4D Euclidean Geometry and Fundamental Riddles in Physics

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.

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 I show that the constraint is obsolete.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 can be one axis of proper space for you. There is no twin paradox if we take cosmic time as the parameter. There is no “character paradox” if we take the 4D vector “flow of proper time” (see Sect. 3) into account. An object’s proper time can flow backward with respect to the observer without being an antiparticle (see Sect. 5.15). Montanus calculated the precession of Mercury’s perihelion in ER [10] and other effects [11] but failed to formulate Maxwell’s equations in ER because of a missing minus sign [10].

Almeida [12] studied geodesics. Gersten [13] showed that the Lorentz transformation is equal to an SO(4) rotation. There is a website about ER: https://euclideanrelativity.com/. Theorists still reject ER because: (1) Dark energy and non-locality make cosmology and QM work. (2) The SO(4) symmetry in ER seems to exclude waves. (3) Paradoxes seem to arise. This paper marks a turning point. It shows that dark energy and non-locality are obsolete concepts, SO(4) is compatible with waves, and projections avoid paradoxes.

The postulates of ER: (1) All energy moves through 4D Euclidean space (ES) at the speed $c$. Physically, one axis of ES is experienced as proper time. (2) The laws of physics have the same form in each object’s reference frame. Its reference frame is spanned by its proper space and proper time. Unlike coordinate space and coordinate time in SR, proper space and proper time are assembled to a Euclidean manifold. The order of the two postulates is reversed. Absoluteness comes first. My first postulate is stronger than in SR: $c$ is both absolute and universal. My second postulate is not limited to inertial frames. To improve readability, all observers are male. To make up for it, Mother Nature is female.

Figure 1 left illustrates the reference frames of two objects in ES. Each object experiences that axis in which it moves at the speed $c$ as its proper time. It experiences the other three axes as its proper space. ES is not observable because one axis is experienced as proper time. Proper space and proper time make up an object’s physical reality. There are as many realities as there are objects. Mathematically, proper space and proper time can be interpreted as two orthogonal projections from ES. Physically, projecting an object from ES to an observer’s proper space (or proper time) is equivalent to measuring its coordinates in his proper space (or else proper time). Figure 1 right illustrates two approaches to describing nature. We must not play SR/GR and ER off against each other. ER solves riddles that are rooted in 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]. ER is very powerful. It can 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 $x_1^{\text{'}}, x_2^{\text{'}}, x_3^{\text{'}}, t^{\text{'}}$ in K’ by

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

(2a)

$$x_2^{\text{'}} = x_2, x_3^{\text{'}} = x_3,$$

(2b)

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

(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. (2a–c) are correct: They transform the coordinates from K to K’. There are covariant equations that transform the coordinates from K’ to K. Physically, there is an issue in SR and also in GR: The empirical concepts of SR/GR fail to solve fundamental riddles. There are coordinate-free formulations of both SR and GR [16, 17], but there is no absolute time in SR/GR and thus no “holistic view” (I repeat the very important definition: 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 will 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 Sect. 3) and solves 15 riddles (see Sect. 5). Thus, the issue 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 astronomers wondered about the retrograde loops of some 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 the geocentric model to SR/GR is not perfect: Heliocentrism and geocentrism exclude each other. ER and SR/GR complement each other. Even so, the analogy is close: (1) After taking some other planet as the center of the Universe (or after a transformation in SR/GR), the view is still geocentric (or else egocentric). (2) Retrograde loops are obsolete in heliocentrism, but they make geocentrism work. Dark energy and non-locality are obsolete in ER, but they make cosmology and QM work. (3) Heliocentrism provides a view from beyond Earth. ER provides a view from beyond any observer’s space and time. (4) The geocentric model was a dogma in the old days. SR and GR are dogmata nowadays. Have physicists not learned from history? Does history repeat itself?

3. The Physics of Euclidean Relativity

ER cannot be derived from measurement instructions because the proper coordinates of other objects cannot be measured. We start with the Minkowski metric of SR

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

(3)

where $\mathrm{d}\tau$ is an infinitesimal distance in proper time $\tau$, whereas all ${\mathrm{d}x}_i$ ($i=1, 2, 3$) and $\mathrm{d}t$ 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 an empirical spacetime because its coordinates are construed by an observer and thus not immanent in rulers and clocks. Rulers measure proper length. Clocks measure proper time. We introduce ER by its metric

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

(4)

where $\mathrm{d}\theta$ 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 an object’s (!) proper space $d_1, d_2, d_3$ and proper time $\tau$. Observers are objects too. The fourth coordinate is $\tau$. The invariant is $\theta$. The metric tensor is the identity matrix. I prefer the indices 1–4 to 0–3 to stress the SO(4) symmetry. An object’s proper space $d_1, d_2, d_3$ and proper time $\tau$ span its reference frame $d_1, d_2, d_3, d_4$ in 4D Euclidean space (ES), where $d_4=c\tau$. The orientation of its reference frame in absolute ES is relative. ES is experienced as a Euclidean spacetime (EST). EST is a natural spacetime because its coordinates are measured by (and thus immanent in) rulers and clocks. Intrinsic rulers and clocks of all objects measure distances in natural spacetime. Do not confuse ER with a Wick rotation [18], where $t$ is imaginary and $\tau$ is the invariant.

Montanus [9] distinguished “absolute Euclidean spacetime” (AEST) from a “relative Euclidean spacetime” (REST). His AEST is my ES. His REST is my EST. Montanus [9,10,11] promoted AEST and disqualified REST. He rejected the idea of four fully symmetric axes. My ES diagrams show: Whatever is proper time for one object, it can be one axis of proper space (or even a mix of proper space and proper time) for another object.

Each object is free to label the axes of its reference frame. We assume: It labels the axis of its current 4D motion as $d_4$. Because of my first postulate, it thus always moves in the $d_4$ axis at the speed $c$. Note that, as in Minkowski diagrams, an object’s reference frame is not attached to the object. The object always moves in the time axis of its reference frame. In ER, an object’s 4D motion can rotate with respect to absolute ES. In this case, its reference frame performs the same rotation as if the frame’s origin were gimbal-mounted to the origin of ES, so that the object always moves in the $d_4$ axis. Because of length contraction at the speed $c$ (see Sect. 4), the object’s $d_4$ axis disappears for itself and is experienced as proper time. Objects moving in the $d_4^{\text{'}}$ axis at the speed $c$ experience $d_4^{\text{'}}$ as proper time. Each object experiences its 4D motion as proper time. There is a 4D vector “flow of proper time” $\mathit{\tau }$ for each object. Information hidden in $\mathit{\tau }$ is not available in SR/GR.

$$\tau = d_4/c, {\tau }^{\text{'}} = d_4^{\text{'}}/c,$$

(5)

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

(6)

where $\mathit{u}$ is an object’s 4D velocity. In ER, speed is not defined as $v_i={\mathrm{d}x}_i/\mathrm{d}t$ ($i=1, 2, 3$) but as $u_{\mu }=\mathrm{d}{d}_{\mu }/\mathrm{d}\theta$ ($\mu =1, 2, 3, 4$). Thus, Eq. (4) is nothing but my first postulate

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

(7)

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

Because of $t\ne \theta$, there is neither a continuous transition between Eqs. (3) and (4) nor between SR/GR and ER. 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 an object-related parameter. GR is locally equivalent to SR. 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 cosmic evolution parameter. Only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. In my Conclusions, I will explain why this is fine.

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 Ls (light seconds) in $ct$. Clock “b” moved 0.6 Ls in $x_1$ and 0.8 Ls in $c{t}^{\text{'}}$. 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, which is equal to cosmic time for either clock. Both clocks display “1.0”.

We now assume that an observer R (or B) moves with clock “r” (or else “b”). In SR and only from the perspective of R, clock “b” is at $c{t}^{\text{'}}=0.8 \mathrm{L}\mathrm{s}$ when “r” is at $ct=1.0 \mathrm{L}\mathrm{s}$ (see Figure 2 left). Thus, “b” is slow with respect to “r” in $t^{\text{'}}$ (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 that axis in which a clock is slow. Thus, both SR and ER describe time dilation correctly if ER yields the same Lorentz factor as SR. In Sect. 4, I will show that this is the case.

There is also a difference regarding the synchronization of clocks: In SR, observer R is able to synchronize clock “b” to his clock “r” (same value of $ct$ in Figure 2 left). If he does, the two clocks are not synchronized for observer B. In ER and independently of observers, 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).

But why does ER provide a holistic view? Eq. (4) is symmetric in all $d_{\mu }$ ($\mu =1, 2, 3, 4$). I call them “pure distances” because ES itself does not distinguish between “spatial” and “temporal”. Only objects/observers experience three axes as spatial and one as temporal. R and B experience different axes as temporal. This is why Figure 2 left works for R but not for B: A second Minkowski diagram is required, where $x_1^{\text{'}}$ and $c{t}^{\text{'}}$ are orthogonal. Here the view is multi-egocentric. In contrast, Figure 2 right works for R and for B at once (at the same cosmic time): Not only are $d_1$ and $d_4$ orthogonal but also $d_1^{\text{'}}$ and $d_4^{\text{'}}$. ES is independent of observers and thus absolute. Here the view is universal and thus holistic.

Regarding waves, I was misled by editors who insisted that the SO(4) symmetry of ES is incompatible with waves. Indeed, it can easily be shown that SO(4) is incompatible with waves that propagate and oscillate as a function of one coordinate. However, SO(4) is compatible with waves that propagate and oscillate as a function of the parameter cosmic time $\theta$. This is because Eq. (4) can be rewritten as

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

(4*)

which is of the same form as Eq. (3). I claim: All objects are waves that propagate through and oscillate in ES as a function of the parameter $\theta$. I call them “wavematters”. I will give evidence of my claim in Sects. 5.13 and 5.14, where it solves two riddles in QM.

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 $d_1^{\text{'}}, d_2^{\text{'}}, d_3^{\text{'}}$). We use “3D space” as a synonym of proper space. The proper time of R (or B) relates to $d_4$ (or else $d_4^{\text{'}}$) according to Eq. (5). Both rockets start at the same point P and at the same cosmic time ${\theta }_0$. They 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 $d_1^{\text{'}}$). The ES diagrams in Figure 3 must fulfill my two postulates and the initial conditions (same P, same ${\theta }_0$). This is achieved by rotating the red and the blue frame with respect to each other. Do not confuse ES diagrams with Minkowski diagrams. In ES diagrams, objects maintain proper length and clocks display proper time. To improve readability, a rocket’s width is drawn in $d_4$ (or $d_4^{\text{'}}$) although it is in the $d_2, d_3$ plane (or else $d_2^{\text{'}}, d_3^{\text{'}}$ plane).

We now assume that $N$ rockets “${\mathrm{r}}_i$” are launched from P at the same cosmic time ${\theta }_0$, where “${\mathrm{r}}_1$” is equal to “r”. We also assume: The 4D vector ${\mathit{\tau }}_i$ of “${\mathrm{r}}_i$” ($2\le i\le N$) is rotated with respect to ${\mathit{\tau }}_{i-1}$ of “${\mathrm{r}}_{i-1}$” by $\pi /2-\phi$. This implies that “${\mathrm{r}}_i$” recedes from “${\mathrm{r}}_{i-1}$” in the 3D space of “${\mathrm{r}}_{i-1}$” at the speed $v_{3\mathrm{D}}$. If $N(\pi /2-\phi )>\pi /2$, some rockets move backward in $d_4$. If one rocket “${\mathrm{r}}_i$” rotates by $\pi$, it stands still in the 3D space of “${\mathrm{r}}_1$” and its 4D vector ${\mathit{\tau }}_i$ is reversed with respect to the 4D vector ${\mathit{\tau }}_1$ of “${\mathrm{r}}_1$”. This example shows that ER does not compete with SR. Our assumptions are not valid in SR: There is no “same cosmic time ${\theta }_0$” and no 4D vector $\mathit{\tau }$ in SR. We can draw all “${\mathrm{r}}_i$” in a Minkowski diagram (launched at the same coordinate time), but our example is outside the scope of SR.

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 rocket “b” for observer R (or else B). In a first step, we project “b” in Figure 3 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)

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

(9)

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

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

(10)

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

(11)

where $d_{4,\mathrm{B}}$ (or $d_{4,\mathrm{B}}^{\text{'}}$) is the distance that B moved in $d_4$ (or else $d_4^{\text{'}}$). With $d_{4,\mathrm{B}}^{\text{'}}=d_{4,\mathrm{R}}$ (R and B cover the same distance in ES but in different 4D 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: $\gamma$ is recovered in ER if we project ES to the axes $d_1$ and $d_4$ of an observer. This result is significant: It tells us that ER predicts the same relativistic effects as SR. The rockets serve as an example. All other objects are orthogonally projected the same way. For an overview of orthogonal projections, the reader is referred to geometry textbooks [19, 20].

We now transform the proper coordinates of observer R (unprimed) to the ones of B (primed). R cannot measure the proper coordinates of B, and vice versa, but we can always calculate them from ES diagrams. Figure 3 right tells us how to calculate the 4D motion of R in the proper coordinates of B. The transformation is a 4D rotation by the angle $\phi$.

$$d_{1,\mathrm{R}}^{\text{'}}\left(\theta \right) = d_{4,\mathrm{R}}\left(\theta \right) \cos \phi = d_{4,\mathrm{R}}\left(\theta \right) {\displaystyle \frac{v_{3\mathrm{D}}}{c}},$$

(13a)

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

(13b)

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

(13c)

Up next, I show that not only the Lorentz factor is recovered in ER but also gravitational time dilation. 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). Clock “b” accelerates in the 3D space of “r” toward Earth while maintaining the speed $c$ in ES. Because of Eq. (7), all accelerations are transversal in ES. The speed $u_{1,\mathrm{b}}$ of “b” in $d_1$ increases at the expense of its speed $u_{4,\mathrm{b}}$ in $d_4$.

Initially, our two clocks shall be very far away from Earth. Eventually, clock “b” falls freely toward Earth. The kinetic energy of “b” (mass $m$) in the $d_1$ axis of “r” is

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

(14)

where $G$ is the gravitational constant, $M$ is the mass of Earth, and $R=d_{1,\mathrm{E}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}}-d_{1,\mathrm{b}}$ is the distance of “b” to Earth in the 3D space of “r”. By applying Eq. (7), we get

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

(15)

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

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

(16)

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

(17)

where ${\gamma }_{\mathrm{g}\mathrm{r}}=(1-2GM/(R{c}^2{\left)\right)}^{-0.5}$ is the same dilation factor as in GR. Eq. (17) tells us: ${\gamma }_{\mathrm{g}\mathrm{r}}$ is recovered in ER if we project ES to the $d_4$ axis of an observer. This result is significant: It tells us that ER predicts the same gravitational time dilation as GR. However, there is a big difference: In GR, curved spacetime replaces gravitational fields. In ER, gravitational fields celebrate a comeback. Any acceleration rotates an object’s $\mathit{\tau }$ and curves its worldline in flat ES. Action at a distance is not an issue if any variation in gravitational field strength also spreads at the speed $c$. Gravitational waves [21] support the idea of GR that gravity is a feature of spacetime. I invite theorists to show two things: (1) Gravitational waves are compatible with ER too. (2) ER can be derived from the variation of an action like GR [22]. Here I showed that ER yields the same $\gamma$ and ${\gamma }_{\mathrm{g}\mathrm{r}}$ as SR/GR.

Summary of time dilation: In SR, a uniformly moving clock “b” is slow with respect to “r” in the time axis of “b”. In GR, an accelerating clock “b” or else a clock “b” in a more curved spacetime is slow with respect to “r” in the time axis of “b”. In ER, a clock “b” is slow with respect to “r” in the time axis of “r” (!) if the 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}}$ of “b” differs from the 4D vector $\mathit{\tau }$ of “r”. Since both $\gamma$ and ${\gamma }_{\mathrm{g}\mathrm{r}}$ are recovered in ER, the Hafele–Keating experiment [23] supports ER too. GPS works in ER just as well as in SR/GR.

Three problems tell us how to read ES diagrams (see Figure 5). Problem 1: Two objects “r” and “b” move through ES at the speed $c$. “r” moves in $d_4$. “b” emits a radio signal at $d_4^{\text{'}}=1.0 \mathrm{L}\mathrm{s}$. The signal recedes radially from “b” in all axes as a function of $\theta$, but it cannot catch up with “r” in the $d_4$ axis. Can the radio signal and “r” collide in the 3D space of “r” if they do not collide in ES? Problem 2: A rocket moves along a guide wire. In ES, both objects move at the speed $c$. The wire moves in $d_4$. As the rocket covers distance in $d_1$, its speed in $d_4$ is 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 $c$. The sun moves in $d_4$. As Earth covers distance in $d_1$ and $d_2$, its speed in $d_4$ is less than $c$. Doesn’t the sun escape from Earth’s orbit?

The questions in the last paragraph seem to disclose geometric paradoxes in ER. The fallacy lies in the assumption that all four axes $d_{\mu }$ ($\mu =1, 2, 3, 4$) would be spatial at once. This is not the case. Only three axes of ES are experienced as spatial and one as temporal. We solve all three problems by projecting ES to the 3D space of that object which moves in $d_4$ at the speed $c$. In its 3D space, it is always at rest. The radio signal collides with “r” in the 3D space of “r” if there is $d_{i,\mathrm{r}}=d_{i,\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}}$ ($i=1, 2, 3$) at one instant in cosmic time $\theta$. Thus, a collision is possible even if there is $d_{4,\mathrm{r}}\ne d_{4,\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}}$. In our example (see Figure 5 left), the signal collides with “r” once $\theta =1.6 \mathrm{s}$ have elapsed since “r” started from the origin. Collisions in 3D space do not show up as collisions in ES. Here is why ES diagrams do not contract physics: ES diagrams do not show events but each object’s flow of proper time. The wire does not spatially escape from the rocket. The sun does not spatially escape from Earth’s orbit. In Figure 4, Earth does not spatially escape from clock “b”.

5. Outlining the Solutions to 15 Fundamental Riddles

In Sects. 5.1 through 5.15, ER solves 15 riddles and declares four concepts obsolete. I will focus on riddles that do not involve gravitational fields (except for Sect. 5.5).

5.1. The Nature of Time

Proper time $\tau$ is what a clock measures. Cosmic time $\theta$ is the total distance covered in ES divided by $c$. Any clock always displays both quantities: its $\tau$ and $\theta$. An observed clock’s 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}}$ can differ from the observer’s 4D vector $\mathit{\tau }$. If it does, the observed clock is slow with respect to the observer’s clock in his time axis.

5.2. Time’s Arrow

“Time’s arrow” is a synonym of time flowing forward only. Why does time flow forward only? Here is the answer: Covered distance cannot decrease but only increase.

5.3. The Factor $c^2$

in the Energy Term $m{c}^2$

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

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

(18)

where $E_{\mathrm{k}\mathrm{i}\mathrm{n},3\mathrm{D}}$ is its kinetic energy in an observer’s coordinate space and $m{c}^2$ is its energy at rest. The term $m{c}^2$ can be derived 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 $d_4^{\text{'}}$. The factor $c^2$ is a hint that it moves through ES at the speed $c$. In SR, there is also

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

(19)

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

5.4. Length Contraction and Time Dilation

In SR, length contraction and time dilation can be traced back to Einstein’s instruction for synchronizing clocks. In ER, these relativistic effects are natural effects: They stem from projecting distances in ES to the axes $d_1$ and $d_4$ of an observer.

5.5. Gravitational Time Dilation

In GR, gravitational time dilation stems from curved spacetime. In ER, this relativistic stems from projecting curved worldlines in 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 Cosmic Microwave Background (CMB)

In the inflationary Lambda-CDM model, the Big Bang occurred “everywhere” (space inflated from a singularity). In Sects. 5.6 through 5.12, I outline an ER-based model of cosmology, in which the Big Bang is locatable: It injected a huge amount of energy into ES at an 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 became less concentrated, 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 receded at a very high speed $v_{3\mathrm{D}}$ (Doppler redshift, see Sect. 5.11). (3) We still observe the CMB today because the radiation 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 Hubble–Lemaître Law

Earth and a galaxy G recede from the origin O of ES at the speed $c$ (see Figure 6 left). While doing so, G recedes from the $d_4$ axis at the speed $v_{3\mathrm{D}}$. Because of the 4D Euclidean geometry, $v_{3\mathrm{D}}$ relates to $D$ as $c$ relates to the radius $r$ of an expanding 4D hypersphere. All energy is within this hypersphere. Some energy is within its 3D hypersurface. The 4D motion of energy can change either continuously by a transversal acceleration (scattering, gravitational field) or discontinuously (photon emission, pair production).

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

(20)

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. (20) turns into

$$v_{3\mathrm{D}}=D_0{\displaystyle \frac{c}{r_0}}=H_0{D}_0,$$

(21)

where $H_0=c/r_0=1/{\theta }_0$ is the Hubble constant, $D_0=D r_0/r$, and $r_0$ is today’s radius of the 4D hypersphere. Eq. (21) is an improved Hubble–Lemaître law [25, 26]. Cosmologists are aware of $\theta$ and $H_{\theta }$. They are not yet aware that the 4D geometry is Euclidean, that $\theta$ is absolute, and that $v_{3\mathrm{D}}$ is equal to $H_0 D_0$ (not to $H_0 D$). Out of two galaxies, the one farther away recedes faster, but each galaxy maintains its recession speed $v_{3\mathrm{D}}$. Time dilation results from Eq. (7): Since G moves in $d_1$ at the speed $v_{3\mathrm{D}}$, it moves in $d_4$ at the speed $(c^2-v_{3\mathrm{D}}^2{)}^{0.5}$. Thus, a clock in G is slow with respect to a clock on Earth in the axis $d_4$ by the factor $c/(c^2-v_{3\mathrm{D}}^2{)}^{0.5}=\gamma$. The $d_4$ values of Earth and an energy $\mathsf{\Delta }E$ (emitted by G at the time $\theta$) never match. Can $\mathsf{\Delta }E$ and Earth collide in the 3D space of Earth if they do not collide in ES? As in Figure 5 left, collisions in 3D space do not show up as collisions in ES. $\mathsf{\Delta }E$ collides with Earth once $\mathsf{\Delta }E$ has covered the same distance in $d_1$ as Earth in $d_4$.

5.8. The Flat Universe

Two orthogonal projections from flat ES make up an observer’s physical reality. This is why he experiences two independent structures: flat 3D space and time.

5.9. 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 3D hypersurface. In ER, cosmic inflation is an obsolete concept.

5.10. Cosmic Homogeneity (Horizon Problem)

How can the universe be so homogeneous if there are causally disconnected regions? In the Lambda-CDM model, region A at $x_1=+r_0$ and region B at $x_1=-r_0$ are causally disconnected unless we postulate a cosmic inflation. Without inflation, information could not have covered $2r_0$ since the Big Bang. In the ER-based model, we use natural concepts: Region A is at $d_1=+r_0$ (see Figure 6 left). Region B is at $d_1=-r_0$ (not shown in Figure 6 left). For A and for B, their $d_4^{\text{'}}$ axis (equal to Earth’s $d_1$ axis) disappears because of length contraction at the speed $c$. Since A and B overlap spatially in their 3D space, they are causally connected. Note that their opposite 4D vectors “flow of proper time” do not affect causal connectivity as long as A and B overlap spatially.

5.11. The Hubble Tension

Up next, I explain the 10 percent deviation in the published values of $H_0$ (known as the “Hubble tension”). We compare CMB measurements with calibrated distance ladder measurements. 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, but 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 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 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 stretched by 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 the $d_1$ axis, Earth moved the same distance $D$ but in the $d_4$ axis (first postulate). There is

$${\displaystyle \frac{1}{H_{\theta }}}={\displaystyle \frac{r}{c}}= {{\displaystyle \frac{r_0-D}{c}}={\displaystyle \frac{1}{H}}}_0-{\displaystyle \frac{D}{c}}.$$

(22)

For a short distance of $D=400 \mathrm{k}\mathrm{p}\mathrm{c}$, Eq. (22) 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 $z$ versus magnitude, which is like plotting $v_{3\mathrm{D}}$ versus $D$, its value of $H_0$ is roughly 10 percent too high. Team B’s value of $H_0$ is not correct because Eq. (21) tells us: We must plot $v_{3\mathrm{D}}$ versus $D_0$ to get a straight line (blue points in Figure 7). Ignoring the 4D Euclidean geometry in distance ladder measurements leads to an overestimation of $H_0$ by 10 percent. This solves the Hubble tension.

We cannot measure $D_0$. Observable magnitudes relate to $D$ and not to $D_0$. Thus, the easiest way to fix the calculation of team B is to rewrite Eq. (21) as

$$v_{3\mathrm{D},0}=D{\displaystyle \frac{c}{r_0}}=H_{0}D,$$

(23)

where $v_{3\mathrm{D},0}$ is today’s 3D speed of a star ${\mathrm{S}}_0$ that happens to be at the same distance $D$ today at which the supernova of S occurred (see Figure 6 right). I kindly ask team B to recalculate $H_0$ after converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$ by combining Eqs. (22), (23), and (20) to

$$H_{\theta }=H_0{\displaystyle \frac{c}{c-H_{0}D}}={\displaystyle \frac{H_0}{1-\frac{v_{3\mathrm{D},0}}{c}}},$$

(24)

$$v_{3\mathrm{D},0}={\displaystyle \frac{v_{3\mathrm{D}}}{1+\frac{v_{3\mathrm{D}}}{c}}}.$$

(25)

Because of Eq. (23), we also get a straight line by applying Eq. (25) and plotting $v_{3\mathrm{D},0}$ versus $D$. In addition, Figure 7 tells us: The more high-redshift data are included in team B’s calculation, the more the Hubble tension increases. The moment of the supernova is irrelevant to team B’s calculation. In the Lambda-CDM model, all that counts is the duration of the light’s journey to Earth ($z$ increases during the journey). In the ER-based model, all that counts is the moment of the supernova. Wavelengths are redshifted by the Doppler effect ($z$ is constant during the journey). Space is not expanding. Energy recedes from the location of the Big Bang in ES. In ER, expanding space is an obsolete concept.

5.12. Dark Energy

Team B can fix the error in its value of $H_0$ by applying Eq. (25). I now disclose another systematic error, but it is inherent in the Lambda-CDM model. It stems from assuming an accelerating expansion of space and is fixed only by applying the ER-based model unless we postulate a dark energy. Most cosmologists [31, 32] believe in an accelerating expansion because the recession speeds $v_{3\mathrm{D}}$ deviate from a straight line if we plot $v_{3\mathrm{D}}$ versus $D$ and because the deviations increase with $D$. An accelerating expansion of space would indeed stretch each wavelength even further and explain the deviations.

In ER, the cause of the deviations is far less speculative: The longer ago a supernova occurred, the more $H_{\theta }$ deviates from $H_0$, and thus the more $v_{3\mathrm{D}}$ deviates from $v_{3\mathrm{D},0}$. If a star ${\mathrm{S}}_0$ happens to be at the same distance of $D=400 \mathrm{M}\mathrm{p}\mathrm{c}$ today at which the supernova of S occurred, Eq. (25) tells us: ${\mathrm{S}}_0$ recedes more slowly (27,064 km/s, the shortest arrow in Figure 6 right) from $d_4$ than S (29,750 km/s). It does so because of the 4D Euclidean geometry: The 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}}$ of ${\mathrm{S}}_0$ deviates less from $\mathit{\tau }$ of Earth than ${\mathit{\tau }}^{\mathit{\text{'}}\mathit{\text{'}}}$ of S deviates from $\mathit{\tau }$. As long as cosmologists are not aware of ER, they hold a “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 because of a larger $H_{\theta }$ in Eq. (20) and not because of a dark energy.

The Hubble tension and dark energy are solved exactly the same way: In Eq. (21), we must not confuse $D_0$ with $D$. Because of Eq. (20) 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 else accelerating) is only virtual 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 an illusion that stems from interpreting astronomical observations in the wrong concepts. Most galaxies recede from Earth, but they do so uniformly in a non-expanding space. In ER, dark energy is an obsolete concept.

The Hubble tension and dark energy are solved by taking the 4D Euclidean geometry into account, and the 4D vector $\mathit{\tau }$ in particular. These results cast doubt on the Lambda-CDM model. GR works well as long as $\mathit{\tau }$ is not crucial, but it is crucial for high-redshift supernovae. Space is not driven by dark energy. Galaxies are driven by their momentum and maintain their recession speed $v_{3\mathrm{D}}$ with respect to Earth. Because of various effects (scattering, gravitational field, photon emission, pair production), some energy deviates from a radial motion in ES while maintaining the speed $c$. Gravitational attraction enables near-by galaxies to move toward our galaxy. Table 1 compares two models of cosmology. Note that “the Universe” (Lambda-CDM model) and “universe” (ER-based model) are not the same thing. Each observer experiences three axes of ES as his universe. Cosmology benefits from ER. In Sects. 5.13 and 5.14, I show that QM also benefits from ER.

5.13. The Wave–Particle Duality

The wave–particle duality was first discussed by Niels Bohr and Werner Heisenberg [34]. It has bothered physicists ever since. In some experiments, objects behave like waves. In others, the same objects behave like particles (known as the “wave–particle duality”). One object cannot be both because a wave’s energy spreads out in space, whereas a particle’s energy is always localized in space. We overcome the duality by introducing another natural concept: All objects are “wavematters” (pure energy) that propagate through and oscillate in ES as a function of the parameter $\theta$. In an observer’s view, wavematters reduce to wave packets if not tracked or else to particles if tracked.

In Figure 8, observer R moves in the $d_4$ axis at the speed $c$. Three wavematters ${\mathrm{W}\mathrm{M}}_1$, ${\mathrm{W}\mathrm{M}}_2$, and ${\mathrm{W}\mathrm{M}}_3$ move in different 4D directions at the speed $c$. To improve readability, a wavematter’s oscillation is drawn in the $d_1, d_4$ plane although it can oscillate in any axis that is orthogonal to its propagation axis. ${\mathrm{W}\mathrm{M}}_1$ does not move relative to R. Thus, it is automatically tracked and reduces to a particle (${\mathrm{P}}_1$). In the 3D space of R, ${\mathrm{W}\mathrm{M}}_2$ and ${\mathrm{W}\mathrm{M}}_3$ reduce to wave packets (${\mathrm{W}}_2,{\mathrm{W}}_3$) if not tracked or else to particles (${\mathrm{P}}_2,{\mathrm{P}}_3$) if tracked. In the 3D space of R, ${\mathrm{W}}_2$ moves at a speed less than $c$. Thus, ${\mathrm{W}}_2$ is what Louis de Broglie called a “matter wave” [35]. Erwin Schrödinger formulated his Schrödinger equation to describe matter waves [36]. In the 3D space of R, both ${\mathrm{W}}_3$ and ${\mathrm{P}}_3$ move at the speed $c$. Thus, ${\mathrm{W}\mathrm{M}}_3$ is the only wavematter that reduces for R to an electromagnetic wave packet or else to a photon. Light gives us a good idea of how wavematters move through ES.

Remarks: (1) “Wavematter” is not just a new word for the wave–particle duality. It is a new concept, which tells us where there the duality stems from and that it is experienced by observers only. It also discloses that particles, matter waves, photons, and electromagnetic waves all stem from a common concept. (2) In today’s physics, there is no “photon’s view”. In ER, we can assign a 3D space and a proper time to each wavematter. In its view, its 4D motion disappears because of length contraction at the speed $c$. In its 3D space, it is always at rest and reduces to a particle. (3) In a particle, a wavematter’s energy condenses to mass. Einstein taught that energy and mass are equivalent [37]. Wavematters suggest that, likewise, a wave’s polarization and a particle’s spin are equivalent.

In double-slit experiments, light creates an interference pattern on a screen if it is not tracked through which slit single portions of energy are passing. The same applies if material objects, such as electrons, are sent through the double-slit [38]. Here light and matter behave like waves. In experiments on the photoelectric effect, an electron is released from a metal surface only if the energy of an incoming photon exceeds the binding energy of that electron. To release it, the photon must interact with it. The interaction reveals the position of photon and electron. They are tracked. Here light and matter behave like particles. Since an observer automatically tracks all objects that are slow in his 3D space, he classifies all slow objects—and thus all macroscopic objects—as matter. To improve readability, most of my ES diagrams do not show wavematters but how they appear to observers.

5.14. Non-Locality

It was Erwin Schrödinger who coined the word “entanglement” in his comment [39] on the Einstein–Podolsky–Rosen paradox [40]. The three authors argued that QM would not provide a complete description of reality. Schrödinger’s neologism does not solve the paradox, but it demonstrates our difficulties in comprehending QM. John Bell [41] showed that QM is incompatible with local hidden-variable theories. Meanwhile, it has been confirmed in several experiments [42,43,44] that entanglement violates locality in an observer’s 3D space. Entanglement has been interpreted as a non-local effect ever since.

Up next, I show that ER is able to “untangle” entanglement. There is no violation of locality in ES, where all four axes are fully symmetric. In Figure 9, observer R moves in the $d_4$ axis at the speed $c$. There are two pairs of entangled wavematters. One pair was created at the point P and moves in opposite directions $\pm d_4^{\text{'}}$ (equal to the axes $\pm d_1$ of R) at the speed $c$. The other pair was created at the point Q and moves in opposite directions $\pm d_4^{\text{'}\text{'}}$ at the speed $c$. In the 3D space of R, the first pair (green) reduces to two entangled photons. The second pair (blue) reduces to two entangled material objects (for instance, electrons). R has no idea how two entangled objects are able to “communicate” in no time.

In the photons’ view (or electrons’ view), the $d_4^{\text{'}}$ axis (or else the $d_4^{\text{'}\text{'}}$ axis) disappears because of length contraction at the speed $c$. Thus, each pair stays together in its respective 3D space. Entangled objects have never been spatially separated in their view, but their proper time flows in opposite 4D directions. This is how two entangled objects are able to communicate in no time. Note that their opposite 4D vectors “flow of proper time” do not affect local communication as long as the twins stay together spatially. There is a “spooky action at a distance” (attributed to Einstein) in an observer’s view only.

The horizon problem and entanglement are solved exactly the same way: An observed region’s (or an observed object’s) 4D vector ${\mathit{\tau }}^{\mathbf{\text{'}}}$ and its 3D space can differ from the observer’s 4D vector $\mathit{\tau }$ and his 3D space. All of this is possible but only in ES, where all four axes are fully symmetric. The SO(4) symmetry of ES solves entanglement. It explains the entanglement of photons just as well as the entanglement of material objects, such as atoms or electrons [45]. Any measurement on one entangled twin will terminate its existence or tilt the axis of its 4D motion. In either case, the twins will not move in opposite 4D directions anymore. The entanglement is destroyed. In ER, non-locality is an obsolete concept.

5.15. The Baryon Asymmetry

In the Lambda-CDM model, almost all matter was created shortly after $\theta =0$, when the temperature was high enough to enable pair production. But this process creates equal amounts of particles and antiparticles, and the process of annihilation annihilates equal amounts of particles and antiparticles. So, why do we observe more baryons than antibaryons (known as the “baryon asymmetry”)? In an observer’s view, wavematters reduce to wave packets or else to particles. Pair production creates particles and antiparticles, which annihilate each other very soon. Thus, there is one source of long-lived particles (reduction of wavematters), one source of short-lived particles (pair production), but only one source of short-lived antiparticles (pair production). This solves the baryon asymmetry.

ER also tells us why an antiparticle’s proper time seems “to flow backward”: Proper time flows in opposite 4D directions for any two wavematters created in pair production. The antiparticle’s 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}\mathit{\text{'}}}$ is reversed with respect to the particle’s 4D vector ${\mathit{\tau }}^{\mathit{\text{'}}}$. In the antiparticle’s view, its proper time flows forward. Note that galaxies moving in $-d_4$ (not shown in Figure 6 left) are not made up of antimatter. Only their flow of proper time is reversed with respect to galaxies moving in $+d_4$. Their physical charges are not reversed. ER predicts that any two wavematters created in pair production are entangled. This gives us a chance to falsify ER. All scientific theories must be falsifiable [46].

6. Conclusions

ER tells us that there is a 4D vector “flow of proper time” $\mathit{\tau }$ for each object. Any acceleration rotates an object’s $\mathit{\tau }$ and curves its worldline in flat ES. The 4D vector $\mathit{\tau }$ is crucial for objects that are very far away or entangled. These objects must be described in natural concepts. ER solves 15 fundamental riddles, such as the nature of time, the Hubble tension, the wave–particle duality, and the baryon asymmetry. It is very unlikely that 15 solutions in different areas of physics are 15 coincidences. Physics had solved some of these riddles but only by adding obsolete concepts. In ER, cosmic inflation, expanding space, dark energy, and non-locality are obsolete concepts. All of them are perfect examples of where to apply Occam’s razor. Occam shaves off obsolete concepts. No exceptions.

It was a wise decision to award Albert Einstein the Nobel Prize for his theory of the photoelectric effect [47] and not for SR/GR. ER penetrates to a deeper level. Einstein, one of the most brilliant physicists ever, did not realize that the fundamental metric of nature is Euclidean. He sacrificed absolute space and time. ER restores absolute time, but it sacrifices the absolute nature of particles, matter waves, photons, and electromagnetic waves. 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 fine: We can always calculate these proper coordinates from ES diagrams as I showed in Eqs. (13a–c). 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 can be obsolete. I was often told that physics is all about observing. I disagree. We cannot observe quarks, can we? Regrettably, physicists have applied empirical concepts—which work well in our everyday life—to the very far and to the very small. This is why cosmology and QM benefit the most from ER. ER is a physical theory because it solves fundamental riddles in physics.

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). Since ER solves riddles in QM, it is likely that quantum gravity is another obsolete concept. More studies are required to understand gravitational effects in ER. (2) Riddles often disappear once the symmetry is matched. The symmetry group of natural spacetime is SO(4). (3) The new invariant $\theta$ finally puts an end to all discussions about time travel. Does any other theory solve the riddle of time’s arrow as beautifully as ER? (4) 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 four obsolete concepts. (5) Greek philosopher Plato anticipated ER in his Allegory of the Cave [48]: Mankind experiences projections and cannot observe any reality beyond.

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 observers), and QM. Together they describe Mother Nature from the very far to the very small. Introducing a holistic view to physics is probably the most significant achievement of this paper. I showed that SR/GR do not provide a holistic view. All observers’ views taken together do not make a holistic view because they still do not provide absolute time. Physics got stuck in its own concepts. Only in natural concepts does Mother Nature disclose her secrets. Everyone is welcome to solve even more riddles by describing her in natural concepts.