Natural Spacetime: Describing Nature in Natural Concepts

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. In SR, space and time are merged into a flat spacetime described by the Minkowski metric. SR is often presented in Minkowski space time [3]. Predicting the lifetime of muons [4] is one example that supports SR. In GR, a curved spacetime is described by the Einstein tensor. 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 QM but not GR.

The postulates of ER: (1) All energy moves through Euclidean spacetime (ES) at the speed of light $c$. (2) The laws of physics have the same form in each observer’s reality. An observer’s reality is created by orthogonally projecting ES to his proper space and proper time. In SR/GR, these two projections are reassembled to a non-Euclidean spacetime. The reassembly is not a topic of my paper. It is a fact that spacetime in SR/GR is non-Euclidean. Information is lost in projections. This implies that ER goes beyond SR/GR and that we face mysteries if we ignore ES. My first postulate is stronger than the second postulate of SR: $c$ is absolute and universal. My second postulate refers neither to ES nor to inertial frames but to observers’ realities. In addition to the postulates, I introduce three natural concepts: Proper space/time replaces coordinate space/time. “Pure energy” (see Section 5.13) replaces wave/particle. I give one example where a process replaces force/field. To improve readability, all observers are male. To make up for it, Mother Nature is female.

Because of the two orthogonal projections to an observer’s proper space and proper time, I suggest that we call ES the “master reality”. Figure 1 left illustrates how to relate an observer’s reality to the master reality. Figure 1 right illustrates where to apply ER and where to apply SR/GR. ER describes the master reality, which is beyond each observer’s reality, and also how his reality relates to ES. SR/GR describe each observer’s reality and also how the realities of two observers relate to each other. Note that ER describes nature but not an observer’s reality. Because of the different realities, ER does not compete with SR/GR. Thus, it is not necessary (and not possible because empirical concepts are excluded in ER) to show that ER solves those mysteries that are solved by SR/GR. Those mysteries remain solved by SR/GR. The benefit of ER is that it solves 15 additional mysteries.

In 1969, Newburgh and Phipps [8] pioneered ER. Montanus [9] added a constraint: A pure time interval must be a pure time interval for all observers. According to Montanus [10], this constraint is required to avoid the twin paradox and a “character paradox” (confusion of photons, particles, antiparticles). I show that the constraint is obsolete. Whatever is proper time for me, it may be one axis of proper space for you. There is no twin paradox if we consider cosmic time as the parameter. There is no character paradox if we consider “pure energy”. Montanus [11] tried to describe kinematics in ES using the Lagrange formalism. Montanus [10] tried to set up Maxwell’s equations in ES but failed because of a plus sign. He overlooked that the SO(4) symmetry of ES is incompatible with waves.

Almeida [12] studied various geodesics in ES. Gersten [13] showed that the Lorentz transformation is an SO(4) rotation (see Section 3). van Linden maintains a website about ER (https://euclideanrelativity.com/). Most physicists reject ER because dark energy and non-locality make cosmology and QM work, ER excludes waves, and paradoxes turn up if ER is believed to describe an observer’s reality. This paper marks a turning point: I disclose an issue in SR/GR. I justify the exclusion of waves. I avoid paradoxes by projecting ES.

It is instructive to contrast Newton’s physics, Einstein’s physics, and ER. In Newton’s physics, all energy moves through 3D Euclidean space as a function of independent time. There is no speed limit for matter. In Einstein’s physics, all energy moves through a non-Euclidean spacetime. The 3D speed of matter is $v_{3\mathrm{D}}<c$. In ER, all energy moves through ES. The 4D speed of all energy is $c$. Newton’s physics [14] shaped Kant’s philosophy [15]. I am convinced that ER will trigger a reformation of physics and philosophy.

2. Disclosing an Issue in Special and General Relativity

The fourth coordinate in SR is an observer’s coordinate time $t$. In § 1 of SR, Einstein gives an instruction for synchronizing clocks at the points P and Q. At $t_{\mathrm{P}}$, a light pulse is sent from P to Q. At $t_{\mathrm{Q}}$, it is reflected at Q. At $t_{\mathrm{P}}^{*}$, it is back at P. The clocks synchronize if

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

(1)

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

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

(2a)

$$x_2^{\prime } = x_2, x_3^{\prime } = x_3,$$

(2b)

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

(2c)

where K’ moves relative to K in $x_1$ at the constant speed $v_{3\mathrm{D}}$ and $\gamma =(1-v_{3\mathrm{D}}^2/c^2{)}^{-0.5}$ is the Lorentz factor. Mathematically, Eqs. (1) and (2a–c) are correct for observers in K. There are covariant equations for observers in K’. Physically, there is an issue in SR and also in GR: The empirical concepts of SR/GR fail to solve fundamental mysteries. The scope of SR/GR is limited to observers’ realities, just as the scope of Newton’s physics is limited to speeds far less than $c$. There are coordinate-free formulations of SR [16] and GR [17], but there is no absolute time in SR/GR and thus no holistic view (I repeat my definition: view that is universal for all objects at the same instant in time). The view in SR/GR is multi-egocentric: SR/GR work for all observers, but each observer’s view is egocentric. All observers’ views taken together do not make a holistic view because they still do not provide absolute time. Without absolute time, observers do not always agree on what is past and what is future. Physics paid a high price for dismissing absolute time: ER restores absolute time (see Section 3) and solves 15 mysteries (see Section 5). Thus, the issue in SR/GR is real.

The issue in SR/GR is not about making wrong predictions. It has much in common with the issue in the geocentric model: In either case, there is no holistic view. Geocentrism is the egocentric view of mankind. In the old days, it was natural to believe that all celestial bodies would orbit Earth. Only the astronomers wondered about the retrograde loops of planets and claimed that Earth orbits the sun. In modern times, engineers have improved rulers and clocks. Today, it is natural to believe that it would be fine to describe nature as accurately as possible but in the empirical concepts of one or more observers. The human brain is smart, but it often takes itself as the center/measure of everything.

The analogy of SR/GR to the geocentric model is stunningly close: (1) It holds despite all covariances. After a transformation in SR/GR (or after appointing another planet as the center of the Universe), the view is again egocentric (or else geocentric). (2) Dark energy and non-locality make cosmology and QM work, but they are obsolete in ER (see Section 5). The retrograde loops make the geocentric model work, but they are obsolete in the heliocentric model. (3) ER is rejected today. The heliocentric model was rejected in the old days. Have physicists not learned from history? Does history repeat itself?

3. The Physics of Euclidean Relativity

The Minkowski metric in SR is often 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 $\tau$, whereas $\mathrm{d}t$ and ${\mathrm{d}x}_i$ ($i=1, 2, 3$) are infinitesimal distances in an observer’s coordinate space $x_1, x_2, x_3$ and coordinate time $t$. Coordinate spacetime $x_1, x_2, x_3, t$ is empirical because its four coordinates are related to an observer. They are not immanent in rulers and clocks. Rulers measure proper length. Clocks measure proper time. I introduce ER by defining its Euclidean metric

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

(4)

where $\mathrm{d}\theta$ is an infinitesimal distance in cosmic time $\theta$, whereas all ${\mathrm{d}d}_i$ ($i=1, 2, 3$) and ${\mathrm{d}d}_4=c \mathrm{d}\tau$ are infinitesimal distances in ES. The roles of $\theta$ and $\tau$ are switched in Eq. (4): The new invariant is absolute, cosmic time $\theta$. The new fourth coordinate is proper time $\tau$. The new metric tensor is the identity matrix. I prefer the indices 1–4 to 0–3 to stress the full symmetry in all four coordinates. I select the symbol $\theta$ because the initial of the Greek letter “theta” is t as in “time”. An object’s proper space $d_1, d_2, d_3$ and proper time $\tau$ span Euclidean spacetime $d_1, d_2, d_3, d_4$ (ES) with $d_4=c\tau$. ES is natural because its four coordinates $d_{\mu }$ ($\mu =1, 2, 3, 4$) are related to the objects of nature. They are measured by and thus immanent in rulers and clocks. Do not confuse Eq. (4) with a “Wick rotation” [18], where proper time $\tau$ is the invariant and coordinate time $t$ is imaginary.

Each object is free to label the axes of ES. We assume that it labels the axis of its current 4D motion as $d_4$. Since it does not move in its proper space, it moves in the $d_4$ axis at the speed $c$ (my first postulate). Because of length contraction at the speed $c$ (see Section 4), the $d_4$ axis disappears for itself and is experienced as proper time. Objects moving in the $d_4^{\prime }$ axis at the speed $c$ experience the $d_4^{\prime }$ axis as proper time. Each object experiences its own 4D motion as proper time. In other words: An object’s proper time flows in the direction of its 4D motion. Thus, there is a relative 4D vector “flow of proper time” $\mathit{\tau }$.

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

(5)

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

(6)

where $\mathit{u}$ is an object’s 4D velocity in ES. There is $u_{\mu }=\mathrm{d}{d}_{\mu }/\mathrm{d}\theta$, where $\theta$ is cosmic time. Speed is not a spatial coordinate divided by the fourth coordinate but any coordinate divided by the invariant. Thus, Eq. (4) is not a random metric but my first postulate

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

(7)

ES is orthogonally projected to an observer’s proper space and proper time. Thus, we can always construe an observer’s reality from ES but not vice versa. This is not an issue because SR describes nature in empirical concepts $x_1\left(\tau \right), x_2\left(\tau \right), x_3\left(\tau \right),t\left(\tau \right)$, where $\tau$ is the parameter and $t$ is coordinate time. On the contrary, ER describes nature in natural concepts $d_1\left(\theta \right), d_2\left(\theta \right), d_3\left(\theta \right), d_4\left(\theta \right)$, where $\theta$ is the parameter and $d_4$ relates to $\tau$. Only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. This is not an issue either as I explain in my Conclusions.

It is instructive to contrast the three concepts of time. Coordinate time $t$ is a subjective measure of time: An observer uses his clock as the master clock. Proper time $\tau$ is an objective measure of time: Clocks measure $\tau$ independently of observers. Cosmic time $\theta$ is the total distance covered in ES (length of a worldline) divided by $c$. By taking $\theta$ as the parameter, all observers will agree on what is past and what is future. Since $\theta$ is absolute, there is no twin paradox in ER. Twins are the same age in cosmic time.

We consider two identical clocks “r” (red clock) and “b” (blue clock). In SR, “r” moves in the $ct$ axis. Clock “b” starts at $x_1=0$ and moves in the $x_1$ axis at a constant speed of $v_{3\mathrm{D}}=0.6 c$. Figure 2 left shows the instant when either clock moved 1.0 s in $ct$. Clock “b” moved 0.6 Ls (light seconds) in $x_1$ and 0.8 Ls in $c{t}^{\prime }$. It displays “0.8”. In ER, “r” moves in the $d_4$ axis. Figure 2 right shows the 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 $d_4$.

We now assume that an observer R (or B) is moving with the clock “r” (or else “b”). In SR and only from R’s perspective, clock “b” is at $c{t}^{\prime }=0.8 \mathrm{L}\mathrm{s}$ when “r” is at $ct=1.0 \mathrm{L}\mathrm{s}$ (see Figure 2 left). Thus, “b” is slow with respect to “r” in $t^{\prime }$ (of B). In ER and independently of observers, clock “b” is at $d_4=0.8 \mathrm{L}\mathrm{s}$ when “r” is at $d_4=1.0 \mathrm{L}\mathrm{s}$ (see Figure 2 right). Thus, “b” is slow with respect to “r” in $d_4$ (of R). In SR and ER, “b” is slow with respect to “r”, but time dilation occurs in different axes. Experiments do not disclose the axis in which a clock is slow. Thus, SR and ER may claim that they describe time dilation correctly.

But why does ER provide a holistic view? Well, ES is independent of observers and thus absolute. This justifies the name “master reality”. Only the projections are relative. Absolute ES shows up in its rotational symmetry: Figure 2 right works for R and for B “at once” (at the same instant in cosmic time), that is, it provides a universal view. The view in Figure 2 left is not universal because a second Minkowski diagram is required for B, where $x_1^{\prime }$ and $c{t}^{\prime }$ are orthogonal. Absolute ES shows up in Eq. (4) too: All four $d_{\mu }$ ($\mu =1, 2, 3, 4$) are interchangeable. Only observers experience distance as spatial or temporal.

Gersten [13] showed that the Lorentz transformation is an SO(4) rotation in a “mixed space” $x_1, x_2, x_3, {ct}^{\prime }$, where only ${ct}^{\prime }$ is primed. The four mixed coordinates $x_1, x_2, x_3, {ct}^{\prime }$ rotate to $x_1^{\prime }, x_2^{\prime }, x_3^{\prime }, ct$. I will not repeat the derivation. I consider it my task to promote ER. However, a mixed space is physically pointless. In ER, unmixed $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }, d_4^{\prime }$ rotate with respect to $d_1, d_2, d_3, d_4$. 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 $ct$ in Figure 2 left). If he does, these clocks are not synchronized for the moving clock. In ER, clocks with the same 4D vector $\mathit{\tau }$ are always synchronized, whereas clocks with different $\mathit{\tau }$ and ${\mathit{\tau }}^{\mathit{\text{'}}}$ are never synchronized (different values of $d_4$ in Figure 2 right).

4. Geometric Effects in Euclidean Relativity

We consider two identical rockets “r” (red rocket) and “b” (blue rocket). Let observer R (or B) be in the rear end of “r” (or else “b”). The 3D space of R (or B) is spanned by $d_1, d_2, d_3$ (or else $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$). We use “3D space” as a synonym of proper space. The proper time of R (or B) relates to $d_4$ (or else $d_4^{\prime }$) according to Eq. (5). Both rockets start at the point P and move relative to each other at the constant speed $v_{3\mathrm{D}}$. R and B are free to label the axis of relative motion in 3D space. R (or B) labels it as $d_1$ (or else $d_1^{\prime }$). The ES diagrams in Figure 3 must fulfill my two postulates and the initial condition (starting at the same point P). This is achieved by rotating the red and the blue frame with respect to each other. To improve readability, a rocket’s width is drawn in $d_4$ (or $d_4^{\prime }$). Do not confuse the ES diagrams with Minkowski diagrams. In ES diagrams, objects maintain proper length and clocks display proper time. Figure 3 bottom shows the projection to the 3D space of R (or B).

We now assume that $N$ rockets “${\mathrm{r}}_i$” are launched from the point P (“${\mathrm{r}}_1$” is equal to “r”). Each rocket “${\mathrm{r}}_i$” ($2\le i\le N$) recedes from “${\mathrm{r}}_{i-1}$” to the right in the 3D space of “${\mathrm{r}}_{i-1}$” at the speed $v_{3\mathrm{D}}$. In ES, each rocket “${\mathrm{r}}_i$” rotates with respect to “${\mathrm{r}}_{i-1}$” by an angle $\pi /2-\phi$ because of the SO(4) symmetry. If $N(\pi /2-\phi )>\pi /2$, some of the rockets move backward in $d_4$. If one rocket rotates with respect to “${\mathrm{r}}_1$” by the angle $\pi$, it even stands still in the 3D space of “${\mathrm{r}}_1$”, and its proper time flows in ${-d}_4$. This outcome does not contradict SR because ES diagrams do not display coordinate time but proper time. The relative speeds $v_{3\mathrm{D}}$ do not add up. Rather, adding 4D vectors to the 4D vector $\mathit{\tau }$ of “${\mathrm{r}}_1$” may result in the reversed 4D vector $-\mathit{\tau }$. It may even result in $\mathit{\tau }$ if the last rocket rotates by the angle $2\pi$. The 4D vector $-\mathit{\tau }$ is the key to solving entanglement (see Section 5.14).

Up next, we verify: (1) Rotating the red and the blue frame with respect to each other causes length contraction. (2) The fact that proper time flows in different 4D directions for R and for B causes time dilation. Let $L_{\mathrm{b},\mathrm{R}}$ (or $L_{\mathrm{b},\mathrm{B}}$) be the length of the rocket “b” for the observer R (or else B). In a first step, we project “b” in Figure 3 top left to the $d_1$ axis.

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

(8)

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

(9)

where $\gamma =(1-v_{3\mathrm{D}}^2/c^2{)}^{-0.5}$ is the same Lorentz factor as in SR. For R, the rocket “b” contracts to $L_{\mathrm{b},\mathrm{R}}$. Which distances will R observe in $d_4$? We mentally continue the rotation of the rocket “b” until $v_{3\mathrm{D}}=c$, that is, until “b” serves as a ruler for R in $d_4$. In his 3D space, this ruler contracts to a point: The $d_4$ axis disappears for R because of length contraction at the speed $c$. In a second step, we project “b” in Figure 3 top left to the $d_4$ axis.

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

(10)

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

(11)

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

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

(12)

where $d_{4,\mathrm{R}}$ is the distance that R moved in $d_4$. Eqs. (9) and (12) tell us: $\gamma$ is recovered in ER if we project ES to the axes $d_1$ and $d_4$ of an observer. The rockets serve as an example. All other objects are projected the same way to an observer’s reality. For an overview of orthogonal projections, the reader is referred to geometry textbooks [19,20].

Up next, we transform the proper coordinates of observer R to those of B. We recall that R (or B) is in the rear end of the rocket “r” (or else “b”). We refer to Figure 3 again, but we now calculate the 4D motion of R and of B as a function of the parameter $\theta$. R and B start at the point P. The starting time is ${\theta }_0$. R cannot measure the proper coordinates of B, and vice versa, but we can calculate them from the ES diagrams in Figure 3.

$$d_{1,\mathrm{B}}^{\prime }\left(\theta \right) = d_{1,\mathrm{B}}^{\prime }\left({\theta }_0\right),$$

(13a)

$$d_{2,\mathrm{B}}^{\prime }\left(\theta \right) = d_{2,\mathrm{B}}^{\prime }\left({\theta }_0\right), d_{3,\mathrm{B}}^{\prime }\left(\theta \right) = d_{3,\mathrm{B}}^{\prime }\left({\theta }_0\right),$$

(13b)

$$d_{4,\mathrm{B}}^{\prime }\left(\theta \right) = d_{4,\mathrm{B}}^{\prime }\left({\theta }_0\right)+c (\theta -{\theta }_0).$$

(13c)

To transform the proper coordinates of R (unprimed) to the proper coordinates of B (primed), we calculate R’s 4D motion in the blue frame of Figure 3 top right.

$$d_{1,\mathrm{R}}^{\prime }\left(\theta \right) = d_{4,\mathrm{R}}\left(\theta \right) \cos \phi = d_{4,\mathrm{R}}\left(\theta \right) v_{3\mathrm{D}}/c,$$

(14a)

$$d_{2,\mathrm{R}}^{\prime }\left(\theta \right) = d_{2,\mathrm{R}}\left(\theta \right), d_{3,\mathrm{R}}^{\prime }\left(\theta \right) = d_{3,\mathrm{R}}\left(\theta \right),$$

(14b)

$$d_{4,\mathrm{R}}^{\prime }\left(\theta \right) = d_{4,\mathrm{R}}\left(\theta \right) \sin \phi = d_{4,\mathrm{R}}\left(\theta \right) {\gamma }^{-1}.$$

(14c)

To understand how an acceleration manifests itself in ES, we return to our two clocks. Clock “r” and Earth move in the $d_4$ axis of “r” at the speed $c$ (see Figure 4), but clock “b” accelerates in the $d_1$ axis of “r” toward Earth while maintaining the speed $c$. 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 [21] support the idea of GR that gravity is a feature of spacetime. In ER, the SO(4) symmetry of ES is incompatible with waves. This is fine because wave is an empirical concept and thus described by SR/GR. However, a natural concept of force and field has yet to be defined, which manifests itself as a force or a field in an observer’s reality. “Process” is a promising natural concept of force and field. A typical process is the transfer of energy or momentum [22]. As an example, we now recover gravitational time dilation in ER. Let us consider the process “transfer of potential energy to kinetic energy”. Initially, our clocks “r” and “b” are very far away from Earth. Eventually, “b” falls freely toward Earth (see Figure 4). The kinetic energy of “b” in $d_1$ is

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

(15)

where $m$ is the mass of “b”, $G$ is the gravitational constant, $M$ is the mass of Earth, and $R$ is the distance of “b” to Earth’s center. By applying Eq. (7), we obtain

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

(16)

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

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

(17)

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

(18)

where ${\gamma }_{\mathrm{g}\mathrm{r}}=(1-2GM/(R{c}^2{\left)\right)}^{-0.5}$ is the same dilation factor as in GR. Eq. (18) tells us: ${\gamma }_{\mathrm{g}\mathrm{r}}$ is recovered in ER if we project ES to the $d_4$ axis of an observer. I derived ${\gamma }_{\mathrm{g}\mathrm{r}}$ from a process. More studies are required to confirm that process is the natural concept of force and field. Since field is an empirical concept, there are no field equations in ER.

Summary of time dilation: In SR, a uniformly moving clock “b” is slow with respect to “r” in the time dimension of “b”. In GR, an accelerating clock “b” or a clock “b” in a stronger gravitational field is slow with respect to “r” in the time dimension of “b”. In ER, a clock “b” is slow with respect to “r” in the time dimension of “r” (!) if the 4D vectors $\mathit{\tau }$ of “r” and ${\mathit{\tau }}^{\mathit{\text{'}}}$ of “b” are not the same. Since both dilation factors $\gamma$ and ${\gamma }_{\mathrm{g}\mathrm{r}}$ are recovered in ER, the results of the Hafele–Keating experiment [23] do not only support SR/GR but also ER. Thus, GPS satellites work in ER as well as in SR/GR.

Three instructive problems teach us how to read ES diagrams correctly (see Figure 5). Problem 1: In billiards, the blue ball “b” approaches the red ball “r”. In ES, both balls move at the speed $c$. Let the red ball move in its $d_4$ axis. As the blue ball covers distance in $d_1$, its speed in $d_4$ must be less than $c$. How can the balls collide if their $d_4$ values do not match? Problem 2: A rocket moves along a guide wire. In ES, both objects move at the speed $c$. Let the wire move in its $d_4$ axis. As the rocket covers distance in $d_1$, its speed in $d_4$ must be less than $c$. Doesn’t the wire escape from the rocket? Problem 3: Earth orbits the sun. In ES, both objects move at the speed $c$. Let the sun move in its $d_4$ axis. As Earth covers distance in $d_1, d_2$, its speed in $d_4$ must be less than $c$. Doesn’t the sun escape from Earth?

The questions in the last paragraph seem to disclose paradoxes in ER. The fallacy lies in the assumption that all four dimensions of ES would be spatial. We solve all problems by projecting ES to the 3D space of the object that moves in $d_4$ at the speed $c$. In its 3D space, it is at rest. We see the solutions in the ES diagrams, too, if we read them correctly: In Figure 5 left, “r” and “b” collide if $d_{i,\mathrm{r}}=d_{i,\mathrm{b}}$ ($i=1, 2, 3$) and if the same proper time has elapsed for both balls ($d_{4,\mathrm{r}}=d_{4,\mathrm{b}}^{\prime }$). Thus, a collision in 3D space does not show up as a collision in ES. This is reasonable because only three axes of ES are experienced as spatial. For the same reason, the wire (or the sun) does not spatially escape from the rocket (or else Earth). ES is not a 4D space but a 4D spacetime. Objects moving in 4D space at the speed $c$ would be causally disconnected. ES diagrams are flow diagrams showing the flow of proper time. Spacetime diagrams in SR/GR are event diagrams showing events (collisions etc.).

5. Outlining the Solutions to 15 Fundamental Mysteries

We recall that ES is the master reality and that cosmic time $\theta$ is the correct parameter for a holistic view. In Sects. 5.1 through 5.15, I outline the solutions to 15 fundamental mysteries of physics and declare four concepts of today’s physics obsolete.

5.1. The Mystery of Time

Proper time $\tau$ is what a clock measures. Cosmic time $\theta$ is the total distance covered in ES divided by $c$. An observer’s clock always displays both quantities: his $\tau$ and $\theta$. An observed clock’s 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}}$ may differ from an observer’s 4D vector $\mathit{\tau }$.

5.2. The Mystery of Time’s Arrow

Time’s arrow is a synonym for “time moving only forward”. The arrow emerges from the fact that any covered distance cannot decrease but only increase.

5.3. The Mystery of the Factor $c^2$ in the Energy Term $m{c}^2$

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

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

(19)

where $E_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is its kinetic energy in an observer’s 3D space and $m{c}^2$ is called its “energy at rest”. We can derive the term $m{c}^2$ from SR, but SR does not tell us why there is a factor $c^2$ in the energy of objects that move at a speed less than $c$. ER is eye-opening: An object is never “at rest”. From its perspective, $E_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is zero and $m{c}^2$ is its kinetic energy in $d_4^{\prime }$. The factor $c^2$ is a hint that it moves through ES at the speed $c$. In SR, there is also

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

(20)

where $p$ is the total momentum of an object and $p_{3\mathrm{D}}$ is its momentum in an observer’s 3D space. Again, ER is eye-opening: From the object’s perspective, $p_{3\mathrm{D}}$ is zero and $mc$ is its momentum in $d_4^{\prime }$. The factor $c$ is a hint that it moves through ES at the speed $c$.

5.4. The Mystery of Length Contraction and Time Dilation

In SR, length contraction and time dilation can be traced back to Einstein’s instruction for synchronizing clocks, but this is just a measurement instruction. ER discloses that they stem from projecting absolute ES to the axes $d_1$ and $d_4$ of an observer.

5.5. The Mystery of Gravitational Time Dilation

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

5.6. The Mystery of the Cosmic Microwave Background (CMB)

In Sects. 5.6 through 5.12, I outline an ER-based model of cosmology. As a mathematical manifold, ES exists independent of time. In particular, ES is neither inflating nor expanding. For some reason, there was a Big Bang. In the inflationary Lambda-CDM model, the Big Bang occurred “everywhere” (space inflated from a singularity). In the ER-based model, the Big Bang is locatable: A huge amount of energy was injected into ES at some origin O. Cosmic time $\theta$ is the total time that has elapsed since the Big Bang. At $\theta =0$, all energy started moving radially away from O. The Big Bang was a singularity in providing energy and radial momentum. Shortly after $\theta =0$, energy was highly concentrated. While it was moving away from O, plasma particles were created in the projection to any 3D space. Recombination radiation was emitted that we still observe as CMB today [24].

The ER-based model must be able to answer these questions: (1) Why is the CMB so isotropic? (2) Why is the temperature of the CMB so low? (3) Why do we still observe the CMB today? Here are some possible answers: (1) The CMB is so isotropic because it has been scattered equally in the 3D space $d_1,d_2,d_3$ of Earth. (2) The temperature of the CMB is so low because the plasma particles had a very high recession speed $v_{3\mathrm{D}}$ (see Section 5.7) shortly after $\theta =0$. (3) We still observe the CMB today because it reaches Earth after having covered the same distance in $d_1, d_2, d_3$ (multiple scattering) as Earth in $d_4$.

5.7. The Mystery of the Hubble–Lemaître Law

In Figure 6 left, Earth and a galaxy G recede from the origin O of ES. In Earth’s 3D space, G recedes from Earth at the speed $v_{3\mathrm{D}}$. According to my first postulate, $v_{3\mathrm{D}}$ relates to the 3D distance $D$ of G to Earth as $c$ relates to the radius $r$ of a 4D hypersphere.

$$v_{3\mathrm{D}} = D c/r = H_{\theta } D,$$

(21)

where $H_{\theta }=c/r=1/\theta$ is the Hubble parameter. If we observe G today at the cosmic time ${\theta }_0$, the recession speed $v_{3\mathrm{D}}$ and $c$ remain unchanged. Thus, Eq. (21) turns into

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

(22)

where $H_0=c/r_0=1/{\theta }_0$ is the Hubble constant, $D_0=D r_0/r$ is today’s 3D distance of G to Earth, and $r_0$ is today’s radius of the 4D hypersphere. Eq. (22) is the correct Hubble–Lemaître law [25,26]. Cosmologists are aware of the Hubble parameter and of the quantity “cosmic time”. They are not yet aware that the 4D geometry is Euclidean, that $v_{3\mathrm{D}}$ is equal to $H_0 D_0$ rather than to $H_0 D$, and that there is no acceleration. Out of any two galaxies, the one farther away recedes faster, but each galaxy maintains its 3D speed $v_{3\mathrm{D}}$. How can we see G’s light if it never hits Earth in the ES diagram? The answer is again that ES is not a 4D space but a 4D spacetime. A collision in 3D space does not show up as a collision in ES.

5.8. The Mystery of the Flat Universe

ES is orthogonally projected to an observer’s proper space and proper time. Thus, he experiences two seemingly discrete structures: flat 3D space and time.

5.9. The Mystery of Cosmic Inflation

Most cosmologists [27,28] believe that an inflation of space shortly after the Big Bang explains the isotropic CMB, the flat universe, and large-scale structures. The latter inflated from quantum fluctuations. I just showed that ER explains the first two effects. ER even explains large-scale structures if the impacts of quantum fluctuations have been expanding like the 4D hypersphere. In ER, cosmic inflation is an obsolete concept.

5.10. The Mystery of Cosmic Homogeneity (Horizon Problem)

How can the universe be so homogeneous if there are causally disconnected regions of space? In the Lambda-CDM model, a region A at $x_1=+r_0$ and a region B at $x_1=-r_0$ are causally disconnected unless we postulate a cosmic inflation. Without it, information could not have covered $2r_0$ since the Big Bang. ER solves the problem without a cosmic inflation: In Figure 6 left, A is at $d_1=+r_0$ and B is at $d_1=-r_0$ (not shown). From A’s or B’s perspective, their $d_4^{\prime }$ axis (equal to Earth’s $d_1$ axis) disappears because of length contraction at the speed $c$. A and B are causally connected because they overlap spatially in either reality. Their opposite 4D vectors ${+\mathit{\tau }}^{\mathit{\text{'}}}$ and ${-\mathit{\tau }}^{\mathit{\text{'}}}$ do not affect causal connectivity.

5.11. The Mystery of the Hubble Tension

Up next, I explain why the published values of the Hubble constant $H_0$ do not match each other (also known as the “Hubble tension”). I compare CMB measurements (Planck space telescope) with calibrated distance ladder measurements (Hubble space telescope). According to team A [29], 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 [30], there is $H_0=73.04\pm 1.04 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B made efforts to minimize the error margins in the distance measurements. However, there is a systematic error in team B’s calculation of $H_0$, which arises from assuming a wrong cause of the redshifts.

We assume that team A’s value of $H_0$ is correct. We simulate the supernova of a star $\mathrm{S}$ that occurred at a distance of $D=400 \mathrm{M}\mathrm{p}\mathrm{c}$ from Earth (see Figure 6 right). The recession speed $v_{3\mathrm{D}}$ of $\mathrm{S}$ is calculated from measured redshifts. The redshift parameter $z=\mathsf{\Delta }\lambda /\lambda$ tells us how each wavelength $\lambda$ of the supernova’s light is either stretched by an expanding space (team B) or else Doppler-redshifted by receding objects (ER-based model). The supernova occurred at the cosmic time $\theta$ (arc called “past”), but we observe it at the cosmic time ${\theta }_0$ (arc called “present”). While the supernova’s light moved the distance $D$ in $d_1$, Earth moved the same distance $D$ but in $d_4$ (my first postulate). There is

$$1/H_{\theta } = r/c = {(r_0-D)/c = 1/H}_0-D/c.$$

(23)

For a very short distance of $D=400 \mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (23) tells us that $H_{\theta }$ deviates from $H_0$ by only 0.009 percent. When plotting $v_{3\mathrm{D}}$ versus $D$ for distances from 0 Mpc to 500 Mpc in steps of 25 Mpc (red points in Figure 7), the slope of a straight-line fit through the origin is roughly 10 percent greater than $H_0$. Since team B calculates $H_0$ from relating $z$ to magnitude, which is like plotting $v_{3\mathrm{D}}$ versus $D$, its value of $H_0$ is roughly 10 percent too high. This solves the Hubble tension. Team B’s value is not correct because, according to Eq. (22), we must plot $v_{3\mathrm{D}}$ versus $D_0$ (!) to get a straight line (blue points in Figure 7). Ignoring that the 4D geometry is Euclidean leads to an overestimation of the Hubble constant.

Since we cannot measure $D_0$ (observable magnitudes relate to $D$ and not to $D_0$), the easiest way to fix the calculation of team B is to rewrite Eq. (22) as

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

(24)

where $v_{3\mathrm{D},0}$ is today’s 3D speed of another star ${\mathrm{S}}_0$ (see Figure 6 right) that happens to be at the same distance $D$ today at which the supernova of star $\mathrm{S}$ occurred. I kindly ask team B to recalculate $H_0$ after converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$ with Eqs. (23), (24), and (21).

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

(25)

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

(26)

By applying Eq. (26) and plotting $v_{3\mathrm{D},0}$ versus $D$, all red points in Figure 7 drop down to the blue points. Figure 7 also tells us: The more high-redshift data are included in team B’s calculation, the more the $H_0$ tension increases. The moment of the supernova is irrelevant to team B’s calculation of $H_0$. In the Lambda-CDM model, all that counts is the duration of the light’s journey to Earth. The parameter $z$ continuously increases during the journey. In the ER-based model, all that counts is the moment of the supernova. Wavelengths are initially redshifted by the Doppler effect. The parameter $z$ remains constant during the journey. Space is not expanding. Energy is receding from the origin O of ES (location of the Big Bang). In ER, expanding space is an obsolete concept.

5.12. The Mystery of Dark Energy

Team B can fix the systematic error in its calculation of $H_0$ by converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$ according to Eq. (26). I now reveal another systematic error, but it is inherent in the Lambda-CDM model. It stems from assuming an accelerating expansion of space and can be fixed only by replacing this model with the ER-based model unless we postulate a dark energy. Most cosmologists [31,32] believe in an accelerating expansion because the calculated recession speeds $v_{3\mathrm{D}}$ deviate from a straight line in the Hubble diagram (if $v_{3\mathrm{D}}$ is plotted versus $D$) and because the deviations increase with $D$. An accelerating expansion would indeed stretch each wavelength even further and explain the deviations.

In ER, the explanation of the deviations is less speculative: The older the redshift data are, the more $H_{\theta }$ deviates from $H_0$, and the more $v_{3\mathrm{D}}$ deviates from $v_{3\mathrm{D},0}$. If another star ${\mathrm{S}}_0$ (see Figure 6 right) happens to be at the same distance of $D=400 \mathrm{M}\mathrm{p}\mathrm{c}$ today at which the supernova of star $\mathrm{S}$ occurred, Eq. (26) tells us: ${\mathrm{S}}_0$ recedes more slowly (27,064 km/s) from Earth than $\mathrm{S}$ (29,750 km/s). It does so because of the geometry. As long as cosmologists are not aware that the 4D geometry is Euclidean, they hold dark energy [33] responsible for an accelerating expansion of space. Dark energy has not been confirmed. It is a stopgap for an effect that the Lambda-CDM model cannot explain. Older supernovae recede faster not because of an accelerating expansion but because of a larger $H_{\theta }$ in Eq. (21).

The Hubble tension and dark energy are solved exactly the same way: In Eq. (22), we must not confuse $D_0$ with $D$. Because of Eq. (21) and because of $H_{\theta }=c/{(r}_0-D)$, the recession speed $v_{3\mathrm{D}}$ is not proportional to $D$ but to $D/(r_0-D)$. This is why the red points in Figure 7 run away from a straight line. Any expansion of space (uniform or accelerating) is only virtual. There is no accelerating expansion of space even if the Nobel Prize in Physics 2011 was given “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae”. This particular prize was given for something that does not really exist. In the Lambda-CDM model, the word “Universe” implies space, but space is not expanding. Most galaxies do recede from Earth, yet they do so uniformly in a non-expanding ES. In ER, dark energy is an obsolete concept.

These results cast doubt on the Lambda-CDM model but not on GR. In other words: Empirical concepts must not be applied in high-redshift cosmology. Galaxies are driven by their momentum. Because of physical interactions, some energy accelerated transversally while maintaining the speed $c$. This enables near-by galaxies to move toward Earth. Table 1 compares two models of cosmology. Note that “Universe” and “universe” are not the same thing. Observers may experience different universes. I showed that natural concepts prove useful in cosmology. Up next, they also prove useful in QM.

5.13. The Mystery of the Wave–Particle Duality

The wave–particle duality was first discussed by Niels Bohr and Werner Heisenberg [34] and has bothered physicists ever since. Electromagnetic waves are oscillations of an electromagnetic field, which propagate through an observer’s 3D space at the speed $c$. In some experiments, objects behave like waves. In other experiments, the very same objects behave like particles (also known as the “wave–particle duality”). In today’s physics, one object cannot be wave and particle at once because the energy of a wave is distributed in space, whereas the energy of a particle is always localized in space.

In order to solve the duality, we make use of two natural concepts: Proper space/time replaces coordinate space/time. “Pure energy” replaces wave/particle. Pure energy can be visualized by what I call “wavematter” (see Figure 8). In an observer’s reality (external view), a wavematter appears as a wave packet or as a particle. As a wave, it propagates in his $x_1$ axis at the speed $c$ and it oscillates in his axes $x_2$ and $x_3$ (electromagnetic field). Since here we talk about an observer’s reality, the wave propagates and oscillates as a function of coordinate time. In its own reality (internal view), the axis of the wavematter’s 4D motion disappears because of length contraction at the speed $c$. It deems itself particle at rest. “Wavematter” is not a substitute word for the duality. Rather, it visualizes a natural concept of energy that takes the internal view of photons into account.

Wave and particle are empirical concepts, just like coordinate space and coordinate time. They are relative too: What I deem wave, deems itself particle at rest. For each wavematter, its pure energy “condenses” (concentrates) to what we call “mass”. Einstein taught us that energy and mass are equivalent [35]. Likewise, a wave’s polarization and a particle’s spin are equivalent. My neologism “wavematter” phrases this equivalence.

In a double-slit experiment, wavematters pass through a double-slit and produce an interference pattern on a screen. An observer deems them wave packets as long as he does not track through which slit each wavematter is passing. Here the external view applies. The photoelectric effect is different. Of course, I can externally witness how a photon releases an electron from a metal surface, but the physical effect is all up to the photon: The electron is released only if the photon energy exceeds the electron’s binding energy. Here the internal view of the photon is the crucial view. The photon behaves like a particle.

The wave–particle duality is also observed in matter, such as electrons [36]. Electrons are wavematters too. They behave like waves as long as they are not tracked. Once they are tracked, they behave like particles. Since an observer automatically tracks objects that are slow in his 3D space, he deems all slow (and thus all macroscopic) objects matter rather than waves. To improve readability, I do not sketch any wavematters in the ES diagrams. I sketch what they are deemed by observers (clocks, rockets, galaxies, etc.).

5.14. The Mystery of Quantum Entanglement

The word “entanglement” was coined by Erwin Schrödinger in his comment [37] on the Einstein–Podolsky–Rosen paradox [38]. These authors argued that QM would not provide a complete description of reality. Schrödinger’s neologism did not solve the paradox, but it demonstrates our difficulties in comprehending QM. John Bell [39] showed that QM is incompatible with local hidden-variable theories. Meanwhile, it has been confirmed in several experiments [40,41,42] that entanglement violates locality in an observer’s 3D space. Quantum entanglement has been considered a non-local effect ever since.

Up next, I show that there is no violation in four dimensions. All we need to untangle entanglement is ER: Non-locality becomes obsolete because all four $d_{\mu }$ ($\mu =1, 2, 3, 4$) are interchangeable. Figure 9 illustrates two wavematters that were created at once at a point P. They move away from each other in opposite 4D directions $\pm d_4^{\prime }$ at the speed $c$. It turns out that they are automatically entangled. For an observer moving in any direction other than $\pm d_4^{\prime }$ (external view), the two wavematters are spatially separated. The observer has no idea how they are able to “communicate” with each other in no time.

For each wavematter (internal view), the $d_4^{\prime }$ axis disappears because of length contraction at the speed $c$. In their common (!) 3D space spanned by $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$, either of them is at the very same position as its twin. From the internal view, the twins have never been spatially separated, but their proper time flows in opposite 4D directions. While the twins stay together spatially, they “communicate” with each other in no time. Their opposite 4D vectors ${+\mathit{\tau }}^{\mathit{\text{'}}}$ and ${-\mathit{\tau }}^{\mathit{\text{'}}}$ do not affect local “communication”. There is a “spooky action at a distance” (phrase attributed to Einstein) from the external view only.

This time, the horizon problem and entanglement are solved exactly the same way: An observed region’s (or object’s) 4D vector ${\mathit{\tau }}^{\mathbf{\text{'}}}$ and its 3D space may differ from an observer’s 4D vector $\mathit{\tau }$ and his 3D space. This is possible only if all four $d_{\mu }$ ($\mu =1,2,3,4$) are interchangeable. ER also explains the entanglement of matter, such as electrons [43]. In an observer’s 3D space, electrons move at a speed $v_{3\mathrm{D}}<c$. In their $\pm d_4^{\mathrm{\text{'}}}$ axis, electrons always 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.

5.15. The Mystery of the Baryon Asymmetry

In the Lambda-CDM model, almost all matter was created shortly after the Big Bang. Only then was the temperature high enough to enable pair production. But pair production creates equal amounts of baryons and antibaryons. So, why do we observe more baryons than antibaryons today (also known as the “baryon asymmetry”)? ER scores again: Energy manifests itself as wavematters, and each wavematter deems itself particle at rest. I solve this mystery at the end because it requires my concept of wavematter.

But why do wavematters not deem themselves antiparticles at rest? Well, antiparticles are not the opposite of particles but particles with the opposite electric charge. Being an antiparticle is relative: What I deem antiparticle, deems itself particle. Thus, the “character paradox” [10] is reasonable and must not be considered a paradox. The baryon asymmetry is only virtual. It disappears as soon as we describe nature in natural concepts. Moreover, ER tells us why time seems to flow backward for antiparticles. This is because proper time flows in opposite 4D directions for any two wavematters created in pair production. According to Section 5.14, wavematters moving in opposite 4D directions should be entangled. This gives us a chance to falsify ER. Scientific theories must be falsifiable [44].

6. Conclusions

ER solves many unsolved mysteries (time’s arrow, Hubble tension, wave–particle duality, baryon asymmetry) and other mysteries that have already been solved but only by adding obsolete concepts (cosmic inflation, expanding space, dark energy, non-locality). This is a perfect example of where to apply Occam’s razor. It shaves off obsolete concepts. Period. SR/GR are considered two of the greatest achievements of physics because they have been confirmed over and over. I showed that SR/GR do not provide a holistic view. Physics got stuck in its own concepts. The stagnation in physics is of its own making.

It was a wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect [45] and not for SR/GR. I showed that ER penetrates to a deeper level. Einstein—one of the most brilliant physicists ever—failed to realize that the fundamental metric chosen by Mother Nature is Euclidean. Einstein sacrificed absolute space and time. ER restores absolute, cosmic time, but it sacrifices the absolute nature of wave and particle. For the first time ever, mankind understands the nature of time: Cosmic time is the total distance covered in ES divided by $c$. The human brain is able to imagine that we move through ES at the speed $c$. With that said, conflicts of mankind become all so small.

Is ER a physical or a metaphysical theory? This is a very good question because only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. I now explain why this is not an issue: We can always calculate these proper coordinates from ES diagrams as I showed in Eqs. (13a–14c). Measuring is an observer’s source of knowledge, but ER tells us not to interpret too much into whatever we measure. Measurements are wedded to observers, whose concepts may be obsolete. Unfortunately, physics has applied empirical concepts—which work well in daily life—to the very large and the very small. This is why high-redshift cosmology and QM profit the most from ER. ER is a physical theory because it solves fundamental mysteries of physics.

I conclude: (1) ER describes the master reality ES. (2) SR/GR describe each observer’s reality. (3) Because of the different realities, ER does not compete with SR/GR. (4) ER provides new information that is hidden in absolute time and thus not available in SR/GR. (5) In ER, cosmic inflation, expanding space, dark energy, and non-locality are obsolete concepts. (6) ER solves 15 mysteries purely geometrically. It is very unlikely that 15 solutions in different areas of physics are 15 coincidences. Only in natural concepts does Mother Nature disclose her secrets. The key to understanding nature is beyond observing.

Final remarks: (1) I only touched on gravity. We must not reject ER because gravity is still an issue. GR seems to solve gravity, but GR is incompatible with QM unless we add another speculative concept (quantum gravity). More studies are required to understand gravitational effects in ER. (2) I only touched on processes. I gave one example in Section 4. More studies are required to confirm that process is the natural concept of force and field. (3) Mysteries often disappear if we match the symmetry. SO(4) is the symmetry group in cosmology and QM. (4) The invariant $\theta$ puts an end to all speculations about time travel. Does any other theory solve the mystery of time’s arrow as beautifully as ER? (5) Physics does not ask: Why is my reality a projection? Nor does it ask: Why is it a wave function? Projections are far less speculative than postulating cosmic inflation and expanding space and dark energy and non-locality. (6) It looks like Plato’s Allegory of the Cave [46] is correct: Mankind experiences projections that are blurred—because of QM.

The primary question behind my theory is: How does all our insight fit together without adding highly speculative concepts? I trust that this very question leads us to the truth. I laid the groundwork for ER and showed how powerful it is. Paradoxes are only virtual. The true pillars of physics are ER, SR/GR (for describing all observers’ realities), and QM. Together they describe Mother Nature from the very large to the very small. Introducing a holistic view to physics is probably my most valuable contribution. All observers’ views taken together do not make a holistic view because they still do not provide absolute time. Everyone is welcome to solve even more mysteries by applying ER.

Comments: It takes open-minded, courageous editors and peer reviewers to evaluate a theory that heralds a paradigm shift. Whoever adheres to established concepts paralyzes the scientific progress. I did not surrender when top journals rejected my theory. Interestingly, I was never given any solid arguments that would disprove my theory. Rather, I was asked to try a different journal. Were the editors dazzled by the success of SR/GR? Did they underestimate the benefits of ER? I was told that 15 solved mysteries are too much to be trustworthy. I disagree. The natural concepts of ER herald a paradigm shift, and paradigm shifts often solve many mysteries at once. We must invest a considerable amount of time to get ourselves acquainted with the new concepts of ER. It seems to me that most editors are afraid of considering new concepts that are off the mainstream. Even good friends refused to support me. Anyway, each setback inspired me to work out the benefits of ER even better. Finally, I succeeded in disclosing a physical issue in SR/GR and also in formulating a holistic theory of spacetime, which shows that Einstein’s general relativity is not as general as it seems.Some physicists have difficulties in accepting ER because the SO(4) symmetry of ES is incompatible with waves. ER is not disputing waves but limiting their occurrence to an observer’s reality. A well-known preprint archive suspended my submission privileges. I was penalized because I disclosed an issue in Einstein’s SR/GR. The editor-in-chief of a top journal replied: “Publishing is for experts only.” One editor could not imagine that the Hubble tension is solved without GR. Another editor rejected my manuscript because it would demand too much from the peer reviewers. I do not blame anyone. Paradigm shifts are always hard to accept. These comments shall encourage young scientists to stand up for promising ideas even if opposing the mainstream is very hard work. Peer reviewers considered my theory “unscholarly research”, “fake science”, and “too simple to be true”. Simplicity and truth are not mutually exclusive. Beauty is when they go hand in hand together.