   L s at the instant when “r” is at d 4 = 1.0   L s (the same axis d 4 ). In ER, time dilation with respect to “r” occurs in d 4 of R. Thus, “b” is slow with respect to “r” in both SR and ER, but the axis, in which the dilation occurs, is different! Experiments do not disclose this axis.
But why does ER provide a holistic view? Well, the Euclidean metric in Eq. (4) is very special: All coordinates have the same sign. Thus, all four dimensions are treated the same. Thus, describing nature in ER does not depend on which axis is considered proper time. This is why ES diagrams are rotationally symmetric and why they provide a holistic view: The ES diagram in Figure 1 works for R and B at once. In SR, we would have to draw a second Minkowski diagram for B, in which x 1 ′ and c t ′ are orthogonal.
Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time τ . I will not repeat the derivation. The reader is referred to his paper. My task is to turn ER into an accepted theory by solving 15 mysteries. Gersten (2003) showed that the Lorentz transformation is an SO(4) rotation in a “mixed space” x 1 ,   x 2 ,   x 3 ,   c t ′ , where c t ′ is the only primed coordinate. A “mixed space” is physical nonsense. It is another hint that SR has an issue. A Lorentz transformation rotates mixed x 1 ,   x 2 ,   x 3 ,   c t ′ to x 1 ′ ,   x 2 ′ ,   x 3 ′ ,   c t . In ER, unmixed d 1 ′ ,   d 2 ′ ,   d 3 ′ ,   d 4 ′ rotate with respect to d 1 ,   d 2 ,   d 3 ,   d 4 (see Sect. 4).
There is also a big difference in the synchronization of clocks: In SR, each observer is able to synchronize a uniformly moving clock to his clock (same value of c t in Figure 1 left). If he does, the two clocks are not synchronized from the perspective of the moving clock. In ER, clocks with the same 4D vector τ are always synchronized, while clocks with different τ and τ ' are never synchronized (different values of d 4 in Figure 1 right).

4. Geometric Effects in 4D Euclidean Spacetime

We consider two identical rockets “r” (red rocket) and “b” (blue rocket). Let observer R (or B) be in the rear end of rocket “r” (or else “b”). We use 3D space and proper space as synonyms. The 3D space of R (or B) is spanned by d 1 ,   d 2 ,   d 3 (or else d 1 ′ ,   d 2 ′ ,   d 3 ′ ). The proper time of R (or B) relates to d 4 (or else d 4 ′ ). Both rockets started at a point P and move relative to each other at the constant speed v 3 D . We are free to label the axis of motion in 3D space. We label it d 1 (or d 1 ′ ). The ES diagrams in Figure 2 must fulfill my two postulates and the initial condition (same point P). We accomplish this by rotating the red and the blue frame with respect to each other. Figure 2 bottom shows the projection to the 3D space of R (or B). We draw 2D rockets but are aware that their width is in d 2 ,   d 3 (or d 2 ′ ,   d 3 ′ ).
We now verify: (1) The fact that the red and the blue frame are rotated with respect to each other causes length contraction. (2) The fact that proper time flows in different 4D directions for R and for B causes time dilation. Let L i , j be the length of rocket i for observer j . In a first step, we project the blue rocket in Figure 2 top left to the d 1 axis.
sin 2 ⁡ φ + cos 2 ⁡ φ   =   ( L b , R / L b , B ) 2 + ( v 3 D / c ) 2   =   1 ,
L b , R   =   γ − 1   L b , B   ( length   contraction ) ,
where γ = ( 1 − v 3 D 2 / c 2 ) − 0.5 is the same Lorentz factor as in SR. For R, rocket “b” contracts by the factor γ − 1 . Which distances will R observe in his d 4 axis? We mentally continue the rotation of “b” in Figure 2 top left until it points vertically down and serves as R’s ruler in the d 4 axis. In the projection to the 3D space of R, this ruler contracts to zero: The d 4 axis disappears for R because of length contraction at the speed c .
In a second step, we project the blue rocket in Figure 2 top left to the d 4 axis.
sin 2 ⁡ φ + cos 2 ⁡ φ   =   ( d 4 , B / d 4 , B ′ ) 2 + ( v 3 D / c ) 2   =   1 ,
d 4 , B   =   γ − 1   d 4 , B ′ ,
where d 4 , B (or d 4 , B ′ ) is the distance that B moved in d 4 (or else d 4 ′ ). With d 4 , B ′ = d 4 , R (R and B cover the same distance in ES but in different directions), we calculate
d 4 , R   =   γ   d 4 , B     ( time   dilation )
where d 4 , R is the distance that R moved in d 4 . Eqs. (9) and (12) tell us: SR works so well because γ is recovered if we project ES to d 1 and to d 4 . This is not a surprise. Weyl (1928) showed that the Lorentz group is generated by 4D rotations.
To understand how an acceleration manifests itself in ES, we return to our two clocks “r” and “b”. We assume that “r” and Earth move in the d 4 axis of “r” at the speed c and that “b” accelerates in the d 1 axis of “r” toward Earth (Figure 3). Because of Eq. (7), the speed u 1 , b of “b” in d 1 increases at the expense of its speed u 4 , b in d 4 .
Gravitational waves support the idea of GR that gravitation is a feature of spacetime (Abbott et al. 2016). Like classical physics, ER takes gravitation as a force that has not yet been unified with the other forces. I claim that curved geodesics in flat ES replace curved spacetime in GR. To support my claim, I now show in ER: Clock “b” is slow with respect to “r” if “b” experiences the gravitational force of a mass M —whether or not “b” is in free fall toward M . It is the same equivalence that motivated Einstein to formulate GR. Let the mass M be Earth. Initially, “r” and “b” move in d 4 far away from Earth. Eventually, “b” is sent in free fall toward Earth in d 1 (Figure 3). The kinetic energy of “b” in d 1 is
1 2 m u 1 , b 2   =   G M m / r ,
where m is the mass of “b”, G is the gravitational constant, and r is the distance of clock “b” to Earth’s center. By applying Eq. (7), we obtain
u 4 , b 2   =   c 2 − u 1 , b 2   =   c 2 − 2 G M / r .
With u 4 , b = d d 4 , b / d t (“b” moves in the d 4 axis at the speed u 4 , b ) and c = d d 4 , r / d t (“r” moves in the d 4 axis at the speed c ), we calculate
d d 4 , b 2   =   ( c 2 − 2 G M / r )   ( d d 4 , r / c ) 2 ,
d d 4 , r   =   γ gr   d d 4 , b   ( gravitational   time   dilation ) ,
where γ gr = ( 1 − 2 G M / ( r c 2 ) ) − 0.5 is the same dilation factor as in GR. Since γ gr does not depend on u 1 , b , clock “b” is slow with respect to “r”—whether or not “b” is in free fall toward Earth. In ER, gravitational time dilation stems from projecting curved geodesics to the d 4 axis of an observer. Eq. (16) tells us: GR works so well because γ g r is recovered if we project ES to d 4 . Since γ and γ gr are recovered, the Hafele–Keating experiment (1972) also supports ER. GPS satellites work in ER as well as in GR! In SR/ER, a moving clock is slow with respect to an observer. In GR/ER, a clock in a stronger gravitational field is slow with respect to an observer. In SR/GR, an observed clock is slow in its own flow of proper time. In ER, an observed clock is slow in the observer’s flow of proper time.
Three instructive problems demonstrate how to draw and how to read ES diagrams correctly (Figure 4). Problem 1: In billiards, the blue ball is hit toward the red ball. In ES, both balls move at the speed c . We assume that the red ball moves in its d 4 axis. As the blue ball covers distance in d 1 , its speed in d 4 must be less than c . How can the two balls ever collide if their d 4 values do not match? Problem 2: Some rocket moves along a guide wire. In ES, rocket and wire move at the speed c . We assume that the wire moves in its d 4 axis. As the rocket covers distance in d 1 , its speed in d 4 must be less than c . Doesn’t the wire escape from the rocket? Problem 3: Earth orbits the sun. In ES, they both move at the speed c . We assume that the sun moves in its d 4 axis. As Earth covers distance in d 1 and d 2 , its speed in d 4 must be less than c . Doesn’t the sun escape from the orbital plane?
The questions in the last paragraph only seem to disclose geometric paradoxes in ER. The fallacy lies in the assumption that all four axes of ES would be spatial. According to Eq. (5), d 4 relates to proper time. In ES, each object moves in the direction of its own flow of proper time. Thus, an object moving in d 4 ′ covers less distance in d 4 . In Figure 4, we solve all problems by projecting ES to the 3D space of the object that moves in d 4 at the speed c . In its 3D space, it is at rest. We see the solutions in the ES diagrams, too, if we read them correctly: For instance, the two balls “r” and “b” in the left ES diagram collide if d i , r = d i , b ( i = 1 ,   2 ,   3 ) and if the same proper time (!) has elapsed for both balls ( d 4 , r = d 4 , b ′ ). We must take into account that proper time flows in a unique direction for each ball.

5. Solving 15 Fundamental Mysteries of Physics

We recall: (1) An observer’s reality is created by projecting ES orthogonally to his proper space and to his proper time. (2) There is a unique 4D vector τ for each object. (3) Cosmic time t is the correct parameter for a holistic view. In Sects. 5.1 through 5.15, ER solves 15 mysteries and declares five concepts of today’s physics obsolete.

5.1. Solving the Mystery of Time

Proper time τ is what clocks measure ( d 4 divided by c ). Cosmic time t is the total distance covered in ES divided by c . For each clock, its own proper time is always equal to cosmic time. An observed clock is slow in the observer’s flow of proper time τ .

5.2. Solving the Mystery of Time’s Arrow

The arrow of time is a synonym for “time moving only forward”. The arrow of cosmic time emerges from the fact that the total distance covered in ES is steadily increasing.

5.3. Solving the Mystery of the Factor c 2 in m c 2

In SR, if forces are absent, the total energy E of an object is given by
E   =   γ m c 2   =   E kin , 3 D + m c 2 ,
where E kin , 3 D is its kinetic energy in an observer’s 3D space and m c 2 is its energy at rest. SR does not tell us why there is a factor c 2 in the energy of objects that in SR do not move at the speed c . ER provides the missing clue: The object is never at rest, but it moves in its d 4 ′ axis. From the object’s perspective, E kin , 3 D is zero and m c 2 is its kinetic energy in d 4 ′ . The factor c 2 is a hint that it moves through ES at the speed c . In SR, there is also
E 2   =   p 2   c 2   =   p 3 D 2   c 2 + m 2   c 4 ,
where p is the total momentum of an object and p 3 D is its momentum in an observer’s 3D space. Again, ER is eye-opening: From the object’s perspective, p 3 D is zero and m c is its momentum in d 4 ′ . The factor c is a hint that it moves through ES at the speed c .

5.4. Solving the Mystery of Length Contraction and Time Dilation

In SR, length contraction and time dilation can be derived from the Lorentz transformation, but their physical cause remains in the dark. ER discloses that length contraction and time dilation stem from projecting ES to an observer’s reality.

5.5. Solving the Mystery of Gravitational Time Dilation

In GR, gravitational time dilation stems from a curved spacetime. ER discloses that gravitational time dilation stems from projecting curved geodesics in flat ES to the d 4 axis of an observer. Eq. (7) tells us: If an object accelerates in his proper space, it automatically decelerates in his proper time. Curved geodesics in flat ES replace curved spacetime in GR. More studies will be necessary to understand other gravitational effects in ER.

5.6. Solving the Mystery of the Cosmic Microwave Background

In Sects. 5.6 through 5.11, I outline an ER-based model of cosmology. Distances are just like numbers. In particular, they are not inflating/expanding. For some reason, there was a Big Bang. In the inflationary Lambda-CDM model based on GR, the Big Bang occurred “everywhere” because space inflated from a singularity. In the ER-based model, the Big Bang can be localized: It injected a huge amount of energy into ES all at once at an origin O, the only natural reference point. The Big Bang occurred at the cosmic time t = 0 . It was a singularity in terms of providing energy and radial momentum. Initially, all this energy receded radially from O at the speed c . Because of forces and spontaneous effects, some energy departed from its radial motion while maintaining the speed c . Today, all energy is confined to a 4D hypersphere. A lot of energy is confined to its expanding 3D hypersurface. An observer experiences only three axes of the 4D hypersphere as spatial.
Shortly after the Big Bang, energy was highly concentrated in ES. In the projection to any 3D space, a very hot and dense plasma was created. While the plasma was expanding, it cooled down. Cosmic recombination radiation (CRR) was emitted that we still observe as cosmic microwave background (CMB) today (Penzias and Wilson 1965). At temperatures of 3,000 K, hydrogen atoms formed. The universe became increasingly transparent for the CRR. In the Lambda-CDM model, this stage was reached about 380,000 years “after” the Big Bang. In the ER-based model, these are 380,000 light years “away from” the Big Bang. The number needs to be recalculated if there was no cosmic inflation.
In the ES diagrams shown in Figure 5, Earth moves vertically at the speed c . The ER-based model must be able to answer these questions: (1) Why do we still observe the CMB today? (2) Why is the CMB nearly isotropic? (3) Why is the temperature of the CMB very low? Here are some possible answers: (1) The CRR has been scattered multiple times in d 1 ,   d 2 ,   d 3 . Some of the scattered CRR reaches an observer on Earth as CMB (in the projection to his 3D space) after having covered the same distance in d 1 ,   d 2 ,   d 3 as Earth in d 4 . The cross section for scattering is low, but the fluence of the CRR is high. (2) The CMB is nearly isotropic because the CRR was created and scattered equally in d 1 , d 2 , d 3 . (3) The temperature of the CMB is very low because the plasma particles had a very high recession speed v 3 D (see Sect. 5.7) shortly after the Big Bang.

5.7. Solving the Mystery of the Hubble–Lemaître law

According to my first postulate, all celestial bodies move through ES at the speed c . Let v 3 D be the 3D speed at which a galaxy G recedes from Earth in 3D space. Figure 5 left tells us: At the cosmic time t (the time elapsed since the Big Bang), v 3 D relates to the 3D distance D of G to Earth as c relates to the radius r of the 4D hypersphere.
v 3 D   =   D c / r   =   H t   D ,
where H t = c / r = 1 / t is the Hubble parameter. If we observe G today at the cosmic time t = t 0 , the recession speed v 3 D and c remain unchanged. Thus, Eq. (19) turns into
v 3 D   =   D 0   c / r 0   =   H 0   D 0 ,
where D 0 is today’s 3D distance of G to Earth, r 0 is today’s radius of the 4D hypersphere, and H 0 = c / r 0 = 1 / t 0 is the Hubble constant. Eq. (20) is the Hubble–Lemaître law (Hubble 1929; Lemaître 1927): The farther a galaxy is, the faster it recedes from Earth. Cosmologists are aware that H t is a parameter. They are not yet aware of the 4D Euclidean geometry shown in Figure 5 left. Only ER tells us that Eqs. (19) and (20) stem from this simple geometry and that we must consider D 0 = r 0   D / r in Eq. (20) rather than D .

5.8. Solving the Mystery of the Flat Universe

For each observer, ES is projected orthogonally to his proper space and to his proper time. Thus, he experiences two seemingly discrete structures: flat 3D space and time.

5.9. Solving the Mystery of Cosmic Inflation

Most cosmologists believe that an inflation of space shortly after the Big Bang (Linde 1990; Guth 1997) would explain the isotropic CMB, the flatness of the universe, and large-scale structures (inflated from quantum fluctuations). I just showed that ER explains the first two effects. ER also explains the third effect if the impacts of the quantum fluctuations have been expanding at the speed c . In ER, cosmic inflation is an obsolete concept.

5.10. Solving the Mystery of the Hubble Tension

There are various methods for calculating H 0 . I explain why the calculated values do not match (also known as the “Hubble tension”). I compare CMB measurements (Planck space telescope) with distance ladder measurements (Hubble space telescope). According to team A (Aghanim et al. 2020), there is H 0 = 67.66 ± 0.42   k m / s / M p c . According to team B (Riess et al. 2018), there is H 0 = 73.52 ± 1.62   k m / s / M p c . Team B made efforts to minimize the error margins in the distance measurements, but assuming a wrong cause of the redshifts gives rise to a systematic error in team B’s calculation of H 0 .
Let us assume that team A’s value of H 0 is correct. We simulate the supernova of a star S that occurred at a distance of D = 400   M p c from Earth (Figure 5 right). The recession speed v 3 D of S is calculated from measured redshifts. The redshift parameter z = Δ λ / λ tells us how each wavelength λ of the supernova’s light is either passively stretched by an expanding space (team B)—or else how each wavelength λ is redshifted by the Doppler effect of actively receding objects (ER-based model). The supernova occurred at the cosmic time t (arc called “past”), but we observe the supernova at the cosmic time t 0 (arc called “present”). While the supernova’s light was moving the distance D in the d 1 axis, Earth moved the same distance D but in the d 4 axis (first postulate). Team B receives redshift data from a cosmic time t < t 0 when there was r < r 0 and H t > H 0 . There is
1 / H t   =   r / c   =   ( r 0 − D ) / c   =   1 / H 0 − D / c .
For a very short distance of D = 400   k p c , Eq. (21) tells us that H t deviates from H 0 by just 0.009 percent. However, when plotting v 3 D versus D for distances from 0 Mpc to 500 Mpc in steps of 25 Mpc (red points in Figure 6), the slope of a straight-line fit through the origin is roughly 10 percent greater than H 0 . Since team B calculates H 0 from similar plots (magnitude versus z ), its value of H 0 is roughly 10 percent too high. This solves the Hubble tension. Team B’s value is not correct because, according to Eq. (20), we must not plot v 3 D versus D . We must plot v 3 D versus D 0 (blue points in Figure 6) to get a straight line.
Since we are not able to measure D 0 (observable magnitudes relate to D rather than to D 0 ), the easiest way to fix the calculation of team B is to rewrite Eq. (20) as
v 3 D , 0   =   D   c / r 0   =   H 0   D ,
where v 3 D , 0 is today’s 3D speed of another star S 0 (Figure 5 right) that happens to be at the same distance D today at which the supernova of star S occurred. I kindly ask team B to recalculate H 0 after converting all v 3 D to v 3 D , 0 . Eq. (21) tells us how to do so.
H t   =   H 0   c   /   ( c − H 0   D )   =   H 0   /   ( 1 − v 3 D , 0 / c ) ,
v 3 D , 0   =   v 3 D   /   ( 1 + v 3 D / c )
By applying Eq. (24), all red points in Figure 6 drop down to the points marked in blue. Of course, team B is well aware that the supernova’s light was emitted in the past, but all that counts in the Lambda-CDM model is the timespan during which the light is moving to Earth. Along the way, each wavelength is continuously stretched by expanding space. The parameter z increases during the journey. In the ER-based model, all that counts is the moment when the supernova occurred. Each wavelength is initially redshifted by the Doppler effect. The parameter z remains constant during the journey. It is tied up when the supernova occurs. Space is not expanding. A 3D hypersurface made up of energy (!) is expanding in ES. In ER, expansion of space is an obsolete concept.

5.11. Solving the Mystery of Dark Energy

Team B can fix the systematic error in its calculation of H 0 by converting all v 3 D to v 3 D , 0 according to Eq. (24). I now reveal another systematic error, but it is inherent in the Lambda-CDM model itself. It stems from assuming an accelerating expansion of space. It can be fixed only by replacing GR with ER unless we insist on the existence of dark energy. Perlmutter et al. (1998) and Riess et al. (1998) advocate an accelerating expansion of space because the calculated recession speeds deviate from Eq. (20) and the deviations increase with distance. An acceleration would stretch each wavelength even further.
In ER, these deviations are much easier to understand: The older the redshift data are, the more H t deviates from H 0 , and the more v 3 D deviates from v 3 D , 0 . If another star S 0 (Figure 5 right) happens to be at the same distance of D = 400   M p c today at which the supernova of star S occurred, Eq. (24) tells us that S 0 recedes more slowly (27,064 km/s) from Earth than S (29,750 km/s). As long as cosmologists are not aware of the 4D Euclidean geometry, they attribute the deviations from Eq. (20) to an accelerating expansion of space caused by dark energy, but dark energy has never been observed. It is a stopgap for an effect that the Lambda-CDM model cannot explain.
For D > 500   M p c , the data marked red in Figure 6 run away from the straight line. The Hubble tension and dark energy are solved with the same clue: In Eq. (20), we must not confuse D 0 with D . Because of Eq. (19) and H t = c / ( r 0 − D ) , the recession speed v 3 D is not proportional to D but to D / ( r 0 − D ) . The illusion of an accelerating expansion stems from confusing D 0 with D (see Figure 6). Any expansion of space—uniform or else accelerating—is only virtual. There is no accelerating expansion of space even if a Nobel Prize in Physics was given “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae” (The Nobel Foundation 2011). There are two misconceptions in these words of praise: (1) In the Lambda-CDM model, Universe implies space, but space is not expanding. (2) All but the nearest galaxies recede from Earth, but they recede uniformly at the speed v 3 D . In ER, dark energy is an obsolete concept.
Radial momentum provided by the Big Bang drives all galaxies away from the origin O of ES. They are driven by themselves rather than by dark energy. Table 1 compares two models of cosmology. Be aware that “Universe” (uppercase) in the Lambda-CDM model is not the same as “universe” (lowercase) in the ER-based model. In the next two sections, I show that ER is compatible with QM. Since quantum gravity is meant to make GR compatible with QM, I also conclude: In ER, quantum gravity is an obsolete concept.

5.12. Solving the Mystery of the Wave–Particle Duality

The wave–particle duality was first discussed by Bohr and Heisenberg (Heisenberg 1969) and has bothered physicists ever since. Electromagnetic waves are oscillations of an electromagnetic field, which propagate through an observer’s 3D space at the speed c . In some experiments, objects behave like waves. In other experiments, the very same objects behave like particles (also known as the “wave–particle duality”). In today’s physics, one object cannot be wave and particle at once because waves distribute energy in space over time, while the energy of particles is localized in space at a given time.
Up next, we solve the duality. All we need is ER and a generalized concept of energy: All energy is “wavematter”, which may appear as a wave packet or as a particle depending on the perspective. In an observer’s reality (external view, Figure 7), a wavematter may appear as a wave packet or as a particle. As a wave, it propagates in his x 1 axis at the speed c and it oscillates in his axes x 2 (electric field) and x 3 (magnetic field). Propagating and oscillating occur as a function of coordinate time t . In its own reality (internal or in-flight view), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed c . It deems itself particle at rest. Be aware that “wavematter” is not just a new word for the duality. It takes into account that there is an internal view of photons. There is no such view in SR/GR because four dimensions are required that are treated the same.
Like coordinate space and coordinate time, waves and particles are subjective concepts construed by an observer: What I deem wave, deems itself particle at rest. Albert Einstein (1905c) taught that energy is equivalent to mass. The equivalence shows itself in the wave–particle duality: Since each wavematter moves through ES at the speed c , the axis of its 4D motion disappears for itself. From its perspective (that is, in its own reality), all of its energy “condenses” to what we call “mass” in a particle at rest.
In a double-slit experiment, wavematters pass through a double-slit and produce an interference pattern on a screen. An observer deems them wave packets as long as he does not track through which slit each wavematter is passing. Here the external view applies. The photoelectric effect is different. Of course, one can externally witness how one photon releases an electron from a metal surface. However, the physical effect—do I have enough energy to release an electron?—is all up to the photon. Only if the photon energy exceeds the binding energy of an electron is this electron released. Here the photon’s internal view applies. This is why it behaves like a particle.
The duality is also observed in matter, such as electrons ((Jönsson 1961). An electron is a wavematter, too. If the electron is not tracked, it behaves like a wave. If the electron is tracked, it behaves like a particle. Since an observer automatically tracks objects that are slow in his 3D space, he deems all slow objects—and thus all macroscopic objects—matter rather than waves. To improve readability, I do not draw wavematters in my ES diagrams. I draw what they are deemed by an observer: clocks, rockets, celestial bodies, etc.

5.13. Solving the Mystery of Non-Locality

The term “entanglement” was coined by Schrödinger (1935) in his comment on the Einstein–Podolsky–Rosen paradox (Einstein et al. 1935). These three authors argued that QM would not provide a complete description of reality. Schrödinger’s word creation did not solve the paradox but demonstrates our difficulties in comprehending QM. Bell (1964) showed that local hidden-variable theories are not compatible with QM. In experiments (Freedman and Clauser 1972; Aspect et al. 1982; Bouwmeester et al. 1997), entanglement violates locality. Ever since, entanglement has been considered a non-local effect.
Up next, we untangle entanglement without the concept of non-locality. All we need is ER: Four dimensions that are treated the same make non-locality obsolete. Figure 8 illustrates two wavematters that were created at once at a point P. They move away from each other in opposite directions ± d 4 ′ at the speed c . It turns out that these wavematters are automatically entangled. For an observer moving in any direction other than ± d 4 ′ (external view), the two wavematters are spatially separated objects. The observer cannot understand how they are able to communicate with each other in no time.
For each wavematter (internal view), the ± d 4 ′ axis disappears because of length contraction at the speed c . In their common (!) proper space spanned by d 1 ′ ,   d 2 ′ ,   d 3 ′ , either of them is at the same position as its twin. From the internal view, the twins have never been separated, but their proper time flows in opposite 4D directions. The twins communicate with each other in no time because they remain spatially united in their proper space. There is a “spooky action at a distance” from the external view only. Entanglement occurs because an observer’s proper space may be different from an observed object’s proper space. This is possible only if four dimensions are treated the same. ER also explains the entanglement of electrons or atoms. In an observer’s proper space, they move at a speed v 3 D < c . In their ± d 4 ′ axis, they move at the speed c . Any measurement tilts the axis of 4D motion of one twin and destroys the entanglement. In ER, non-locality is an obsolete concept.

5.14. Solving the Mystery of Spontaneous Effects

In spontaneous emission, a photon is emitted by an excited atom. Prior to the emission, the photon energy moves with the atom. After the emission, this energy moves by itself. Today’s physics cannot explain how this energy is boosted to the speed c in no time. In ES, both atom and photon move at the speed c . Thus, there is no need to boost any energy to the speed c . All it takes is energy whose 4D motion at the speed c flips spontaneously into an observer’s 3D space. In absorption, a photon is spontaneously absorbed by an atom. Today’s physics cannot explain how this energy is slowed down to the atom’s speed in no time. In ES, both photon and atom move at the speed c . Thus, there is no need to slow down any energy. Similar arguments apply to pair production and annihilation. Spontaneous effects are another clue that all energy moves through ES at the speed c .

5.15. Solving the Mystery of the Baryon Asymmetry

In the Lambda-CDM model, almost all matter was created shortly after the Big Bang. Only then was the temperature high enough to enable pair production. However, baryons and antibaryons should have annihilated each other again because the energy density was also very high. Since we observe more baryons than antibaryons today (also known as the “baryon asymmetry”), it is assumed that more baryons were created shortly after the Big Bang (Canetti et al. 2012). However, pair production should create baryons and antibaryons equally. Right here, the ER-based model scores again: Since each wavematter deems itself particle at rest, the Big Bang injected a huge number of particles into ES. The baryon asymmetry was caused by the Big Bang and is not affected by pair production.
But why do wavematters not deem themselves antiparticles? Well, antiparticles are not the opposite of particles. Antiparticles have the opposite electric charge and seem to flow backward in time because proper time flows in opposite 4D directions for any two wavematters created in pair production. There is a reasonable character paradox: What I deem antiparticle, deems itself particle. According to ER, any two wavematters created in pair production should be entangled. This is one option of how to falsify ER.

6. Conclusions

ER solves mysteries that have not been solved in 100+ years or that have been solved but with concepts that are obsolete in ER: cosmic inflation, expansion of space, dark energy, quantum gravity, non-locality. Today’s physics needs these concepts to make cosmology and QM work, but Occam’s razor shaves them off. I advise physics to let go of all obsolete concepts by accepting ER. I demonstrated that waves and particles are subjective concepts construed by an observer. A master reality with wavematters (described by ER) is beyond each observer’s reality with waves and particles (described by SR/GR). Electrodynamics is a theory of an observer’s reality. It is not a theory of the master reality.
Unfortunately, most physicists consider SR/GR two of the greatest achievements of physics just because they have been confirmed many times over. I showed that SR/GR do not provide a holistic view, and I suspect that the stagnation in today’s physics is due to this constraint. Physics got stuck in its own concepts. 15 solved mysteries tell us that there is a lot more physics beyond SR/GR. It is very unlikely that 15 solutions in various (!) fields of physics are just 15 coincidences. Only in 4D Euclidean spacetime does Mother Nature disclose her secrets. If we think of each observer’s reality as an oversized stage, the key to understanding cosmology and QM is beyond the stage curtain.
It was a wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect (Einstein 1905a) and not for SR/GR. ER penetrates to a deeper level. Einstein—one of the most brilliant physicists ever—failed to realize that the fundamental metric chosen by Mother Nature is Euclidean. Nature chose a simple but beautiful setting for life: 4D Euclidean spacetime. Einstein sacrificed absolute space and time. I sacrifice the absoluteness of waves and particles, but I restore absolute time (cosmic time is absolute, while the 4D vector “flow of proper time” is relative). Space remains relative (an observer experiences three out of four dimensions as his proper space/universe). For the first time, mankind understands the nature of time: Cosmic time is the total distance covered in ES divided by c . The human brain is able to imagine that we move through 4D Euclidean spacetime at the speed of light. With that said, conflicts of mankind become all so small.
Is ER a physical or a metaphysical theory? This is a very good question because only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. Physics is the science of describing the universe and its interior. Our primary source of knowledge is observing, but an observer has access to just one active perspective that gives rise to mysteries. ER tells us: If we limit physics to observing (to just one active perspective), we cannot solve the Hubble tension—and we solve other mysteries only by adding obsolete concepts. ER describes nature from all perspectives at once. By taking the past and the present into account at once, ER solves the Hubble tension and dark energy. By taking the perspectives of all objects into account at once, ER solves non-locality. ER is a physical theory because it helps us understand what we observe.
Final remarks: (1) I addressed gravitation only briefly, but I ask you once more to be patient and fair. We should not reject ER just because gravitational effects are not yet fully understood. It is promising that ER predicts the same gravitational lensing and the same perihelion precession of Mercury’s orbit as GR (Montanus 2023). (2) To cherish the beauty of ER, we must give ourselves a push and accept that an observer’s reality is a projection. We must not ask in physics: Why is it a projection? Nor must we ask: Why is it a probability function? In my opinion, an inflating or expanding space is at least as speculative as a projection. (3) It looks like Plato was right with his Allegory of the Cave (see Politeia, 514a): Mankind experiences a projection that is blurred—because of QM.
It is not by chance that the author of this paper is an experimental physicist. It seems to me that SR and GR are not suspicious to theorists. Several prominent theorists told me that ER would be nonsense. I laid the groundwork for ER and showed how powerful it is. Paradoxes are only virtual. The true pillars of physics are ER and QM. Together, they describe the very large and the very small. Requesting a holistic view of nature is what I consider the most innovative part of this paper. Now everyone is welcome to solve even more mysteries in ER. May ER get the broad acceptance that it deserves!

Funding

No funds, grants, or other support was received.

Comments

It takes open-minded, courageous editors and reviewers to evaluate a theory that comes with a paradigm shift. Whoever adheres to established concepts is paralyzing the scientific progress. I did not surrender when my paper was rejected by several journals. Interestingly, I was never given conclusive arguments. Rather, I was asked to try a different journal. Were the editors dazzled by the success of SR/GR? Did they underestimate the benefits of ER? Even friends refused to support me. However, each setback inspired me to work out the benefits of ER even better. Finally, I succeeded in disclosing an issue in SR/GR and in formulating a new theory that is even more general than GR. These comments shall encourage young scientists to stand up for promising ideas, but be aware that opposing the mainstream is exhausting. Here are some statements that I received from top journals: “Unscholarly research.” “Fake science.” “Too simple to be true.” Well, the math of GR is obsolete in ER just as the retrograde loops of planets are obsolete in the heliocentric model. The editor-in-chief of a top journal replied: “Publishing is for experts only.” arXiv suspended my submission privileges. Simple and true are not mutually exclusive. Beauty is when they go hand in hand.

Acknowledgments

I would like to thank Siegfried W. Stein for his contributions to Sect. 5.10 and for the Figs. 2 and 5 (partly). After several unsuccessful submissions, he eventually decided to withdraw his co-authorship. I also thank Matthias Bartelmann, Dirk Rischke, Jürgen Struckmeier, and Andreas Wipf for some valuable comments. In particular, I thank all reviewers and editors for the precious time that they spent on my manuscript.

Conflicts of Interest

The author has no competing interests to declare.

Ethical Approval

Not applicable.

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Figure 1. Minkowski diagram and ES diagram for two clocks “r” (red) and “b” (blue). Left: In SR, “b” is slow with respect to “r” in t ′ . Coordinate time is relative (“b” is not at the same positions in c t and c t ′ ). Right: In ER, “b” is slow with respect to “r” in d 4 . Cosmic time is absolute (“r” is in d 4 at the same position as “b” in d 4 ′ ). Only the ES diagram is rotationally symmetric.
Figure 1. Minkowski diagram and ES diagram for two clocks “r” (red) and “b” (blue). Left: In SR, “b” is slow with respect to “r” in t ′ . Coordinate time is relative (“b” is not at the same positions in c t and c t ′ ). Right: In ER, “b” is slow with respect to “r” in d 4 . Cosmic time is absolute (“r” is in d 4 at the same position as “b” in d 4 ′ ). Only the ES diagram is rotationally symmetric.
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Figure 2. ES diagrams and 3D projections for two rockets “r” (red) and “b” (blue). Top: Both rockets move in different 4D directions at the speed c . Bottom left: Projection to the 3D space of R. Rocket “b” contracts to L b , R . Bottom right: Projection to the 3D space of B. Rocket “r” contracts to L r , B .
Figure 2. ES diagrams and 3D projections for two rockets “r” (red) and “b” (blue). Top: Both rockets move in different 4D directions at the speed c . Bottom left: Projection to the 3D space of R. Rocket “b” contracts to L b , R . Bottom right: Projection to the 3D space of B. Rocket “r” contracts to L r , B .
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Figure 3. ES diagram for two clocks “r” (red) and “b” (blue). Clock “r” and Earth move in the d 4 axis of “r” at the speed c . Clock “b” accelerates in the d 1 axis of “r” toward Earth.
Figure 3. ES diagram for two clocks “r” (red) and “b” (blue). Clock “r” and Earth move in the d 4 axis of “r” at the speed c . Clock “b” accelerates in the d 1 axis of “r” toward Earth.
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Figure 4. Solving three instructive problems. Each snapshot shows one instant in cosmic time, which serves as the parameter in ER. The left ES diagram shows ten snapshots at once. Left: The blue ball “b” is hit toward the red ball “r”. In the projection, the two balls collide. Center: Some rocket moves along a guide wire. In the projection, the wire does not escape from the rocket. Right: Earth orbits the sun. In the projection, the sun does not escape from the orbital plane.
Figure 4. Solving three instructive problems. Each snapshot shows one instant in cosmic time, which serves as the parameter in ER. The left ES diagram shows ten snapshots at once. Left: The blue ball “b” is hit toward the red ball “r”. In the projection, the two balls collide. Center: Some rocket moves along a guide wire. In the projection, the wire does not escape from the rocket. Right: Earth orbits the sun. In the projection, the sun does not escape from the orbital plane.
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Figure 5. Solving the mysteries 5.6, 5.7, 5.10, and 5.11. The circular arcs are part of an expanding 3D hypersurface. Left: The galaxy G recedes from Earth at the 3D speed v 3 D . Right: The supernova of a star S occurred at a distance of D = 400   M p c from Earth. If another star S 0 happens to be at the same distance D today, S 0 recedes more slowly from Earth than S .
Figure 5. Solving the mysteries 5.6, 5.7, 5.10, and 5.11. The circular arcs are part of an expanding 3D hypersurface. Left: The galaxy G recedes from Earth at the 3D speed v 3 D . Right: The supernova of a star S occurred at a distance of D = 400   M p c from Earth. If another star S 0 happens to be at the same distance D today, S 0 recedes more slowly from Earth than S .
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Figure 6. Hubble diagram for simulated supernovae at distances up to 1250 Mpc. The horizontal axis is D or else D 0 . Only Eq. (20) yields a straight line. Eq. (19) does not because H t is not a constant.
Figure 6. Hubble diagram for simulated supernovae at distances up to 1250 Mpc. The horizontal axis is D or else D 0 . Only Eq. (20) yields a straight line. Eq. (19) does not because H t is not a constant.
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Figure 7. Illustration of a wavematter. In an observer’s reality (external view, coordinate spacetime!), a wavematter may appear as a wave packet or as a particle. As a wave, it propagates and oscillates as a function of coordinate time. In its own reality (internal view), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed c . It deems itself particle at rest.
Figure 7. Illustration of a wavematter. In an observer’s reality (external view, coordinate spacetime!), a wavematter may appear as a wave packet or as a particle. As a wave, it propagates and oscillates as a function of coordinate time. In its own reality (internal view), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed c . It deems itself particle at rest.
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Figure 8. Two wavematters moving in ± d 4 ′ are spatially separated objects for an observer who moves in any direction other than ± d 4 ′ (external view). For each wavematter (internal view), the ± d 4 ′ axis disappears. In their common proper space, both wavematters remain spatially united.
Figure 8. Two wavematters moving in ± d 4 ′ are spatially separated objects for an observer who moves in any direction other than ± d 4 ′ (external view). For each wavematter (internal view), the ± d 4 ′ axis disappears. In their common proper space, both wavematters remain spatially united.
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Table 1. Comparing two different models of cosmology
Table 1. Comparing two different models of cosmology
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