Euclidean Relativity Solves 15 Mysteries of Physics

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. In SR, he merges them into a flat spacetime described by an indefinite distance function. SR is often presented in Minkowski space time [3] because it illustrates the invariance of the spacetime interval very well. The lifetime of muons [4] is an example that supports SR. In GR, curved spacetime is described by a pseudo-Riemannian metric. The deflection of starlight [5] and the high accuracy of GPS [6] are two examples that support GR. Quantum field theory [7] 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. My first postulate is stronger than the second postulate of SR: $c$ is absolute and universal. My second postulate is restricted to each observer’s reality rather than to inertial frames. ER also comes with a generalized concept of energy: All energy in ES is made up of quanta that may appear as wave packets and particles for different observers.

In 1969, Newburgh and Phipps introduced ER [8]. Montanus distinguished absolute from relative Euclidean spacetime (AEST, REST) [9]. AEST is ES. REST is ES in an object’s “ES diagram” (see Sect. 3). Montanus also verified gravitational lensing and the perihelion precession of elliptical orbits in ES [10,11]. He even tried to formulate electrodynamics in ES [10,11], but he overlooked that the SO(4) symmetry of ES is not compatible with waves. Almeida studied geodesics in ES [12]. Gersten showed that the Lorentz transformation is an SO(4) rotation in a “mixed space” (see Sect. 3) [13]. van Linden reviews ER models [14]. Physicists still reject ER because: (1) They expect waves to be covered. (2) Dark energy and non-locality make cosmology and QM work. (3) ER faces several paradoxes if not applied properly (see Sect. 4). This paper marks a turning point: I disclose a physical issue in SR/GR; I explain why there are no waves in ER; 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 4D Euclidean space. The 4D speed of all energy is $u_{4\mathrm{D}}=c$. Newton’s physics [15] reformed philosophy. Chances are that ER will reform physics and philosophy.

2. Disclosing an Issue in Special and General Relativity

In SR [1], there are two concepts of time: subjective coordinate time $t$ and objective proper time $\tau$. The fourth coordinate in SR is $t$. In § 1 of SR, Einstein gives an instruction of how to synchronize 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 $x_1^{\text{'}}, x_2^{\text{'}}, x_3^{\text{'}}, t^{\text{'}}$ in K’ by

$$x_1^{\text{'}} = \gamma (x_1-v_{3\mathrm{D}} t) , x_2^{\text{'}} = x_2 , x_3^{\text{'}} = x_3,$$

(2a)

$$t^{\text{'}} = \gamma (t-v_{3\mathrm{D}} x_1/c^2),$$

(2b)

where K’ moves relative to K in $x_1$ at the 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. Both theories describe nature from the perspective of just one observer each (one group of observers, to be exact). In SR, the group consists of observers who do not move relative to each other. In GR, the group consists of observers who share the same gravitational field. Because of this constraint (“just one observer each”), there is no holistic view of nature. In particular, two groups will not always agree on what is past and what is future. Physicists have been playing down the constraint by surrendering the concept of simultaneity. I show that the constraint is a physical issue. By replacing SR/GR with ER, we will solve 15 mysteries of physics. However, I do not disprove SR/GR. Both theories do their job very well in an observer’s reality.

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 Middle Ages, 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 Middle Ages, it was not accepted to challenge the geocentric model. Today, most editors do not accept papers that challenge SR/GR. Have physicists not learned from history? Does history repeat itself?

3. Basic Physics of Euclidean Relativity

The indefinite distance function in SR [1] 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 ${\mathrm{d}x}_i$ are infinitesimal distances in “coordinate spacetime” $x_1, x_2, x_3, t$. This concept is construed because all four coordinates are “extrinsic concepts” (concepts that are not immanent in rulers and clocks). An observer defines them from his proper coordinates. We rearrange Eq. (3) and get

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

(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 ES. In Eq. (4), the roles of $t$ and $\tau$ have switched: The fourth coordinate in ER is an object’s proper time $\tau$ (what any clock measures), and $t$ is the new invariant “cosmic time”. I keep the symbol $t$ to point out that Eqs. (3) and (4) are equivalent. I use the indices “1, 2, 3, 4” rather than “0, 1, 2, 3” to point out the full symmetry in ES. In ER, any object’s proper space $d_1, d_2, d_3$ and its proper time $\tau$ span “natural spacetime”, which is ES if we interpret $c\tau$ as $d_4$. This concept is natural because all four coordinates are “intrinsic concepts” (concepts that are immanent in rulers and clocks). We must not confuse the switch with the “Wick rotation” [16], which just replaces $t$ with $it$, but keeps $\tau$ as the invariant.

We are free to label the four axes of ES for each object. We always take $d_4$ as that axis in which the object moves at the speed $c$. My “ES diagrams” illustrate ES from an object’s perspective. In these diagrams, the vertical axis is always $d_4$. An object keeps moving in $d_4$ during its lifetime. An “object’s reality” is created by projecting ES orthogonally to its proper space $d_1, d_2, d_3$ and to its proper time ${\tau =d}_4/c$. For any two objects, $d_4$ and $d_4^{\text{'}}$ (proper time $\tau$ and $\tau \text{'}$) may label different 4D directions. We specify

$$\tau = d_4/c,$$

(5)

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

(6)

where $\mathit{\tau }$ is the 4D vector “proper flow of time” of an object and $\mathit{u}$ is its 4D velocity. The four components of $\mathit{u}$ are $u_i=\mathrm{d}{d}_i/\mathrm{d}t$. 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 reformulates the principle of relativity. It also defines that each “observer’s reality” is created by projecting ES orthogonally to his proper space and to his proper time. These two concepts are set equal to coordinate space and coordinate time in SR (or parameterized in GR) and reassembled to a non-Euclidean spacetime. It may sound tricky, but it just reflects that spacetime in SR/GR is customized to observers. Because of two unequal projections, there is no smooth transition from ER to SR/GR, and vice versa. We do not integrate the differentials in Eq. (3). We take an object’s $d_1\left(t\right), d_2\left(t\right), d_3\left(t\right), \tau \left(t\right)$ for granted rather than an observer’s $x_1\left(\tau \right), x_2\left(\tau \right), x_3\left(\tau \right),t\left(\tau \right)$.

ES is the origin of each observer’s reality. This is why I call ES the “master reality”. An observer’s reality is unique because the two orthogonal projections depend on his position in ES. In SR/GR, spacetime is relative. In ER, natural spacetime is absolute, but the ES diagrams and the two projections to an observer’s reality are relative. However, the SO(4) symmetry of ES is not compatible with waves. This is fine because ER tells us that wave and particle are subjective concepts: What I deem wave, deems itself particle at rest (see Sect. 5.12). We must distinguish between the master reality ES described by ER (without waves) and an observer’s reality described by SR/GR (with waves).

It is instructive to contrast coordinate time $t$, proper time $\tau$, and cosmic time $t$. Coordinate time $t$ is an extrinsic measure of time: It is equal to $\tau =\left|\mathit{\tau }\right|$ for the observer only. Proper time $\tau$ is an intrinsic measure of time: It is independent of observers. Cosmic time $t$ is invariant and thus absolute: It is the total distance covered in ES (length of a geodesic) divided by $c$. By taking $\tau$ as the fourth coordinate and the parameter $t$ as absolute time, it is possible for all observers to agree on what is past and what is future. However, time is just a subordinate quantity in ER: Only by covering distance is time passing by. Thus, I suggest to define new units for distance and speed and to measure time in these new units. In some diagrams, I project ES to an “observer’s 3D space” (his proper space). We are free to label the axis of motion in his 3D space. We often take $d_1$ as this axis.

Let us compare SR with ER. We consider two identical clocks “r” (red clock) and “b” (blue clock). In SR, “r” is at rest: It moves only in the axis $ct$ at $x_1=0$. Clock “b” starts at $x_1=0$, but it moves in the axis $x_1$ at the constant speed of $v_{3\mathrm{D}}=0.6 c$. Figure 1 left shows that 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 $ct\text{'}$ (time dilation). Thus, “b” displays “0.8”. ER is different: Figure 1 right shows that instant when either clock moved 1.0 s in its proper time. Both clocks display “1.0”. Clock “b” moved 0.6 Ls in $d_1$ and 0.8 Ls in $d_4$.

Now watch out as this paragraph demystifies time dilation: Let observer R be with clock “r”. Let observer B be with clock “b”. In SR, $t$ belongs to R and $t^{\text{'}}$ belongs to B. Observer R calculates (Lorentz transformation) that clock “b” displays $t^{\text{'}}=0.8 \mathrm{s}$. Thus, “b” is slow with respect to “r” in $t^{\text{'}}$. Time dilation in SR thus occurs in $t^{\text{'}}$, which belongs to B. In ER, $d_4$ belongs to R and $d_4^{\text{'}}$ belongs to B. Observer R measures (in his unprimed coordinate $d_4$) that clock “b” is at the position of $d_4=0.8 \mathrm{L}\mathrm{s}$. Thus, “b” is slow with respect to “r” in $d_4$. Time dilation in ER thus occurs in $d_4$, which belongs to R. In both SR and ER, “b” is slow with respect to “r”. Coordinate time $t$ or $t^{\text{'}}$ are construed coordinates, whereas proper time $\tau$ or $\tau \text{'}$ ($d_4$ or $d_4^{\text{'}}$) are natural/measurable quantities.

Gersten showed that the Lorentz transformation is an SO(4) rotation in $x_1, x_2, x_3, {ct}^{\text{'}}$ [13]. He calls these coordinates “mixed space” because ${ct}^{\text{'}}$ is the only primed coordinate. Such a mixed space does not make sense physically. We may consider it another hint that the concept of coordinate spacetime in SR has a physical issue. The Lorentz transformation rotates the mixed coordinates $x_1, x_2, x_3, {ct}^{\text{'}}$ to $x_1^{\text{'}}, x_2^{\text{'}}, x_3^{\text{'}}, ct$. In ER, the unmixed coordinates $d_1^{\text{'}}, d_2^{\text{'}}, d_3^{\text{'}}, d_4^{\text{'}}$ appear rotated 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). But if he does, the clocks aren’t synchronized from the perspective of the moving clock. In ER, clocks with the same 4D vector $\mathit{\tau }$ are always synchronized, whereas clocks with different 4D vectors $\mathit{\tau }$ and $\mathit{\tau }\mathit{\text{'}}$ are never synchronized (different values of $d_4$ in Figure 1 right). Thus, the synchronization of clocks in ER is not as tricky as in SR.

4. Geometric Effects in 4D Euclidean Space

We consider two identical rockets “r” (red rocket) and “b” (blue rocket) and assume: There is an observer R (or B) in the rear end of rocket “r” (or else rocket “b”) who uses $d_1, d_2, d_3, d_4$ (or else $d_1^{\text{'}}, d_2^{\text{'}}, d_3^{\text{'}}, d_4^{\text{'}}$) as his coordinates. $d_1, d_2, d_3$ (or $d_1^{\text{'}}, d_2^{\text{'}}, d_3^{\text{'}}$) span the 3D space of R (or else B). $d_4$ (or $d_4^{\text{'}}$) relates to the proper time of R (or else B). The rockets started at the same point P and move relative to each other at the constant 3D speed $v_{3\mathrm{D}}$. All 3D motion is in $d_1$ (or else $d_1^{\text{'}}$). The ES diagrams (Figure 2 top) must fulfill my two postulates and the requirement that both rockets started at the same point P. We achieve this only by rotating the two reference frames with respect to each other. The projection to the 3D space of R (or else B) is shown in Figure 2 bottom. For a better visualization, the rockets are drawn in 2D although their width is in the axes $d_2{, d}_3$ and $d_2^{\text{'}}, d_3^{\text{'}}$.

We now verify: (1) The reference frames of R and of B are rotated with respect to each other causing length contraction. (2) Proper time flows in different 4D directions for R and for B causing time dilation. Let $L_{i,\mathrm{R}}$ (or $L_{i,\mathrm{B}}$) be the length of rocket $i$ as measured by R (or else B). In a first step, we project the blue rocket in Figure 2 top left to the axis $d_1$.

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

(8)

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

(9)

where $\gamma =(1-v_{3\mathrm{D}}^2/c^2{)}^{-0.5}$ is the same Lorentz factor as in SR. Rocket “b” appears contracted to R by the factor ${\gamma }^{-1}$. But which distances will R observe in his axis $d_4$? For the answer, we mentally continue the rotation of rocket “b” in Figure 2 top left until it is pointing vertically down ($\phi =0°$) and serves as R’s ruler in the axis $d_4$. In the projection to the 3D space of R, this ruler contracts to zero: The axis $d_4$ disappears for R.

In a second step, we project the blue rocket in Figure 2 top left to the axis $d_4$.

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

(10)

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

(11)

where $d_{4,\mathrm{B}}$ (or $d_{4,\mathrm{B}}^{\text{'}}$) is the distance that B moved in $d_4$ (or else $d_4^{\text{'}}$). With $d_{4,\mathrm{B}}^{\text{'}}=d_{4,\mathrm{R}}$ (R and B cover the same distance in ES, but in different directions), we calculate

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

(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 showed that the Lorentz group is generated by 4D rotations [17].

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 axis $d_4$ of “r” at the speed $c$ and that “b” accelerates in the axis $d_1$ of “r” towards 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 [18] support the idea of GR that gravitation would be a feature of spacetime. However, particle physics is still considering gravitation a force, which has not yet been unified with the other 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 same axis $d_4$. At some time, “b” is sent in free fall towards Earth in the axis $d_1$ of “r” (Figure 3). The kinetic energy of “b” with the mass $m$ is

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

(13)

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 get

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

(14)

With $u_{4,\mathrm{b}}={\mathrm{d}d}_{4,\mathrm{b}}/\mathrm{d}t$ (“b” moves in the axis $d_4$ at the speed $u_{4,\mathrm{b}}$) and $c={\mathrm{d}d}_{4,\mathrm{r}}/\mathrm{d}t$ (“r” moves in the axis $d_4$ at the speed $c$), we calculate

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

(15)

$$\mathrm{d}{d}_{4,\mathrm{r}} = {\gamma }_{\mathrm{g}\mathrm{r}} \mathrm{d}{d}_{4,\mathrm{b}} (\mathrm{gravitational} \mathrm{time} \mathrm{dilation}),$$

(16)

where ${\gamma }_{\mathrm{g}\mathrm{r}}=(1-2GM/(r{c}^2{\left)\right)}^{-0.5}$ is the same dilation factor as in GR. 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 just as well as in GR! When “b” returns to “r”, the time displayed by “b” is behind the time displayed by “r”. This dilation stems from projecting curved geodesics. In GR, it stems from a curved spacetime. Let us summarize 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 time. In ER, an observed clock is slow in the observer’s flow of time. Since $\gamma$ and ${\gamma }_{\mathrm{g}\mathrm{r}}$ are recovered in ER, the Hafele–Keating experiment [19] is in line with ER, too.

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 axis $d_4$. 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’s 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 axis $d_4$. How can he ever detect the reflection? Problem 3: Earth revolves around the sun. We assume that the sun moves in its axis $d_4$. As Earth covers distance in $d_1, d_2, d_4$, 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. The fallacy in all problems lies in the assumption that there would be four observable (spatial) dimensions. Just three distances 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 axis $d_4$” 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.

5. Solving 15 Fundamental Mysteries of Physics

I recall: (1) Each observer’s reality is created by projecting ES. (2) In SR/GR, the four axes of such a reality are reassembled to a non-Euclidean spacetime. Since information is lost in each projection, the performance of SR/GR is limited. In Sects. 5.1 through 5.15, ER solves 15 mysteries of physics and declares five concepts of physics obsolete.

5.1. Solving the Mystery of Time

Cosmic time is the total distance covered in ES divided by $c$. Proper time is what any clock measures (distance $d_4$ divided by $c$). There is no definition of coordinate time other than “what I read on my clock” (with special emphasis on “I” and “my”).

5.2. Solving the Mystery of Time’s Arrow

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

5.3. Solving the Mystery of the Factor c² in mc²

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

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

(17)

where $E_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is its kinetic energy in 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: $E_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is an object’s kinetic energy in the axes $d_1, d_2, d_3$ of the observer, $m{c}^2$ is its kinetic energy in his axis $d_4$, and $\gamma m{c}^2$ is the sum of both energies. Eq. (17) tells us: All energy moves through ES at the speed $c$. 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 3D space. After dividing Eq. (18) by $c^2$, we recognize the vector addition of an object’s momentum $p_{3\mathrm{D}}$ in the axes $d_1, d_2, d_3$ of the observer and its momentum $mc$ in his axis $d_4$.

5.4. Solving the Mystery of Length Contraction and Time Dilation

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.

5.5. Solving the Mystery of Gravitational Time Dilation

ER discloses that gravitational time dilation stems from projecting curved geodesics in ES to the axis $d_4$ of an observer. If an object accelerates in his proper space, it automatically decelerates in his proper time. Of course, more research will be necessary that covers gravitation and gravitational effects in ER. In GR, gravitational time dilation stems from a curved spacetime. In Sects. 5.6 through 5.11, I will show that an ER-based model of cosmology outperforms the GR-based Lambda-CDM model.

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 localize 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 was a singularity in provided energy. Initially, all energy receded radially from O at the speed $c$. Thus, the Big Bang also provided radial momentum. Today, all energy is confined to a 4D hypersphere with the radius $r$. A lot of energy is confined to its 3D hypersurface, which is expanding at the speed $c$. Interactions (such as the isotropic emission of photons or transversal acceleration) caused some energy to depart from its radial motion while keeping the speed $c$.

Shortly after the Big Bang, energy was highly concentrated in ES. In the projection to any reality, a very hot and dense plasma was created. While this plasma was expanding, it cooled down. During plasma recombination, radiation was emitted, which we observe as cosmic microwave background (CMB) today [20]. At temperatures of roughly 3,000 K, hydrogen atoms formed. The universe became more and more transparent for the CMB. In the Lambda-CDM model, this stage was reached 380,000 years “after” the Big Bang. In the ER-based model, these are 380,000 light years “away from” the Big Bang. If there was no cosmic inflation (see Sect. 5.9), the value “380,000” needs to be recalculated.

In Figure 5, nature is described from the perspective of Earth (Earth moves vertically!). A lot of energy moves radially: It keeps the radial momentum provided by the Big Bang. The CMB in Figure 5 left moves transversally to $d_4$. It cannot move in $d_4$ because it already moves in $d_1$ at the speed $c$. Now we interpret three observations: (1) The CMB is nearly isotropic because it was created equally in the 3D space $d_1,d_2,d_3$ of an observer’s reality. (2) The temperature of the CMB is very low because of a very high recession speed $v_{3\mathrm{D}}^{\text{'}}$ (see Sect. 5.10) of all involved plasma particles. (3) We still observe the CMB today because it started moving at a very low speed $c^{\text{'}}\ll c$ in a very dense medium.

5.7. Solving the Mystery of the Hubble–Lemaître law

The speed $v_{3\mathrm{D}}$ at which a galaxy G recedes from Earth in 3D space today (Figure 5 left) relates to their 3D distance $D$ as $c$ relates to today’s 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 and $t$ is the cosmic time elapsed since the Big Bang. Eq. (19) is the Hubble–Lemaître law [21,22]: The farther a galaxy, the faster it is receding from Earth. Cosmologists are already aware that $H_t$ is a parameter rather than a constant. They are not yet aware of the 4D Euclidean geometry.

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

It is assumed that a cosmic inflation of space in the early universe [23,24] caused the isotropic CMB, the flatness of the universe, and large-scale structures (inflated from quantum fluctuations). I just demonstrated that ER explains the first two observations. ER also explains the third observation if we assume that the impacts of quantum fluctuations have been expanding in ES at the speed $c$. In ER, cosmic inflation is an obsolete concept.

5.10. Solving the Mystery of the Hubble Tension

There are several methods for calculating the Hubble constant $H_0=c/r_0$, where $r_0$ is today’s radius of the 4D hypersphere. Up next, I explain why the calculated values of $H_0$ do not match (known as the “Hubble tension”). I compare measurements of the CMB using the Planck space telescope with calibrated distance ladder techniques using the Hubble space telescope. According to team A [25], there is $H_0=67.66\pm 0.42 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. According to team B [26], there is $H_0=73.52\pm 1.62 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B made efforts to minimize the error margins in the distance measurements. I will show that misinterpreting the redshift data causes a systematic error in team B’s calculation of $H_0$. We assume that the value of team A is correct. We now simulate a supernova S’ at the distance of $D=400 \mathrm{M}\mathrm{p}\mathrm{c}$. If this supernova occurred today (S in Figure 5 right), we would calculate from Eq. (19)

$$v_{3\mathrm{D}} = H_0 D = \mathrm{27,064} \mathrm{k}\mathrm{m}/\mathrm{s},$$

(20)

$$z = \mathsf{\Delta }\lambda /{\lambda }_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}} \approx v_{3\mathrm{D}}/c = 0.0903,$$

(21)

where the redshift parameter $z$ tells us how each emitted wavelength ${\lambda }_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ of the supernova’s light 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 S’ occurred and an arc called “present” when its light arrives on Earth. Team B receives data from a time $t^{\text{'}}=1/H_{t^{\text{'}}}$ when there was $r^{\text{'}}<r_0$ and $H_{t^{\text{'}}}>H_0$. Because of my first postulate, Earth moved the same distance $D$, but in the axis $d_4$, when the light of S’ arrives. Thus, there is

$$1/H_{t^{\text{'}}} = r^{\text{'}}/c = {(r_0-D)/c = 1/H}_0-D/c,$$

(22)

$$H_{t^{\text{'}}} = 74.37 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}.$$

(23)

Since team B is not aware of Eq. (22), it concludes that 74.37 km/s/Mpc would be the value of $H_0$. In truth, team B ends up with a value $H_{t^{\text{'}}}$ of the past. For a short distance of $D=400 \mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (22) tells us that $H_{t^{\text{'}}}$ deviates from $H_0$ by only 0.009 percent. But when plotting $v_{3\mathrm{D}}^{\text{'}}$ versus $D$ for long distances (50 Mpc, 100 Mpc, ..., 450 Mpc), the slope $H_{t^{\text{'}}}$ is 8 to 9 percent higher than $H_0$, which solves the Hubble tension. I ask team B to recalculate $H_0$ after converting all $v_{3\mathrm{D}}^{\text{'}}$ to today’s value $v_{3\mathrm{D}}$. Eq. (22) tells us how to do so:

$$H_{t^{\text{'}}} = H_0 c / (c-H_0 D) = H_0 / (1-v_{3\mathrm{D}}/c),$$

(24)

$$v_{3\mathrm{D}} = v_{3\mathrm{D}}^{\text{'}} / (1+v_{3\mathrm{D}}^{\text{'}}/c).$$

(25)

Of course, team B is well aware of the fact that the supernova’s light was emitted in the past. But in the Lambda-CDM model, all that counts 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 expanding space. Thus, the redshift parameter $z$ is increasing during the journey to Earth. That moment when the supernova occurred is irrelevant. In the ER-based model, that 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 $t^{\text{'}}$ by the Doppler effect. During the journey to Earth, the redshift parameter $z^{\text{'}}$ remains constant. It is tied up in a “package” when the supernova occurs and sent to Earth, where it is measured. A 3D hypersurface (defined by its contained energy!) is expanding 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 $H_0$ within the Lambda-CDM model by converting all $v_{3\mathrm{D}}^{\text{'}}$ to $v_{3\mathrm{D}}$ according to Eq. (25). Now I reveal another systematic error that is inherent in the Lambda-CDM model itself. It has to do with assuming an accelerating expansion of space, and it can only be fixed by replacing that model with the ER-based model of cosmology (unless we postulate a dark energy). Today’s cosmologists [27,28] favor an accelerating expansion of space because the calculated recession speeds deviate from the values predicted by Eq. (19). The deviations increase with distance, and an accelerating expansion of space would stretch each ${\lambda }_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{t}}$ even more.

The ER-based model gives a simpler explanation for the deviations from the Hubble–Lemaître law: $H_{t^{\text{'}}}=1/t^{\text{'}}$ from any past is higher than $H_0$. The older the redshift data, the more does $H_{t^{\text{'}}}$ deviate from $H_0$, and the more does $v_{3\mathrm{D}}^{\text{'}}$ deviate from $v_{3\mathrm{D}}$. If a supernova S (small white circle in Figure 5 right) occurred today at the same distance of 400 Mpc as S’, the supernova S would recede slower (27,064 km/s) than S’ (29,748 km/s) just because $H_{t^{\text{'}}}$ deviates from $H_0$. As long as we are not familiar with the 4D Euclidean geometry, higher redshifts are attributed to an accelerating expansion of space. Now that we know the 4D geometry, we can attribute higher redshifts to data from deeper pasts.

In the ER-based model, all redshifts stem from the Doppler effect of receding galaxies. Because the Lorentz factor is recovered in the projections from ES, the equations of SR remain valid in an observer’s reality. Thus, there is

$$\frac{v_{3\mathrm{D}}}{c} = \frac{{\left(1+z\right)}^2-1}{{\left(1+z\right)}^2+1},$$

(26)

where $z$ is the observed redshift. While the supernova’s light moved $D$ in the axis $d_1$, Earth moved the same $D$ in the axis $d_4$ (Figure 5 right). Let $r^{\text{'}}$ be the radius when the light was created. From Eq. (19) and $r^{\text{'}}=r_0-D$, we calculate $v_{3\mathrm{D}}^{\text{'}}$ at the time $t^{\text{'}}$.

$$v_{3\mathrm{D}}^{\text{'}} = v_{3\mathrm{D}}{ r}_0/r^{\text{'}} = v_{3\mathrm{D}} / (1-D/r_0).$$

(27)

Figure 6 shows the distance modulus $\mu$ of 16 low-redshift and 24 high-redshift supernovae versus $v_{3\mathrm{D}}^{\text{'}}/c$. Low-redshift data were published by Hamuy et al [29], high-redshift data by Perlmutter et al [27]. I considered those supernovae that had been studied by both [27] and [30]. For all 40 supernovae, I calculated $v_{3\mathrm{D}}$ from Eq. (26). Then I used Eq. (27), $D={10}^{0.2\mu +1}$, and $r_0=14.25 \mathrm{G}\mathrm{p}\mathrm{c}$ to calculate $v_{3\mathrm{D}}^{\text{'}}$.

Linear regression yields the blue straight line in Figure 6. The equation is given by

$$v_{3\mathrm{D}}^{\text{'}} = H_0^{*} D,$$

(28)

where $H_0^{*}$ is a true constant. The offset “44” in Figure 6 relates to $H_0^{*}\approx 48 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$ (see Appendix B). $H_0^{*}$ is lower than $H_0$ in the Lambda-CDM model, but it is not the task of ER to recover a value that stems from a different spacetime. Only in ER do all 40 supernovae (including the high redshifts) fit very well to a straight line. Eq. (28) is the correct Hubble–Lemaître law. Space is not expanding, but energy is receding. The term “dark energy” [31] was coined to explain an accelerating expansion of space. There is no expansion of space. In ER, dark energy is an obsolete concept. It has never been observed anyway.

Any expansion of space (uniform as well as accelerating) is only virtual. There is no accelerating expansion of the Universe 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” [32]. This praise comes with two misconceptions: (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 . In each observer’s reality, there only seems to be an accelerating expansion of space.

Radial momentum provided by the Big Bang drives all galaxies away from the origin O. They are driven by themselves rather than by dark energy. If the 3D hypersurface has always been expanding at the speed $c$, the time elapsed since the Big Bang is $1/H_0^{*}$, which is 20.4 billion years rather than 13.8 billion years [33]. The new estimate would explain the existence of stars as old as 14.5 billion years [34]. Table 1 compares two models of cosmology. Be aware that “Universe” (capitalized) in the Lambda-CDM model is not the same as “universe” in the ER-based model. In the next two sections, I will demonstrate that ER is compatible with QM. Since “quantum gravity” is meant to make GR compatible with QM, I 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 Niels Bohr and Werner Heisenberg [35] and has bothered physicists ever since. Electromagnetic waves are oscillations of an electromagnetic field, which propagate through 3D space at the speed $c$. In some experiments, objects behave like waves. In other experiments, the same objects behave like particles. Up next, I explain how the very same object (here: an electromagnetic wave packet) can be deemed both wave and particle. From an observer’s perspective, it is a wave. From its own perspective, it is a particle. The following arguments hold for gravitational waves, too, if the electromagnetic field is replaced with a gravitational field.

To understand the duality, we use a generalized concept of energy: All energy in ES is made up of quanta that may appear as wave packets and particles for different observers. In Figure 7, such an energy quantum “wp” (wave packet) is illustrated. If I observe “wp” (external view, coordinate spacetime), I deem it wave: It propagates in my axis $x_1$ at the speed $c$, and it oscillates in my axes $x_2$ and $x_3$ (electromagnetic field). Propagating and oscillating occur in coordinate time $t$. However, “wp” has features of a particle, too: From its own perspective (internal view or “in-flight view”, not available in SR/GR), the axis of its 4D motion disappears because of length contraction at the speed $c$. Thus, “wp” deems itself particle at rest. The four dimensions of space enable this internal view.

Only the SO(4) symmetry of ES tells us that wave and particle are subjective concepts: What I deem wave, deems itself particle at rest. Albert Einstein once demonstrated that energy is equivalent to mass [36]. This equivalence manifests itself in the wave–particle duality: Since each wave packet moves through ES at the speed $c$, the axis of its 4D motion disappears for itself. That is to say: From its own perspective (in its own reality), all of its energy “condenses” to what we call “mass” in a particle at rest.

In a double-slit experiment, coherent energy quanta pass through a double-slit and produce some interference pattern on a screen. An observer deems them waves as long as he does not track through which slit each energy quantum is passing. Thus, he is a typical external observer. The photoelectric effect is quite different. Of course, one can externally witness how one photon releases one electron from a metal surface. But the physical effect (“Do I have enough energy to release one electron?”) is all up to the photon’s view. Only if the photon’s energy exceeds the binding energy of an electron is this electron released. Thus, we must interpret the photoelectric effect from the internal view of the photon. Here its view is crucial! The photon behaves like a particle.

The wave–particle duality is also observed in matter, such as electrons [37]. Electrons are energy quanta, too. From the internal view (if I track a single electron), this electron is a particle: Which slit will it pass through? From the external view (if I observe an electron without tracking it), this electron behaves like a wave. Because I automatically track slow objects (slow for me), I deem all macroscopic objects matter rather than waves. This argument justifies drawing solid rockets and celestial bodies in my ES diagrams.

5.13. Solving the Mystery of Non-Locality

The term “entanglement” [38] was coined by Erwin Schrödinger in his comment on the Einstein–Podolsky–Rosen paradox [39]. These three physicists argued that QM would not provide a complete description of reality. Schrödinger’s word creation did not solve the paradox, but it demonstrates our difficulties in comprehending QM. John Bell proved that QM is not compatible with local hidden-variable theories [40]. Several experiments have confirmed that entanglement violates the concept of locality [41,42,43]. Ever since has entanglement been considered a non-local effect.

Now I show how to untangle entanglement without the concept of non-locality. All we have to do is discuss it in ES: The fourth dimension of space makes non-locality obsolete. Figure 8 displays two wave packets that were created at once at a point P and are now moving away from each other in opposite directions $\pm d_4^{\text{'}}$ at the speed $c$. These wave packets are entangled. For an observer moving in a direction other than $\pm d_4^{\text{'}}$ (external view), they appear as two distinct objects. The observer cannot understand how the two wave packets communicate with each other in no time.

For each wave packet in Figure 8 (internal view), the axis $\pm d_4^{\text{'}}$ disappears because of length contraction at the speed $c$. In their common (!) proper space spanned by $d_1^{\text{'}}, d_2^{\text{'}}, d_3^{\text{'}}$, either one deems itself at the very same position as its twin. From either perspective, they are one object, which 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 four dimensions of space. ER also explains entanglement of electrons or atoms. They move at a speed $v_{3\mathrm{D}}<c$ in my proper space, but in their axis $\pm d_4^{\text{'}}$ they move at the speed $c$. Any measurement tilts the axis of 4D motion of one twin and thus 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’s energy moved 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 $90°$) 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 $c$.

5.15. Solving the Mystery of the Baryon Asymmetry

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. But the density was also very high so that baryons and antibaryons should have annihilated each other again. Since we do observe a lot more baryons than antibaryons today (known as the “baryon asymmetry”), it is assumed that an excess of baryons must have been produced in the early Universe [44]. However, such an asymmetry in pair production has never been observed.

ER solves the baryon asymmetry: Because each energy quantum 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 energy quanta deem themselves particles rather than antiparticles? The answer is that antiparticles are not the opposite of particles. An antiparticle is a particle, too, but with the opposite electric charge. Antiparticles seem to flow backward in time because proper time flows in opposite directions for any two quanta created in pair production. As they move in opposite directions at the speed $c$, they are automatically entangled.

6. Conclusions

ER solves mysteries, which SR/GR either have not solved in 100+ years or which have been solved, but only 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 we should drop concepts that prove to be obsolete (Occam’s razor). On the other hand, there are waves (electromagnetic, gravitational), but the SO(4) symmetry of ES is not compatible with waves. I thus suggest that we distinguish between the master reality ES described by ER (without waves) and an observer’s reality described by SR/GR (with waves). I demonstrated that there is a lot more physics beyond an observer’s reality. The true pillars of physics are ER and QM.

SR/GR have been confirmed many times over. Thus, they are considered two of the greatest achievements of physics. I showed that their performance is limited, and I suspect that this limitation causes today’s stagnation in physics. Physicists feel comfortable with SR/GR, but if we think of an observer’s reality as an oversized stage, the keys to cosmology and QM are beyond the curtain of this stage. Only in natural spacetime does nature disclose her secrets. We must not think that an impressive confirmation of ER would still be missing. A holistic view of nature is a necessary requirement for solving 15 fundamental mysteries, such as the Hubble tension, dark energy, and non-locality.

It was a wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect [45] rather than 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 nature is Euclidean. He sacrificed absolute space and time. I sacrifice the absoluteness of waves and particles, but I do restore absolute time (cosmic time). For the first time, mankind understands the nature of time: Time is distance covered in ES divided by the speed $c$. The human brain is able to imagine that we move through 4D space at the speed of light. With that said, conflicts of mankind become all so small.

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 elliptical orbits as GR [11]. (2) The beauty of ER is its symmetry. But to cherish ER we must give ourselves a push by accepting 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? (3) It looks like Plato was right with his Allegory of the Cave [46]: 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. The construed concepts of spacetime in SR/GR are not suspicious to theorists. This paper lays the groundwork for ER. Everyone is welcome to join in! May ER now get the broad acceptance that it deserves.