Only in Natural Spacetime Does Nature Disclose Her Secrets

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. SR is based on a flat spacetime with an indefinite distance function. SR is often interpreted in Minkowski space time, which visualizes relativistic effects very well [3]. Predicting the lifetime of muons [4] is one example that supports SR. GR is based on a curved spacetime with a pseudo-Riemannian metric. The deflection of starlight during a solar eclipse [5] and the very 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 is moving 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. I also work with a generalized concept of energy: In each observer’s reality, all objects are “wavematter” (electromagnetic wave packet and matter in one).

I am not the first author to investigate ER. Montanus described ER [8], but he tried to formulate electrodynamics in proper time [9] and overlooked that the SO(4) symmetry of ES is not compatible with waves. I will clarify why it is fine that there are no waves in ES: ER and SR/GR describe different realities. Almeida studied world lines in ES [10]. Gersten showed that the Lorentz transformation is equivalent to an SO(4) rotation [11]. van Linden gives an overview of ER models [12]. No conclusive theory of ER has yet been published. Physicists reject ER because of three reasons: (1) They expect waves to be covered by ER. (2) Customized concepts, such as dark energy and non-locality, make cosmology and QM work. (3) ER faces some paradoxes if not applied properly. This paper marks a turning point: I explain why ER is a powerful theory even if it does not cover waves; I disclose an issue in SR/GR; I avoid paradoxes by projecting ES to an observer’s reality.

It is instructive to contrast Newton’s physics, Einstein’s physics, and ER. In Newton’s physics, all energy is moving through 3D Euclidean space as a function of an independent time. The speed of matter is $v_{3\mathrm{D}}\ll c$. In Einstein’s physics, all energy is moving through 4D non-Euclidean spacetime. The speed of matter is $v_{3\mathrm{D}}<c$. In ER, all energy is moving through 4D Euclidean space. The 4D speed of everything is $u_{4\mathrm{D}}=c$. The concepts of space and time in Newton’s physics [13] once inspired the philosophy of Immanuel Kant [14]. Chances are that will ER reform both physics and philosophy.

2. Disclosing an Issue in Special and General Relativity

There are two concepts of time in SR [1]: an observer-related coordinate time $t$ and proper time $\tau$ of each observer/object. The fourth coordinate in SR is $t$. In § 1 of SR, Albert Einstein gives an instruction of how to synchronize two clocks at P and Q. At “P time” $t_{\mathrm{P}}$, a light pulse is sent from P towards Q. At “Q time” $t_{\mathrm{Q}}$, the pulse is reflected at Q towards P. At “P time” $t_{\mathrm{P}}^{*}$, it is back at P. The clock at Q synchronizes to the clock at P 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)

$$t^{\prime }=\gamma (t-v_{3\mathrm{D}}{x}_1/c^2),$$

(2b)

where K’ moves relative to K in $x_1$ at a 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 describing his reality. Because of the relativity postulate, there are similar equations for an observer B in K’ describing his reality. Physically, there is an issue: No device is able to measure $x_i$ or $t$. Rulers and clocks measure proper distance $d_i$ and proper time $\tau$. One observer sets his proper space $d_1,d_2,d_3$ and his proper time $\tau$ equal to coordinate space $x_1,x_2,x_3$ and coordinate time $t$, respectively. However, rulers and clocks are not aware of these construed concepts. Thus, they cannot display what is calculated in Eqs. (2a–b). Rules and clocks are also not aware of those $x_i$ that parameterize spacetime in GR. Thus, they cannot display what is calculated in GR either.

The issue in SR/GR is comparable to the issue in the geocentric model: In either case, there is no holistic view, but just one perspective. In the Middle Ages, it was natural to believe that all celestial bodies would revolve around Earth. Only 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 coordinate space and coordinate time (or else the parameterization in GR) would be general concepts. How could such “extrinsic concepts” (concepts that are not immanent in rulers and clocks) be general? The human brain is very powerful, but it often deems itself the center and the measure of everything in the universe.

The analogy with the geocentric model holds despite the covariance of SR/GR. It is always possible to transform the equations for a second observer. But then again, they are true for just one observer (just one planet). The analogy goes deeper than we might expect. Retrograde loops are obsolete, but only in the holistic view of the heliocentric model. Dark energy and non-locality turn out to be obsolete, but only in the holistic view of ER. I repeat: SR/GR are not wrong. They work well for each observer, but just like the geocentric model they do miss the big picture. Issues in the geocentric model were ignored for a long time. Several editors informed me that they do not accept manuscripts about issues in 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 a distance in $\tau$ and $\mathrm{d}t$ is the related distance in $t$. Coordinate space $x_1,x_2,x_3$ and coordinate time $t$ span “coordinate spacetime”. This spacetime is a construed concept because all four coordinates are nothing but constructs of one observer. We may rearrange Eq. (3) if it makes sense—I will demonstrate that it does—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 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 displays). The variable $t$ becomes the new invariant “cosmic time”. I keep the symbol $t$ to stress the equivalence of Eqs. (3) and (4). In ER, the proper coordinates $d_1,d_2,d_3,\tau$ of any object span “natural spacetime”, which is ES if we take $c\tau$ as $d_4$. This spacetime 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” [15], which replaces $t$ with $it$, but keeps $\tau$ as the invariant.

The metric in Eq. (4) is Euclidean. Because of the symmetry, we are free to label the four axes of an object’s reference frame. We always take $d_4$ as that axis in which it is moving at the speed $c$. The axes $d_1,d_2,d_3$ span its proper space. Thus, 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)

All four coordinates $d_i$ are dimensions of space. Each observer’s reality is created by projecting ES orthogonally to his proper space $d_1,d_2,d_3$ and to his proper time ${\tau =d}_4/c$. These four axes are set equal to $x_1,x_2,x_3,t$ in SR (or they are parameterized in GR) and reassembled to a non-Euclidean spacetime. It sounds tricky, but it just reflects that physics has customized space and time to one observer rather than to any object! The two projections reflect that space and time are treated differently in SR/GR. Because information is lost in each projection, ER cannot be derived from SR or GR.

I like to call ES the “master reality” because it is the origin of each observer’s reality. However, the SO(4) symmetry of ES is not compatible with waves. This is perfectly fine because waves are observer-related features: What I deem wave, deems itself matter (see Sect. 5.12 for the details). We must distinguish between an observer’s reality (with waves) and the master reality (without waves). SR/GR describe an observer’s reality. ER describes the master reality. There are features in SR/GR that do not appear in ER. Thus, ER does not include SR/GR. Rather, ER and SR/GR apply to different realities, but the realities described by SR/GR are construed from the master reality described by ER. What we can say is that ER penetrates to a deeper level. In Sect. 5, I will demonstrate that ER improves our understanding of cosmology and quantum mechanics.

It is very instructive to contrast coordinate time $t$, proper time $\tau$, and cosmic time $t$. Coordinate time $t$ is equal to $\tau =\left|\mathit{\tau }\right|$ for the observer only. Thus, it is an extrinsic measure of time. Proper time $\tau$ is an intrinsic measure of time: It is independent of observers. Cosmic time $t$ is the total distance covered in ES (total length of the world line) divided by $c$. Cosmic time is invariant and thus absolute. Proper time and cosmic time are subordinate quantities derived from distance and $c$. Only by covering distance is time passing by for each object. I thus suggest to measure distance in “light seconds”, $c$ in its own new unit to be given, and time in “light seconds per this new unit”. I use Cartesian coordinates in all diagrams. Below some diagrams, I project ES to an observer’s 3D space. We are free to label the axis of relative motion in 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, clock “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 $v_{3\mathrm{D}}=0.6c$. Figure 1 left shows that very instant when both clocks 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 $ct\prime$ (time dilation). Thus, “b” displays “0.8”. ER is different: Figure 1 right shows that very instant when both clocks moved 1.0 s in the proper time of each clock. Clock “b” moved 0.6 Ls in $d_1$. According to Eq. (7), it also moved 0.8 Ls in $d_4$. In total, “b” moved 1.0 Ls. Thus, both clocks display “1.0”.

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 $t^{\prime }$ belongs to B. Observer R calculates (Lorentz transformation) that clock “b” displays $t^{\prime }=0.8\mathrm{s}$. Thus, “b” is slow with respect to “r” in $t^{\prime }$. Time dilation in SR thus occurs in $t^{\prime }$, which belongs to B. In ER, $d_4$ belongs to R and $d_4^{\prime }$ 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 SR and ER, “b” is slow with respect to “r”. However, coordinate time $t$ and $t^{\prime }$ are construed coordinates, whereas proper time $\tau$ and $d_4$ are physical (measurable) quantities.

Gersten showed that the Lorentz transformation is equivalent to an SO(4) rotation in $x_1,x_2,x_3,{ct}^{\prime }$ [11]. He calls these coordinates “mixed space” because ${ct}^{\prime }$ is the only primed coordinate. Such a “mixed space” does not make sense physically, but we can take it as a hint that coordinate spacetime is not adequate to describe nature. In SR, the Lorentz transformation rotates the mixed coordinates $x_1,x_2,x_3,{ct}^{\prime }$ to $x_1^{\prime },x_2^{\prime },x_3^{\prime },ct$. In ER, the unmixed coordinates $d_1^{\prime },d_2^{\prime },d_3^{\prime },d_4^{\prime }$ appear rotated with respect to $d_1,d_2,d_3,d_4$ (see Sect. 4).

There is also a huge difference in the synchronization of clocks: In SR, each observer is able to synchronize a moving clock to his clock (same value of $t$ in Figure 1 left). But if he does, the two clocks aren’t synchronized from the perspective of the moving clock. In GR, the same applies to a clock in a gravitational field. 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).

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 $d_1^{\prime },d_2^{\prime },d_3^{\prime },d_4^{\prime }$) as his coordinates. $d_1,d_2,d_3$ (or $d_1^{\prime },d_2^{\prime },d_3^{\prime }$) span the 3D space of R (or else B). $d_4$ (or $d_4^{\prime }$) relates to the proper time of R (or else B). The rockets started at the same point P and are moving relative to each other at a constant speed $v_{3\mathrm{D}}$. All 3D motion is in $d_1$ (or else $d_1^{\prime }$). 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 $d_2^{\prime },d_3^{\prime }$.

We now confirm: (1) The reference frames of R and B are rotated with respect to each other causing length contraction. (2) The time of R and the time of B flow in different 4D directions 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)

$$L_{\mathrm{b},\mathrm{R}}={\gamma }^{-1}{L}_{\mathrm{b},\mathrm{B}}\left(\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right),$$

(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 $d_4$.

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

(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

$$d_{4,\mathrm{R}}=\gamma d_{4,\mathrm{B}}\left(\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right),$$

(12)

where $d_{4,\mathrm{R}}$ is the distance that R moved in $d_4$. Eqs. (9) and (12) tell us: Predictions made by SR are correct because the Lorentz factor $\gamma$ is recovered in the projections.

To understand how an acceleration in 3D space manifests itself in ES, we now assume that clock “b” accelerates in the axis $d_1$ of clock “r” towards Earth (Figure 3). We also assume that “r” and Earth move in the axis $d_4$ at the speed $c$. Because of Eq. (7), the speed $u_{1,\mathrm{b}}$ of “b” in the axis $d_1$ increases at the expense of its speed $u_{4,\mathrm{b}}$ in the axis $d_4$.

Gravitational waves [16] 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 world lines in ES replace curved spacetime in GR. To support my claim, I now use ES coordinates to calculate gravitational time dilation. Let “r” and “b” be two identical clocks, which are far away from Earth. Initially, they are next to each other and move in the same axis $d_4$ at the speed $c$. At some time, clock “b” is sent in free fall towards Earth in the axis $d_1$ of clock “r”. The kinetic energy of clock “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)

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

(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: Predictions made by GR are correct because the dilation factor ${\gamma }_{\mathrm{g}\mathrm{r}}$ is recovered in the projection. Thus, GPS satellites do their job in ER as well as in GR! If clock “b” returns to clock “r”, the time displayed by “b” will be behind the time displayed by “r”. This dilation is due to projecting curved world lines. In GR, it is due to a curved spacetime. Here is a short summary of how time dilation manifests in SR, GR, and ER: In SR and ER, a moving clock is slow with respect to an observer. In GR and ER, a clock in a gravitational field is slow with respect to an observer. In SR/GR an observed clock is slow in its proper time. In ER, an observed clock is slow in the observer’s proper time.

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. Because information is lost in each projection, the performance of SR/GR must be limited. In this section, I show: ER solves 15 mysteries of physics, and it 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 displays (distance $d_4$ divided by $c$). By contrast, there is no definition of coordinate time other than “what I read on my clock” (attributed to Einstein himself).

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 $c^2$ in $m{c}^2$

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 is moving 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 world lines 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. In GR, gravitational time dilation is due to a curved spacetime. However, ER and GR do not compete against each other. GR describes an observer’s reality. ER describes the master reality. Of course, more detailed studies will be necessary that address gravitation and gravitational effects in ER.

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. Euclidean space exists just like all 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 ER, the Big Bang can be localized: It injected a huge amount of energy into a non-inflating ES all at once at what I call “origin O”, the only natural reference point in ES. The Big Bang was a singularity in provided energy. Initially, all energy was receding radially from O at the speed $c$. That is, the Big Bang also provided an overall radial momentum. Today, all energy is confined to a 4D hypersphere with the radius $r$, and 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 [17]. 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 ER, 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” still needs to be recalculated in ER.

Figure 5 left shows the ES diagram for observers on Earth (here Earth is moving in $d_4$). A lot of energy moves radially: It keeps the radial momentum provided by the Big Bang. The CMB moves transversally to $d_4$. It cannot move in $d_4$ because it already moves in $d_1$ at the speed $c$. Now I 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 $v_{3\mathrm{D}}^{\prime }$ (see Sect. 5.10 for the details) of all involved plasma particles. (3) We still observe the CMB today because it started moving at a very low speed $c^{\prime }\ll c$ 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 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 and $t$ is the cosmic time elapsed since the Big Bang. Eq. (19) is the Hubble–Lemaître law [18,19]: 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 [20,21] 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 do not match. I consider measurements of the CMB made with the Planck space telescope and compare them with calibrated distance ladder techniques using the Hubble space telescope. According to team A [22], 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 [23], there is $H_0=73.52\pm 1.62\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$.

Team B made great efforts to minimize the error margin by optimizing the distance measurements. I will show that misinterpreting the redshift data causes a systematic error in team B’s calculation of $H_0$. Let us 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

$$v_{3\mathrm{D}}=H_{0}D=\mathrm{27,064}\mathrm{k}\mathrm{m}/\mathrm{s},$$

(20)

$$z=\mathsf{\Delta }\lambda /{\lambda }_0\approx v_{3\mathrm{D}}/c=0.0903,$$

(21)

where the redshift parameter $z$ tells us how any wavelength ${\lambda }_0$ of the supernova’s light is either passively stretched by an expanding space (team B), or how each ${\lambda }_0$ is redshifted by the Doppler effect of objects that are actively receding in ES (ER-based model). In Figure 5 right, there is an arc “past” when the supernova S’ occurred and an arc “present” when its light arrives on Earth. Because all energy is moving through ES at the speed $c$, Earth moved the same distance $D$, but in the axis $d_4$, when the light of S’ arrives. Thus, team B is receiving data from a time $t^{\prime }=1/H_{t^{\prime }}$ when there was $r^{\prime }<r_0$ and $H_{t^{\prime }}>H_0$.

$$1/H_{t^{\prime }}=r^{\prime }/c={(r_0-D)/c=1/H}_0-D/c,$$

(22)

$$H_{t^{\prime }}=74.37\mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}.$$

(23)

Thus, team B is calculating $H_{t^{\prime }}$ rather than $H_0$ because it does not take Eq. (22) into account. For a short distance of $D=400\mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (22) tells us that $H_{t^{\prime }}$ deviates from $H_0$ by only 0.009 percent. When plotting $v_{3\mathrm{D}}^{\prime }$ versus $D$ for long distances (50 Mpc, 100 Mpc, ..., 450 Mpc), the slope $H_{t^{\prime }}$ is indeed 8 to 9 percent higher than $H_0$. I kindly ask team B to adjust the calculated speed $v_{3\mathrm{D}}^{\prime }$ to today’s speed $v_{3\mathrm{D}}$ by converting Eq. (22) to

$$H_{t^{\prime }}=H_{0}c/(c-H_{0}D)=H_0/(1-v_{3\mathrm{D}}/c),$$

(24)

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

(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 traveling from the supernova to Earth. Along the way, its wavelength is passively stretched by expanding space. The moment when the supernova occurred is irrelevant. In the ER-based model, that moment is relevant, but the timespan is irrelevant. The wavelength of the light is initially redshifted by the Doppler effect. During the journey to Earth, the parameter $z^{\prime }$ remains constant. It is tied up in some “package” when the supernova occurs and then sent to Earth, where it is measured. A 3D hypersurface (made of energy!) is expanding rather than space. In ER, expansion of space is an obsolete concept.

5.11. Solving the Mystery of Dark Energy

The systematic error made by team B can be fixed within the Lambda-CDM model by adjusting $v_{3\mathrm{D}}^{\prime }$ to today’s speed $v_{3\mathrm{D}}$ according to Eq. (25). Up next, I reveal a 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 this model with the ER-based model of cosmology (unless we believe in dark energy). Today’s cosmologists favor an accelerating expansion of space because the calculated recession speeds deviate from the values predicted by Eq. (19). Deviations increase with distance and are attributed to an accelerating expansion of space, which would stretch each ${\lambda }_0$ even more.

The ER-based model gives a simpler explanation for the deviations from the Hubble–Lemaître law: $H_{t^{\prime }}=1/t^{\prime }$ from any past is higher than $H_0$. The older the redshift data, the more does $H_{t^{\prime }}$ deviate from $H_0$, and the more does $v_{3\mathrm{D}}^{\prime }$ 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 $H_{t^{\prime }}$ 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.

Perlmutter et al [24] and Riess et al [25] interpret data from high-redshift supernovae as an accelerating expansion of space. In ER, 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 $r^{\prime }$ be the radius when the light was created. From Eq. (19) and $r^{\prime }=r_0-D$, we calculate $v_{3\mathrm{D}}^{\prime }$ at the time $t^{\prime }$.

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

(27)

Figure 6 shows the distance modulus $\mu$ of 16 low-redshift and 24 high-redshift supernovae versus $v_{3\mathrm{D}}^{\prime }/c$. Low-redshift data were published by Hamuy et al [26], high-redshift data by Perlmutter et al [24]. I considered those supernovae that had been studied by both [24] and [27]. 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 $v_{3\mathrm{D}}^{\prime }$.

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

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

(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” [28] 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.

Thus, 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”. This praise contains two misconceptions: (1) In the Lambda-CDM model, “Universe” also implies space, but space is not expanding at all. (2) There is receding energy, but it is moving uniformly in ES 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 [29]. The new estimate would explain the existence of stars as old as 14.5 billion years [30]. 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 [31] 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 same objects behave like matter (particles). Up next, I explain how one and the same object can be deemed both wave and matter. From an observer’s perspective, each object is wave or matter depending on its 3D speed. From its own perspective, it is always matter.

We will work with a generalized concept of energy (Figure 7): In each observer’s reality, all objects are “wavematter” (electromagnetic wave packet and matter in one). If I observe a wavematter WM in my reality (external view, coordinate spacetime!), I deem it wave (if its speed is $v_{3\mathrm{D}}=c$), or matter ($v_{3\mathrm{D}}\ll c$), or either one ($v_{3\mathrm{D}}<c$). If I deem WM wave, it propagates in my axis $x_1$ and it oscillates in my axes $x_2$ and $x_3$ (electromagnetic field). Propagating and oscillating occur in coordinate time $t$. However, WM has features of a particle, too: From its own perspective (internal view or in-flight view), the axis of its 4D motion disappears because of length contraction at the speed $c$. Thus, WM deems itself matter at rest. Be aware that “wavematter” is not just another word for the duality, but a generalized concept of energy that discloses why there is a duality.

Only the SO(4) symmetry of ES tells us: What I deem wave, deems itself matter. Einstein demonstrated that energy is equivalent to mass [32]. This equivalence manifests itself in the wave–particle duality: Because each wavematter is moving through ES at the speed of light $c$, its 4D motion is suppressed for itself. From its own perspective (in its own reality), its energy “condenses” to mass in matter at rest. Waves and thus the wave–particle duality are observer-related features. They appear only in an observer’s reality.

In a double-slit experiment, an observer detects coherent waves that pass through a double-slit and produce some pattern of interference on a screen. He deems all of these wavematters waves because he is not tracking through which slit each wavematter passes. Thus, he is an 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 up to the photon’s view. Only if the photon’s energy exceeds the binding energy of an electron is this electron released. Thus, the photoelectric effect must be interpreted from the internal view of the photon. Here its view is crucial! It behaves like a particle.

The wave–particle duality is also observed in matter, such as electrons [33]. According to my generalized concept of energy, electrons are wavematter, too. From the internal view (if I were the electron), the electron is a particle: Which slit will I pass through? From the external view (if I do not track single electrons), electrons behave like waves. Because I automatically track slow objects (slow for me), I deem all macroscopic wavematters matter. This argument justifies drawing solid rockets and celestial bodies.

5.13. Solving the Mystery of Quantum Entanglement

The term “entanglement” [34] was coined by Erwin Schrödinger in his comment on the Einstein–Podolsky–Rosen paradox [35]. 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 [36]. Several experiments have confirmed that quantum entanglement violates the concept of locality [37,38,39]. Ever since has it 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 solve it in ES: The fourth dimension of space makes non-locality obsolete. Figure 8 displays two wavematters that were created at once at a point P and are now moving away from each other in opposite directions $\pm d_4^{\prime }$ at the speed $c$. These two wavematters are entangled. If they are observed by a third wavematter moving in a direction other than $\pm d_4^{\prime }$, they appear as two objects (external view). The third wavematter cannot understand how the other two wavematters communicate with each other in no time.

And here is the internal view: For each entangled wavematter in Figure 8, the axis $\pm d_4^{\prime }$ 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 one of them 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. The different positions in ${\mathrm{d}}_4^{\prime }$ are irrelevant: The twins stay together in their proper space even if their proper time flows in opposite directions. Entanglement occurs because observer and observed objects may experience different proper spaces and different 4D vectors $\mathit{\tau }$ and $\mathit{\tau }\mathit{\text{'}}$. 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 $\pm d_4^{\prime }$ 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 was moving with the atom. After the emission, this energy is moving 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 are moving at the speed $c$. So, 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—and this energy speeds off at once. In absorption, a photon is spontaneously absorbed by an atom. Today’s physics cannot explain how the photon’s energy is slowed down to the atom’s speed in no time. In ES, both photon and atom are moving at the speed $c$. So, there is no need to slow down any energy. Similar arguments apply to pair production and to annihilation. Spontaneous effects are another clue that energy is always moving 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 [40]. However, such an asymmetry in pair production has never been observed.

ER offers a unique solution to the baryon asymmetry: Since each wavematter deems itself matter, there was matter in ES immediately after the Big Bang. Today, there is much less antimatter than matter because antimatter is created in pair production only. One may ask: Why does wavematter not deem itself antimatter? The answer is that energy has two faces: wave and matter. Antimatter is not the opposite of matter; antimatter has the opposite electric charge. Antimatter seems to flow backward in time because proper time flows in opposite directions for any two wavematters created in pair production. As they move in opposite directions at the speed $c$, they are automatically entangled.

6. Conclusions

ER solves mysteries that have not been solved in 100+ years or that have been solved by adding several customized concepts: cosmic inflation, expansion of space, dark energy, quantum gravity, and non-locality. These concepts are obsolete in ER, but they are needed in today’s physics to make cosmology and QM work. On the other hand, electromagnetic and gravitational waves are facts in today’s physics, but they do not appear in ES because of its SO(4) symmetry. ER and SR/GR do not compete against each other. We must distinguish between an observer’s reality (with waves) and the master reality (without waves). SR/GR describe an observer’s reality. ER describes the master reality.

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 coordinate spacetime in SR (or else of parameterized spacetime in GR) as an oversized stage, ER tells us: The keys to cosmology and QM are beyond the curtain. Only in natural spacetime does nature disclose her secrets. The deflection of starlight is an impressive confirmation of GR [5]. We would be mistaken if we believe that an impressive confirmation of ER is still missing. I demonstrated that ER is very powerful: It improves our understanding of cosmology and quantum mechanics.

It was a very wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect [41] rather than for SR/GR. ER penetrates to a deeper level. Einstein, one of the most brilliant physicists ever, did not realize that the fundamental metric chosen by nature is Euclidean. Einstein sacrificed absolute space and time. I sacrifice the absoluteness of waves and matter, but I restore absolute (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 are moving through 4D space at the speed of light. With that said, conflicts of mankind become all so small.

Final remarks: (1) I ask you once more to be patient and fair. For instance, I have only briefly addressed gravitation. But I am confident that ER will improve our understanding of gravitation, too. We should not reject ER just because detailed studies about gravitation are still missing. (2) The beauty of ER is its symmetry. However, you will cherish ER only if you give yourself a little push by accepting that an observer’s reality is only 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 [42]: 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. In SR and GR, spacetime is a construed concept, which is not suspicious to theorists. This paper lays the groundwork for ER. Everyone is welcome to join in! May ER get the broad acceptance that it deserves.