Preprint

Article

Altmetrics

Downloads

10563

Views

10771

Comments

1

Markolf H. Niemz

Markolf H. Niemz

This version is not peer-reviewed

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.* Any (!) object’s proper space *d*_{1}, *d*_{2}, *d*_{3} and proper time *τ* span natural spacetime, which is 4D Euclidean space (ES) if we interpret *cτ* as *d*_{4}. 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**. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts. We must learn to distinguish between an observer’s reality (described by SR/GR) and the master reality ES (described by ER). ER solves 15 mysteries at once.

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. While it is true that many attempts to falsify SR/GR have failed, we must not reject all attempts. Scientific theories must be falsifiable (Popper, 1935). I finally 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 space.

To sum it all up: Predictions made by SR/GR are correct, but ER penetrates to a deeper level. I apologize for my many preprint versions, but I received almost no support. It was 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 of physics.

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 spacetime because it illustrates the invariance of the spacetime interval very well (Minkowski, 1910). Predicting the lifetime of muons (Rossi & 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.

Newburgh and Phipps (1969) pioneered ER. Montanus (1991) described an “absolute Euclidean spacetime” with a preferred frame of reference, where a pure time interval is a pure time interval for all observers. Montanus (2023) claims that a preferred frame would avoid the twin paradox in ER, collisions of particles at a distance, and a “character paradox” (confusion of photons, particles, and antiparticles). As we will see, a preferred frame is not required. There is no twin paradox in ER, there are no collisions at a distance in the projections from ES, and the character paradox is reasonable. Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time $\tau $. Montanus (2023) even tried to formulate Maxwell’s equations in ER, but he 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 Sect. 3). van Linden (2023) runs a website about various ER models. However, physicists are still opposed to ER because dark energy and non-locality make cosmology and QM work, waves are excluded in ER, and paradoxes may turn up. This paper marks a turning point: I disclose an issue in SR/GR, I justify the exclusion of waves, and 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 ${v}_{3\mathrm{D}}\ll c$. In Einstein’s physics, all energy moves through 4D non-Euclidean spacetime. The speed of matter is ${v}_{3\mathrm{D}}<c$. In ER, all energy moves through ES. The 4D speed of all energy is ${u}_{4\mathrm{D}}=c$. Newton’s physics (Newton, 1687) influenced Kant’s philosophy (Kant, 1781). Will ER reform both physics and philosophy?

In SR (Einstein, 1905b), there are two concepts of time: coordinate time $t$ and proper time $\tau $. 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” ${t}_{\mathrm{P}}$, a light pulse is sent from P to Q. At “Q time” ${t}_{\mathrm{Q}}$, it is reflected. At “P time” ${t}_{\mathrm{P}}^{*}$, it is back at P. The clocks synchronize if

$${t}_{\mathrm{Q}}-{t}_{\mathrm{P}}={t}_{\mathrm{P}}^{*}-{t}_{\mathrm{Q}}$$

In § 3 of SR, Einstein derives the Lorentz transformation. The coordinates ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}t$ of an event in a system K are transformed to the coordinates ${x}_{1}^{\prime},\text{}{x}_{2}^{\prime},\text{}{x}_{3}^{\prime},\text{}{t}^{\prime}$ in K’ by
where K’ moves relative to K in ${x}_{1}$ at a constant speed ${v}_{3\mathrm{D}}$, while $\gamma =(1-{v}_{3\mathrm{D}}^{2}/{c}^{2}{)}^{-0.5}$ 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, 15 fundamental mysteries of physics are solved. Thus, the issue is real. I show that the scope of SR/GR is rather limited. Their concepts of spacetime work well except for cosmology and QM.

$${x}_{1}^{\prime}=\gamma \text{}({x}_{1}-{v}_{3\mathrm{D}}\text{}t),\text{}{x}_{2}^{\prime}={x}_{2}\text{},\text{}{x}_{3}^{\prime}={x}_{3},$$

$${t}^{\prime}=\gamma \text{}(t-{v}_{3\mathrm{D}}\text{}{x}_{1}/{c}^{2}),$$

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?

The indefinite distance function in SR (Einstein, 1905b) is usually written as
where $\mathrm{d}\tau $ is an infinitesimal distance in $\tau $, while $\mathrm{d}t$ and ${\mathrm{d}x}_{i}$ ($i=1,\text{}2,\text{}3$) are infinitesimal distances in coordinate spacetime ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}t$. This spacetime is construed because coordinate space ${x}_{1},\text{}{x}_{2},\text{}{x}_{3}$ and coordinate time $t$ are subjective concepts: They are not immanent in rulers and clocks but defined by an observer. Rulers measure proper distance ${d}_{\mu}$ ($\mu =1,\text{}2,\text{}3,\text{}4$). Clocks measure proper time $\tau $. We may rearrange Eq. (3) and obtain
where ${\mathrm{d}d}_{i}={\mathrm{d}x}_{i}$ ($i=1,\text{}2,\text{}3$) and ${\mathrm{d}d}_{4}=c\text{}\mathrm{d}\tau $ are infinitesimal distances in ES. The roles of $t$ and $\tau $ are switched: The fourth coordinate in ER is an object’s proper time $\tau $ (measured by itself) multiplied by $c$. The new invariant is cosmic time $t$. I retain the symbol $t$ to stress the equivalence of Eqs. (3) and (4). The indices 1 to 4 point out the full symmetry. Any (!) object’s proper space ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ and proper time $\tau $ span natural spacetime, which is ES if we interpret $c\tau $ as ${d}_{4}$. This spacetime is natural because all ${d}_{\mu}$ ($\mu =1,\text{}2,\text{}3,\text{}4$) are objective concepts: They are immanent in rulers and clocks. We must not confuse ER with a Wick rotation (Wick, 1954), which replaces $t$ with $it$ and keeps $\tau $ invariant.

$${{c}^{2}\text{}\mathrm{d}\tau}^{2}={{c}^{2}\text{}\mathrm{d}t}^{2}-\mathrm{d}{x}_{1}^{2}-\mathrm{d}{x}_{2}^{2}-\mathrm{d}{x}_{3}^{2},$$

$${{c}^{2}\text{}\mathrm{d}t}^{2}=\mathrm{d}{d}_{1}^{2}+\mathrm{d}{d}_{2}^{2}+\mathrm{d}{d}_{3}^{2}+\mathrm{d}{d}_{4}^{2},$$

“ES diagrams” show ES from an object’s perspective. For each object, we are free to label the four axes of ES. We always take ${d}_{4}$ as the axis in which the object itself moves at the speed $c$. During its lifetime, the object keeps moving in ${d}_{4}$ (always drawn vertically). An “object’s reality” is created by projecting ES orthogonally to its proper space and to its proper time. For any two objects, $\tau $ and $\tau \prime $ may flow in different 4D directions.
where $\mathit{\tau}$ is the 4D vector “flow of proper time” of an object and $\mathit{u}$ is its 4D velocity. For all objects, there is ${u}_{\mu}=\mathrm{d}{d}_{\mu}/\mathrm{d}t$ (cosmic time $t$). Thus, Eq. (4) matches my first postulate

$$\tau ={d}_{4}/c,\text{}\tau \prime ={d}_{4}^{\prime}/c,$$

$$\mathit{\tau}={d}_{4}\text{}\mathit{u}/{c}^{2},\text{}\mathit{\tau}\mathit{\text{'}}={d}_{4}^{\prime}\text{}\mathit{u}\mathit{\text{'}}/{c}^{2},$$

$${u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}+{u}_{4}^{2}={c}^{2}.$$

My second postulate revises the principle of relativity, and it defines an observer’s reality: It 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 $\tau $, in particular, cannot factor in an object’s 4D vector $\mathit{\tau}$. Since replacing coordinate time with cosmic time is a discontinuous operation, there is no continuous transition between SR/GR and ER. We take an object’s ${d}_{1}\left(t\right),\text{}{d}_{2}\left(t\right),\text{}{d}_{3}\left(t\right),\text{}{d}_{4}\left(t\right)$ for granted rather than an observer’s ${x}_{1}\left(\tau \right),\text{}{x}_{2}\left(\tau \right),\text{}{x}_{3}\left(\tau \right),t\left(\tau \right)$.

Since ES is “beyond” (prior to) projecting, I call it the “master reality” (master of each observer’s reality). Spacetime in SR/GR is relative. ES is absolute. All ES diagrams and the projections are relative. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts (see Sect. 5.12). We must learn to distinguish between an observer’s reality with waves and particles (described by SR/GR) and the master reality ES with wavematters (described by ER).

It is instructive to contrast the three concepts of time. Coordinate time $t$ is a subjective measure of time: It is equal to $\tau =\left|\mathit{\tau}\right|$ for the observer only. Proper time $\tau $ 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 invariant and thus absolute, there is no twin paradox in ER. Twins share the same age in cosmic time. In ER, time is a subordinate quantity: Only by covering distance is time passing by. I suggest that we define a standard unit for speed and that we measure time in compound units.

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 $ct$ 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 ${v}_{3\mathrm{D}}=0.6\text{}c$. 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 $ct\prime $. 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 observer R (or B) now be with clock “r” (or else “b”). In the blue frame of Figure 1 left, “b” displays ${t}^{\prime}=1.0\text{}\mathrm{s}$ at the instant when “r” displays $t=0.8\text{}\mathrm{s}$ (dashed line). In the red frame of Figure 1 left, “b” displays ${t}^{\prime}=0.8\text{}\mathrm{s}$ at the instant when “r” displays $t=1.0\text{}\mathrm{s}$ (solid line). In SR, time dilation with respect to “r” thus occurs in ${t}^{\prime}$ of B. In the red frame of Figure 1 right, “b” is at ${d}_{4}=0.8\text{}\mathrm{L}\mathrm{s}$ at the instant when “r” is at ${d}_{4}=1.0\text{}\mathrm{L}\mathrm{s}$ (same axis ${d}_{4}$). In ER, time dilation with respect to “r” thus occurs in ${d}_{4}$ of R. In both SR and ER, “b” is slow with respect to “r”. However, ${t}^{\prime}=0.8\text{}\mathrm{s}$ is calculated only (B measures time in $\tau \prime $), while ${d}_{4}=0.8\text{}\mathrm{L}\mathrm{s}$ is measurable (${d}_{4}$ relates to $\tau $). Rotate either graph in Figure 1 to see that only ER provides a holistic view: The ES diagram lives up to R and B at once. A new Minkowski diagram is required for B, where ${x}_{1}^{\prime}$ and $c{t}^{\prime}$ are orthogonal.

Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time $\tau $. 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” ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}{ct}^{\prime}$, where ${ct}^{\prime}$ 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 ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}{ct}^{\prime}$ to ${x}_{1}^{\prime},\text{}{x}_{2}^{\prime},\text{}{x}_{3}^{\prime},\text{}ct$. In ER, unmixed ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime},\text{}{d}_{4}^{\prime}$ rotate with respect to ${d}_{1},\text{}{d}_{2},\text{}{d}_{3},\text{}{d}_{4}$ (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 $t$ 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 $\mathit{\tau}$ are always synchronized, while clocks with different $\mathit{\tau}$ and $\mathit{\tau}\mathit{\text{'}}$ are never synchronized (different values of ${d}_{4}$ in Figure 1 right).

We consider two identical rockets “r” (red rocket) and “b” (blue rocket) and assume that there is an observer R (or B) in the rear end of rocket “r” (or else “b”). His ES diagram is ${d}_{1},\text{}{d}_{2},\text{}{d}_{3},\text{}{d}_{4}$ (or else ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime},\text{}{d}_{4}^{\prime}$). The 3D space of R (or else B) is spanned by ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ (or else ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime}$). We use “3D space” and “proper space” as synonyms. The proper time of R (or else B) relates to ${d}_{4}$ (or else ${d}_{4}^{\prime}$). The rockets started at the same point P and move relative to each other at the constant 3D speed ${v}_{3\mathrm{D}}$. We are free to label the axis of motion in 3D space. Here, it is ${d}_{1}$. The ES diagrams in Figure 2 top must fulfill my two postulates and the initial condition (starting point P). This is achieved by rotating the ES diagrams with respect to each other. Figure 2 bottom shows the projection to the 3D space of R (or else B). The rockets are drawn in 2D although their width is in ${d}_{2}$ or ${d}_{3}$ (${d}_{2}^{\prime}$ or ${d}_{3}^{\prime}$).

We now verify: (1) The fact that the ES diagrams of R and of B 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.
where $\gamma =(1-{v}_{3\mathrm{D}}^{2}/{c}^{2}{)}^{-0.5}$ is the same Lorentz factor as in SR. For R, rocket “b” contracts by the factor ${\gamma}^{-1}$. 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 $c$.

$${\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi =({L}_{\mathrm{b},\mathrm{R}}/{L}_{\mathrm{b},\mathrm{B}}{)}^{2}+({v}_{3\mathrm{D}}/c{)}^{2}=1,$$

$${L}_{\mathrm{b},\mathrm{R}}={\gamma}^{-1}\text{}{L}_{\mathrm{b},\mathrm{B}}\text{(length contraction),}$$

In a second step, we project the blue rocket in Figure 2 top left to the ${d}_{4}$ axis.
where ${d}_{4,\mathrm{B}}$ (or ${d}_{4,\mathrm{B}}^{\prime}$) is the distance that B moved in ${d}_{4}$ (or else ${d}_{4}^{\prime}$). With ${d}_{4,\mathrm{B}}^{\prime}={d}_{4,\mathrm{R}}$ (R and B cover the same distance in ES but in different directions), we calculate
where ${d}_{4,\mathrm{R}}$ is the distance that R moved in ${d}_{4}$. Eqs. (9) and (12) tell us: SR works so well because $\gamma $ is recovered when projecting 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.

$${\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi =({d}_{4,\mathrm{B}}/{d}_{4,\mathrm{B}}^{\prime}{)}^{2}+({v}_{3\mathrm{D}}/c{)}^{2}=1,$$

$${d}_{4,\mathrm{B}}={{\gamma}^{-1}\text{}d}_{4,\mathrm{B}}^{\prime},$$

$${d}_{4,\mathrm{R}}=\gamma \text{}{d}_{4,\mathrm{B}}\text{(time dilation),}$$

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 ${u}_{1,\mathrm{b}}$ of “b” in ${d}_{1}$ increases at the expense of its speed ${u}_{4,\mathrm{b}}$ in ${d}_{4}$.

Gravitational waves support the idea of GR that gravitation is a feature of spacetime (Abbott et al., 2016). However, classical physics considers gravitation a force that has not yet been unified with the other three forces of physics. I claim that curved geodesics in ES replace curved spacetime in GR. To support my claim, we now calculate gravitational time dilation in ES. Let “r” and “b” be two identical clocks far away from Earth. Initially, they move next to each other in the ${d}_{4}$ axis of “r”. At some point, “b” is sent in free fall toward Earth in the ${d}_{1}$ axis of “r” (Figure 3). The kinetic energy of “b” with the mass $m$ is
where $G$ is the gravitational constant, $M$ is the mass of Earth, and $r$ is the distance of clock “b” to Earth’s center. By applying Eq. (7), we obtain

$${\frac{1}{2}mu}_{1,\mathrm{b}}^{2}=GMm/r,$$

$${u}_{4,\mathrm{b}}^{2}={c}^{2}-{u}_{1,\mathrm{b}}^{2}={c}^{2}-2GM/r.$$

With ${u}_{4,\mathrm{b}}={\mathrm{d}d}_{4,\mathrm{b}}/\mathrm{d}t$ (“b” moves in the ${d}_{4}$ axis at the speed ${u}_{4,\mathrm{b}}$) and $c={\mathrm{d}d}_{4,\mathrm{r}}/\mathrm{d}t$ (“r” moves in the ${d}_{4}$ axis at the speed $c$), we calculate
where ${\gamma}_{\mathrm{g}\mathrm{r}}=(1-2GM/(r{c}^{2}{\left)\right)}^{-0.5}$ is the same dilation factor as in GR. It does not depend on ${u}_{1,\mathrm{b}}$. Eq. (16) tells us: GR works so well because ${\gamma}_{\mathrm{g}\mathrm{r}}$ is recovered when projecting ES to ${d}_{4}$. Thus, GPS satellites do their job in ER as well as in GR! When “b” returns to “r”, clock “b” is behind clock “r”. This dilation stems from projecting curved geodesics. In GR, it stems from a curved spacetime. We sum up time dilation: 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 flow of proper time. In ER, an observed clock is slow in the observer’s flow of proper time. Since both $\gamma $ and ${\gamma}_{\mathrm{g}\mathrm{r}}$ are recovered, the experiment by Hafele and Keating (1972) also supports ER.

$$\mathrm{d}{d}_{4,\mathrm{b}}^{2}=({c}^{2}-2GM/r)\text{}(\mathrm{d}{d}_{4,\mathrm{r}}/c{)}^{2},$$

$$\mathrm{d}{d}_{4,\mathrm{r}}={\gamma}_{\mathrm{g}\mathrm{r}}\text{}\mathrm{d}{d}_{4,\mathrm{b}}\text{}\left(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\text{}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\text{}\mathrm{d}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right),$$

Three instructive examples (Figure 4) demonstrate how to project from ES to 3D space. Problem 1: A 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 moves along the wire, its speed in ${d}_{4}$ must be slower than $c$. Wouldn’t the wire eventually be outside the rocket? Problem 2: A mirror passes a rocket. An observer in the rocket tip sends a light pulse to the mirror and tries to detect the reflection. In ES, all objects move at the speed $c$ but in different directions. We assume that the observer moves in his ${d}_{4}$ axis. How can he ever detect the reflection? Problem 3: Earth revolves around the sun. We assume that the sun moves in its ${d}_{4}$ axis. As Earth covers a distance in ${d}_{1}$ and ${d}_{2}$, its speed in ${d}_{4}$ must be slower than $c$. Wouldn’t the sun escape from the orbital plane of Earth?

The questions in the last paragraph seem to imply that there are geometric paradoxes in ER, but there aren’t any. The fallacy in all problems lies in the assumption that all four spatial dimensions would be observable. Just three of them are observable! All problems are solved by projecting ES to 3D space (Figure 4 bottom). These projections tell us what an observer’s reality is like because “suppressing the ${d}_{4}$ axis” is equivalent to “length contraction makes ${d}_{4}$ disappear”. The suppressed axis ${d}_{4}$ is experienced as time. We easily verify in an observer’s 3D space: The guide wire remains within the rocket; the light pulse is reflected back to the observer; the sun remains in the orbital plane of Earth.

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 $\mathit{\tau}$ 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 it declares five concepts of today’s physics obsolete.

Proper time $\tau $ 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 $\mathit{\tau}$.

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

in $m{c}^{2}$In SR, if forces are absent, the total energy $E$ of an object is given by
where ${E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ 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 never move at the speed $c$. ER provides this missing clue: The object is not at rest, but it moves in its ${d}_{4}^{\prime}$ axis. From its own perspective, its ${E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is zero, and $m{c}^{2}$ is its kinetic (!) energy in ${d}_{4}^{\prime}$. The factor ${c}^{2}$ is a hint that it moves through ES at the speed $c$. In SR, there is also
where $p$ is the total momentum of an object and ${p}_{3\mathrm{D}}$ is its momentum in an observer’s 3D space. Again, ER is eye-opening: From the object’s perspective, its ${p}_{3\mathrm{D}}$ is zero, and $mc$ is its momentum in ${d}_{4}^{\prime}$. The factor $c$ is a hint that it moves through ES at the speed $c$. Eqs. (17) and (18) are not valid in ER. In ER, there is $E=m{c}^{2}$ and $p=mc$ for each object.

$$E=\gamma m{c}^{2}={E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}+m{c}^{2},$$

$${E}^{2}={p}^{2}{c}^{2}={p}_{3\mathrm{D}}^{2}{c}^{2}+{m}^{2}{c}^{4},$$

ER discloses that length contraction and time dilation stem from projecting ES to an observer’s reality. 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 gravitational time dilation stems from projecting curved geodesics in flat ES to the ${d}_{4}$ axis of an observer. If an object accelerates in his proper space, it automatically decelerates in his proper time. I am aware that more studies will be necessary to explain other gravitational effects. In GR, gravitational time dilation stems from a curved spacetime. In the next six sections, I show that ER outperforms GR in cosmology.

In this section, I outline an ER-based model of cosmology. There is no need to create ES. Space exists just like numbers. For some reason, there was a Big Bang. In the GR-based Lambda-CDM model, the Big Bang occurred “everywhere” because space inflated from a singularity. In the ER-based model, we can locate the Big Bang: It injected a huge amount of energy into a non-inflating and non-expanding ES all at once at what I call “origin O”, the only natural reference point. The Big Bang occurred at the cosmic time $t=0$ and 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 physical interactions (scattering, transversal acceleration, spontaneous emission), some energy departed from its radial motion while maintaining the speed $c$. Today, all energy is confined to a 4D hypersphere, while a significant amount of energy is confined to its 3D hypersurface.

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 & 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 Figure 5, nature is described from the perspective of Earth (Earth moves vertically). From this perspective, the CRR cannot move in ${d}_{4}$ because it moves in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ 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 so low? Here are some possible answers: (1) The CRR has been scattered multiple times in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ (Figure 5 left shows ${d}_{1}$). Some of the scattered CRR reaches Earth as CMB (in the projection to its 3D space) after having covered the same distance in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ as Earth in ${d}_{4}$. The cross section for scattering is low, but the fluence of the CRR is high. (2) The CRR was created and scattered equally in ${d}_{1},{d}_{2},{d}_{3}$. (3) The very low temperature is due to a very high recession speed (see Sect. 5.10) of all plasma particles.

The speed ${v}_{3\mathrm{D}}$ at which a galaxy G recedes from Earth in 3D space (Figure 5 left) relates to its 3D distance $D$ as $c$ relates to the radius $r$ of the 4D hypersphere.
where ${H}_{t}=c/r=1/t$ is the Hubble parameter and $t$ is the cosmic time elapsed since the Big Bang. If we compare galaxies as they appear to us today at $t={t}_{0}$, Eq. (19) reduces to
where ${v}_{3\mathrm{D},0}$ is today’s recession speed, ${H}_{0}=c/{r}_{0}=1/{t}_{0}$ is the Hubble constant, and ${r}_{0}$ is today’s radius of the 4D hypersphere. 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 not a constant. They are not yet aware of the 4D Euclidean geometry.

$${v}_{3\mathrm{D}}=Dc/r={H}_{t}\text{}D,$$

$${v}_{3\mathrm{D},0}={H}_{0}\text{}D,$$

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

Most cosmologists believe that an inflation of space in the early universe (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 quantum fluctuations have been expanding at the speed $c$. **In ER, cosmic inflation is an obsolete concept.**

There are various methods for calculating ${H}_{0}$. Up next, I explain why the calculated values do not match (also known as the “Hubble tension”). I compare CMB measurements using the Planck space telescope with distance ladder measurements using the Hubble space telescope. Team A (Aghanim et al., 2020) calculates ${H}_{0}=67.66\pm 0.42\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B (Riess et al., 2018) calculates ${H}_{0}=73.52\pm 1.62\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B made efforts to minimize the error margins in the distance measurements, but I show that a misinterpretation of the redshifts causes a systematic error in its calculation of ${H}_{0}$. We assume that team A’s value of ${H}_{0}$ is correct. We simulate the supernova of a star $\mathrm{S}$ that occurred at a distance of $D=400\text{}\mathrm{M}\mathrm{p}\mathrm{c}$ from Earth (Figure 5 right). The measurable redshift parameter
tells us how each wavelength ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ emitted by the supernova is either passively stretched by an expanding space (team B), or how each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is redshifted by the Doppler effect of actively receding objects (ER-based model). In Figure 5 right, there is an arc called “past” when the supernova occurred and an arc called “present” when the supernova’s light arrives on Earth. While this light was moving the distance $D$ in the ${d}_{1}$ axis, Earth moved the same distance $D$ in the ${d}_{4}$ axis (first postulate). Thus, team B receives redshift data from a cosmic time $t<{t}_{0}$ when there was $r<{r}_{0}$ and ${H}_{t}>{H}_{0}$.

$$z=\mathsf{\Delta}\lambda /{\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$$

$$1/{H}_{t}=r/c={({r}_{0}-D)/c=1/H}_{0}-D/c.$$

In our simulated case ($D=400\text{}\mathrm{M}\mathrm{p}\mathrm{c}$), there is ${H}_{t}=74.37\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. For a very short distance of $D=400\text{}\mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (22) tells us that ${H}_{t}$ deviates from ${H}_{0}$ by only 0.009 percent. However, when plotting ${v}_{3\mathrm{D}}$ versus $D$ for long distances (50 Mpc, 100 Mpc, 150 Mpc, ... 450 Mpc), the slope of a fitted straight line is 8 to 9 percent greater than ${H}_{0}$. Since team B is calculating ${H}_{0}$ from the slope of such plots, its value of ${H}_{0}$ is 8 to 9 percent too high. This solves the Hubble tension. I kindly ask team B to recalculate ${H}_{0}$ after converting all ${v}_{3\mathrm{D}}$ of the past to today’s values ${v}_{3\mathrm{D},0}$. Eq. (22) tells us how to do so:

$${H}_{t}={H}_{0}\text{}c\text{}/\text{}(c-{H}_{0}\text{}D)={H}_{0}\text{}/\text{}(1-{v}_{3\mathrm{D},0}/c),$$

$${v}_{3\mathrm{D},0}={v}_{3\mathrm{D}}\text{}/\text{}(1+{v}_{3\mathrm{D}}/c).$$

Of course, team B is well aware of the fact 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 ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is continuously stretched by an expanding space. The redshift parameter $z$ increases during the journey to Earth. That moment when the supernova occurred is irrelevant. In the ER-based model, that very moment is relevant, but the timespan is irrelevant. Each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is initially redshifted at the cosmic time $t$ by the Doppler effect. During the journey to Earth, the redshift parameter $z$ remains constant. This parameter is tied up in a “package” when the supernova occurs and sent to Earth, where it is measured. Space is not expanding. A 3D hypersurface (made of energy!) is receding in 4D space. **In ER, expansion of space is an obsolete concept.**

Team B can fix the systematic error in its calculation of ${H}_{0}$ within the Lambda-CDM model by converting all ${v}_{3\mathrm{D}}$ to ${v}_{3\mathrm{D},0}$ 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 that model with the ER-based model of cosmology unless we postulate a dark energy. Perlmutter et al. (1998) and Riess et al. (1998) favor an accelerating expansion of space because the calculated recession speeds deviate from Eq. (20). These deviations increase with distance, and an accelerating expansion of space would stretch each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ even further.

The ER-based model provides a simpler explanation for the deviations from Eq. (20): We must consider Eq. (19). The older the redshift data are, the more ${H}_{t}$ deviates from ${H}_{0}$, and the more ${v}_{3\mathrm{D}}$ deviates from ${v}_{3\mathrm{D},0}$. If another star ${\mathrm{S}}_{0}$ (small circle in Figure 5 right) is at the same distance (400 Mpc) today as $\mathrm{S}$ was in the past, it recedes more slowly from Earth (27,064 km/s) than $\mathrm{S}$ (29,748 km/s) according to Eq. (24). 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. Turner (1998) coined the term “dark energy” to designate a cause for the accelerating expansion of space, but dark energy has never been observed. Dark energy is a stopgap for an effect that the Lambda-CDM model is not able to explain. Now that we are aware of the 4D Euclidean geometry, the deviations from Eq. (20) can be attributed to deeper pasts. **In ER, dark energy is an obsolete concept.**

Any expansion of space—uniform or accelerating—is only virtual. The illusion of a non-linear expansion stems from Eq. (19): The deeper in the past a supernova occurred, the greater are both ${H}_{t}$ and $D$! There is no accelerating expansion of the Universe even if the 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). Actually, there are two misconceptions in these words of praise: (1) In the Lambda-CDM model, the term “Universe” implies space, but space is not expanding at all. (2) There is receding energy, but it recedes uniformly at the speed $c$.

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.**

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, these objects behave like particles. Up next, I explain how the very same object appears as a wave packet or as a particle depending on the perspective. From an observer’s perspective, it may appear as a wave packet or as a particle. From its own perspective, it is a particle at rest.

Based on the wave–particle duality, I suggest that we introduce a generalized concept of energy: All energy is “wavematter”, which may appear as wave packets or as particles in an observer’s reality (Figure 6). In my reality (external view, coordinate spacetime!), such a wavematter may appear as a wave packet or as a particle. As a wave, it propagates in my ${x}_{1}$ axis at the speed $c$, and it oscillates in my axes ${x}_{2}$ (electric field) and ${x}_{3}$ (magnetic field). Both propagating and oscillating occur as a function of coordinate time $t$. In its own reality (internal view, “in-flight view”), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed $c$. Thus, it deems itself particle at rest (energy at rest). “Wavematter” is more than just another word for the duality. The internal view requires at least four spatial dimensions and is thus not available in SR/GR.

.Like coordinate space and coordinate time, waves and particles are subjective concepts (defined by an observer): What I deem wave, deems itself particle at rest. 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 own 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. Thus, he is an external observer. 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’s view. Only if its energy exceeds the binding energy of an electron is this electron released. The internal view of the photon is crucial for this effect. Thus, the photon behaves like a particle.

The duality is also observed in matter, such as electrons (Jönsson, 1961). Electrons are wavematters too. From the internal view (if the electron is tracked), this electron is a particle: Which slit will it pass through? From the external view (if the electron is not tracked), this electron behaves like a wave. Since I automatically track all slow objects (slow for me), I deem macroscopic objects matter/particles rather than waves. This argument justifies the drawing of solid rockets and celestial bodies in the ES diagrams.

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 no local hidden variable theory is compatible with QM. In many experiments (Freedman & 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 have to do is discuss it in ES. Four spatial dimensions make non-locality obsolete. Figure 7 displays two wavematters that were created at once at a point P. They are now moving away from each other in opposite directions $\pm {d}_{4}^{\prime}$ at the speed $c$. It turns out that these wavematters are automatically entangled. For an observer moving in any direction other than $\pm {d}_{4}^{\prime}$ (external view), they are two distinct objects. The observer cannot understand how these two wavematters are able to communicate with each other in no time.

For each wavematter in Figure 7 (internal view), the $\pm {d}_{4}^{\prime}$ axis disappears because of length contraction at the speed $c$. In their common (!) proper space spanned by ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime}$, either of them deems itself at the very same position as its twin. From either perspective, they are one object that has never been separated. This is how they communicate with each other in no time. There is no “spooky action at a distance”. The twins stay together in their proper space even if their proper time flows in opposite directions. Entanglement occurs because an observer’s proper space may be different from an observed object’s proper space. This is possible only if there are at least four spatial dimensions. ER explains the entanglement of electrons or atoms too. In my proper space, they move at a speed ${v}_{3\mathrm{D}}<c$. In their $\pm {d}_{4}^{\prime}$ 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.**

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 from ES whose 4D motion “swings completely” (rotates by an angle of $90\xb0$) into an observer’s 3D space. This energy speeds off at the speed $c$. 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 to annihilation. Spontaneous effects are another clue that all energy moves through ES at the speed $c$.

According to the Lambda-CDM model, almost all matter in the Universe was created shortly after the Big Bang. Only then was the temperature high enough to enable the pair production of baryons and antibaryons. However, the energy density was also very high so that the baryons and antibaryons should have annihilated each other again. Since we observe more baryons than antibaryons today (also known as the “baryon asymmetry”), it is assumed that an excess of baryons was produced in the early Universe (Canetti et al., 2012). However, an asymmetry in pair production has never been observed.

ER solves the baryon asymmetry: Since each wavematter deems itself particle, there were particles in ES immediately after the Big Bang. There are much less antiparticles than particles today because antiparticles are created in pair production only. One may ask: Why do wavematters not deem themselves antiparticles? Antiparticles are not the opposite of particles but particles with the opposite electric charge. They seem to flow backward in time because proper time flows in opposite directions for any two wavematters created in pair production. These two wavematters are automatically entangled.

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. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts emerging from a construed spacetime in SR/GR. Beyond an observer’s reality with waves and particles, there is the master reality ES with wavematters.

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 limitation. 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 natural spacetime does Mother Nature disclose her secrets. If we think of each observer’s reality as an oversized stage, the keys to cosmology and QM are beyond the stage curtain. The true pillars of physics are ER and QM. Together, they describe the very large and the very small.

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: fully symmetric, 4D Euclidean space. Einstein sacrificed absolute space and time. I sacrifice the absoluteness of waves and particles, but I restore absolute time. 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 all move through 4D space at the speed of light. With that said, conflicts of mankind become all so small.

No funds, grants, or other support was received.

I would like to thank Siegfried W. Stein for his contributions to Sect. 5.10 and for the Figures 2 and 4 (partly), and 5 (partly). After several unsuccessful submissions, he 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 editors and reviewers for the precious time that they spent on my manuscript.

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. I publish these comments to encourage young scientists to stand up for promising ideas, but opposing the mainstream can be exhausting. Here are some statements that I received from top journals: “Unscholarly research.” “This is fake science.” “Too simple to be true.” Simplicity and truth are not mutually exclusive! The editor-in-chief of a top journal replied that publishing is “for experts only”. A well-known preprint archive suspended my submission privileges.

The author has no competing interests to declare.

Not applicable.

- Abbott, B. P.; et al. (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116(6), 061102. [CrossRef]
- Aghanim, N.; et al. (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6. [CrossRef]
- Almeida, J. B. (2001). An alternative to Minkowski space-time. [CrossRef]
- Ashby, N. (2003). Relativity in the global positioning system. Living Reviews in Relativity, 6(1), 1–42. [CrossRef]
- Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental test of Bell’s inequalities using time-varying analyzers. Physical Review Letters, 49(25), 1804–1807. 1807. [CrossRef]
- Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics, 1(3), 195–200. [CrossRef]
- Bouwmeester, D.; et al. (1997). Experimental quantum teleportation. Nature, 390, 575–579. [CrossRef]
- Canetti, L., Drewes, M., & Shaposhnikov, M. (2012). Matter and antimatter in the universe. New Journal of Physics, 14, 095012. [CrossRef]
- Dyson, F. W., Eddington, A. S., & Davidson, C. (1920). A determination of the deflection of light by the sun’s gravitational field, from observations made at the total eclipse of May 29, 1919. Philosophical Transactions of the Royal Society A, 220, 291–333. 29 May. [CrossRef]
- Einstein, A. (1905a). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik, 322(6), 132–148. [CrossRef]
- Einstein, A. (1905b). Zur Elektrodynamik bewegter Körper. Annalen der Physik, 322(10), 891–921. [CrossRef]
- Einstein, A. (1905c). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 323(13), 639–641. [CrossRef]
- Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354(7), 769–822. [CrossRef]
- Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777–780. [CrossRef]
- Freedman, S. J., & Clauser, J. F. (1972). Experimental test of local hidden-variable theories. Physical Review Letters, 28(14), 938–941. [CrossRef]
- Gersten, A. (2003). Euclidean special relativity. Foundations of Physics, 33(8), 1237–1251. 1251. [CrossRef]
- Guth, A. H. (1997). The inflationary universe. Perseus Books.
- Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: Predicted relativistic time gains. Science, 177, 166–168. [CrossRef]
- Heisenberg, W. (1969). Der Teil und das Ganze. Piper.
- Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences of the United States of America, 15(3), 168–173. [CrossRef]
- Jönsson, C. (1961). Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Zeitschrift für Physik, 161, 454–474. [CrossRef]
- Kant, I. (1781). Kritik der reinen Vernunft. Hartknoch.
- Lemaître, G. (1927). Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Annales de la Société Scientifique de Bruxelles A, 47, 49–59.
- Linde, A. (1990). Inflation and quantum cosmology. Academic Press.
- Minkowski, H. (1910). Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Mathematische Annalen, 68, 472–525. [CrossRef]
- Montanus, J. M. C. (1991). Special relativity in an absolute Euclidean space-time. Physics Essays, 4(3), 350–356.
- Montanus, J. M. C. (2001). Proper-time formulation of relativistic dynamics. Foundations of Physics, 31(9), 1357–1400. [CrossRef]
- Montanus, H. (2023, September 23). Proper Time as Fourth Coordinate. ISBN 978-90-829889-4-9. Available online: https://greenbluemath.nl/proper-time-as-fourth-coordinate/ (accessed on 10 January 2024).
- Newburgh, R. G., & Phipps Jr., T. E. (1969). A space–proper time formulation of relativistic geometry. Physical Sciences Research Papers (United States Air Force), no. 401.
- Newton, I. Philosophiae naturalis principia mathematica. Joseph Streater, 1687.
- Penzias, A. A., & Wilson, R. W. (1965). A measurement of excess antenna temperature at 4080 Mc/s. The Astrophysical Journal, 142, 419–421. [CrossRef]
- Perlmutter, S.; et al. (1998). Measurements of Ω and Λ from 42 high-redshift supernovae. [CrossRef]
- Popper, K. Logik der Forschung. Julius Springer, 1935.
- Riess, A. G.; et al. (1998). Observational evidence from supernovae for an accelerating universe and a cosmological constant. The Astronomical Journal, 116(3), 1009–1038. [CrossRef]
- Riess, A. G.; et al. (2018). Milky Way Cepheid standards for measuring cosmic distances and application to Gaia DR2. The Astrophysical Journal, 861(2), 126. [CrossRef]
- Rossi, B., & Hall, D. B. (1941). Variation of the rate of decay of mesotrons with momentum. Physical Review, 59(3), 223–228. [CrossRef]
- Ryder, L. H. Quantum field theory. Cambridge University Press, 1985.
- Schrödinger, E. (1935). Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23, 807–812. [CrossRef]
- The Nobel Foundation The Nobel Prize in Physics 2011. 2011. 2011. Available online: https://www.nobelprize.org/prizes/physics/2011/summary/ (accessed on 10 January 2024).
- Turner, M. S. (1998). Dark matter and dark energy in the universe. [CrossRef]
- van Linden, R. Euclidean relativity. 2023. Available online: https://euclideanrelativity.com (accessed on 10 January 2024).
- Weyl, H. (1928). Gruppentheorie und Quantenmechanik. Hirzel.
- Wick, G. C. (1954). Properties of Bethe-Salpeter wave functions. Physical Review, 96(4), 1124–1134. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

Submitted:

06 February 2024

Posted:

07 February 2024

Read the latest preprint version here

Alerts

Markolf H. Niemz

Markolf H. Niemz

This version is not peer-reviewed

Submitted:

06 February 2024

Posted:

07 February 2024

Read the latest preprint version here

Alerts

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.* Any (!) object’s proper space *d*_{1}, *d*_{2}, *d*_{3} and proper time *τ* span natural spacetime, which is 4D Euclidean space (ES) if we interpret *cτ* as *d*_{4}. 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**. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts. We must learn to distinguish between an observer’s reality (described by SR/GR) and the master reality ES (described by ER). ER solves 15 mysteries at once.

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. While it is true that many attempts to falsify SR/GR have failed, we must not reject all attempts. Scientific theories must be falsifiable (Popper, 1935). I finally 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 space.

To sum it all up: Predictions made by SR/GR are correct, but ER penetrates to a deeper level. I apologize for my many preprint versions, but I received almost no support. It was 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 of physics.

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 spacetime because it illustrates the invariance of the spacetime interval very well (Minkowski, 1910). Predicting the lifetime of muons (Rossi & 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.

Newburgh and Phipps (1969) pioneered ER. Montanus (1991) described an “absolute Euclidean spacetime” with a preferred frame of reference, where a pure time interval is a pure time interval for all observers. Montanus (2023) claims that a preferred frame would avoid the twin paradox in ER, collisions of particles at a distance, and a “character paradox” (confusion of photons, particles, and antiparticles). As we will see, a preferred frame is not required. There is no twin paradox in ER, there are no collisions at a distance in the projections from ES, and the character paradox is reasonable. Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time $\tau $. Montanus (2023) even tried to formulate Maxwell’s equations in ER, but he 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 Sect. 3). van Linden (2023) runs a website about various ER models. However, physicists are still opposed to ER because dark energy and non-locality make cosmology and QM work, waves are excluded in ER, and paradoxes may turn up. This paper marks a turning point: I disclose an issue in SR/GR, I justify the exclusion of waves, and 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 ${v}_{3\mathrm{D}}\ll c$. In Einstein’s physics, all energy moves through 4D non-Euclidean spacetime. The speed of matter is ${v}_{3\mathrm{D}}<c$. In ER, all energy moves through ES. The 4D speed of all energy is ${u}_{4\mathrm{D}}=c$. Newton’s physics (Newton, 1687) influenced Kant’s philosophy (Kant, 1781). Will ER reform both physics and philosophy?

In SR (Einstein, 1905b), there are two concepts of time: coordinate time $t$ and proper time $\tau $. 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” ${t}_{\mathrm{P}}$, a light pulse is sent from P to Q. At “Q time” ${t}_{\mathrm{Q}}$, it is reflected. At “P time” ${t}_{\mathrm{P}}^{*}$, it is back at P. The clocks synchronize if

$${t}_{\mathrm{Q}}-{t}_{\mathrm{P}}={t}_{\mathrm{P}}^{*}-{t}_{\mathrm{Q}}$$

In § 3 of SR, Einstein derives the Lorentz transformation. The coordinates ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}t$ of an event in a system K are transformed to the coordinates ${x}_{1}^{\prime},\text{}{x}_{2}^{\prime},\text{}{x}_{3}^{\prime},\text{}{t}^{\prime}$ in K’ by
$${x}_{1}^{\prime}=\gamma \text{}({x}_{1}-{v}_{3\mathrm{D}}\text{}t),\text{}{x}_{2}^{\prime}={x}_{2}\text{},\text{}{x}_{3}^{\prime}={x}_{3},$$
where K’ moves relative to K in ${x}_{1}$ at a constant speed ${v}_{3\mathrm{D}}$, while $\gamma =(1-{v}_{3\mathrm{D}}^{2}/{c}^{2}{)}^{-0.5}$ 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, 15 fundamental mysteries of physics are solved. Thus, the issue is real. I show that the scope of SR/GR is rather limited. Their concepts of spacetime work well except for cosmology and QM.

$${t}^{\prime}=\gamma \text{}(t-{v}_{3\mathrm{D}}\text{}{x}_{1}/{c}^{2}),$$

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?

The indefinite distance function in SR (Einstein, 1905b) is usually written as
$${{c}^{2}\text{}\mathrm{d}\tau}^{2}={{c}^{2}\text{}\mathrm{d}t}^{2}-\mathrm{d}{x}_{1}^{2}-\mathrm{d}{x}_{2}^{2}-\mathrm{d}{x}_{3}^{2},$$
where $\mathrm{d}\tau $ is an infinitesimal distance in $\tau $, while $\mathrm{d}t$ and ${\mathrm{d}x}_{i}$ ($i=1,\text{}2,\text{}3$) are infinitesimal distances in coordinate spacetime ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}t$. This spacetime is construed because coordinate space ${x}_{1},\text{}{x}_{2},\text{}{x}_{3}$ and coordinate time $t$ are subjective concepts: They are not immanent in rulers and clocks but defined by an observer. Rulers measure proper distance ${d}_{\mu}$ ($\mu =1,\text{}2,\text{}3,\text{}4$). Clocks measure proper time $\tau $. We may rearrange Eq. (3) and obtain
$${{c}^{2}\text{}\mathrm{d}t}^{2}=\mathrm{d}{d}_{1}^{2}+\mathrm{d}{d}_{2}^{2}+\mathrm{d}{d}_{3}^{2}+\mathrm{d}{d}_{4}^{2},$$
where ${\mathrm{d}d}_{i}={\mathrm{d}x}_{i}$ ($i=1,\text{}2,\text{}3$) and ${\mathrm{d}d}_{4}=c\text{}\mathrm{d}\tau $ are infinitesimal distances in ES. The roles of $t$ and $\tau $ are switched: The fourth coordinate in ER is an object’s proper time $\tau $ (measured by itself) multiplied by $c$. The new invariant is cosmic time $t$. I retain the symbol $t$ to stress the equivalence of Eqs. (3) and (4). The indices 1 to 4 point out the full symmetry. Any (!) object’s proper space ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ and proper time $\tau $ span natural spacetime, which is ES if we interpret $c\tau $ as ${d}_{4}$. This spacetime is natural because all ${d}_{\mu}$ ($\mu =1,\text{}2,\text{}3,\text{}4$) are objective concepts: They are immanent in rulers and clocks. We must not confuse ER with a Wick rotation (Wick, 1954), which replaces $t$ with $it$ and keeps $\tau $ invariant.

“ES diagrams” show ES from an object’s perspective. For each object, we are free to label the four axes of ES. We always take ${d}_{4}$ as the axis in which the object itself moves at the speed $c$. During its lifetime, the object keeps moving in ${d}_{4}$ (always drawn vertically). An “object’s reality” is created by projecting ES orthogonally to its proper space and to its proper time. For any two objects, $\tau $ and $\tau \prime $ may flow in different 4D directions.
$$\mathit{\tau}={d}_{4}\text{}\mathit{u}/{c}^{2},\text{}\mathit{\tau}\mathit{\text{'}}={d}_{4}^{\prime}\text{}\mathit{u}\mathit{\text{'}}/{c}^{2},$$
where $\mathit{\tau}$ is the 4D vector “flow of proper time” of an object and $\mathit{u}$ is its 4D velocity. For all objects, there is ${u}_{\mu}=\mathrm{d}{d}_{\mu}/\mathrm{d}t$ (cosmic time $t$). Thus, Eq. (4) matches my first postulate

$$\tau ={d}_{4}/c,\text{}\tau \prime ={d}_{4}^{\prime}/c,$$

$${u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}+{u}_{4}^{2}={c}^{2}.$$

My second postulate revises the principle of relativity, and it defines an observer’s reality: It 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 $\tau $, in particular, cannot factor in an object’s 4D vector $\mathit{\tau}$. Since replacing coordinate time with cosmic time is a discontinuous operation, there is no continuous transition between SR/GR and ER. We take an object’s ${d}_{1}\left(t\right),\text{}{d}_{2}\left(t\right),\text{}{d}_{3}\left(t\right),\text{}{d}_{4}\left(t\right)$ for granted rather than an observer’s ${x}_{1}\left(\tau \right),\text{}{x}_{2}\left(\tau \right),\text{}{x}_{3}\left(\tau \right),t\left(\tau \right)$.

Since ES is “beyond” (prior to) projecting, I call it the “master reality” (master of each observer’s reality). Spacetime in SR/GR is relative. ES is absolute. All ES diagrams and the projections are relative. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts (see Sect. 5.12). We must learn to distinguish between an observer’s reality with waves and particles (described by SR/GR) and the master reality ES with wavematters (described by ER).

It is instructive to contrast the three concepts of time. Coordinate time $t$ is a subjective measure of time: It is equal to $\tau =\left|\mathit{\tau}\right|$ for the observer only. Proper time $\tau $ 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 invariant and thus absolute, there is no twin paradox in ER. Twins share the same age in cosmic time. In ER, time is a subordinate quantity: Only by covering distance is time passing by. I suggest that we define a standard unit for speed and that we measure time in compound units.

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 $ct$ 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 ${v}_{3\mathrm{D}}=0.6\text{}c$. 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 $ct\prime $. 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 observer R (or B) now be with clock “r” (or else “b”). In the blue frame of Figure 1 left, “b” displays ${t}^{\prime}=1.0\text{}\mathrm{s}$ at the instant when “r” displays $t=0.8\text{}\mathrm{s}$ (dashed line). In the red frame of Figure 1 left, “b” displays ${t}^{\prime}=0.8\text{}\mathrm{s}$ at the instant when “r” displays $t=1.0\text{}\mathrm{s}$ (solid line). In SR, time dilation with respect to “r” thus occurs in ${t}^{\prime}$ of B. In the red frame of Figure 1 right, “b” is at ${d}_{4}=0.8\text{}\mathrm{L}\mathrm{s}$ at the instant when “r” is at ${d}_{4}=1.0\text{}\mathrm{L}\mathrm{s}$ (same axis ${d}_{4}$). In ER, time dilation with respect to “r” thus occurs in ${d}_{4}$ of R. In both SR and ER, “b” is slow with respect to “r”. However, ${t}^{\prime}=0.8\text{}\mathrm{s}$ is calculated only (B measures time in $\tau \prime $), while ${d}_{4}=0.8\text{}\mathrm{L}\mathrm{s}$ is measurable (${d}_{4}$ relates to $\tau $). Rotate either graph in Figure 1 to see that only ER provides a holistic view: The ES diagram lives up to R and B at once. A new Minkowski diagram is required for B, where ${x}_{1}^{\prime}$ and $c{t}^{\prime}$ are orthogonal.

Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time $\tau $. 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” ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}{ct}^{\prime}$, where ${ct}^{\prime}$ 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 ${x}_{1},\text{}{x}_{2},\text{}{x}_{3},\text{}{ct}^{\prime}$ to ${x}_{1}^{\prime},\text{}{x}_{2}^{\prime},\text{}{x}_{3}^{\prime},\text{}ct$. In ER, unmixed ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime},\text{}{d}_{4}^{\prime}$ rotate with respect to ${d}_{1},\text{}{d}_{2},\text{}{d}_{3},\text{}{d}_{4}$ (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 $t$ 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 $\mathit{\tau}$ are always synchronized, while clocks with different $\mathit{\tau}$ and $\mathit{\tau}\mathit{\text{'}}$ are never synchronized (different values of ${d}_{4}$ in Figure 1 right).

We consider two identical rockets “r” (red rocket) and “b” (blue rocket) and assume that there is an observer R (or B) in the rear end of rocket “r” (or else “b”). His ES diagram is ${d}_{1},\text{}{d}_{2},\text{}{d}_{3},\text{}{d}_{4}$ (or else ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime},\text{}{d}_{4}^{\prime}$). The 3D space of R (or else B) is spanned by ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ (or else ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime}$). We use “3D space” and “proper space” as synonyms. The proper time of R (or else B) relates to ${d}_{4}$ (or else ${d}_{4}^{\prime}$). The rockets started at the same point P and move relative to each other at the constant 3D speed ${v}_{3\mathrm{D}}$. We are free to label the axis of motion in 3D space. Here, it is ${d}_{1}$. The ES diagrams in Figure 2 top must fulfill my two postulates and the initial condition (starting point P). This is achieved by rotating the ES diagrams with respect to each other. Figure 2 bottom shows the projection to the 3D space of R (or else B). The rockets are drawn in 2D although their width is in ${d}_{2}$ or ${d}_{3}$ (${d}_{2}^{\prime}$ or ${d}_{3}^{\prime}$).

We now verify: (1) The fact that the ES diagrams of R and of B 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.
$${\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi =({L}_{\mathrm{b},\mathrm{R}}/{L}_{\mathrm{b},\mathrm{B}}{)}^{2}+({v}_{3\mathrm{D}}/c{)}^{2}=1,$$
$${L}_{\mathrm{b},\mathrm{R}}={\gamma}^{-1}\text{}{L}_{\mathrm{b},\mathrm{B}}\text{(length contraction),}$$
where $\gamma =(1-{v}_{3\mathrm{D}}^{2}/{c}^{2}{)}^{-0.5}$ is the same Lorentz factor as in SR. For R, rocket “b” contracts by the factor ${\gamma}^{-1}$. 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.
$${\mathrm{sin}}^{2}\phi +{\mathrm{cos}}^{2}\phi =({d}_{4,\mathrm{B}}/{d}_{4,\mathrm{B}}^{\prime}{)}^{2}+({v}_{3\mathrm{D}}/c{)}^{2}=1,$$
where ${d}_{4,\mathrm{B}}$ (or ${d}_{4,\mathrm{B}}^{\prime}$) is the distance that B moved in ${d}_{4}$ (or else ${d}_{4}^{\prime}$). With ${d}_{4,\mathrm{B}}^{\prime}={d}_{4,\mathrm{R}}$ (R and B cover the same distance in ES but in different directions), we calculate
where ${d}_{4,\mathrm{R}}$ is the distance that R moved in ${d}_{4}$. Eqs. (9) and (12) tell us: SR works so well because $\gamma $ is recovered when projecting 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.

$${d}_{4,\mathrm{B}}={{\gamma}^{-1}\text{}d}_{4,\mathrm{B}}^{\prime},$$

$${d}_{4,\mathrm{R}}=\gamma \text{}{d}_{4,\mathrm{B}}\text{(time dilation),}$$

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 ${u}_{1,\mathrm{b}}$ of “b” in ${d}_{1}$ increases at the expense of its speed ${u}_{4,\mathrm{b}}$ in ${d}_{4}$.

Gravitational waves support the idea of GR that gravitation is a feature of spacetime (Abbott et al., 2016). However, classical physics considers gravitation a force that has not yet been unified with the other three forces of physics. I claim that curved geodesics in ES replace curved spacetime in GR. To support my claim, we now calculate gravitational time dilation in ES. Let “r” and “b” be two identical clocks far away from Earth. Initially, they move next to each other in the ${d}_{4}$ axis of “r”. At some point, “b” is sent in free fall toward Earth in the ${d}_{1}$ axis of “r” (Figure 3). The kinetic energy of “b” with the mass $m$ is
where $G$ is the gravitational constant, $M$ is the mass of Earth, and $r$ is the distance of clock “b” to Earth’s center. By applying Eq. (7), we obtain

$${\frac{1}{2}mu}_{1,\mathrm{b}}^{2}=GMm/r,$$

$${u}_{4,\mathrm{b}}^{2}={c}^{2}-{u}_{1,\mathrm{b}}^{2}={c}^{2}-2GM/r.$$

With ${u}_{4,\mathrm{b}}={\mathrm{d}d}_{4,\mathrm{b}}/\mathrm{d}t$ (“b” moves in the ${d}_{4}$ axis at the speed ${u}_{4,\mathrm{b}}$) and $c={\mathrm{d}d}_{4,\mathrm{r}}/\mathrm{d}t$ (“r” moves in the ${d}_{4}$ axis at the speed $c$), we calculate
$$\mathrm{d}{d}_{4,\mathrm{r}}={\gamma}_{\mathrm{g}\mathrm{r}}\text{}\mathrm{d}{d}_{4,\mathrm{b}}\text{}\left(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\text{}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\text{}\mathrm{d}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right),$$
where ${\gamma}_{\mathrm{g}\mathrm{r}}=(1-2GM/(r{c}^{2}{\left)\right)}^{-0.5}$ is the same dilation factor as in GR. It does not depend on ${u}_{1,\mathrm{b}}$. Eq. (16) tells us: GR works so well because ${\gamma}_{\mathrm{g}\mathrm{r}}$ is recovered when projecting ES to ${d}_{4}$. Thus, GPS satellites do their job in ER as well as in GR! When “b” returns to “r”, clock “b” is behind clock “r”. This dilation stems from projecting curved geodesics. In GR, it stems from a curved spacetime. We sum up time dilation: 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 flow of proper time. In ER, an observed clock is slow in the observer’s flow of proper time. Since both $\gamma $ and ${\gamma}_{\mathrm{g}\mathrm{r}}$ are recovered, the experiment by Hafele and Keating (1972) also supports ER.

$$\mathrm{d}{d}_{4,\mathrm{b}}^{2}=({c}^{2}-2GM/r)\text{}(\mathrm{d}{d}_{4,\mathrm{r}}/c{)}^{2},$$

Three instructive examples (Figure 4) demonstrate how to project from ES to 3D space. Problem 1: A 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 moves along the wire, its speed in ${d}_{4}$ must be slower than $c$. Wouldn’t the wire eventually be outside the rocket? Problem 2: A mirror passes a rocket. An observer in the rocket tip sends a light pulse to the mirror and tries to detect the reflection. In ES, all objects move at the speed $c$ but in different directions. We assume that the observer moves in his ${d}_{4}$ axis. How can he ever detect the reflection? Problem 3: Earth revolves around the sun. We assume that the sun moves in its ${d}_{4}$ axis. As Earth covers a distance in ${d}_{1}$ and ${d}_{2}$, its speed in ${d}_{4}$ must be slower than $c$. Wouldn’t the sun escape from the orbital plane of Earth?

The questions in the last paragraph seem to imply that there are geometric paradoxes in ER, but there aren’t any. The fallacy in all problems lies in the assumption that all four spatial dimensions would be observable. Just three of them are observable! All problems are solved by projecting ES to 3D space (Figure 4 bottom). These projections tell us what an observer’s reality is like because “suppressing the ${d}_{4}$ axis” is equivalent to “length contraction makes ${d}_{4}$ disappear”. The suppressed axis ${d}_{4}$ is experienced as time. We easily verify in an observer’s 3D space: The guide wire remains within the rocket; the light pulse is reflected back to the observer; the sun remains in the orbital plane of Earth.

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 $\mathit{\tau}$ 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 it declares five concepts of today’s physics obsolete.

Proper time $\tau $ 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 $\mathit{\tau}$.

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

in $m{c}^{2}$In SR, if forces are absent, the total energy $E$ of an object is given by
where ${E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ 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 never move at the speed $c$. ER provides this missing clue: The object is not at rest, but it moves in its ${d}_{4}^{\prime}$ axis. From its own perspective, its ${E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is zero, and $m{c}^{2}$ is its kinetic (!) energy in ${d}_{4}^{\prime}$. The factor ${c}^{2}$ is a hint that it moves through ES at the speed $c$. In SR, there is also
where $p$ is the total momentum of an object and ${p}_{3\mathrm{D}}$ is its momentum in an observer’s 3D space. Again, ER is eye-opening: From the object’s perspective, its ${p}_{3\mathrm{D}}$ is zero, and $mc$ is its momentum in ${d}_{4}^{\prime}$. The factor $c$ is a hint that it moves through ES at the speed $c$. Eqs. (17) and (18) are not valid in ER. In ER, there is $E=m{c}^{2}$ and $p=mc$ for each object.

$$E=\gamma m{c}^{2}={E}_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}+m{c}^{2},$$

$${E}^{2}={p}^{2}{c}^{2}={p}_{3\mathrm{D}}^{2}{c}^{2}+{m}^{2}{c}^{4},$$

ER discloses that length contraction and time dilation stem from projecting ES to an observer’s reality. 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 gravitational time dilation stems from projecting curved geodesics in flat ES to the ${d}_{4}$ axis of an observer. If an object accelerates in his proper space, it automatically decelerates in his proper time. I am aware that more studies will be necessary to explain other gravitational effects. In GR, gravitational time dilation stems from a curved spacetime. In the next six sections, I show that ER outperforms GR in cosmology.

In this section, I outline an ER-based model of cosmology. There is no need to create ES. Space exists just like numbers. For some reason, there was a Big Bang. In the GR-based Lambda-CDM model, the Big Bang occurred “everywhere” because space inflated from a singularity. In the ER-based model, we can locate the Big Bang: It injected a huge amount of energy into a non-inflating and non-expanding ES all at once at what I call “origin O”, the only natural reference point. The Big Bang occurred at the cosmic time $t=0$ and 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 physical interactions (scattering, transversal acceleration, spontaneous emission), some energy departed from its radial motion while maintaining the speed $c$. Today, all energy is confined to a 4D hypersphere, while a significant amount of energy is confined to its 3D hypersurface.

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 & 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 Figure 5, nature is described from the perspective of Earth (Earth moves vertically). From this perspective, the CRR cannot move in ${d}_{4}$ because it moves in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ 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 so low? Here are some possible answers: (1) The CRR has been scattered multiple times in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ (Figure 5 left shows ${d}_{1}$). Some of the scattered CRR reaches Earth as CMB (in the projection to its 3D space) after having covered the same distance in ${d}_{1},\text{}{d}_{2},\text{}{d}_{3}$ as Earth in ${d}_{4}$. The cross section for scattering is low, but the fluence of the CRR is high. (2) The CRR was created and scattered equally in ${d}_{1},{d}_{2},{d}_{3}$. (3) The very low temperature is due to a very high recession speed (see Sect. 5.10) of all plasma particles.

The speed ${v}_{3\mathrm{D}}$ at which a galaxy G recedes from Earth in 3D space (Figure 5 left) relates to its 3D distance $D$ as $c$ relates to the radius $r$ of the 4D hypersphere.
where ${H}_{t}=c/r=1/t$ is the Hubble parameter and $t$ is the cosmic time elapsed since the Big Bang. If we compare galaxies as they appear to us today at $t={t}_{0}$, Eq. (19) reduces to
where ${v}_{3\mathrm{D},0}$ is today’s recession speed, ${H}_{0}=c/{r}_{0}=1/{t}_{0}$ is the Hubble constant, and ${r}_{0}$ is today’s radius of the 4D hypersphere. 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 not a constant. They are not yet aware of the 4D Euclidean geometry.

$${v}_{3\mathrm{D}}=Dc/r={H}_{t}\text{}D,$$

$${v}_{3\mathrm{D},0}={H}_{0}\text{}D,$$

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

Most cosmologists believe that an inflation of space in the early universe (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 quantum fluctuations have been expanding at the speed $c$. **In ER, cosmic inflation is an obsolete concept.**

There are various methods for calculating ${H}_{0}$. Up next, I explain why the calculated values do not match (also known as the “Hubble tension”). I compare CMB measurements using the Planck space telescope with distance ladder measurements using the Hubble space telescope. Team A (Aghanim et al., 2020) calculates ${H}_{0}=67.66\pm 0.42\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B (Riess et al., 2018) calculates ${H}_{0}=73.52\pm 1.62\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B made efforts to minimize the error margins in the distance measurements, but I show that a misinterpretation of the redshifts causes a systematic error in its calculation of ${H}_{0}$. We assume that team A’s value of ${H}_{0}$ is correct. We simulate the supernova of a star $\mathrm{S}$ that occurred at a distance of $D=400\text{}\mathrm{M}\mathrm{p}\mathrm{c}$ from Earth (Figure 5 right). The measurable redshift parameter
tells us how each wavelength ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ emitted by the supernova is either passively stretched by an expanding space (team B), or how each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is redshifted by the Doppler effect of actively receding objects (ER-based model). In Figure 5 right, there is an arc called “past” when the supernova occurred and an arc called “present” when the supernova’s light arrives on Earth. While this light was moving the distance $D$ in the ${d}_{1}$ axis, Earth moved the same distance $D$ in the ${d}_{4}$ axis (first postulate). Thus, team B receives redshift data from a cosmic time $t<{t}_{0}$ when there was $r<{r}_{0}$ and ${H}_{t}>{H}_{0}$.

$$z=\mathsf{\Delta}\lambda /{\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$$

$$1/{H}_{t}=r/c={({r}_{0}-D)/c=1/H}_{0}-D/c.$$

In our simulated case ($D=400\text{}\mathrm{M}\mathrm{p}\mathrm{c}$), there is ${H}_{t}=74.37\text{}\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. For a very short distance of $D=400\text{}\mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (22) tells us that ${H}_{t}$ deviates from ${H}_{0}$ by only 0.009 percent. However, when plotting ${v}_{3\mathrm{D}}$ versus $D$ for long distances (50 Mpc, 100 Mpc, 150 Mpc, ... 450 Mpc), the slope of a fitted straight line is 8 to 9 percent greater than ${H}_{0}$. Since team B is calculating ${H}_{0}$ from the slope of such plots, its value of ${H}_{0}$ is 8 to 9 percent too high. This solves the Hubble tension. I kindly ask team B to recalculate ${H}_{0}$ after converting all ${v}_{3\mathrm{D}}$ of the past to today’s values ${v}_{3\mathrm{D},0}$. Eq. (22) tells us how to do so:
$${H}_{t}={H}_{0}\text{}c\text{}/\text{}(c-{H}_{0}\text{}D)={H}_{0}\text{}/\text{}(1-{v}_{3\mathrm{D},0}/c),$$

$${v}_{3\mathrm{D},0}={v}_{3\mathrm{D}}\text{}/\text{}(1+{v}_{3\mathrm{D}}/c).$$

Of course, team B is well aware of the fact 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 ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is continuously stretched by an expanding space. The redshift parameter $z$ increases during the journey to Earth. That moment when the supernova occurred is irrelevant. In the ER-based model, that very moment is relevant, but the timespan is irrelevant. Each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ is initially redshifted at the cosmic time $t$ by the Doppler effect. During the journey to Earth, the redshift parameter $z$ remains constant. This parameter is tied up in a “package” when the supernova occurs and sent to Earth, where it is measured. Space is not expanding. A 3D hypersurface (made of energy!) is receding in 4D space. **In ER, expansion of space is an obsolete concept.**

Team B can fix the systematic error in its calculation of ${H}_{0}$ within the Lambda-CDM model by converting all ${v}_{3\mathrm{D}}$ to ${v}_{3\mathrm{D},0}$ 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 that model with the ER-based model of cosmology unless we postulate a dark energy. Perlmutter et al. (1998) and Riess et al. (1998) favor an accelerating expansion of space because the calculated recession speeds deviate from Eq. (20). These deviations increase with distance, and an accelerating expansion of space would stretch each ${\lambda}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ even further.

The ER-based model provides a simpler explanation for the deviations from Eq. (20): We must consider Eq. (19). The older the redshift data are, the more ${H}_{t}$ deviates from ${H}_{0}$, and the more ${v}_{3\mathrm{D}}$ deviates from ${v}_{3\mathrm{D},0}$. If another star ${\mathrm{S}}_{0}$ (small circle in Figure 5 right) is at the same distance (400 Mpc) today as $\mathrm{S}$ was in the past, it recedes more slowly from Earth (27,064 km/s) than $\mathrm{S}$ (29,748 km/s) according to Eq. (24). 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. Turner (1998) coined the term “dark energy” to designate a cause for the accelerating expansion of space, but dark energy has never been observed. Dark energy is a stopgap for an effect that the Lambda-CDM model is not able to explain. Now that we are aware of the 4D Euclidean geometry, the deviations from Eq. (20) can be attributed to deeper pasts. **In ER, dark energy is an obsolete concept.**

Any expansion of space—uniform or accelerating—is only virtual. The illusion of a non-linear expansion stems from Eq. (19): The deeper in the past a supernova occurred, the greater are both ${H}_{t}$ and $D$! There is no accelerating expansion of the Universe even if the 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). Actually, there are two misconceptions in these words of praise: (1) In the Lambda-CDM model, the term “Universe” implies space, but space is not expanding at all. (2) There is receding energy, but it recedes uniformly at the speed $c$.

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.**

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, these objects behave like particles. Up next, I explain how the very same object appears as a wave packet or as a particle depending on the perspective. From an observer’s perspective, it may appear as a wave packet or as a particle. From its own perspective, it is a particle at rest.

Based on the wave–particle duality, I suggest that we introduce a generalized concept of energy: All energy is “wavematter”, which may appear as wave packets or as particles in an observer’s reality (Figure 6). In my reality (external view, coordinate spacetime!), such a wavematter may appear as a wave packet or as a particle. As a wave, it propagates in my ${x}_{1}$ axis at the speed $c$, and it oscillates in my axes ${x}_{2}$ (electric field) and ${x}_{3}$ (magnetic field). Both propagating and oscillating occur as a function of coordinate time $t$. In its own reality (internal view, “in-flight view”), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed $c$. Thus, it deems itself particle at rest (energy at rest). “Wavematter” is more than just another word for the duality. The internal view requires at least four spatial dimensions and is thus not available in SR/GR.

.Like coordinate space and coordinate time, waves and particles are subjective concepts (defined by an observer): What I deem wave, deems itself particle at rest. 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 own 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. Thus, he is an external observer. 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’s view. Only if its energy exceeds the binding energy of an electron is this electron released. The internal view of the photon is crucial for this effect. Thus, the photon behaves like a particle.

The duality is also observed in matter, such as electrons (Jönsson, 1961). Electrons are wavematters too. From the internal view (if the electron is tracked), this electron is a particle: Which slit will it pass through? From the external view (if the electron is not tracked), this electron behaves like a wave. Since I automatically track all slow objects (slow for me), I deem macroscopic objects matter/particles rather than waves. This argument justifies the drawing of solid rockets and celestial bodies in the ES diagrams.

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 no local hidden variable theory is compatible with QM. In many experiments (Freedman & 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 have to do is discuss it in ES. Four spatial dimensions make non-locality obsolete. Figure 7 displays two wavematters that were created at once at a point P. They are now moving away from each other in opposite directions $\pm {d}_{4}^{\prime}$ at the speed $c$. It turns out that these wavematters are automatically entangled. For an observer moving in any direction other than $\pm {d}_{4}^{\prime}$ (external view), they are two distinct objects. The observer cannot understand how these two wavematters are able to communicate with each other in no time.

For each wavematter in Figure 7 (internal view), the $\pm {d}_{4}^{\prime}$ axis disappears because of length contraction at the speed $c$. In their common (!) proper space spanned by ${d}_{1}^{\prime},\text{}{d}_{2}^{\prime},\text{}{d}_{3}^{\prime}$, either of them deems itself at the very same position as its twin. From either perspective, they are one object that has never been separated. This is how they communicate with each other in no time. There is no “spooky action at a distance”. The twins stay together in their proper space even if their proper time flows in opposite directions. Entanglement occurs because an observer’s proper space may be different from an observed object’s proper space. This is possible only if there are at least four spatial dimensions. ER explains the entanglement of electrons or atoms too. In my proper space, they move at a speed ${v}_{3\mathrm{D}}<c$. In their $\pm {d}_{4}^{\prime}$ 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.**

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 from ES whose 4D motion “swings completely” (rotates by an angle of $90\xb0$) into an observer’s 3D space. This energy speeds off at the speed $c$. 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 to annihilation. Spontaneous effects are another clue that all energy moves through ES at the speed $c$.

According to the Lambda-CDM model, almost all matter in the Universe was created shortly after the Big Bang. Only then was the temperature high enough to enable the pair production of baryons and antibaryons. However, the energy density was also very high so that the baryons and antibaryons should have annihilated each other again. Since we observe more baryons than antibaryons today (also known as the “baryon asymmetry”), it is assumed that an excess of baryons was produced in the early Universe (Canetti et al., 2012). However, an asymmetry in pair production has never been observed.

ER solves the baryon asymmetry: Since each wavematter deems itself particle, there were particles in ES immediately after the Big Bang. There are much less antiparticles than particles today because antiparticles are created in pair production only. One may ask: Why do wavematters not deem themselves antiparticles? Antiparticles are not the opposite of particles but particles with the opposite electric charge. They seem to flow backward in time because proper time flows in opposite directions for any two wavematters created in pair production. These two wavematters are automatically entangled.

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. However, the SO(4) symmetry of ES is incompatible with waves. This is fine because waves and particles are subjective concepts emerging from a construed spacetime in SR/GR. Beyond an observer’s reality with waves and particles, there is the master reality ES with wavematters.

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 limitation. 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 natural spacetime does Mother Nature disclose her secrets. If we think of each observer’s reality as an oversized stage, the keys to cosmology and QM are beyond the stage curtain. The true pillars of physics are ER and QM. Together, they describe the very large and the very small.

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: fully symmetric, 4D Euclidean space. Einstein sacrificed absolute space and time. I sacrifice the absoluteness of waves and particles, but I restore absolute time. 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 all move through 4D space at the speed of light. With that said, conflicts of mankind become all so small.

No funds, grants, or other support was received.

I would like to thank Siegfried W. Stein for his contributions to Sect. 5.10 and for the Figures 2 and 4 (partly), and 5 (partly). After several unsuccessful submissions, he 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 editors and reviewers for the precious time that they spent on my manuscript.

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. I publish these comments to encourage young scientists to stand up for promising ideas, but opposing the mainstream can be exhausting. Here are some statements that I received from top journals: “Unscholarly research.” “This is fake science.” “Too simple to be true.” Simplicity and truth are not mutually exclusive! The editor-in-chief of a top journal replied that publishing is “for experts only”. A well-known preprint archive suspended my submission privileges.

The author has no competing interests to declare.

Not applicable.

- Abbott, B. P.; et al. (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters, 116(6), 061102. [CrossRef]
- Aghanim, N.; et al. (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6. [CrossRef]
- Almeida, J. B. (2001). An alternative to Minkowski space-time. [CrossRef]
- Ashby, N. (2003). Relativity in the global positioning system. Living Reviews in Relativity, 6(1), 1–42. [CrossRef]
- Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental test of Bell’s inequalities using time-varying analyzers. Physical Review Letters, 49(25), 1804–1807. 1807. [CrossRef]
- Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics, 1(3), 195–200. [CrossRef]
- Bouwmeester, D.; et al. (1997). Experimental quantum teleportation. Nature, 390, 575–579. [CrossRef]
- Canetti, L., Drewes, M., & Shaposhnikov, M. (2012). Matter and antimatter in the universe. New Journal of Physics, 14, 095012. [CrossRef]
- Dyson, F. W., Eddington, A. S., & Davidson, C. (1920). A determination of the deflection of light by the sun’s gravitational field, from observations made at the total eclipse of May 29, 1919. Philosophical Transactions of the Royal Society A, 220, 291–333. 29 May. [CrossRef]
- Einstein, A. (1905a). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik, 322(6), 132–148. [CrossRef]
- Einstein, A. (1905b). Zur Elektrodynamik bewegter Körper. Annalen der Physik, 322(10), 891–921. [CrossRef]
- Einstein, A. (1905c). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 323(13), 639–641. [CrossRef]
- Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354(7), 769–822. [CrossRef]
- Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777–780. [CrossRef]
- Freedman, S. J., & Clauser, J. F. (1972). Experimental test of local hidden-variable theories. Physical Review Letters, 28(14), 938–941. [CrossRef]
- Gersten, A. (2003). Euclidean special relativity. Foundations of Physics, 33(8), 1237–1251. 1251. [CrossRef]
- Guth, A. H. (1997). The inflationary universe. Perseus Books.
- Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: Predicted relativistic time gains. Science, 177, 166–168. [CrossRef]
- Heisenberg, W. (1969). Der Teil und das Ganze. Piper.
- Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences of the United States of America, 15(3), 168–173. [CrossRef]
- Jönsson, C. (1961). Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Zeitschrift für Physik, 161, 454–474. [CrossRef]
- Kant, I. (1781). Kritik der reinen Vernunft. Hartknoch.
- Lemaître, G. (1927). Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Annales de la Société Scientifique de Bruxelles A, 47, 49–59.
- Linde, A. (1990). Inflation and quantum cosmology. Academic Press.
- Minkowski, H. (1910). Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Mathematische Annalen, 68, 472–525. [CrossRef]
- Montanus, J. M. C. (1991). Special relativity in an absolute Euclidean space-time. Physics Essays, 4(3), 350–356.
- Montanus, J. M. C. (2001). Proper-time formulation of relativistic dynamics. Foundations of Physics, 31(9), 1357–1400. [CrossRef]
- Montanus, H. (2023, September 23). Proper Time as Fourth Coordinate. ISBN 978-90-829889-4-9. Available online: https://greenbluemath.nl/proper-time-as-fourth-coordinate/ (accessed on 10 January 2024).
- Newburgh, R. G., & Phipps Jr., T. E. (1969). A space–proper time formulation of relativistic geometry. Physical Sciences Research Papers (United States Air Force), no. 401.
- Newton, I. Philosophiae naturalis principia mathematica. Joseph Streater, 1687.
- Penzias, A. A., & Wilson, R. W. (1965). A measurement of excess antenna temperature at 4080 Mc/s. The Astrophysical Journal, 142, 419–421. [CrossRef]
- Perlmutter, S.; et al. (1998). Measurements of Ω and Λ from 42 high-redshift supernovae. [CrossRef]
- Popper, K. Logik der Forschung. Julius Springer, 1935.
- Riess, A. G.; et al. (1998). Observational evidence from supernovae for an accelerating universe and a cosmological constant. The Astronomical Journal, 116(3), 1009–1038. [CrossRef]
- Riess, A. G.; et al. (2018). Milky Way Cepheid standards for measuring cosmic distances and application to Gaia DR2. The Astrophysical Journal, 861(2), 126. [CrossRef]
- Rossi, B., & Hall, D. B. (1941). Variation of the rate of decay of mesotrons with momentum. Physical Review, 59(3), 223–228. [CrossRef]
- Ryder, L. H. Quantum field theory. Cambridge University Press, 1985.
- Schrödinger, E. (1935). Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23, 807–812. [CrossRef]
- The Nobel Foundation The Nobel Prize in Physics 2011. 2011. 2011. Available online: https://www.nobelprize.org/prizes/physics/2011/summary/ (accessed on 10 January 2024).
- Turner, M. S. (1998). Dark matter and dark energy in the universe. [CrossRef]
- van Linden, R. Euclidean relativity. 2023. Available online: https://euclideanrelativity.com (accessed on 10 January 2024).
- Weyl, H. (1928). Gruppentheorie und Quantenmechanik. Hirzel.
- Wick, G. C. (1954). Properties of Bethe-Salpeter wave functions. Physical Review, 96(4), 1124–1134. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

Theoretical evidence for principles of special relativity based on isotropic and uniform four-dimensional space

Takuya Yamashita

,

2023

Concerning the origin of the spacetime with regard to a multi-rest physical structure

Ahmed Isam

,

2021

Spacetime Quantization Illustrates General Relativity and Quantum Theory

Yazeed Alharbi

,

2021

© 2024 MDPI (Basel, Switzerland) unless otherwise stated