Natural Spacetime: Describing Nature in Natural Concepts

2. Disclosing an Issue in Special and General Relativity

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

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

(1)

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

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

(2a)

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

(2b)

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

(2c)

where K’ moves relative to K in $x_1$ at the constant speed $v_{3\mathrm{D}}$ and $\gamma =(1-v_{3\mathrm{D}}^2/c^2{)}^{-0.5}$ is the Lorentz factor. Mathematically, Eqs. (1) and (2a–c) are correct for observers in K. There are covariant equations for observers in K’. Physically, there is an issue in SR and also in GR: The empirical concepts of SR/GR fail to solve fundamental mysteries. There are coordinate-free formulations of both SR [16] and GR [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 and GR work for all observers, but each observer’s view is egocentric. All observers’ views taken together do not make a holistic view because they still do not provide absolute time. Without absolute time, observers do not always agree on what is past and what is future. Physics has paid a high price for dismissing absolute time: ER restores absolute time (see Sect. 3) and solves 15 fundamental mysteries (see Sect. 5). Thus, the issue in SR/GR is real.

The issue in SR/GR is not about making wrong predictions. It has much in common with the issue in the geocentric model: In either case, there is no holistic view. Geocentrism is the egocentric view of mankind. In the old days, it was natural to believe that all celestial bodies would orbit Earth. Only 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 SR/GR to the geocentric model is not perfect: Heliocentrism and geocentrism exclude each other, whereas ER and SR/GR complement each other. Even so, the analogy is close: (1) After a transformation in SR/GR (or after choosing another center of the Universe), the view is again egocentric (or else geocentric). (2) Retrograde loops make geocentrism work, but heliocentrism declares retrograde loops obsolete. Dark energy and non-locality make cosmology and QM work, but ER declares dark energy and non-locality obsolete. (3) The geocentric model was a dogma in the old days. SR/GR are dogmata today. 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 non-Euclidean 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 $\mathrm{d}t$ and ${\mathrm{d}x}_i$ ($i=1, 2, 3$) are infinitesimal distances in an observer’s coordinate space $x_1, x_2, x_3$ and coordinate time $t$. Coordinate spacetime $x_1, x_2, x_3, t$ is empirical because its four coordinates are construed by an observer and thus not immanent in rulers and clocks. Rulers measure proper length. Clocks measure proper time. We introduce ER by defining its 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 cosmic time $\theta$, whereas all ${\mathrm{d}d}_i$ ($i=1, 2, 3$) and ${\mathrm{d}d}_4=c \mathrm{d}\tau$ are infinitesimal “pure distances”. Now the role of proper time $\tau$ is reversed: The new fourth coordinate is proper time $\tau$; the new invariant is absolute, cosmic time $\theta$; the new metric tensor is the identity matrix. I choose the symbol $\theta$ because the initial of the Greek letter theta is “t” as in “time”. I prefer the indices 1–4 to 0–3 to stress the full symmetry in all four coordinates. Any object’s proper space $d_1, d_2, d_3$ and its proper time $\tau$ span Euclidean spacetime $d_1, d_2, d_3, d_4$ (ES), where $d_4=c\tau$. This spacetime is natural because its four coordinates $d_{\mu }$ ($\mu =1, 2, 3, 4$) are measured by and are thus immanent in rulers and clocks. Intrinsic rulers and clocks of all objects measure out natural spacetime! Eq. (4) is not a Wick rotation [18], where $t$ is imaginary and $\tau$ is the invariant.

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

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

(5)

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

(6)

where $\mathit{u}$ is an object’s 4D velocity in ES. Information is lost if the 4D vector $\mathit{\tau }$ is ignored, as in SR/GR. There is $u_{\mu }=\mathrm{d}{d}_{\mu }/\mathrm{d}\theta$ for all $\mu =1, 2, 3, 4$. Thus, speed is not a spatial coordinate divided by $\tau$ but any coordinate divided by $\theta$. All processes are a function of the parameter $\theta$. Eq. (4) is not a random metric but my first postulate

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

(7)

It is instructive to contrast the three concepts of time. Coordinate time $t$ is a subjective measure of time: An observer uses his clock as the master clock. Proper time $\tau$ is an objective measure of time: Clocks measure $\tau$ independently of observers. Cosmic time $\theta$ is the total distance covered in ES (length of a worldline) divided by $c$. By taking $\theta$ as the parameter, all observers will agree on what is past and what is future. Since $\theta$ is absolute, there is no twin paradox in ER. Twins are the same age in cosmic time. A finite $c$ implies that none of the four coordinates can be equal to absolute time, as in Newton’s physics. However, a finite $c$ does not imply that there is no parameter “absolute time”.

Different concepts disable a unification of 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 proper time $\tau$ is the parameter and $t$ is coordinate time. GR describes nature locally the same as SR. Only 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 cosmic time $\theta$ is the parameter and $d_4$ relates to $\tau$. Only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. In Sect. 6, I 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 $c{t}^{\prime }$. It displays “0.8”. In ER, “r” moves in the $d_4$ axis. Figure 2 right shows the instant when either clock moved 1.0 s in its proper time, 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 $c{t}^{\prime }=0.8 \mathrm{L}\mathrm{s}$ when “r” is at $ct=1.0 \mathrm{L}\mathrm{s}$ (see Figure 2 left). Thus, “b” is slow with respect to “r” in $t^{\prime }$ (of B). In ER and independently of observers, clock “b” is at $d_4=0.8 \mathrm{L}\mathrm{s}$ when “r” is at $d_4=1.0 \mathrm{L}\mathrm{s}$ (see Figure 2 right). Thus, “b” is slow with respect to “r” in $d_4$ (of R). In SR and ER, “b” is slow with respect to “r”, but time dilation occurs in different axes. Experiments do not disclose the axis in which a clock is slow. This is why both SR and ER describe time dilation correctly—provided that they yield the same Lorentz factor. As I show in Sect. 4, they do.

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

Gersten [13] showed that the Lorentz transformation can be considered an SO(4) rotation in a mixed space $X_1, X_2, X_3, T^{\prime }$, in which only the $T^{\prime }$ axis is primed. In ER, unmixed $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }, d_4^{\prime }$ of an observed object are rotated with respect to an observer’s $d_1, d_2, d_3, d_4$ (see Sect. 4). There is also a noteworthy difference in the synchronization of clocks: In SR, each observer is able to synchronize a uniformly moving clock to his clock (same value of $ct$ in Figure 2 left). If he does, these clocks are not synchronized for the moving clock. In ER, clocks with the same 4D vector $\mathit{\tau }$ are always synchronized, whereas clocks with different $\mathit{\tau }$ and ${\mathit{\tau }}^{\prime }$ are never synchronized (different values of $d_4$ in Figure 2 right).

4. Geometric Effects in Euclidean Relativity

We consider two identical rockets “r” (red rocket) and “b” (blue rocket). Let observer R (or B) be in the rear end of “r” (or else “b”). The 3D space of R (or B) is spanned by $d_1, d_2, d_3$ (or else $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$). We use “3D space” as a synonym of proper space. The proper time of R (or B) relates to $d_4$ (or else $d_4^{\prime }$) according to Eq. (5). Both rockets start at the 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 $d_1^{\prime }$). 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 $d_4^{\prime }$). Figure 3 bottom shows the projection to 3D space.

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 (“anti-rocket”) 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 instant in coordinate time), but our example is outside the scope of SR. Likewise, an example in empirical concepts is outside the scope of ER.

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 top left to the $d_1$ axis.

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

(8)

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

(9)

where $\gamma =(1-v_{3\mathrm{D}}^2/c^2{)}^{-0.5}$ is the same Lorentz factor as in SR. For R, rocket “b” contracts to $L_{\mathrm{b},\mathrm{R}}$. But why do we calculate the same $\gamma$ despite the Euclidean metric? Well, there is no length contraction in ES but in the projection to $d_1$ only. 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 a point: The $d_4$ axis disappears for R because of length contraction at the speed $c$. In a second step, we project “b” in Figure 3 top left to the $d_4$ axis.

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

(10)

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

(11)

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

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

(12)

where $d_{4,\mathrm{R}}$ is the distance that R moved in $d_4$. Eqs. (9) and (12) tell us: $\gamma$ is recovered in ER if we project ES to the axes $d_1$ and $d_4$ of an observer. 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 those of observer B (primed). R cannot measure the proper coordinates of B, and vice versa, but we can calculate them from ES diagrams. For example, Figure 3 top right tells us how to calculate the 4D motion of R in the proper coordinates of B and as a function of the parameter $\theta$. The transformation is identical to a rotation in ES by the angle $\phi$.

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

(13a)

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

(13b)

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

(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 “r” and “b”. Clock “r” and Earth move in the $d_4$ axis of “r” at the speed $c$ (see Figure 4). Clock “b” accelerates in the $d_1$ axis of “r” toward Earth while maintaining the speed $c$. Because of Eq. (7), the speed $u_{1,\mathrm{b}}$ of “b” in the $d_1$ axis increases at the expense of its speed $u_{4,\mathrm{b}}$ in the $d_4$ axis.

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

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

(14)

where $m$ is the mass of “b”, $G$ 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 current 3D distance of “b” to Earth’s center in the 3D space of “r”. By applying Eq. (7), the equivalent to my first postulate, we obtain

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

(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)

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

(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. 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 field strength also spreads at the speed $c$. Gravitational waves [21] support the idea of GR that gravity is a feature of spacetime. In ER, the SO(4) symmetry of ES is incompatible with waves. This is fine because we must apply SR/GR whenever we use empirical concepts. The field equations of GR can be derived from the variation of an action [22]. As an experimental physicist, I invite theorists to show the same for ER. I consider it my task to describe nature in natural concepts. Here I demonstrated 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 ${\mathit{\tau }}^{\prime }$ 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 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 $d_4^{\prime }=1.0 \mathrm{L}\mathrm{s}$. The signal recedes radially from “b” in all four dimensions and as a function of $\theta$, but it cannot catch up with “r” in the $d_4$ axis. Can “r” and the signal collide in 3D space if they don’t 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 paradoxes in ER. The fallacy lies in the assumption that all four axes of ES would be spatial at once. This is not the case. An observer experiences three out of four axes of ES as spatial and the remaining axis as temporal. We solve all problems by projecting ES to the 3D space of the object that moves in $d_4$ at the speed $c$. In its 3D space, it is 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. 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” when $\theta =1.6 \mathrm{s}$ have elapsed since “r” and “b” 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 Mysteries

We recall that there is a 4D vector $\mathit{\tau }$ in ER, which is not available in SR/GR. In Sects. 5.1 through 5.15, ER solves 15 mysteries and declares four concepts obsolete.

5.1. The Mystery of Time

Proper time $\tau$ is what a clock measures. Cosmic time $\theta$ is the total distance covered in ES divided by $c$. Any clock always displays both quantities: its proper time $\tau$ and $\theta$. An observed clock’s 4D vector ${\mathit{\tau }}^{\prime }$ may differ from an 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. The Mystery of Time’s Arrow

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

5.3. The Mystery of the Factor $c^2$

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 $d_4^{\prime }$. The factor $c^2$ is a hint that it moves through ES at the speed $c$. In SR, there is also

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

(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 $d_4^{\prime }$. The factor $c$ is a hint that it moves through ES at the speed $c$.

5.4. The Mystery of Length Contraction and Time Dilation

In SR, length contraction and time dilation can be traced back to Einstein’s instruction for synchronizing clocks. ER gives us a non-empirical explanation. It discloses that these effects stem from projecting worldlines in ES to the axes $d_1$ and $d_4$ of an observer.

5.5. The Mystery of Gravitational Time Dilation

In GR, gravitational time dilation stems from curved spacetime. ER discloses that this 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 Mystery of 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 was moving away from O, plasma particles were created in the projection to any 3D space. Recombination radiation was emitted that we still observe as CMB today [24].

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

5.7. The Mystery of the Hubble–Lemaître Law

In Figure 6 left, Earth and a galaxy G recede from the origin O of ES. In Earth’s 3D space, G recedes from Earth at the speed $v_{3\mathrm{D}}$. According to my first postulate, $v_{3\mathrm{D}}$ relates to the 3D distance $D$ of G to Earth as $c$ relates to the radius $r$ of a 4D hypersphere. All energy is within this hypersphere. Most energy is within its 3D hypersurface. Because of physical interactions, some energy accelerated transversally while maintaining the speed $c$.

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

(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, $D_0=D r_0/r$ is today’s 3D distance of G to Earth, and $r_0$ is today’s radius of the 4D hypersphere. Eq. (21) is the improved Hubble–Lemaître law [25,26]. Cosmologists are aware of $\theta$ and $H_{\theta }$. They are not 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 $H_0 D$). Out of two galaxies, the one farther away recedes faster, but each galaxy maintains its speed $v_{3\mathrm{D}}$. The $d_4$ values of Earth and an energy $\mathsf{\Delta }E$ (emitted by G at the time $\theta$) never match. Can Earth and $\mathsf{\Delta }E$ collide in the 3D space of Earth if they don’t collide in ES? The answer is the same as for Figure 5 left: Collisions in 3D space do not show up as collisions in ES. Earth and $\mathsf{\Delta }E$ collide when $\mathsf{\Delta }E$ has covered the same distance in $d_1$ as Earth in $d_4$.

5.8. The Mystery of the Flat Universe

An observer’s view is created by orthogonally projecting ES to his proper space and to his proper time. Thus, he experiences two discrete structures: flat space and time.

5.9. The Mystery of Cosmic Inflation

Most cosmologists [27,28] believe that an inflation of space shortly after the Big Bang explains the isotropic CMB, the flat universe, and large-scale structures. The latter inflated from quantum fluctuations. I just showed that ER explains the first two effects. ER even explains large-scale structures if the impacts of quantum fluctuations have been expanding like the 3D hypersurface. In ER, cosmic inflation is an obsolete concept.

5.10. The Mystery of Cosmic Homogeneity (Horizon Problem)

How can the universe be so homogeneous if there are causally disconnected regions? 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 $d_4^{\prime }$ 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 Mystery of the Hubble Tension

Up next, I explain the 10 percent deviation in the published values of $H_0$ (also known as the “Hubble tension” or the “$H_0$ tension”). Let us compare CMB measurements (Planck space telescope) with calibrated distance ladder measurements (Hubble space telescope). According to team A [29], there is $H_0=67.66\pm 0.42 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. According to team B [30], there is $H_0=73.04\pm 1.04 \mathrm{k}\mathrm{m}/\mathrm{s}/\mathrm{M}\mathrm{p}\mathrm{c}$. Team B made efforts to minimize the error margins in the distance measurements. However, there is a systematic error in team B’s calculation of $H_0$, which arises from assuming a wrong cause of the redshifts.

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

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

(22)

For a very 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 relating $z$ to magnitude, which is like plotting $v_{3\mathrm{D}}$ versus $D$, its value of $H_0$ is roughly 10 percent too high. This solves the Hubble tension. Team B’s value is not correct because, according to Eq. (21), we must plot $v_{3\mathrm{D}}$ versus $D_0$ (!) to get a straight line (blue points in Figure 7). Ignoring that the 4D geometry is Euclidean leads to an overestimation of the Hubble constant.

Since we cannot measure $D_0$ (observable magnitudes relate to $D$ and not to $D_0$), the easiest way to fix the calculation of team B is to rewrite Eq. (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 another star ${\mathrm{S}}_0$ (see Figure 6 right) that happens to be at the same distance $D$ today at which the supernova of star $\mathrm{S}$ occurred. I kindly ask team B to recalculate $H_0$ after converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$ with Eqs. (22), (23), and (20).

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

(24)

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

(25)

By applying Eq. (25) and plotting $v_{3\mathrm{D},0}$ versus $D$, we also get a straight line according to Eq. (23). In addition, Figure 7 tells us: The more high-redshift data are included in team B’s calculation, the more the $H_0$ tension increases. The moment of the supernova is irrelevant to team B’s calculation. 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. The Mystery of Dark Energy

Team B can fix the systematic error in its calculation of $H_0$ by converting all $v_{3\mathrm{D}}$ to $v_{3\mathrm{D},0}$ according to Eq. (25). I now reveal another systematic error, but it is inherent in the Lambda-CDM model. It stems from assuming an accelerating expansion of space and can be fixed only by replacing this model with the ER-based model unless we postulate a dark energy. Most cosmologists [31,32] believe in an accelerating expansion because the calculated recession speeds $v_{3\mathrm{D}}$ deviate from a straight line in the Hubble diagram (if $v_{3\mathrm{D}}$ is plotted versus $D$) and because the deviations increase with $D$. An accelerating expansion would indeed stretch each wavelength even further and explain the deviations.

In ER, the explanation of the deviations is less speculative: The older the redshift data are, the more $H_{\theta }$ deviates from $H_0$, and the more $v_{3\mathrm{D}}$ deviates from $v_{3\mathrm{D},0}$. If another star ${\mathrm{S}}_0$ (see Figure 6 right) happens to be at the same distance of $D=400 \mathrm{M}\mathrm{p}\mathrm{c}$ today at which the supernova of star $\mathrm{S}$ occurred, Eq. (25) tells us: ${\mathrm{S}}_0$ recedes more slowly (27,064 km/s) from Earth than $\mathrm{S}$ (29,750 km/s). It does so because of the 4D Euclidean geometry: The 4D vector ${\mathit{\tau }}^{\prime }$ of ${\mathrm{S}}_0$ deviates less from Earth’s 4D vector $\mathit{\tau }$ than the 4D vector ${\mathit{\tau }}^{\prime \prime }$ of $\mathrm{S}$. As long as cosmologists are not aware of ER, they hold dark energy [33] responsible for an accelerating expansion of space. Dark energy has not been confirmed. It is a stopgap for an effect that the Lambda-CDM model cannot explain. Older supernovae recede faster not because of an accelerating expansion but because of a larger $H_{\theta }$ in Eq. (20).

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 spacetime. In ER, dark energy is an obsolete concept.

The mysteries of the $H_0$ 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 itself is correct as long as $\mathit{\tau }$ is not crucial, but $\mathit{\tau }$ is crucial for high-redshift supernovae. Space is not driven by dark energy. Galaxies are driven by their momentum. Because of physical interactions, some energy accelerated transversally while maintaining the speed $c$. This enables near-by galaxies to move toward Earth. Two models of cosmology are compared in Table 1. Note that “the Universe” is not the same as “his universe”. Each observer experiences three out of four axes of ES as “his universe”. Cosmology benefits from ER. Up next, I show that QM also benefits from ER.

5.13. The Mystery of the Wave–Particle Duality

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

Natural concepts solve the wave–particle duality: Proper space/time (pure distance) replace coordinate space/time. “Wavematters” (pure energy) replace wave/particle. Figure 8 illustrates a single wavematter. In an observer’s view (“external view”), each wavematter “reduces” (collapses) to a wave packet if not tracked or else a particle if tracked. As a wave packet, it propagates in the observer’s $x_1$ axis at the speed $c$ and it oscillates in his axes $x_2$ and $x_3$ (electromagnetic field). It does so in his coordinate space as a function of his coordinate time. In the wavematter’s view (“internal view”), the axis of its 4D motion disappears because of length contraction at the speed $c$. In its 3D space, it is always at rest. Thus, it deems itself particle at rest. Note that there are neither waves nor particles in ES. The wave–particle duality is experienced by observers only.

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

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

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

5.14. The Mystery of Non-Locality

It was Erwin Schrödinger who coined the word “entanglement” in his comment [37] on the Einstein–Podolsky–Rosen paradox [38]. 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 [39] showed that QM is incompatible with local hidden-variable theories. Meanwhile, it has been confirmed in several experiments [40,41,42] that entanglement violates locality in an observer’s 3D space. Entanglement has been considered a non-local effect ever since.

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

For the entangled wavematters (internal view), their $d_4^{\prime }$ axis disappears because of length contraction at the speed $c$. Since the twins stay together spatially in their 3D space spanned by $d_1^{\prime }, d_2^{\prime }, d_3^{\prime }$, they are able to communicate with each other in no time. The twins have never been spatially separated in their view, but their proper time flows in opposite 4D directions. 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” (phrase attributed to Einstein) from the external view only.

This time, the horizon problem and entanglement are solved exactly the same way: An observed region’s (or object’s) 4D vector ${\mathit{\tau }}^{\mathbf{\text{'}}}$ and its 3D space may differ from an observer’s 4D vector $\mathit{\tau }$ and his 3D space. This is possible only if all four $d_{\mu }$ ($\mu =1,2,3,4$) are interchangeable. The full symmetry in all four $d_{\mu }$ is the key to solving entanglement. ER also explains the entanglement of matter, such as electrons [43]. Entangled electrons move in opposite 4D directions $\pm d_4^{\mathrm{\text{'}}}$ at the speed $c$. A measurement tilts the axis of 4D motion of one twin and destroys the entanglement. In ER, non-locality is an obsolete concept.

5.15. The Mystery of the Baryon Asymmetry

In the Lambda-CDM model, almost all matter was created shortly after the Big Bang, when the temperature was high enough to enable pair production. But this process creates equal amounts of particles and antiparticles. So, why do we observe more baryons than antibaryons (also known as the “baryon asymmetry”)? ER scores again: On the one hand, all wavematters injected by the Big Bang reduce to waves or else particles (see Sect. 5.13). On the other hand, pair production creates particles and antiparticles. Thus, there are two sources of particles (Big Bang, pair production), but there is just one source of antiparticles (pair production). Besides, antiparticles are often annihilated shortly after their creation. These arguments explain why we observe more baryons than antibaryons.

ER also tells us why it seems that an antiparticle’s time flows backward: Proper time flows in opposite 4D directions for any two wavematters created in pair production. In an observer’s view, the antiparticle’s ${\mathit{\tau }}^{\prime }$ is reversed with respect to the particle’s $\mathit{\tau }$. However, an antiparticle’s proper time moves forward in its own view. Note that any two wavematters should be entangled if they move in opposite 4D directions (see Figure 9). This prediction gives us a chance to falsify ER. Scientific theories must be falsifiable [44].

6. Conclusions

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

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

Is ER a physical or a metaphysical theory? This is a very good question because only in proper coordinates can we access ES, but the proper coordinates of other objects cannot be measured. I now explain why this is fine: We can always calculate these proper coordinates from ES diagrams as I demonstrated 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 may be obsolete. I was told that physics is all about observing. I disagree. We cannot observe the quantum world, can we? Unfortunately, physicists have applied empirical concepts—which work well in everyday life—to the very far 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.

To sum it all up: (1) Acceleration rotates an object’s $\mathit{\tau }$ and curves its worldline in flat ES. (2) Information hidden in $\mathit{\tau }$ solves 15 mysteries. It is extremely unlikely that 15 solutions in different (!) areas of physics are just 15 coincidences. (3) Different concepts disable a unification of SR/GR and ER. Either scope is limited. We must not apply SR/GR but ER whenever $\mathit{\tau }$ is crucial (high-redshift supernovae, entanglement). We must not apply ER but SR/GR whenever we use empirical concepts. Thus, we must not blindly trust SR/GR as if they were dogmata. Only in natural concepts does nature disclose her secrets.

Final remarks: (1) I only touched on gravity. We must not reject ER because gravity is still an issue. GR seems to solve gravity, but GR is incompatible with QM unless we add another speculative concept (quantum gravity). More studies are required to understand gravitational effects in ER. (2) Mysteries often disappear once the symmetry is matched. The symmetry group for describing nature in natural concepts is SO(4). (3) The invariant $\theta$ finally puts an end to all discussions about time travel. Does any other theory solve the mystery 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 less speculative than postulating cosmic inflation and expanding space and dark energy and non-locality. (5) It seems that Greek philosopher Plato anticipated ER with his Allegory of the Cave [46]: 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 an observer’s view), 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. All observers’ views taken together do not make a holistic view because they still do not provide absolute time. Everyone is welcome to solve even more mysteries by applying ER.