Three Steps Turn Euclidean Relativity Into a Pillar of Physics

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. In SR, space and time are fused 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 “distant collisions” (without physical contact) and a “character paradox” (confusion of photons, particles, and antiparticles). I show that the constraint is obsolete. There are no distant collisions once we take projections into account. There is no character paradox once we take the 4D vector “flow of proper time” (see Sect. 3) into account. Not only can the proper time of an antiparticle flow backward with respect to an observer, but also the proper time of a particle or a galaxy (see Sect. 5.15). Montanus calculated the precession of Mercury’s perihelion in ER [10] and other effects [11], but he failed to derive Maxwell’s equations because of a wrong sign [10]. Montanus used coordinate time t as the parameter. The correct parameter in ER is cosmic time θ.

Almeida [12] studied geodesics in ER. Gersten [13] interpreted the Lorentz transformation as an SO(4) rotation. There is also an ER website: https://euclideanrelativity.com/. Previous formulations of ER merely swap coordinate time t with the parameter τ. This is the first paper where we apply three steps to make ER work: (1) The new time coordinate is proper time τ. (2) The new parameter is cosmic time $\theta =1/H_{\theta }$, where $H_{\theta }$ is the Hubble parameter. (3) An observer’s reality is a projection from 4D Euclidean space (ES). Most physicists still reject ER because dark energy and non-locality make today’s cosmology and QM work, the SO(4) symmetry in ER seems to exclude waves, and there seem to be paradoxes in ER. This paper marks a turning point. I show: Dark energy and non-locality are obsolete, SO(4) is compatible with waves, and projections avoid paradoxes.

The two postulates of ER: (1) All energy moves through ES at the speed of light $c$. (2) The laws of physics have the same form in each object’s reference frame. An object’s reference frame is spanned by its proper space and proper time. Unlike coordinate space and coordinate time in SR, proper space and proper time in ER are assembled to a Euclidean spacetime. My first postulate is stronger than the second SR postulate: $c$ is absolute and universal. My second postulate is not limited to inertial frames. In addition, I reversed the order of the postulates. Absoluteness comes first. Relativity comes second.

Fig. 1 illustrates the reference frames of two objects “r” and “b”. 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. Proper space and proper time make up its “reality”. There are as many realities as there are objects. Mathematically, ES is 4D Euclidean space and an object’s proper space/proper time are two orthogonal projections from ES. Physically, three axes of ES are experienced as spatial, one axis as temporal, and projecting an object from ES to an observer’s reality is equivalent to measuring its coordinates.

Figure 1. An object’s proper space and proper time are two orthogonal projections from ES.

Figure 1. An object’s proper space and proper time are two orthogonal projections from 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] influenced many philosophers. I am convinced that ER will reform both physics and philosophy. For a better readability, I refer to an observer as “he”. To compensate, I refer to nature as “she”.

2. Identifying an Issue in Special and General Relativity

In SR, the fourth coordinate of spacetime is 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 $x_1^{\text{'}}, x_2^{\text{'}}, x_3^{\text{'}}, t^{\prime }$ 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)

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

(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. Eqs. (2a–c) transform the coordinates from K to K’. There are covariant equations that transform the coordinates from K’ to K. Mathematically, SR is correct. Physically, there is an issue in SR and also in GR: The concepts of SR/GR fail to solve fundamental mysteries of physics. There are coordinate-free formulations of SR [15] and also of GR [16], but there is no absolute time in SR/GR and thus no “holistic view” (in a holistic view, the spacetime diagram is universal for all observers at once). The view in SR and GR is “multi-egocentric” (SR/GR work for each observer, but there is no universal spacetime diagram). 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 and solves 15 mysteries (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: 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 some planets and claimed that Earth orbits the sun. In modern times, engineers have improved the accuracy of rulers and clocks. Today, it is natural to believe that an observer’s rulers and clocks would be sufficient to describe nature. The reviewer of a top journal rejected ER because “modern physics is the discovery that absolute time plays no role in the phenomena that we observe”. 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: While heliocentrism and geocentrism exclude each other, ER does not conflict with SR/GR. Yet the analogy is close: (1) After taking another 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 today’s cosmology and QM work. (3) Heliocentrism overcomes the limitation of a geocentric view. ER overcomes the limitation of a multi-egocentric view. (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 a man-made spacetime because its coordinates are constructs and thus not inherent in rulers and clocks. Rulers measure proper length. Clocks measure proper time. We introduce ER by the Euclidean 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 the parameter $\theta$ (to be given a meaning later on), whereas all $\mathrm{d}{d}_{\mu }$ ($\mu =1, 2, 3, 4$) are infinitesimal distances in 4D Euclidean space (ES). We prefer the indices 1–4 to 0–3 to stress the SO(4) symmetry. Each object (!) is free to label the axes of its reference frame in ES. Observers are objects too. We assume: Each object labels the axis of its current 4D motion as $d_4$ and the other three axes of ES as $d_1, d_2, d_3$. Because of my first postulate, it thus always moves in the $d_4$ axis at the speed $c$. The orientation of an object’s reference frame in ES can change: If it moves along a curved worldline, the orientation of its reference frame always adapts to this curvature. Because of length contraction at the speed $c$ (see Sect. 4), the $d_4$ axis disappears for itself and is experienced as proper time. An object moving in $d_4^{\text{'}}$ at the speed $c$ experiences $d_4^{\text{'}}$ as proper time. In ER, $\tau$ is the length of a 4D Euclidean vector “flow of proper time” $\mathit{\tau }$.

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

(5)

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

(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)

In other words: 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 ES ($d_4=c\tau$). Absolute ES is experienced as a relative Euclidean spacetime (EST): Each object experiences its 4D motion through ES as its proper time and the other three axes as its proper space. In EST, the fourth coordinate is $\tau$. The invariant parameter is $\theta$. The metric tensor is the identity matrix. EST is a natural spacetime because its coordinates are measured by and thus inherent in rulers and clocks. Intrinsic rulers and clocks of all objects measure distances in EST and not in $x_1, x_2, x_3, t$.

Montanus [9,10,11] merely rearranged Eq. (3) to enforce a Euclidean metric. He did not distinguish between $t$ and $\theta$. He also rejected symmetric axes and promoted a spacetime, where a pure time interval always remains a pure time interval. I show: Whatever is proper time to me, it can be one axis of proper space (or a mix of proper space and proper time) for you. Do not confuse ER with a Wick rotation [17], which keeps $\tau$ invariant.

Because of $t\ne \theta$, there is no continuous transition between Eqs. (3) and (4) nor between SR/GR and ER. This fact underlines that ER provides a unique description of nature. SR describes nature in man-made spacetime $x_1\left(\tau \right), x_2\left(\tau \right), x_3\left(\tau \right), t\left(\tau \right)$, where the parameter $\tau$ is object-related. GR is locally equivalent to SR. ER describes nature in natural spacetime $d_1\left(\theta \right), d_2\left(\theta \right), d_3\left(\theta \right), d_4\left(\theta \right)$, where $\theta$ is what I call the “cosmic evolution parameter”. As I will demonstrate in Sect. 5, the parameter $\theta$ proves more powerful than the parameter $\tau$. 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.

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 $c$. As the invariant in Eq. (4), $\theta$ is a concept of absolute time. This is why I also call it “cosmic time”. In terms of cosmic time, there is no twin paradox. Twins share the same age in cosmic time $\theta$. By referring to $\theta$, observers always agree on what is past and what is future. Regarding causality, a finite $c$ is incompatible with a coordinate “absolute time”, but compatible with a parameter “absolute 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 $v_{3\mathrm{D}}=0.6 c$. Fig. 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 $c{t}^{\prime }$. It displays “0.8”. In ER, “r” moves in the $d_4$ axis. Fig. 2 right shows the instant when either clock moved 1.0 s in its proper time. Both clocks display “1.0”. Since “r” remains at $d_1=0$ and “b” remains at $d_1^{\text{'}}=0$, there is $\tau =\theta$ for “r” and ${\tau }^{\prime }=\theta$ for “b” according to Eq. (4). A uniformly moving clock always displays both its $\tau$ and $\theta$. Yet $\tau$ is not the invariant in ER. Thus, $d_4$ of “r” is not equal to $d_4$ of “b”. In ER, $\theta$ is the invariant. Thus, $d_4$ of “r” is equal to $d_4^{\text{'}}$ of “b”.

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 $c{t}^{\prime }=0.8 \mathrm{Ls}$ when “r” is at $ct=1.0 \mathrm{Ls}$ (see Fig. 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 $d_4=0.8 \mathrm{Ls}$ when “r” is at $d_4=1.0 \mathrm{Ls}$ (see Fig. 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.

“Relativity” has different connotations in SR and ER: In Minkowski spacetime, spatial and temporal distances are relative (they depend on an observer’s frame of reference). In ER, the orientation of an object’s frame of reference in ES is relative (it depends on the object’s 4D vector $\mathit{\tau }$). Absolute ES does not distinguish between spatial and temporal distances (the distances in all four axes are “pure distances”). Only in relative EST does an object experience $d_1, d_2, d_3$ as spatial and $d_4$ as temporal. There is also a great difference regarding clock synchronization: In SR, R can synchronize clock “b” to his clock “r” (same value of $ct$ in Fig. 2 left). If he does, the clocks are not synchronized for B. In ER and independently of observers, clocks with the same $\mathit{\tau }$ are naturally synchronized. Clocks with different $\mathit{\tau }$ cannot be synchronized (different values of $d_4$ in Fig. 2 right).

But why does ER provide a holistic view? Eq. (4) is symmetric in all $d_{\mu }$ ($\mu =1, 2, 3, 4$). R and B experience different axes as temporal. This is why Fig. 2 left works for R, but not for B: A second Minkowski diagram is required, where $x_1^{\text{'}}$ and $c{t}^{\prime }$ are orthogonal. Here the view is multi-egocentric. Fig. 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 $d_4^{\text{'}}$. The ES diagram is independent of observers and thus universal. Here the view is holistic. Note that the Michelson–Morley experiment [18] refutes absolute 3D space (“ether”), but not absolute ES.

Regarding waves, I was misled by editors who insisted that the SO(4) symmetry of ES is incompatible with waves. SO(4) is incompatible with waves that propagate as a function of a coordinate time $t$, but compatible with waves that propagate as a function of the cosmic evolution parameter $\theta$. 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). A great advantage of mathematics is that it remains the same when replacing variables. Maxwell’s equations thus have the same form in ER as in today’s physics except that $\theta$ replaces $t$ and that waves can propagate in one out of four axes. I claim: All objects are “wavematters” (pure energy) that propagate through and oscillate in ES as a function of the parameter $\theta$. In Sect. 5.13, I will give evidence of my claim.

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 Fig. 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. For a better readability, a rocket’s width is drawn in $d_4$ (or $d_4^{\text{'}}$), although its width is in $d_2, d_3$ (or else $d_2^{\text{'}}, d_3^{\text{'}}$).

Up next, we verify: Projecting distances in ES to the axes $d_1$ and $d_4$ of an observer causes length contraction and 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 Fig. 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)

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

(9)

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

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

(10)

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

(11)

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

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

(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 once we project ES to the axes $d_1$ and $d_4$ of an observer. Thus, ER predicts the same relativistic effects as SR. The two rockets only serve as an example. Other objects are projected the same way. For instance, the lifetime of a muon is recovered in ER when R is kept as the observer and the blue rocket is replaced by a muon. For orthogonal projections, the reader is referred to textbooks about geometry [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. Fig. 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 $\phi$.

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

(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. Initially, our two clocks “r” and “b” shall be very far away from Earth (see Fig. 4). Eventually, “b” falls freely toward Earth and accelerates while maintaining the speed $c$ in ES. Earth and “r” keep on moving in the $d_4$ axis at the speed $c$.

Because of Eq. (7), all accelerations in ES are transversal. The speed $u_{1,\mathrm{b}}$ of “b” in $d_1$ increases at the expense of its speed $u_{4,\mathrm{b}}$ in $d_4$. Thus, “b” is slow with respect to “r” in $d_4$. In the gravitational field of Earth, the kinetic energy of “b” (mass $m$) in $d_1$ is

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

(14)

where $G$ is the gravitational constant, $M$ is the mass of Earth, and $R=d_{1,\mathrm{Earth}}-d_{1,\mathrm{b}}$ is the distance of “b” to the center of Earth in the 3D space of “r”. Eq. (7) gives us

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

(15)

We now make use of differential geometry. With $u_{4,\mathrm{b}}=\mathrm{d}{d}_{4,\mathrm{b}}/\mathrm{d}\theta$ (“b” moves in $d_4$ at the speed $u_{4,\mathrm{b}}$) and $c=\mathrm{d}{d}_{4,\mathrm{r}}/\mathrm{d}\theta$ (“r” moves in $d_4$ at the speed $c$), we calculate

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

(16)

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

(17)

where ${\gamma }_{\mathrm{gr}}={(1-2GM/(R{c}^2))}^{-0.5}$ is the same dilation factor as in GR. Eq. (17) tells us: ${\gamma }_{\mathrm{gr}}$ is recovered in ER once we project ES to the $d_4$ axis of an observer. Thus, ER predicts the same relativistic effects as GR. Yet there is a great difference: In GR, gravity is the curvature of spacetime. In ER, gravity makes its comeback as a force. Any acceleration rotates an object’s $\mathit{\tau }$ and curves its worldline in ES. “Action at a distance” is not an issue if field variations propagate at the speed $c$ and as a function of $\theta$. Since ES is projected (reduced) to an observer’s 3D space, the $1/R^2$ law of gravity is maintained.

Clock “b” is slow with respect to “r” in $d_4$. Since “r” displays both its $\tau$ and $\theta$, “b” is slow even with respect to absolute time $\theta$! An accelerated clock displays its $\tau$, but not $\theta$. This is why clocks placed next to each other display different times after being exposed to different gravitational fields. Since ${\gamma }_{\mathrm{gr}}$ does not depend on $u_{1,\mathrm{b}}$, “b” is slow with respect to $\theta$ whether or not it stops moving relative to Earth. I invite theorists to show two things: (1) Gravitational waves [21] are compatible with ER. (2) Variational principles [22] are an alternative to derive ER. Here I showed: $\gamma$ and ${\gamma }_{\mathrm{gr}}$ are recovered in ER.

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 accelerated 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 ${\mathit{\tau }}^{\prime }$ of “b” differs from the 4D vector $\mathit{\tau }$ of “r”. Since both $\gamma$ and ${\gamma }_{\mathrm{gr}}$ are recovered in ER, the Hafele–Keating experiment [23] supports ER too. GPS works in ER just as well as in SR/GR.

Fig. 5 illustrates how to read ES diagrams. Problem 1: Two objects move through ES. “r” moves in $d_4$. “b” emits a radio signal at $d_4^{\text{'}}=1.0 \mathrm{Ls}$. The signal recedes radially from “b” in all axes as a function of $\theta$, but cannot catch up with “r” in the $d_4$ axis. Can the 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. The wire moves in $d_4$. The rocket’s speed in $d_4$ is less than $c$. Doesn’t the wire escape from the rocket? Problem 3: Earth orbits the sun. The sun moves in $d_4$. Earth’s speed in $d_4$ is less than $c$. Doesn’t the sun escape from Earth’s orbit?

The last paragraph seems to reveal paradoxes in ER. The fallacy lies in the assumption that all four axes $d_{\mu }$ 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 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. In Fig. 5 left, the signal collides with “r” in the 3D space of “r” when their positions in $d_1, d_2, d_3, \theta$ (not $d_4$) coincide. This is the case when $\theta =1.6 \mathrm{s}$ has elapsed since “r” and “b” started from the origin. The collision also takes place in the 3D space of the signal (not shown). In the 3D space of the signal, the 4D motion of “r” changes at $\theta =1.0 \mathrm{s}$. Collisions in 3D space do not show up as collisions in ES. This is because events are a function of the parameter $\theta$, which is not an axis in ES diagrams. ES diagrams do not show events, but each object’s flow of proper time. The sun does not spatially escape from Earth’s orbit. Rather, the sun and Earth are aging in different 4D directions.

5. Outlining the Solutions to 15 Fundamental Mysteries

In Sects. 5.1 through 5.15, ER solves 15 mysteries and declares four concepts obsolete.

5.1. The Nature of Time

Clocks measure proper time $\tau$. This fact is true even under acceleration. Cosmic time $\theta$ is the total distance covered in ES divided by $c$. A uniformly moving clock always displays both its $\tau$ and $\theta$. An accelerated clock displays its $\tau$, but not $\theta$. An observed clock’s 4D vector ${\mathit{\tau }}^{\prime }$ 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, the total energy $E$ of an object (mass $m$) in the absence of forces is given by

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

(18)

where $E_{\mathrm{kin},3\mathrm{D}}$ 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, $E_{\mathrm{kin},3\mathrm{D}}$ 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 effect 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 can be localized: It injected a huge amount of energy into ES at an origin O. Cosmic time $\theta$ is absolute time that has elapsed uniformly 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. Ever since the Big Bang, this energy has been moving through ES at the speed $c$. Shortly after $\theta =0$, energy was highly concentrated. While receding from O, it became less concentrated and reduced to plasma particles in 3D space. Recombination radiation was emitted that we 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? Some possible answers: (1) The CMB is so isotropic because it has been scattered equally in all axes $d_1, d_2, d_3$ of the 3D space 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 Fig. 6 left). While doing so, G recedes from the $d_4$ axis at the speed $v_{3\mathrm{D}}$. Distance $D$ (or $D_0$) is the distance of G to Earth in the 3D space of Earth at the time $\theta$ (or else ${\theta }_0$). 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. Most energy is in its 3D hypersurface. The 4D motion of energy can change continuously by a transversal acceleration (scattering, gravitational field) or discontinuously (photon emission, pair production).

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

(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

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

(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]. Cosmology is aware of $\theta$ and $H_{\theta }$. It is 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 is a consequence of 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 the galaxy G is slow with respect to a clock on Earth in $d_4$ by the factor $c/{(c^2-v_{3\mathrm{D}}^2)}^{0.5}=\gamma$. The $d_4$ values of Earth and an energy $\mathrm{\Delta }E$ (emitted by G at the time $\theta$) never match. Can $\mathrm{\Delta }E$ and Earth collide in the 3D space of Earth if they do not collide in ES? As in Fig. 5 left, collisions in 3D space do not show up as collisions in ES. $\mathrm{\Delta }E$ collides with Earth once $\mathrm{\Delta }E$ 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 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 cosmic inflation. Without inflation, information could not have covered $2r_0$ since the Big Bang. The ER-based model applies natural concepts: Region A is at $d_1=+r_0$ (see Fig. 6 left). Region B is at $d_1=-r_0$ (not shown in Fig. 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 show that ER predicts the 10 percent deviation in the published values of $H_0$ (known as the “Hubble tension”). We consider CMB measurements and distance ladder measurements. According to team A [29], there is $H_0=67.66\pm 0.42 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. According to team B [30], there is $H_0=73.04\pm 1.04 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$. Team B made efforts to minimize the error margins in the distance measurements, but there is a systematic error in team B’s value of $H_0$. The error stems 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 $D=400 \mathrm{Mpc}$ from Earth (see Fig. 6 right). The recession speed $v_{3\mathrm{D}}$ of S is calculated from measured redshifts. The redshift parameter $z=\mathrm{\Delta }\lambda /\lambda$ 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 $\theta$, but we observe it today at the cosmic time ${\theta }_0$ (see Fig. 6 right). While the supernova’s light moved the distance $D$ in $-d_1$, Earth moved the same distance $D$, but in $+d_4$ (same speed $c$ according to my first postulate).

$$1/H_{\theta } = r/c = \left(r_0-D\right)/c = 1/H_0-D/c .$$

(22)

For a short distance of $D=400 \mathrm{kpc}$, 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 Fig. 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 Fig. 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 $D_0$ because 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

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

(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 Fig. 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

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

(24)

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

(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, Fig. 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

I now identify another systematic error, but it is inherent in the Lambda-CDM model. It stems from assuming an accelerating expansion of space and is only solved by switching to the ER-based model unless we postulate dark energy. Most cosmologists [31,32] believe in an accelerating expansion because the recession speeds $v_{3\mathrm{D}}$ increasingly deviate from a straight line when $v_{3\mathrm{D}}$ is plotted versus $D$. 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 $D=400 \mathrm{Mpc}$ 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 Fig. 6 right) from $d_4$ than S (29,750 km/s). It does so because of the 4D Euclidean geometry: The 4D vector ${\mathit{\tau }}^{\prime }$ of ${\mathrm{S}}_0$ deviates less from $\mathit{\tau }$ of Earth than ${\mathit{\tau }}^{″}$ of S deviates from $\mathit{\tau }$. As of today, cosmologists hold dark energy [33] responsible for an accelerating expansion of space. Dark energy has not been confirmed. It is a stopgap solution for an effect that the Lambda-CDM model cannot explain. Supernovae occurring earlier in cosmic time recede faster because of a larger $H_{\theta }$ in Eq. (20) and not because of 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 $D/\left(r_0-D\right)$. This is why the red points in Fig. 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 waves are distributed in space and capable of interference, whereas particles are localized in space and not capable of interference. To overcome the duality, we introduce 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 Fig. 8, observer R moves in the $d_4$ axis at the speed $c$. Three wavematters ${\mathrm{WM}}_1$, ${\mathrm{WM}}_2$, and ${\mathrm{WM}}_3$ move in different 4D directions at the speed $c$. For a better 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. ${\mathrm{WM}}_1$ 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, ${\mathrm{WM}}_2$ and ${\mathrm{WM}}_3$ 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, ${\mathrm{WM}}_3$ 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 duality. It is a new concept, which tells us where the duality stems from and that it is experienced by observers only. Isn’t it enriching to learn 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. Albert Einstein taught that energy and mass are equivalent [37]. “Wavematter” is a natural concept that stands for the equivalence of a wave’s energy and a particle’s mass.

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. The photon must interact with that electron to release it. The interaction discloses their current position. 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. For a better 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 Fig. 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 $\pm d_4^{″}$ 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 $d_4^{″}$ 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 ${\mathit{\tau }}^{\prime }$ 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 ${\mathit{\tau }}^{″}$ is reversed with respect to the particle’s 4D vector ${\mathit{\tau }}^{\prime }$. In the antiparticle’s view, its proper time flows forward. ER predicts that any two wavematters created in pair production are entangled. This gives us a chance to falsify ER. Scientific theories must be falsifiable [46]. Note that galaxies moving in $-d_4$ (not shown in Fig. 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.

6. Conclusions

Modern physics lacks two qualities of time: absolute and vectorial. On the one hand, there is the cosmic evolution parameter $\theta$ (absolute time), which separates absolute past, present, and future. There is no absolute time in SR/GR. On the other hand, $\tau$ is the length of a 4D Euclidean vector “flow of proper time” $\mathit{\tau }$. There is no $\mathit{\tau }$ in SR/GR. While SR/GR work for all observers, the 15 mysteries solved in Sect. 5 show that the scope of SR/GR is limited physically. The 4D vector $\mathit{\tau }$ is crucial for objects that are very far away or entangled. Information hidden in $\theta$ and in $\mathit{\tau }$ is not available in SR/GR. It is very unlikely that 15 solutions in different (!) areas of physics are 15 coincidences. Some of the 15 mysteries had been solved without ER, but with concepts that now prove obsolete. ER declares cosmic inflation, expanding space, dark energy, and non-locality obsolete. They are all subject to Occam’s razor. Occam shaves off obsolete concepts. No exceptions.

It was a wise decision to award 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 metric of nature is Euclidean. In fact, his instruction for synchronizing clocks blocks access to ER. He sacrificed absolute space and time. ER restores absolute time, but sacrifices the absolute nature of particles, matter waves, photons, and electromagnetic waves. In retrospect, two unfortunate practices of physicists delayed the formulation of ER: (1) Clocks measure $\tau$, but the construct $t$ is more common in the equations of physics than natural $\tau$. (2) Cosmology is aware of the Hubble parameter $H_{\theta }$, but the parameter $\tau$ is preferred to $\theta =1/H_{\theta }$ in both SR and GR. For the first time ever, mankind now understands the nature of time: Cosmic time $\theta$ is the total distance covered in ES divided by $c$. The human brain is able to imagine that we move 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 man-made concepts—which work well in everyday life—to the very distant and the very small. This is why cosmology and QM benefit the most from ER. ER is a physical theory because it solves fundamental mysteries of physics.

Comments: (1) More studies on gravity are required, but this is no reason to reject ER. GR seems to solve gravity, but GR is incompatible with QM unless we add quantum gravity. Since ER solves mysteries of QM, quantum gravity is probably another obsolete concept. (2) In ES, there are no singularities and thus no black holes. Once again, this argument is no reason to reject ER. I can imagine that projecting several massive objects from ES onto a small region in an observer’s 3D space results in the formation of a supermassive object. (3) Mysteries often disappear if the symmetry is matched. The symmetry group of natural spacetime is SO(4). (4) Absolute time finally puts an end to discussions about time travel. Does any other theory explain time’s arrow as beautifully as ER? (5) Physics does not ask: Why is my reality a projection or a wave function? Projections are far less speculative than cosmic inflation plus expanding space plus dark energy plus non-locality.

It seems as if Plato had anticipated ER in his Allegory of the Cave [48]: Mankind experiences projections and cannot observe any reality beyond. I laid the groundwork for ER and showed how powerful it is. Paradoxes are only virtual. The pillars of physics are ER, SR/GR (for observers), and QM. Together they describe nature from the very distant down to the very small. The key question in science is this: How do we describe nature without adding highly speculative concepts? The answer leads to the truth. Introducing a holistic view to physics is the most significant contribution of this paper. 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. Man-made concepts block our view of nature as a whole. Einstein described nature in a mix of man-made concepts ($t$) and natural concepts ($\tau$). ER describes nature exclusively in natural concepts ($\theta$, $\tau$). Only in natural concepts does Mother Nature reveal her secrets. Everyone is welcome to solve even more mysteries by describing her in natural concepts.