Only Euclidean Relativity Provides a Holistic View of Nature

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. In SR, he merges them into a flat spacetime described by an indefinite distance function. SR is often presented in Minkowski 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.

Two postulates of ER: (1) All energy moves through 4D Euclidean space (ES) at the speed of light $c$. (2) The laws of physics have the same form in each “observer’s reality”, which is created by projecting ES orthogonally to his proper space and to his proper time. To improve readability, I always refer to an observer as “he”. To make up for it, I refer to nature as “she”. My first postulate is stronger than the second SR postulate: $c$ is absolute and universal. My second postulate refers to realities rather than to inertial frames. I also introduce a generalized concept of energy: All energy is “wavematter”, which may appear as a wave packet or as a particle depending on the perspective (see Sect. 5.12).

Newburgh and Phipps (1969) pioneered ER. Montanus (1991) described an absolute Euclidean spacetime with a “preferred frame of reference” (a pure time interval is a pure time interval for all observers). Montanus (2023) claims: Without the preferred frame, we would face the twin paradox, non-contact collisions, and a “character paradox” (confusion of photons, particles, and antiparticles). I will show that the preferred frame is obsolete. Whatever is time for me, it may be space for you. There is no twin paradox. There are no non-contact collisions. The character paradox is reasonable. Montanus (2001) used the Lagrange formalism to set up the kinematic equations in proper time. Montanus (2023) even tried to formulate Maxwell’s equations in ER but wondered about a wrong sign. He overlooked that the SO(4) symmetry of ES is incompatible with waves.

Almeida (2001) investigated geodesics in ES. Gersten (2003) showed that the Lorentz transformation is an SO(4) rotation in a “mixed space” (see Sect. 3). van Linden (2023) runs a website about various ER models. Physicists are still opposed to ER because dark energy and non-locality make cosmology and QM work, waves are excluded, and paradoxes may turn up if ER is not interpreted correctly. This paper marks a turning point: I disclose an issue in SR/GR. I justify the exclusion of waves. I avoid paradoxes by projecting ES.

It is instructive to contrast Newton’s physics, Einstein’s physics, and ER. In Newton’s physics, all energy moves through 3D Euclidean space as a function of independent time. The speed of matter is $v_{3\mathrm{D}}\ll c$. 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?

2. Disclosing an Issue in Special and General Relativity

In SR (Einstein, 1905b), there are two concepts of time: coordinate time $t$ and proper time $\tau$. 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

$$t_{\mathrm{Q}}-t_{\mathrm{P}} = t_{\mathrm{P}}^{*}-t_{\mathrm{Q}}.$$

(1)

In § 3 of SR, Einstein derives the Lorentz transformation. The coordinates $x_1, x_2, x_3, t$ of an event in a system K are transformed to the coordinates $x_1^{\prime }, x_2^{\prime }, x_3^{\prime }, t^{\prime }$ in K’ by

$$x_1^{\prime }=\gamma \left(x_1-v_{3\mathrm{D}} t\right),x_2^{\prime }=x_2,x_3^{\prime }=x_3,$$

(2a)

$$t^{\prime }=\gamma \left(t-v_{3\mathrm{D}} x_1/c^2\right),$$

(2b)

where K’ moves relative to K in $x_1$ at the constant speed $v_{3\mathrm{D}}$ and $\gamma ={(1-v_{3\mathrm{D}}^2/c^2)}^{-0.5}$ 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, we will solve not less than 15 fundamental mysteries of physics. Thus, the issue is real.

The issue in SR/GR is very similar to the issue in the geocentric model: In either case, there is no holistic view but just one active perspective. In the old days, it was natural to believe that all celestial bodies would revolve around Earth. Only the astronomers wondered about the retrograde loops of planets and claimed: Earth revolves around the sun. In modern times, engineers have improved the precision of rulers and clocks. Eventually, it was natural to believe that it would be fine to describe nature as accurately as possible but from just one active perspective. The human brain is very powerful, but unfortunately it often deems itself the center/measure of everything in the universe.

The analogy is strong: (1) It holds despite the covariance of SR/GR. After a transformation (or else after replacing the center Earth), there is again just one active perspective. (2) SR/GR miss the big picture just like the geocentric model. Retrograde loops are obsolete but only in the holistic view of the heliocentric model. Dark energy and non-locality are obsolete but only in the holistic view of ER. (3) In the old days, alternatives to the geocentric model were not taken seriously. Today, alternatives to SR/GR are not taken seriously. Have physicists not learned from history? Does history repeat itself?

3. The Physics of Euclidean Relativity

The indefinite distance function in SR (Einstein, 1905b) is usually written as

$$c^2 \mathrm{d}{\tau }^2=c^2 \mathrm{d}{t}^2-\mathrm{d}{x}_1^2-\mathrm{d}{x}_2^2-\mathrm{d}{x}_3^2,$$

(3)

where $\mathrm{d}\tau$ is an infinitesimal distance in proper time $\tau$, while $\mathrm{d}t$ and $\mathrm{d}{x}_i$ ($i=1, 2, 3$) are infinitesimal distances in coordinate spacetime $x_1, x_2, x_3, t$. This spacetime is construed because coordinate space $x_1, x_2, x_3$ and coordinate time $t$ are subjective concepts: They are not immanent in rulers and clocks but construed by an observer. Rulers measure proper distance. Clocks measure proper time. We may rearrange Eq. (3) if it makes sense:

$$c^2 \mathrm{d}{t}^2=\mathrm{d}{d}_1^2+\mathrm{d}{d}_2^2+\mathrm{d}{d}_3^2+\mathrm{d}{d}_4^2,$$

(4)

where $\mathrm{d}{d}_i=\mathrm{d}{x}_i$ ($i=1, 2, 3$) and $\mathrm{d}{d}_4=c \mathrm{d}\tau$ are infinitesimal distances in 4D Euclidean space (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 indicate the full symmetry. Any (!) object’s proper space $d_1, d_2, d_3$ and proper time $\tau$ span natural spacetime, which is ES if we interpret $c\tau$ as $d_4$. This spacetime is natural because all $d_{\mu }$ ($\mu =1, 2, 3, 4$) are objective concepts: They are immanent in rulers and clocks. ER must not be confused with a Wick rotation (Wick, 1954), where time is imaginary.

For each object, we are free to label the four axes of ES. We always take $d_4$ as the axis in which the object currently moves at the speed $c$. That is, our selected axes of ES are not static. “ES diagrams” map ES from an object’s perspective. In these diagrams, the axis $d_4$ is drawn vertically. It is not observable by the object but experienced as time. 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:

$$\tau =d_4/c,{\tau }^{\prime }=d_4^{\prime }/c,$$

(5)

$$\tau =d_4 \mathit{u}/c^2,{\tau }^{\prime }=d_4^{\prime } {\mathit{u}}^{\prime }/c^2,$$

(6)

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:

$$u_1^2+u_2^2+u_3^2+u_4^2=c^2.$$

(7)

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 $\tau$. 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 $d_1\left(t\right), d_2\left(t\right), d_3\left(t\right), d_4\left(t\right)$ for granted rather than an observer’s $x_1\left(\tau \right), x_2\left(\tau \right), x_3\left(\tau \right),t\left(\tau \right)$.

Since ES is beyond (prior to) the projections, I call it the “master reality”. Spacetime in SR/GR is relative. ES itself is absolute, but the orientations of 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 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 $v_{3\mathrm{D}}=0.6 c$. 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 an observer R (or B) be next to clock “r” (or else “b”). In the blue frame of Figure 1 left, “b” displays $t^{\prime }=1.0 \mathrm{s}$ at the instant when “r” displays $t=0.8 \mathrm{s}$ (dashed line). In the red frame of Figure 1 left, “b” displays $t^{\prime }=0.8 \mathrm{s}$ at the instant when “r” displays $t=1.0 \mathrm{s}$ (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 $d_4=0.8 \mathrm{Ls}$ at the instant when “r” is at $d_4=1.0 \mathrm{Ls}$ (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, $t^{\prime }=0.8 \mathrm{s}$ is calculated only (B measures time in $\tau \prime$), while $d_4=0.8 \mathrm{Ls}$ 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” $x_1, x_2, x_3, c{t}^{\prime }$, where $c{t}^{\prime }$ 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 $x_1, x_2, x_3, c{t}^{\prime }$ to $x_1^{\prime }, x_2^{\prime }, x_3^{\prime }, ct$. In ER, unmixed $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }, d_4^{\prime }$ rotate with respect to $d_1, d_2, d_3, d_4$ (see Sect. 4).

There is also a big difference in the synchronization of clocks: In SR, each observer is able to synchronize a uniformly moving clock to his clock (same value of $t$ 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 $\mathit{\tau }\prime$ are never synchronized (different values of $d_4$ in Figure 1 right).

4. Geometric Effects in 4D Euclidean Space

We consider two identical rockets “r” (red rocket) and “b” (blue rocket). Let observer R (or B) be in the rear end of rocket “r” (or else “b”). His ES diagram is $d_1, d_2, d_3, d_4$ (or else $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }, d_4^{\prime }$). The 3D space of R (or else B) is spanned by $d_1, d_2, d_3$ (or else $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$). 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 speed $v_{3\mathrm{D}}$. We are free to label the axis of motion in 3D space. We label it $d_1$ (or else $d_1^{\prime }$). The ES diagrams in Figure 2 top must fulfill my two postulates and the initial condition (same starting point P). This is achieved by rotating the red and the blue frame with respect to each other. Figure 2 bottom shows the projection to the 3D space of R (or else B). We draw 2D rockets but are aware that their width is in $d_2, d_3$ (or else $d_2^{\prime }, d_3^{\prime }$).

We now verify: (1) The fact that the red and the blue frame are rotated with respect to each other causes length contraction. (2) The fact that proper time flows in different 4D directions for R and for B causes time dilation. Let $L_{i,j}$ be the length of rocket $i$ for observer $j$. In a first step, we project the blue rocket in Figure 2 top left to the $d_1$ axis:

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

(8)

$$L_{\mathrm{b},\mathrm{R}}={\gamma }^{-1} L_{\mathrm{b},\mathrm{B}}\hspace{1em}\hspace{1em}(\mathrm{length} \mathrm{contraction}),$$

(9)

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:

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

(10)

$$d_{4,\mathrm{B}}={\gamma }^{-1} d_{4,\mathrm{B}}^{\prime },$$

(11)

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

$$d_{4,\mathrm{R}} = \gamma d_{4,\mathrm{B}}\hspace{1em}\hspace{1em}(\mathrm{time} \mathrm{dilation}),$$

(12)

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.

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 flat 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” is

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

(13)

where $m$ is the mass of “b”, $G$ 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

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

(14)

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

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

(15)

$$\mathrm{d}{d}_{4,\mathrm{r}}={\gamma }_{\mathrm{gr}} \mathrm{d}{d}_{4,\mathrm{b}}\hspace{1em}\hspace{1em}(\mathrm{gravitational} \mathrm{time} \mathrm{dilation}),$$

(16)

where ${\gamma }_{\mathrm{gr}}={(1-2GM/(r{c}^2))}^{-0.5}$ 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 ${\gamma }_{\mathrm{gr}}$ 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 ${\gamma }_{\mathrm{gr}}$ are recovered, the experiment by Hafele and Keating (1972) also supports ER.

Three instructive problems demonstrate how to draw and how to read ES diagrams correctly (Figure 4). Problem 1: In billiards, the blue ball is hit toward the red ball. In ES, both balls move at the speed $c$. We assume that the red ball moves in its $d_4$ axis. As the blue ball covers distance in $d_1$, its speed in $d_4$ must be less than $c$. How can the two balls ever collide if their $d_4$ values do not match? Problem 2: Some rocket moves along a guide wire. In ES, rocket and wire move at the speed $c$. We assume that the wire moves in its $d_4$ axis. As the rocket covers distance in $d_1$, its speed in $d_4$ must be less than $c$. Doesn’t the wire escape from the rocket? Problem 3: Earth orbits the sun. In ES, they both move at the speed $c$. We assume that the sun moves in its $d_4$ axis. As Earth covers distance in $d_1$ and $d_2$, its speed in $d_4$ must be less than $c$. Doesn’t the sun escape from the orbital plane?

The questions in the last paragraph only seem to disclose geometric paradoxes in ER. The fallacy in all problems lies in the assumption that objects would move in ES as a function of cosmic time. We recall that cosmic time is a parameter. There is no fifth dimension beside the four dimensions $d_1, d_2, d_3, d_4$. Thus, ES is not (!) a temporal sequence of snapshots showing instants in cosmic time. ES is all snapshots at once. To improve readability, the left ES diagram in Figure 4 is the only one in this paper that shows ten snapshots at once. The guide wire does not escape from the rocket, but the wire’s proper time (related to $d_4$) flows in a direction other than the rocket’s proper time (related to $d_4^{\prime }$). All problems are solved by projecting ES to the 3D space of the object marked as “not moving”.

5. Solving 15 Fundamental Mysteries of Physics

We recall: (1) An observer’s reality is created by projecting ES orthogonally to his proper space and to his proper time. (2) There is a unique 4D vector $\mathit{\tau }$ 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.

5.1. Solving the Mystery of Time

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 $\mathit{\tau }$.

5.2. Solving the Mystery of Time’s Arrow

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.

5.3. Solving the Mystery of the Factor $c^2$ in $m{c}^2$

In SR, if forces are absent, the total energy $E$ of an object is given by

$$E=\gamma m{c}^2=E_{\mathrm{kin},3\mathrm{D}}+m{c}^2,$$

(17)

where $E_{\mathrm{kin},3\mathrm{D}}$ 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 do not move at the speed $c$. ER provides the missing clue: The object is not at rest, but it moves in its $d_4^{\prime }$ axis. From its own perspective, its $E_{\mathrm{kin},3\mathrm{D}}$ 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

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

(18)

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 $c$.

5.4. Solving the Mystery of Length Contraction and Time Dilation

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

5.5. Solving the Mystery of Gravitational Time Dilation

In GR, gravitational time dilation stems from a curved spacetime. ER discloses that gravitational time dilation stems from projecting curved geodesics in flat ES to the $d_4$ axis of an observer. Eq. (7) tells us: If an object accelerates in his proper space, it automatically decelerates in his proper time. Thus, curved geodesics in flat ES replace curved spacetime in GR. More studies will be necessary to explain other gravitational effects.

5.6. Solving the Mystery of the Cosmic Microwave Background

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 $d_1, d_2, d_3$ 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 $d_1, d_2, d_3$. Some of the scattered CRR reaches an observer on Earth as CMB (in the projection to his 3D space) after having covered the same total distance in $d_1, d_2, d_3$ as Earth in $d_4$. The cross section for scattering is low, but the fluence of the CRR is high. (2) The CRR was created and scattered equally in $d_1,d_2,d_3$. (3) Shortly after the Big Bang, the plasma particles had a very high recession speed $v_{3\mathrm{D}}$ (see Sect. 5.7).

5.7. Solving the Mystery of the Hubble–Lemaître law

According to my first postulate, all celestial bodies move through ES at the speed $c$. Let $v_{3\mathrm{D}}$ be the 3D speed at which a galaxy G recedes from Earth in 3D space. Figure 5 left tells us: At the cosmic time $t$ (the time elapsed since the Big Bang), $v_{3\mathrm{D}}$ relates to the 3D distance $D$ of G to Earth as $c$ relates to the radius $r$ of the 4D hypersphere:

$$v_{3\mathrm{D}}=Dc/r=H_t D,$$

(19)

where $H_t=c/r=1/t$ is the Hubble parameter. If we observe G today at the cosmic time $t=t_0$, the recession speed $v_{3\mathrm{D}}$ and $c$ remain unchanged. Thus, Eq. (19) turns into

$$v_{3\mathrm{D}}=D_0 c/r_0=H_0 D_0,$$

(20)

where $D_0$ is today’s 3D distance of G to Earth, $r_0$ is today’s radius of the 4D hypersphere, and $H_0=c/r_0=1/t_0$ is the Hubble constant. Eq. (20) is the Hubble–Lemaître law (Hubble, 1929; Lemaître, 1927): The farther a galaxy is, the faster it recedes from Earth. Cosmologists are aware that $H_t$ is a parameter. They are not yet aware of the 4D Euclidean geometry shown in Figure 5 left. Only ER tells us that Eqs. (19) and (20) stem from this simple geometry and that we must consider $D_0=r_0 D/r$ in Eq. (20) rather than $D$!

5.8. Solving the Mystery of the Flat Universe

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

5.9. Solving the Mystery of Cosmic Inflation

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

5.10. Solving the Mystery of the Hubble Tension

There are various methods for calculating $H_0$. I explain why the calculated values do not match (also known as the “Hubble tension”). I compare CMB measurements (Planck space telescope) with distance ladder measurements (Hubble space telescope). According to team A (Aghanim et al., 2020), there is $H_0=67.66\pm 0.42 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. According to team B (Riess et al., 2018), there is $H_0=73.52\pm 1.62 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. Team B made efforts to minimize the error margins in the distance measurements, but assuming a wrong cause of the redshifts gives rise to a systematic error in team B’s calculation of $H_0$.

Let us assume that team A’s value of $H_0$ is correct. We simulate the supernova of a star $\mathrm{S}$ that occurred at a distance of $D=400 \mathrm{Mpc}$ from Earth (Figure 5 right). The recession speed $v_{3\mathrm{D}}$ of $\mathrm{S}$ is calculated from measured redshifts. The redshift parameter $z=\mathsf{\Delta }\lambda /\lambda$ tells us how each wavelength $\lambda$ of the supernova’s light is either passively stretched by an expanding space (team B)—or else how each wavelength $\lambda$ is redshifted by the Doppler effect of actively receding objects (ER-based model). The supernova occurred at the cosmic time $t$ (arc called “past”), but we observe the supernova at the cosmic time $t_0$ (arc called “present”). While the supernova’s light was moving the distance $D$ in the $d_1$ axis, Earth moved the same distance $D$ but in the $d_4$ axis (first postulate). Thus, team B receives redshift data from a cosmic time $t<t_0$ when there was $r<r_0$ and $H_t>H_0$. There is

$$1/H_t=r/c=\left(r_0-D\right)/c=1/H_0-D/c.$$

(21)

For a very short distance of $D=400 \mathrm{kpc}$, Eq. (21) tells us that $H_t$ deviates from $H_0$ by just 0.009 percent. However, when plotting $v_{3\mathrm{D}}$ versus $D$ for distances from 0 Mpc to 500 Mpc in steps of 25 Mpc (red points in Figure 6), the slope of a straight-line fit through the origin is roughly 10 percent greater than $H_0$. Since team B calculates $H_0$ from similar plots (magnitude versus $z$), its value of $H_0$ is roughly 10 percent too high. This solves the Hubble tension. Team B’s value is not correct because, according to Eq. (20), we must not plot $v_{3\mathrm{D}}$ versus $D$. We must plot $v_{3\mathrm{D}}$ versus $D_0$ (blue points in Figure 6) to get a straight line.

Since we are not able to measure $D_0$ (observable magnitudes relate to $D$ rather than to $D_0$), the easiest way to fix the calculation of team B is to rewrite Eq. (20) as

$$v_{3\mathrm{D},0}=D c/r_0=H_0 D$$

(22)

where $v_{3\mathrm{D},0}$ is today’s 3D speed of another star ${\mathrm{S}}_0$ (Figure 5 right) that happens to be at the same distance $D$ today at which the supernova of star $\mathrm{S}$ occurred. I kindly ask team B to recalculate $H_0$ after converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$. Eqs. (21) and (22) tell us how to do so:

$$H_t=H_0 c/\left(c-H_0 D\right)=H_0/\left(1-v_{3\mathrm{D},0}/c\right),$$

(23)

$$v_{3\mathrm{D},0}=v_{3\mathrm{D}}/\left(1+v_{3\mathrm{D}}/c\right).$$

(24)

By applying Eq. (24), all red points in Figure 6 drop down to the points marked in blue. Of course, team B is well aware that the supernova’s light was emitted in the past, but all that counts in the Lambda-CDM model is the timespan during which the light is moving to Earth. Along the way, each wavelength is continuously stretched by expanding space. The parameter $z$ increases during the journey. In the ER-based model, all that counts is the moment when the supernova occurred. Each wavelength is initially redshifted by the Doppler effect. The parameter $z$ remains constant during the journey: It is tied up when the supernova occurs. Space is not expanding. A 3D hypersurface made up of energy (!) is receding in 4D space. In ER, expansion of space is an obsolete concept.

5.11. Solving the Mystery of Dark Energy

Team B can fix the systematic error in its calculation of $H_0$ by converting all $v_{3\mathrm{D}}$ 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 GR with ER unless we insist on the existence of dark energy. Perlmutter et al. (1998) and Riess et al. (1998) advocate an accelerating expansion of space because the calculated recession speeds deviate from Eq. (20) and the deviations increase with distance. An acceleration would stretch each wavelength even further.

In ER, these deviations are much easier to understand: The older the redshift data are, the more $H_t$ deviates from $H_0$, and the more $v_{3\mathrm{D}}$ deviates from $v_{3\mathrm{D},0}$. If another star ${\mathrm{S}}_0$ (Figure 5 right) happens to be at the same distance of $D=400 \mathrm{Mpc}$ today at which the supernova of star $\mathrm{S}$ occurred, Eq. (24) tells us that ${\mathrm{S}}_0$ recedes more slowly (27,064 km/s) from Earth than $\mathrm{S}$ (29,750 km/s). As long as cosmologists are not aware of the 4D Euclidean geometry, they attribute the deviations from Eq. (20) to an accelerating expansion of space caused by dark energy, but dark energy has never been observed. It is a stopgap for an effect that the Lambda-CDM model cannot explain. For $D>500 \mathrm{Mpc}$, the data marked red in Figure 6 run away from the straight line. This is caused by the fact that $v_{3\mathrm{D}}$ is proportional to $D/\left(r_0-D\right)$. In ER, dark energy is an obsolete concept.

In fact, the Hubble tension and dark energy are solved with the same clue: In Eq. (20), we must not confuse $D_0$ with $D$. Any expansion of space—uniform or else accelerating—is only virtual. Eq. (19) helps us understand the illusion of an accelerating expansion: The 3D speed $v_{3\mathrm{D}}$ is not proportional to $D$ but to $D/\left(r_0-D\right)$. There is no accelerating expansion of space even if the Nobel Prize in Physics 2011 was given “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae” (The Nobel Foundation, 2011). There are two misconceptions in these words of praise: (1) The term “Universe” in the Lambda-CDM model does imply space, but space is not expanding. (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.

5.12. Solving the Mystery of the Wave–Particle Duality

The wave–particle duality was first discussed by Bohr and Heisenberg (Heisenberg, 1969) and has bothered physicists ever since. Electromagnetic waves are oscillations of an electromagnetic field, which propagate through an observer’s 3D space at the speed $c$. In some experiments, objects behave like waves. In other experiments, the very same objects behave like particles (also known as the “wave–particle duality”). In today’s physics, one object cannot be wave and particle at once because waves distribute energy in space over time, while the energy of particles is localized in space at a given time.

Up next, we solve the duality. All we need is ER and a generalized concept of energy: All energy is “wavematter”, which may appear as a wave packet or as a particle depending on the perspective. In an observer’s reality (external view, Figure 7), a wavematter may appear as a wave packet or as a particle. As a wave, it propagates in his $x_1$ axis at the speed $c$, and it oscillates in his axes $x_2$ (electric field) and $x_3$ (magnetic field). Propagating and oscillating occur as a function of coordinate time $t$. In its own reality (internal or in-flight view), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed $c$. It deems itself particle at rest. Be aware that “wavematter” is not just a new word for the duality. It takes into account that there is an internal view of photons. There is no such view in SR/GR because it requires at least four spatial dimensions.

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. Here the external view applies. The photoelectric effect is different. Of course, one can externally witness how one photon releases an electron from a metal surface. However, the physical effect—do I have enough energy to release an electron?—is all up to the photon. Here its internal view applies. Only if the photon energy exceeds the binding energy of an electron is this electron released. In this case, both wavematters behave like particles.

The duality is also observed in matter, such as electrons (Jönsson, 1961). An electron is a wavematter too. From the external view (if the electron is not tracked), it behaves like a wave. From the internal view (if the electron is tracked), it behaves like a particle: Which slit will it pass through? Since I automatically track all slow objects (slow in my 3D space), I deem macroscopic objects particles rather than waves. This is why it is fine to draw solid rockets and celestial bodies in an observer’s ES diagrams.

5.13. Solving the Mystery of Non-Locality

The term “entanglement” was coined by Schrödinger (1935) in his comment on the Einstein–Podolsky–Rosen paradox (Einstein et al., 1935). These three authors argued that QM would not provide a complete description of reality. Schrödinger’s word creation did not solve the paradox but demonstrates our difficulties in comprehending QM. Bell (1964) showed that local hidden-variable theories are not compatible with QM. In experiments (Freedman & Clauser, 1972; Aspect et al., 1982; Bouwmeester et al., 1997), entanglement violates locality. Ever since, entanglement has been considered a non-local effect.

Up next, we untangle entanglement without the concept of non-locality. All we need is ER. Four spatial dimensions make non-locality obsolete. Figure 8 shows two wavematters that were created at once at the 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 8 (internal view), the $\pm d_4^{\prime }$ axis disappears because of length contraction at the speed $c$. In their common (!) proper space spanned by $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$, 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, but 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 if there are at least four spatial dimensions. ER explains the entanglement of electrons or atoms too. In an observer’s 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.

5.14. Solving the Mystery of Spontaneous Effects

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

5.15. Solving the Mystery of the Baryon Asymmetry

In the Lambda-CDM model, almost all matter was created shortly after the Big Bang. Only then was the temperature high enough to enable 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 shortly after the Big Bang (Canetti et al., 2012). However, such 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. In particular, there is a reasonable character paradox: What I deem antiparticle, deems itself particle.

6. Conclusions

ER solves mysteries that have not been solved in 100+ years or that have been solved but with concepts that are obsolete in ER: cosmic inflation, expansion of space, dark energy, quantum gravity, non-locality. Today’s physics needs these concepts to make cosmology and QM work, but Occam’s razor shaves them off. 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 understanding cosmology and QM are beyond the stage curtain.

It was a wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect (Einstein, 1905a) and not for SR/GR. ER penetrates to a deeper level. Einstein—one of the most brilliant physicists ever—failed to realize that the fundamental metric chosen by Mother Nature is Euclidean. Nature chose a simple but beautiful setting for life: 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.

Is ER a physical or a metaphysical theory? This is a good question because observing plays an important role in physics, yet just three dimensions of ES are observable at a time. Physics is the discipline of describing the universe and its constituents. The main tasks are observing and modeling. The issue with observing is that each observer by himself is not able to describe nature objectively. He always has a unique (subjective) perspective, which may give rise to mysteries. Unlike SR/GR, ER takes all perspectives into account at once. By taking the different perspectives of the past and the present into account, ER solves the mysteries of the Hubble tension and dark energy. By taking the different perspectives of an observer and entangled objects into account, ER solves the mystery of non-locality. ER helps us understand what we observe. Thus, ER is a physical theory.

Final remarks: (1) I addressed gravitation only briefly, but I ask you once more to be patient and fair. We should not reject ER just because gravitational effects are not yet fully understood. It is promising that ER predicts the same gravitational lensing and the same perihelion precession of Mercury’s orbit as GR (Montanus, 2023). (2) To cherish the beauty of ER, we must give ourselves a push and accept that an observer’s reality is a projection. We must not ask in physics: Why is it a projection? Nor must we ask: Why is it a probability function? In my opinion, an inflating or expanding space is at least as speculative as a projection. (3) It looks like Plato was right with his Allegory of the Cave (see Politeia, 514a): Mankind experiences a projection that is blurred—because of QM.

It is not by chance that the author of this paper is an experimental physicist. It seems that SR and GR are not suspicious to theorists. I laid the groundwork for ER and showed that ER is a conclusive and powerful theory. Paradoxes are only virtual. The true pillars of physics are ER and QM. Together, they describe the very large and the very small. Everyone is welcome to join in! May ER get the broad acceptance that it deserves.