An Issue in Einstein’s Concept of Time

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An Issue in Einstein’s Concept of Time

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Submitted:

10 April 2023

Posted:

11 April 2023

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Abstract

Today’s concept of time is based on Einstein’s theories of special (SR) and general relativity (GR). Many physicists anticipate that GR has an issue since it is not compatible with quantum mechanics. Here we show: SR and GR work well for each observer describing his unique reality, but “Einstein time” (Einstein’s concept of time) has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time hides the big picture! In Euclidean relativity (ER), we replace egocentric Einstein time (coordinate time of an observer) with universal Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light c. Euclidean time is distance covered in ES, divided by c. For each object, Euclidean time flows in a unique 4D direction equal to its current direction of motion. Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and mc². ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

Keywords:

Subject: Physical Sciences - Theoretical Physics

. Euclidean time is distance covered in ES, divided by

c

. For each object, Euclidean time flows in a unique 4D direction equal to its current direction of motion. Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Unlike other ER models, we claim that an observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR, but now we learn that they stem from a projection. ER outperforms SR in explaining time’s arrow and

m c^{2}

. ER outperforms a GR-based cosmology in solving competing Hubble constants and declaring cosmic inflation, expansion of space, and dark energy redundant. Most important, ER is compatible with quantum mechanics: It solves the wave–particle duality and quantum entanglement while declaring non-locality redundant.

Important Remarks

We kindly ask all readers including editors and reviewers to read these preliminary remarks. They help you to avoid those traps that previous reviewers already stepped into. Most readers seem to believe that our theory is just another attempt to identify an issue in Einstein’s theory of special relativity (SR) [1]. Since SR has been experimentally confirmed many times over, our theory is considered a waste of time. What they don’t see: The issue is in Einstein’s concept of time! It affects all of physics including SR, general relativity (GR) [2], and quantum mechanics. We do not dispute any predictions made by SR or GR. Quite the opposite is true: The Lorentz factor is recovered in our theory, and we explain why SR and GR work so well despite the issue in Einstein’s concept of time. Cosmology and quantum mechanics work well only if concepts are added (cosmic inflation, expansion of space, dark energy, non-locality) that we prove to be redundant.

It is because of this issue in Einstein’s concept of time that GR is not compatible with quantum mechanics. We make three changes to the foundations of physics—new concepts of time, distance, and energy—that make relativity compatible with quantum mechanics. Isn’t that reason enough to give our theory of Euclidean relativity (ER) a chance? We must ask this question because one editor informed us that some journals do not consider refutations of SR. Sorry, but why is that? Have SR and GR turned into a dogma that must not be questioned anymore? According to Karl Popper, a theory is scientific only if it is falsifiable [3]. Neither SR nor GR nor ER nor any concept of time is ever set in stone!

Five pieces of advice: (1) Be willing to question Einstein’s concept of time. Do not hold on to it just because you have always worked with it so far. There is progress in science only if we are open for new concepts. (2) Do not take SR and GR for granted when evaluating ER. Neither must we take the geocentric model for granted when evaluating the heliocentric model. Previous reviewers made a systematic error when evaluating ER with concepts of SR and GR. ER is different. In ER, everything is moving at the speed of light. (3) Evaluate ER reasonably. SR comes with Einstein’s concept of time. ER comes with Euclidean time. The Lorentz factor is recovered in ER. So, this argument doesn’t favor SR over ER. Yet ER solves mysteries which SR and GR haven’t solved in 100+ years. (4) Be patient and fair. Do not expect us to address all of physics in this paper. SR and GR have been tested for 100+ years. We must hang on for ER to prove itself, too. (5) Let illustrations inspire you. Artwork assists us to conceive of 4D. Geometric derivations are equivalent to equations.

To sum it all up: Predictions made by SR and GR are correct, but ER penetrates to a deeper level. We do apologize for having published several preprint versions. It was really tricky to figure out why SR and GR make correct predictions despite the issue in Einstein’s concept of time. Sect. 2 is about disclosing this issue. Sect. 3 gives us an intuitive approach to Euclidean time. In Sect. 4, we derive the Lorentz factor and gravitational time dilation. In Sect. 5, we solve 15 mysteries and declare four concepts of today’s physics redundant. In our Conclusions, Occam’s razor knocks out Einstein’s concept of time.

1. Introduction

Today’s concepts of space and time were coined by Albert Einstein. His theory of SR [1] is based on a flat spacetime with an indefinite (not positive-semidefinite) distance function. SR is often interpreted in Minkowski spacetime (MS) because Hermann Minkowski’s geometric interpretation [4] was very successful in explaining relativistic effects. Predicting the lifetime of muons [5] is one example that demonstrates the power of SR. General relativity (GR) [2] includes gravitation and is based on a curved spacetime with a pseudo-Riemannian metric. GR is supported, for example, by the deflection of starlight during a solar eclipse [6] and by the high accuracy of GPS. Quantum field theory [7] unifies classical field theory, SR, and quantum mechanics, but not GR.

We call our theory “Euclidean relativity” and build it on these three postulates: (1) In Euclidean spacetime (ES), all energy is moving at the speed of light. (2) The laws of physics have the same form in each observer’s “reality” (orthogonal projections of ES to his proper 3D space and to his proper flow of time). (3) All energy is “wavematter” (electromagnetic wave packet and matter in one). Our first postulate is stronger than the second SR postulate. The speed of light

c

is both absolute and universal. Everything is moving through ES at the speed

c

. Moving through MS at the speed of light is a pointless concept as objects at rest in 3D space would then move in time at “one second per one second”. Our second postulate is the same as the first SR postulate, except that there is no limitation to inertial frames and that we distinguish ES from an observer’s reality. Our third postulate makes relativity compatible with quantum mechanics.

For all followers of our theory: We loosened our first postulate. Only shortly after the Big Bang was all energy moving radially in ES. Today, some energy is moving transversally because of energy conversion events, such as plasma recombination and supernovae. We were misled by other models, but now our statement is consistent: All energy is confined to an expanding 4D hypersphere; most energy is confined to its 3D hypersurface (see Sect. 5.6). We hold on to our three new concepts of time, distance, and energy.

We aren’t the first physicists to investigate ER: In the early 1990s, Montanus already described ES [8]. He also formulated electrodynamics and gravitational lensing in ES [9]. Almeida compared trajectories in MS with trajectories in ES [10]. Gersten demonstrated that the Lorentz transformation is equivalent to an SO(4) rotation [11]. van Linden studied energy and momentum in ES [12]. Pereira claimed a “hypergeometrical universe”, where matter is made from deformed space [13]. Yet none of these models identifies the issue in Einstein’s concept of time. And they all run into geometric paradoxes discussed in Sect. 4 because they don’t project ES to an observer’s reality. Only Machotka added a “boundedness postulate” to avoid paradoxes [14], but it sounds rather contrived. We overcome such paradoxes by limiting reality in our second postulate. An observer’s reality is only created by projecting ES orthogonally to his proper 3D space and to his proper flow of time.

It is instructive to compare our theory with Newton’s physics and Einstein’s physics. In Newton’s physics, all objects are moving through 3D space as a function of an independent time. The speed of matter is

v_{3 D} ≪ c

. In Einstein’s physics, all objects are moving through 4D spacetime given by 3D space and time, where time is linked to, but different from space (time is measured in seconds). The speed of matter is

v_{3 D} < c

. In our theory, all objects are moving through 4D ES given by four symmetric distances (all distances are measured in light seconds), where time is only a subordinate quantity. The 4D speed of everything is

u_{4 D} = c

. Newton’s physics inspired Kant’s philosophy [15]. Our theory will have a huge impact on modern physics and philosophy. Replacing the concept of time is probably the biggest adjustment since the formulation of quantum mechanics.

2. An Issue in Einstein’s Concept of Time

Today’s concept of time traces back to Albert Einstein. We thus call it “Einstein time”

t

. § 1 of SR [1] is an instruction of how to synchronize two clocks at the positions P and Q. At “P time”

t_{P}

, an observer sends a light pulse from P towards Q. At “Q time”

t_{Q}

, it is reflected at Q towards P. At “P time”

t_{P}^{*}

, it is back at P. Both clocks synchronize if

t_{Q} - t_{P} = t_{P}^{*} - t_{Q} .

(1)

In § 3 of SR [1], Einstein derives the Lorentz transformation for two systems moving relative to each other at a constant speed. The coordinates

x_{1}, x_{2}, x_{3}, t

of an event in a system K are transformed to the coordinates

x_{1}^{'}, x_{2}^{'}, x_{3}^{'}, t^{'}

of that event in a system K’ by

x_{1}^{'} = γ (x_{1} - v_{3 D} t),

(2a)

x_{2}^{'} = x_{2},

(2b)

x_{3}^{'} = x_{3},

(2c)

t^{'} = γ (t - v_{3 D} x_{1} / c^{2}),

(2d)

where the system K’ is moving relative to K in the axis

x_{1}

and at the constant speed

v_{3 D}

. The factor

γ = (1 - v_{3 D}^{2} / c^{2})^{- 0.5},

is the Lorentz factor.

Eqs. (1) and (2a-d) are correct for one observer R in K describing his reality. Because of the relativity postulate, we can write down a similar set of equations for one observer B in K’ describing his reality. So, all theories that are consistent with SR (such as electrodynamics) will be valid for either observer. SR works well for each observer describing his reality, but Einstein time has an issue. It arranges all events in the universe in a 1D line on my watch, yet neither cosmology nor quantum mechanics care about my watch. Einstein time is egocentric: It considers the watch of an observer (“ego”) the center of time, just as the geocentric model considers Earth (“geo”) the center of the solar system. This analogy (and the pun “ego/geo”) should give food for thought to all skeptics.

In order to find an alternative concept of time, we now take a closer look at the effect of time dilation. In § 4 of SR [1], Einstein derives that there is a dilation in Einstein time: The clock of an observer B in K’ is slow with respect to the clock of an observer R in K by the factor

γ

. Time dilation has been experimentally confirmed. So, any alternative concept must recover it and the same

γ

. Now watch out as the next sentences are our entrance to ER: Most physicists aren’t aware that there are two variables in which this time dilation can show up for the same (!) observer R. Einstein and Minkowski assumed that the clock of B is slow with respect to R in

t^{'}

(“proper time” of B). As we explain next, it can also be slow with respect to R in

t

(“coordinate time” of R).

Figure 1 top illustrates a Minkowski diagram of two identical rockets—except for their color—with a proper length of 0.5 Ls (light seconds). They started at the origin and move relative to each other in the axis

x_{1}

at a speed of

0.6 c

. We choose these very high values to visualize relativistic effects. We show that moment when the red rocket has moved 1 s in

t

. Observer R is in the rear end of the red rocket r. His/her view is the red frame with the coordinates

x_{1}

and

t

. Observer B is in the rear end of the blue rocket b. His/her view is the blue frame with the coordinates

x_{1}^{'}

and

t^{'}

. Only for visualization do we draw our rockets in 2D although their width is in the dimensions

x_{2} {, x}_{3}

x_{2}^{'}, x_{3}^{'}

(not displayed in Figure 1). For R, the blue rocket contracts to 0.4 Ls because of length contraction. For B, the rear end of the blue rocket has moved only 0.8 s in

t^{'}

because of time dilation.

It is well known that simultaneity isn’t absolute in SR. In Figure 1 top, R synchronized all clocks inside r and b according to § 2 of SR [1]:

t = 1.0 s

. In this diagram, clocks inside b display a different time for B:

t^{'} = 0.8 s

and

t^{'} = 0.5 s

. Clocks that are synchronized for R aren’t synchronized for B. Yet we must assume that B would also synchronize all clocks inside r and b. To depict the reality of B, we must draw a second Minkowski diagram (not shown here) where clocks inside r aren’t synchronized for R. Since we need two diagrams, we can’t take the measurements of R and B seriously at once. In SR, there is no “at once for both”. Each observer claims just for himself that all clocks are synchronized.

In experimental physics, we are used to take measurements of all observers seriously at once. We can do so if we claim: Each observer measures clocks inside his own rocket as synchronous, while he measures all moving clocks as asynchronous. We get to this “Euclidean time” by replacing the asymmetric axes

x_{1}

and

t

with symmetric distances

d_{1}

and

d_{4}

, and by rotating rocket b thereafter. We then end up with an ES diagram (Figure 1 center) in which the two values “0.8” and “0.5” show up in

d_{4}

(which belongs to R).

In MS,

t

is the coordinate time of R, and

t^{'}

is the proper time of B. In ES, R uses the same variable

d_{4}

for measuring the time of R and for measuring the time of B. In either case (MS and ES), the clock of B is slow with respect to R. In MS, it is slow with respect to R in

t^{'}

(which belongs to B). In ES, it is slow with respect to R in

d_{4}

related to

t

(which belongs to R). Common sense tells us that two identical clocks run the same whether or not they move relative to each other. This is true in ES: Only by observing a moving clock (by projecting

d_{4}^{'}

d_{4}

) does this clock become slow with respect to R.

3. Introducing Euclidean Time and Euclidean Spacetime

MS comes with an indefinite (not positive semidefinite) distance function, which is usually written as

{(c d τ)}^{2} = {(c d t)}^{2} - d x_{1}^{2} - d x_{2}^{2} - d x_{3}^{2},

(3a)

where

τ

is the proper time of an object and

t

is the coordinate time of an observer. We can rearrange the terms in Eq. (3a), so that we end up with a Euclidean metric

{(c d t)}^{2} = d d_{1}^{2} + d d_{2}^{2} + d d_{3}^{2} + d d_{4}^{2},

(3b)

where

d_{i} = x_{i}

for

i = 1,2, 3

and

d_{4} = c τ

. The roles of Einstein time

t

(coordinate time of an observer) and Euclidean time

τ

(proper time of each object) have switched: All invariants are now based on

t

, whereas the fourth dimension in all vectors is now based on

τ

. The switch affects all time-dependent equations of physics and must not be confused with the “Wick rotation”, which replaces

t

i t

, but keeps

τ

as the invariant.

Euclidean time isn’t egocentric (centered in the observer), but universal (centered in each observed object). Because of the symmetry in Eq. (3b), we are free to label the four axes. We assume that each object moves only in its axis

d_{4}

. According to our first postulate, it does so at the speed

c

. Euclidean time is distance covered in ES, divided by

c

τ = d_{4} / c (Euclidean time) .

(4)

Eq. (4) tells us that Euclidean time is only a subordinate quantity derived from covered distance. Time isn’t fundamental to physics as already claimed by other authors [16]. Distance and speed are more significant than time. So, we suggest to choose new units for speed and time.

c

should be specified in its own new unit to be given.

τ

should be specified in “light seconds per this new unit”. We claim that the coordinates

d_{i}

in Eq. (3b) are projected to an observer’s proper 3D space and to his proper flow of time, which is equal to his Einstein time. For each object, we define a 4D vector “flow of time”

τ = d_{4} u / c^{2} (Flow of time),

(5)

where

u

is the Cartesian ES velocity of the object and

u / c

is a unit 4D vector pointing to the current direction of motion of the object. The Cartesian ES velocity

u

has four components

u_{i} = d d_{i} / d t

. From Eq. (3b), we get

u_{1}^{2} + u_{2}^{2} + u_{3}^{2} + u_{4}^{2} = c^{2},

(6)

From Eqs. (5) and (6), we calculate that there is indeed

τ = | τ | = d_{4} / c

t

flows in 1D, whereas

τ

flows in 4D. Einstein time hides that there is a unique 4D vector “flow of time” for each object. Einstein time makes us believe that time would flow in one direction for all objects in the universe!

ES is an open 4D manifold with a Euclidean metric. We can describe ES either in four hyperspherical coordinates (

ϕ_{1}, ϕ_{2}, ϕ_{3}, r

), where each

ϕ_{i}

is a hyperspherical angle and

r

is radial distance from an origin,—or in four symmetric, Cartesian coordinates (

d_{1}, d_{2}, d_{3}, d_{4}

), where each

d_{i}

is axial distance from an origin.

r

and all

d_{i}

are “spatial and temporal distance in one”. Distance isn’t covered as a function of independent time. Only by covering distance is Euclidean time passing by. All distances are measured in light seconds (Ls) by odometers. There is no need to calibrate the odometers because light seconds in ES are absolute. The symmetry of all

d_{i}

supports the idea of natural units. “Space” and “time” in everyday life are only interpretations of distance.

Hyperspherical coordinates are good for grasping the big picture in cosmology. We claim: The Big Bang injected a huge amount of energy into ES all at once at what we take as “origin O”. It also provided an overall radial momentum: Shortly after the Big Bang, all energy moved radially away from O. Today, some energy is moving transversally because of energy conversion events, such as plasma recombination and supernovae.

Cartesian ES coordinates serve as a “master reference frame”: Each observer’s reality is created by projecting the coordinates

d_{i}

orthogonally to his proper 3D space and to his proper flow of time. The coordinates

d_{i}

can be calculated from

d_{1} = r \cos ϕ_{1},

(7a)

d_{2} = r \sin ϕ_{1} \cos ϕ_{2},

(7b)

d_{3} = r \sin ϕ_{1} \sin ϕ_{2} \cos ϕ_{3},

(7c)

d_{4} = r \sin ϕ_{1} \sin ϕ_{2} \sin ϕ_{3},

(7d)

In our ES diagrams, we often choose Cartesian coordinates in which an object starts moving from some origin P other than O. Because of the ES symmetry, we are free to label all four axes. As mentioned before, each object shall move only in its axis

d_{4}

. Below our ES diagrams, we project ES to an observer’s proper 3D space. Here we are free to label the axis that we project to. We assume: There is relative motion only in

d_{1}

and

d_{4}

. So, our ES diagrams display

d_{1}

and

d_{4}

, while our 3D projections display

d_{1}

The projections of

d_{1}, d_{2}, d_{3}, d_{4}

d_{1}, d_{2}, d_{3}

and to

d_{4}

are orthogonal. We don’t replace the concept of space because

d_{1}, d_{2}, d_{3}

are equal to

x_{1}, x_{2}, x_{3}

. We replace the concept of time because there is

t = τ

only for clocks moving in

d_{4}

. If a clock moves in a direction other than

d_{4}

, its distance covered in ES is projected to the axis

d_{4}

of an observer.

4. Geometric Effects in Euclidean Spacetime

We consider the same two rockets as in Figure 1. Observer R (or B) in the rear end of the red rocket r (or else blue rocket b) uses

d_{1}, d_{2}, d_{3}, d_{4}

(or else

d_{1}^{'}, d_{2}^{'}, d_{3}^{'}, d_{4}^{'}

) as coordinates.

d_{1}, d_{2}, d_{3}

span the 3D space of R, and

d_{1}^{'}, d_{2}^{'}, d_{3}^{'}

span the 3D space of B.

d_{4}

relates to the Einstein time of R, and

d_{4}^{'}

relates to the Einstein time of B. The rockets move relative to each other in either 3D space at the constant speed

v_{3 D}

(Figure 2 bottom). As just explained, all 3D motion is in

d_{1}

(or else

d_{1}^{'}

). Our ES diagrams (Figure 2 top) must fulfill these requirements: (1) According to our first postulate, both rockets must move at the speed

c

. (2) Our second postulate must be fulfilled. (3) Both rockets started at the same point P. There is only one way of how to draw our ES diagrams: We must rotate the two reference frames with respect to each other. Only a rotation guarantees full symmetry, so that the laws of physics have the same form in the 3D spaces of R and of B.

We now verify two effects in ES: (1) Since B moves relative to R, the proper 3D space of B is rotated with respect to the proper 3D space of R causing length contraction. (2) Since B moves relative to R, the time of B and the time of R flow in different directions causing time dilation. We define

L_{i, R}

(or

L_{i, B}

) as length of the rocket

i

as measured by the observer R (or else B). In a first step, we project the blue rocket in Figure 2 top left to the axis

d_{1}

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

(8)

L_{b, R} = γ^{- 1} L_{b, B} (Length contraction),

(9)

where

γ = (1 - v_{3 D}^{2} / c^{2})^{- 0.5}

is the same Lorentz factor as in SR. The blue rocket appears contracted to observer R by the factor

γ^{- 1}

, but in different directions. Bottom left: Projection to the 3D space of R. The relative speed is

v_{3 D}

. The blue rocket contracts to

L_{b, R}

. Bottom right: Projection to the 3D space of B. The red rocket contracts to

L_{r, B}

We now ask: Which distances will R observe in his axis

d_{4}

? For the answer, we mentally continue the rotation of the blue rocket in Figure 2 top left until it is pointing vertically down (

φ = 0 °

) and serves as R’s ruler in the axis

d_{4}

. In the projection to the 3D space of R, this ruler contracts to zero: The axis

d_{4}

“is suppressed” (disappears) for R. In a second step, we project the blue rocket in Figure 2 top left to the axis

d_{4}

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

(10)

d_{4, B} = {γ^{- 1} d}_{4, B}^{'},

(11)

where

d_{4, B}

(or

d_{4, B}^{'}

) is the distance that B has moved in

d_{4}

(or else

d_{4}^{'}

). With

d_{4, B}^{'} = d_{4, R}

(full symmetry in ES) and the substitutions

d_{4, B} = c t_{B}

and

d_{4, R} = c t_{R}

, we get

t_{R} = γ t_{B} (Einstein time dilation),

(12)

where

t_{R}

(or

t_{B}

) is the distance that R (or else B) has moved in the Einstein time

t

of R. Eq. (12) tells us that the clock of B is slow with respect to R in the variable

t

, and not in

t^{'}

. There is no Euclidean time dilation because

τ

is absolute (

τ_{R} = τ_{B}

Despite the Euclidean metric in ES, the Lorentz factor

γ

is recovered in Eqs. (9) and (12). This is no surprise because Weyl showed that the Lorentz group is generated by 4D rotations [17]. Gersten [11] demonstrated that the Lorentz transformation is equivalent to an SO(4) rotation in a “mixed space”

x_{1}, x_{2}, x_{3}, {c t}^{'}

. While this is mathematically correct, such a “mixed space” doesn’t make sense physically. Yet it is a hint that Einstein time has an issue! In ER,

c τ

of an observed object (and not

c t^{'}

) is taken as the fourth coordinate of that object. The SO(4) rotation now takes place in

d_{1}, d_{2}, d_{3}, d_{4}

(Figure 2). The Lorentz factor

γ

is recovered in an observer’s reality by projecting ES to

d_{1}, d_{2}, d_{3}

and to

d_{4}

And the Lorentz transformation? It is recovered, together with the Lorentz factor

γ

, once the observer ignores the richness of

τ

and holds on to

t

. Since his selected concept of time (

τ

t

) has no effect on how clocks are running, it also has no effect on the physics involved. SR and all theories based on SR work equally well in either concept of time. Yet if the observer selects

t

, he won’t be able to grasp the big picture in cosmology and quantum mechanics (see Sect. 5). This is why the issue in Einstein time is real!

In order to understand how an acceleration in 3D space manifests itself in ES, let us assume that the blue rocket b in Figure 3 accelerates in the axis

d_{1}

. According to Eq. (6), the speed

u_{1}

of b must then increase at the expense of its speed

u_{4}

. So, b is rotating and moving along a curved trajectory in Cartesian ES coordinates. Any acceleration of an object in 3D space relates to a 4D rotation and a curved trajectory in Cartesian ES coordinates.

. The blue rocket accelerates in the axis

d_{1}

. Bottom: Projection to the 3D space of R. The red rocket is “at rest”. The blue rocket accelerates against the red rocket.

Up next, we demonstrate that the ES geometry can also improve our understanding of gravitation. Let us imagine that Earth is located to the right of the blue rocket in Figure 3 bottom. We assume that the blue rocket is accelerating in the gravitational field of Earth. Eq. (6), which we applied for drawing Figure 3, tells us: If an object accelerates in the axis

d_{1}

of an observer, it automatically decelerates in his axis

d_{4}

(in his flow of time).

Gravitational waves [18] support the idea of GR that gravitation would be a property of spacetime, but they might be predicted by ER, too. Particle physics is still considering gravitation a force that has not yet been unified with the other three forces of physics. We claim: Curved trajectories in Cartesian ES coordinates replace curved spacetime in GR. Eq. (6) is the key equation which relates any motion in

d_{1}, d_{2}, d_{3}

to a motion in

d_{4}

. To support our claim, we now use Cartesian ES coordinates to calculate the Einstein time dilation in the gravitational field of Earth. Let A and B be two identical clocks far away from Earth. They are synchronized, next to each other, and move in the axis

d_{4}

at the speed

c

. Clock B is then allowed to approach Earth in the axis

d_{1}

of A. The kinetic energy of B (mass

m

) is

{\frac{1}{2} m u}_{1, B}^{2} = G M m / r,

(13)

where

u_{1, B}

is the speed of B in the axis

d_{1}

of A,

G

is the gravitational constant,

M

is the mass of Earth, and

r

is the distance of B to Earth’s center. By applying Eq. (6), we get

u_{1, B}^{2} + u_{4, B}^{2} = 2 G M / r + u_{4, B}^{2} = c^{2},

(14)

u_{4, B}^{2} / c^{2} = 1 - 2 G M / (r c^{2}),

(15)

where

u_{4, B}

is the speed of B in the axis

d_{4}

of A. With

u_{4, B} = {d d}_{4, B} / d t_{A}

and

c = {d d}_{4, B} / d t_{B}

(there is no steady axis

d_{4}^{'}

, but B keeps moving at the speed

c

), we get

d t_{B} = {(1 - 2 G M / ({r c}^{2}))}^{0.5} d t_{A},

(16a)

d t_{A} = γ_{g r a v} d t_{B} (Gravitational Einstein time dilation),

(16b)

where

d t_{B}

is the distance that B has moved in the Einstein time

t

of A, while A itself has moved the distance

d t_{A}

. The dilation factor

γ_{g r a v} = (1 - 2 G M / ({r c}^{2}))^{- 0.5}

is the same as in GR [2]. Be aware that

γ_{g r a v}

depends on

r

, but not on

u_{1, B}

. If clock B returns to A, the time displayed by B will be behind the time displayed by A. In GR, this effect is due to a curved spacetime. Applying Eq. (6) in Eq. (14) indicates: In ER, this effect is due to a curved trajectory of B which is projected to the axis

d_{4}

of A.

We finish this section by discussing three instructive paradoxes (Figure 4). They demonstrate the benefit of our concept “distance” and of the projections from ES to an observer’s reality. Problem 1: A rocket moves along a guide wire. In ES, rocket and wire move at the speed

c

. We assume that the wire moves in some axis

d_{4}

. As the rocket moves along the wire, its speed in

d_{4}

must be slower than

c

. Wouldn’t the wire eventually be outside the rocket? Problem 2: A mirror passes a rocket. An observer in the rocket’s tip sends a light pulse to the mirror and tries to detect the reflection. In ES, all objects move at the speed

c

, but in different directions. We assume that the observer moves in some axis

d_{4}

. How can he ever detect the reflection? Problem 3: Earth revolves around the sun. We assume that the sun moves in some axis

d_{4}

. As Earth covers distance in

d_{1}, d_{2}, d_{4}

, its speed in

d_{4}

must be slower than

c

. Wouldn’t the sun escape from the orbital plane of Earth?

The questions in the last paragraph seem to imply that there are geometric paradoxes in ER, but there aren’t. The fallacy in all problems lies in the assumption that there would be four observable (spatial) dimensions. Yet just three distances of ES are observable! We solve all problems by projecting 4D ES orthogonally to 3D space (Figure 4). Then the axis

d_{4}

is suppressed. The projection tells us what an observer’s reality is like because “suppressing

d_{4}

” is equivalent to “length contraction makes

d_{4}

disappear”. Suppressed distance is felt as time. We easily verify in 3D space: The guide wire remains within the rocket; the light pulse is reflected back to the observer; the sun remains in the orbital plane of Earth. Other models [8,9,10,11,12,13] run into paradoxes because they don’t project ES to an observer’s reality.

5. Solving 15 Fundamental Mysteries of Physics

Why should we know about ER and the master frame ES if SR and GR work so well for each observer? In this section, we demonstrate that ER outperforms SR and GR in the understanding of time, time’s arrow,

m c^{2}

, cosmology, and quantum mechanics.

5.1. Solving the Mystery of Time

Euclidean time is distance covered in ES, divided by

c

. Time originates from each object rather than from my watch. Because time can flow in countless 4D directions, the metaphor of “time running in a straight 1D line” is limited in scope. Unlike our precise definition of Euclidean time, there is no definition of Einstein time other than “what I can read on my watch” (attributed to Albert Einstein himself).

5.2. Solving the Mystery of Time’s Arrow

“Time’s arrow” is a synonym for time moving only forward. The arrow emerges from our definition of Euclidean time: The distance covered in ES and

c

always have a positive value. There is no “negative distance” and thus no “going back in time”.

5.3. Solving the Mystery of $m c^{2}$

In SR, where forces are absent, the total energy

E

of an object is given by

E = γ m c^{2} = E_{k i n, 3 D} + m c^{2},

(17)

where

E_{k i n, 3 D}

is an object’s kinetic energy in 3D space and

m c^{2}

is its “energy at rest”. SR doesn’t tell us why there is a

c^{2}

in the energy of objects that in SR never move at the speed of light. ER gives us this missing clue and is thus superior to SR:

m c^{2}

is the kinetic energy of moving through the fourth dimension. The

c^{2}

in Eq. (17) is strong evidence that everything is moving through ES at the speed

c

, while it is at rest in its proper 3D space. There is also

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

(18)

where

p

and

p_{3 D}

are the momenta of an object in ES and in 3D space. Dividing Eq. (18) by

c^{2}

gives us the vector addition of an object’s momentum in its proper 3D space and its momentum

m c

of moving through the fourth dimension.

5.4. Solving the Mystery of Relativistic Effects

In SR, length contraction and time dilation can be derived from the Lorentz transformation, but the physical cause of these relativistic effects remains in the dark. ER discloses that they stem from projecting the master frame ES to an observer’s reality.

5.5. Solving the Mystery of Gravitational Time Dilation

Eq. (16b) tells us: The Einstein time of an object in a gravitational field passes by more slowly with respect to an observer who is very far away from the center of this field. The object’s curved trajectory in Cartesian ES coordinates is projected to the observer’s proper 3D space (here the object accelerates) and to his proper flow of time (here it decelerates). Curved trajectories in Cartesian ES coordinates replace curved spacetime in GR.

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

Now we are ready for our new model of cosmology based on ER. There is no need to create ES. It exists just like numbers. Because of some reason that we don’t know, there was a Big Bang. In today’s model of cosmology, it makes no sense to ask where it occurred: Because space inflated from a singularity, it occurred everywhere. In ES, the Big Bang can be localized at what we take as origin O. The Big Bang injected a huge amount of energy into ES all at once. It also provided an overall radial momentum.

Right after the Big Bang, the concentration of energy was extremely high in ES. In the projection to any reality, a very hot and dense plasma was created. While this plasma was expanding, it cooled down. During plasma recombination, electromagnetic radiation was emitted that we observe as CMB today [19]. At temperatures of roughly 3,000 K, hydrogen atoms formed and the universe became transparent for the CMB. In today’s model of cosmology, this stage was reached about 380,000 years “after” the Big Bang. In ER, these are 380,000 light years “away from” the Big Bang. The value “380,000” still needs to be recalculated because we claim that there was no cosmic inflation (see Sect. 5.9).

Figure 5 left shows the ES diagram for observers on Earth (here Earth is moving in

d_{4}

). Most energy is moving radially: It keeps the radial momentum provided by the Big Bang. The CMB is moving transversally: It can’t move in the axis

d_{4}

because it already moves in

d_{1}

at the speed of light. All energy is confined to an expanding 4D hypersphere; most energy is confined to its 3D hypersurface. We now explain three striking observations regarding the CMB: (1) It is nearly isotropic because it was created equally in

d_{1}, d_{2}, d_{3}

. Cosmic inflation is not needed! (2) The temperature of the CMB is very low because of a very high recession speed

v_{3 D}^{'}

(see Sect. 5.10) and thus a very high Doppler redshift. (3) We observe the CMB today because it started moving at a speed

c^{'} ≪ c

in a very dense medium.

. Left: The CMB is isotropic because it was created equally in

d_{1}, d_{2}, d_{3}

(

d_{2}, d_{3}

not shown here). The CMB has a very low temperature because of a very high

v_{3 D}^{'}

. We observe the CMB today because it started moving at a speed

c^{'} ≪ c

. Right: A supernova S’ occurred when the radius

r^{'}

was smaller than today’s radius

r

. Team B measures S’ in a distance

d_{1} = D

. Earth moved the same

D

, but in

d_{4}

, when the light of S’ arrives. A supernova S occurring today (same

D

) recedes slower than S’

5.7. Solving the Mystery of Hubble’s Law

Figure 5 left shows a galaxy G, which is moving away from the origin O and from Earth. The recession speed

v_{3 D}

relates to the distance

D

c

relates to the radius

r

v_{3 D} = D c / r = H_{0} D (Hubble’s law),

(19)

where

H_{0} = c / r

is the Hubble constant,

c

is in km/s, and

r

is in Mpc. There it is! Eq. (19) is Hubble’s law [20]: The farther a galaxy, the faster it is moving away from Earth.

5.8. Solving the Mystery of the Flat Universe

ES is projected orthogonally to an observer’s proper 3D space. So, this 3D space has no curvature in the fourth dimension. Each observer experiences a flat 3D universe.

5.9. Solving the Mystery of Cosmic Inflation

Many physicists believe that an inflation of space in the early Universe [21,22] would explain the isotropic CMB, the flatness of the observed universe, and large-scale structures (inflated from quantum fluctuations). We showed in Sects. 5.7 and 5.8 that ES can explain the first two of these observations. It also explains the third observation if we only assume that the impacts of early quantum fluctuations have been expanding at the speed of light. Cosmic inflation is a redundant concept.

5.10. Solving the Mystery of Competing Hubble Constants

There are several methods of calculating the Hubble constant

H_{0}

, but unfortunately the results vary from one method to another. Here we consider measurements of the CMB made with the Planck space telescope [23]. We compare them with calibrated distance ladder techniques (distance and redshift of celestial objects) using the Hubble space telescope [24]. We now explain why the values of

H_{0}

obtained by the two teams don’t match within the specified error margins. According to team A [23], there is

H_{0} = 67.66 \pm 0.42 k m / s / M p c

. According to team B [24], there is

H_{0} = 73.52 \pm 1.62 k m / s / M p c

Team B made efforts to minimize the error margin by optimizing the distance measurements. Yet, as we will prove now, misinterpreting the redshift measurements causes a systematic error in team B’s calculation of

H_{0}

. We assume that 67.66 km/s/Mpc would be today’s value of

H_{0}

. We simulate a supernova at a distance of

d_{1} = D = 400 M p c

. If this supernova occurred today (S in Figure 5 right), Eq. (19) would give us the recession speed

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

(20)

z = Δ λ / λ_{0} ≅ v_{3 D} / c = 0.0903,

(21)

where the redshift parameter

z

tells us how any wavelength

λ_{0}

of the supernova’s light is either passively stretched by an expanding space (team B)—or how

λ_{0}

is redshifted by the Doppler effect of objects that are actively receding in ES (our model).

In this and the next paragraph, we demonstrate that team B measures a higher value

z^{'}

, and thus calculates a higher value

v_{3 D}^{'}

, and thus calculates a higher value

H_{0}^{'}

(which is not the same as

H_{0}

). In Figure 5 right, there is one circle called “past”, where the supernova S’ occurred that team B is measuring, and a second circle called “present”, where its light arrives on Earth. Today, this supernova has turned into a neutron star. Because everything is moving at the speed

c

, Earth moved the same distance

D

, but in the axis

d_{4}

, when the light of S’ arrives. Hence, team B is receiving data from an ancient time

τ^{'} = 1 / H_{0}^{'}

when there was a smaller radius

r^{'}

and a larger Hubble constant

H_{0}^{'}

1 / H_{0}^{'} = r^{'} / c = {(r - D) / c = 1 / H}_{0} - D / c,

(22)

H_{0}^{'} = 74.37 k m / s / M p c .

(23)

Because of this higher value and of Eq. (19), all data measured and calculated by team B relate to a higher 3D speed

v_{3 D}^{'} = 29,748 k m / s,

for the same

D

. Because of

z^{'} ≅ v_{3 D}^{'} / c

, this is going to happen: Team B measures a redshift of

z^{'} = 0.0992

, which is indeed higher than 0.0903. Because of this higher value of

z^{'}

, team B will calculate

v_{3 D}^{'} = 29,748 k m / s

from

z^{'} ≅ v_{3 D}^{'} / c

and thus

H_{0}^{'} = 74.37 k m / s / M p c

from Eq. (19). Hence, team B will conclude that 74.37 km/s/Mpc would be today’s value

H_{0}

. In truth, team B ends up with a value

H_{0}^{'}

of the past because it isn’t aware of Eq. (22) and of the ES geometry.

For a shorter distance of

D = 400 k p c

, Eq. (22) tells us that team B’s Hubble constant

H_{0}^{'}

deviates from team A’s Hubble constant

H_{0}

by only 0.009 percent. Yet when plotting

v_{3 D}^{'}

versus

D

for various distances (we chose 50 Mpc, 100 Mpc, 150 Mpc, ..., and 450 Mpc as we didn’t have the raw distance data used by [24]), the resulting slope (team B’s Hubble constant) is 8 to 9 percent higher than team A’s Hubble constant. We kindly ask team B to improve its calculation by eliminating the systematic error in the redshift measurement. It must adjust the calculated speed

v_{3 D}^{'}

to today’s speed

v_{3 D}

by converting Eq. (22) to

H_{0}^{'} = H_{0} c / (c - H_{0} D) = H_{0} / (1 - v_{3 D} / c),

(24)

v_{3 D} = v_{3 D}^{'} / (1 + v_{3 D}^{'} / c),

(25)

We conclude: The redshift is caused by the Doppler effect of objects that are actively receding in ES. Matching the two competing values of

H_{0}

(team B’s published value is indeed 8 to 9 percent higher than team A’s value) is probably the strongest proof of our theory. Team A’s value is correct:

H_{0} = 67 - 68 k m / s / M p c

. If the 3D hypersurface in Figure 5 has always been expanding at the speed

c

, the total time having elapsed since the Big Bang would be equal to

1 / H_{0}

, which is 14.5 billion years rather than 13.8 billion years [25]. The adjusted age would explain the existence of stars as old as 14.5 billion years [26].

Of course, team B is well aware of the fact that the supernova’s light was emitted in the past. Yet in the Lambda-CDM model, all that counts is the timespan

Δ t

during which light is traveling from the supernova to Earth. Along the way, its wavelength is passively stretched by expanding space. So, the total redshift is only developing during the journey to Earth. We can put it this way: The redshift parameter

z^{'}

starts from zero and increases continuously during the journey to Earth. The fact that the supernova occurred long ago in the past at a time

t_{s}

is irrelevant for team B’s calculation.

In ER, the moment

τ_{s}

(when a supernova occurs) is significant, but the timespan

Δ τ

(during which light is traveling to Earth) is irrelevant. The wavelength of the supernova’s light is initially redshifted by the Doppler effect. During its journey to Earth, the parameter

z^{'}

remains constant. Here we can put it this way: The redshift parameter

z^{'}

is tied up at the moment

τ_{s}

“in a package” and sent to Earth, where it is measured. In the Lambda-CDM model, space itself is expanding. In ER, a 3D hypersurface (actively receding energy, not space!) is expanding in ES. Expansion of space is a redundant concept.

5.11. Solving the Mystery of Dark Energy

The CDM model of cosmology assumes an expanding space to explain the distance-dependent recession of celestial objects. Meanwhile, it has been extended to the Lambda-CDM model, where Lambda is the cosmological constant. Cosmologists are now favoring an accelerating expansion [27,28] over a uniform expansion. This is because the calculated recession speeds deviate from values predicted by Eq. (19) if

H_{0}

is taken as an averaged constant. The deviations increase with distance

D

and are compensated by assuming an accelerating expansion of space. Such an acceleration would stretch the wavelength even more and thus increase the recession speeds according to Eq. (21).

Our model gives a much simpler explanation for the deviations from Hubble’s law: Because of

H_{0} = c / r = c / d_{4} = 1 / τ

, the Hubble constant

H_{0}

is a function

H_{0} (τ)

H_{0}^{'}

from every past is higher than today’s value. The older the redshift data are, the more will

H_{0}^{'}

deviate from today’s value

H_{0}

, and the more will

v_{3 D}^{'}

deviate from

v_{3 D}

. The small white circle in Figure 5 right helps us understand these deviations: If a new supernova S occurred today at the same distance

D = 400 M p c

as the shown supernova S’ in the past, S would recede slower (27,064 km/s) than S’ (29,748 km/s) just because of the different values of

H_{0}

and

H_{0}^{'}

. As long as the ES geometry is unknown, higher redshifts are attributed to an accelerating expansion of space. Now that we know about the ES geometry, we can attribute higher redshifts to data from deeper pasts.

We conclude that any expansion of space—uniform as well as accelerating—is only virtual. There is no accelerating expansion of the Universe even if a Nobel Prize was given “for the discovery of the accelerating expansion of the Universe through observations of distant supernovae” [29]. This phrasing actually contains two misconceptions: (1) In the Lambda-CDM model, the term “Universe” implies space, but space isn’t expanding at all. (2) There is a uniform expansion of a 3D hypersurface (actively receding energy), but no “accelerating expansion” whatsoever.

The term “dark energy” [30] was coined to come up with a cause for an accelerating expansion of space. We just explained that there is no expansion of space. So, dark energy is an artifact of Einstein time. Dark energy is a redundant concept. It has never been observed anyway. Radial momentum provided by the Big Bang drives galaxies away from the origin O. They are driven by themselves rather than by dark energy!

Table 1 summarizes huge differences in the meaning of Big Bang, Universe/universe, space, and time. In the Lambda-CDM model, the Big Bang was the beginning of the Universe. In our model, the Big Bang was the injection of energy into ES. In the Lambda-CDM model, Universe (capitalized) is all space, all time, and all energy. In our model, universe is the proper 3D space of one observer. In the Lambda-CDM model, spacetime is curved. In our model, trajectories of objects are curved in Cartesian ES coordinates. There is also a significant difference regarding the underlying theory of relativity: GR isn’t compatible with quantum mechanics; ER is compatible with quantum mechanics.

5.12. Solving the Mystery of the Wave–Particle Duality

We can’t tell which solved mystery is the most important one. Yet the wave–particle duality has certainly kept physicists busy since it was first discussed by Niels Bohr and Werner Heisenberg [31]. The Maxwell equations tell us that electromagnetic waves are oscillations of an electromagnetic field that move through 3D space at the speed of light

c

. In some experiments, objects behave like “waves” (electromagnetic wave packets). But in other experiments, the same objects behave like particles. In today’s physics, an object can’t be both at once because waves distribute energy in space over time, while the energy of particles is localized in space at a given time. This is why we added our third postulate: All energy is “wavematter” (electromagnetic wave packet and matter in one). By combining our concepts of distance and wavematter, we now demonstrate: Waves and particles are actually the same thing (energy), but seen from two perspectives.

Figure 6 illustrates in Cartesian ES coordinates what our new concept of wavematter is all about. If I observe a wavematter (we call it the “external view”), this wavematter comes in four orthogonal dimensions: It propagates in my axis

d_{1}

at some speed

v_{3 D} \leq c

, and it oscillates in my axes

d_{2}

(electric field) and

d_{3}

(magnetic field); propagating and oscillating are functions of Euclidean time

τ

(related to my fourth axis

d_{4}

). So, I can observe how this wavematter is propagating and oscillating: I deem it wave.

From its own perspective (we call it the “internal view” or the “in-flight view”), each wavematter propagates in its axis

d_{4}^{'}

at the speed

c

. Yet because of length contraction at the speed

c

, the axis

d_{4}^{'}

is suppressed for this wavematter. For this reason, its own propagating disappears for itself: It deems itself matter at rest. It still observes the other objects propagating and oscillating in its proper 3D space as it keeps on feeling Euclidean time, while it is invisibly propagating in its axis

d_{4}^{'}

. We conclude that there is an external view and an internal view of each wavematter. In today’s physics, there is no reference frame moving at the speed

c

and thus no internal view of a photon. Be aware that “wavematter” isn’t just another word for the duality, but a generalized concept of energy disclosing why there is a wave–particle duality in an observer’s proper 3D space.

As an example, we now investigate the symmetry in three wavematters

{W M}_{1}

{W M}_{2}

, and

{W M}_{3}

. We assume that they are all moving away from the same point P in ES, but in different directions (Figure 7 top left).

d_{1}, d_{2}, d_{3}, d_{4}

are Cartesian coordinates in which

{W M}_{1}

moves only in

d_{4}

. Hence,

d_{4}

is that axis which

{W M}_{1}

deems time multiplied by

c

, and

d_{1}, d_{2}, d_{3}

span

{W M}_{1}

’s 3D space (Figure 7 bottom left). As the axis

d_{4}

disappears because of length contraction,

{W M}_{1}

deems itself matter at rest (

M_{1}

{W M}_{3}

moves orthogonally to

{W M}_{1}

d_{1}^{'}, d_{2}^{'}, d_{3}^{'}, d_{4}^{'}

are Cartesian coordinates in which

{W M}_{3}

moves only in

d_{4}^{'}

(Figure 7 top right). In this case,

d_{4}^{'}

is that axis which

{W M}_{3}

deems time multiplied by

c

, and

d_{1}^{'}, d_{2}^{'}, d_{3}^{'}

span

{W M}_{3}

’s 3D space (Figure 7 bottom right). As the axis

d_{4}^{'}

disappears because of length contraction,

{W M}_{3}

also deems itself matter at rest (

M_{3}

moves in

d_{4}

. Top right: ES in coordinates where

{W M}_{3}

moves in

d_{4}^{'}

. Bottom left: Projection to

{W M}_{1}

’s 3D space.

{W M}_{1}

deems itself matter at rest (

M_{1}

) and

{W M}_{3}

wave (

W_{3}

). Bottom right: Projection to

{W M}_{3}

’s 3D space.

{W M}_{3}

deems itself matter at rest (

M_{3}

) and

{W M}_{1}

wave (

W_{1}

)

Yet how do

{W M}_{1}

and

{W M}_{3}

move in each other’s view? We must fulfill our first two postulates and the requirement that they both started at the same point P. There is only one way of how to draw our ES diagrams: We must rotate the two reference frames with respect to each other. Only a rotation guarantees full symmetry, so that the laws of physics have the same form in the 3D spaces of

{W M}_{1}

and of

{W M}_{3}

. We can put it this way:

{W M}_{3}

’s 4D motion “swings completely” (rotates by an angle of

90 °

) into

{W M}_{1}

’s 3D space, so that

{W M}_{1}

deems

{W M}_{3}

wave (

W_{3}

). Regarding

{W M}_{2}

, we split its 4D motion into a motion parallel to

{W M}_{1}

’s motion (internal view) and a motion orthogonal to

{W M}_{1}

’s motion (external view). So,

{W M}_{1}

can deem

{W M}_{2}

either matter (

M_{2}

) or wave (

W_{2}

The secret to understanding our new concepts “distance” and “wavematter” is all in Figure 7. Here we see how they go hand in hand: We claim the symmetry of all four Cartesian coordinates in ES and, on top of that, the symmetry of waves and matter. What I deem wave, deems itself matter. Just as distance is spatial and temporal distance in one, so is wavematter wave and matter in one. Here is a compelling reason for this unique claim of our theory: Einstein taught that energy is equivalent to mass. Full symmetry of waves and matter is a consequence of this equivalence. As the axis

d_{4}

disappears because of length contraction, the energy in a propagating wave “condenses” to mass in matter at rest.

In a double-slit experiment, an observer detects coherent waves which pass through a double-slit and produce some pattern of interference on a screen. He observes wavematters from ES whose 4D motion “swings completely” (rotates by an angle of

90 °

) into his proper 3D space. He deems all these wavematters waves because he isn’t tracking through which slit each wavematter is passing. If he did, the interference pattern would disappear immediately. So, he is a typical external observer.

The photoelectric effect is quite different. Of course, one can externally witness how one photon releases one electron from a metal surface. But the physical effect itself (“Do I have enough energy to release one electron?”) is all up to the photon’s view. Only if the photon’s energy exceeds the binding energy of an electron is this electron released. So, we must interpret the photoelectric effect from the internal view of each wavematter. Here its view is crucial! It behaves like a particle, which is commonly called “photon”.

The wave–particle duality is also observed in matter, such as electrons [32]. According to our third postulate, electrons are wavematter, too. From the internal view (if I track them), electrons are particles: “Which slit will I go through?” From the external view (if I don’t track them), electrons behave more like waves. Because I automatically track slow objects, I deem all macroscopic wavematters matter: Their speed in my 3D space is rather low compared with the speed of light thus favoring the internal view. This justifies drawing solid rockets and celestial objects in most of our ES diagrams.

5.13. Solving the Mystery of Quantum Entanglement

The term “entanglement” [33] was coined by Erwin Schrödinger when he published his comment on the Einstein–Podolsky–Rosen paradox [34]. The three authors argued that quantum mechanics wouldn’t provide a complete description of reality. John Bell proved that quantum mechanics is incompatible with local hidden-variable theories [35]. Schrödinger’s word creation didn’t solve the paradox, but demonstrates up to the present day the difficulties that we have in comprehending quantum mechanics. Several experiments have meanwhile confirmed that entangled particles violate the concept of locality [36,37,38]. Ever since has quantum entanglement been considered a non-local effect.

We will now “untangle” quantum entanglement without the concept of non-locality. All we need to do is discuss quantum entanglement in ES. Figure 8 displays two wavematters that were created at once at the same point P and move away from each other in opposite directions at the speed

c

. We claim that these wavematters are entangled. We assume that they are moving in the axes

{+ d}_{4}

and

- d_{4}

, respectively. If they are observed by a third wavematter that is moving in a direction other than

\pm d_{4}

, they are deemed two objects. This third wavematter can’t understand how the entangled wavematters are able to communicate with each other in no time. This is again the external view.

disappears because of length contraction. It deems its twin and itself one object (internal view). For a third wavematter that is moving in a direction other than

{\pm d}_{4}

, the axis

d_{4}

doesn’t disappear. It deems the displayed wavematters two objects (external view)

And here comes the internal (in-flight) view in ES: For each entangled wavematter in Figure 8, the axis

d_{4}

disappears because of length contraction at the speed

c

. That is to say: In the projection to their common 3D space spanned by

d_{1}, d_{2}, d_{3}

, either wavematter deems itself at the very same position as its twin. From either perspective, they are one object that has never been separated. This is why they communicate with each other in no time! Entanglement is another strong evidence that everything is moving through ES at the speed

c

. Our solution to entanglement isn’t limited to photons. Electrons or atoms can be entangled as well. They move at a speed

v_{3 D} < c

in my 3D space, but in their axis

d_{4}

they also move at the speed

c

. We conclude: Even non-locality is a redundant concept.

5.14. Solving the Mystery of Spontaneity

In spontaneous emission, a photon is emitted by an excited atom. Prior to the emission, the photon’s energy was moving with the atom. After the emission, this energy is moving by itself. Today’s physics can’t explain how this energy is boosted to the speed

c

in no time. In ES, both atom and photon are moving at the speed

c

. So, there is no need to boost any energy to the speed

c

. All it takes is energy from ES whose 4D motion “swings completely” (rotates by an angle of

90 °

) into an observer’s proper 3D space—and this energy speeds off at once. In absorption, a photon is spontaneously absorbed by an atom. Today’s physics can’t explain how the photon’s energy is slowed down to the atom’s speed in no time. In ES, both photon and atom are moving at the speed

c

. So, there is no need to slow down any energy. Similar arguments apply to pair production and annihilation. We consider spontaneity another clue that everything is moving through ES at the speed

c

5.15. Solving the Mystery of the Baryon Asymmetry

According to the Lambda-CDM model, almost all matter in the Universe was created shortly after the Big Bang. Only then was the temperature high enough to enable the pair production of baryons and antibaryons. Yet the density was also very high so that baryons and antibaryons should have annihilated each other again. Since we do observe a lot more baryons than antibaryons today (also known as the “baryon asymmetry”), it is assumed that more baryons than antibaryons must have been produced in the early Universe [39]. However, an asymmetry in pair production has never been observed.

Our theory offers a unique solution to the baryon asymmetry: Since each wavematter deems itself matter, there was matter in 3D space right after the Big Bang. Pair production isn’t needed to create matter, and an asymmetry in pair production isn’t needed to explain the baryon asymmetry. There is much less antimatter than matter because antimatter is created only in pair production. One may ask why wavematter doesn’t deem itself antimatter, but this question is missing the point. Energy has two faces: wave and matter. “Antimatter” is matter, too, but with the opposite electric charge.

6. Conclusions

To this day, all attempts to unify GR and quantum mechanics have failed miserably. In Sects. 5.1 through 5.15, ER solves mysteries which SR and GR either haven’t solved in 100+ years—or that have meanwhile been solved, but only by applying concepts (cosmic inflation, expansion of space, dark energy, non-locality) that we proved to be redundant. Now we let Occam’s razor, a powerful tool in science, do its job: Because ER outperforms SR and GR, Occam’s razor knocks out Einstein time and these four redundant concepts. We also conclude: ER is compatible with quantum mechanics. Egocentric Einstein time prevents physicists from grasping the big picture in cosmology and quantum mechanics. We glimpse “behind the curtain” if we only replace Einstein time with Euclidean time.

Since SR and GR have been experimentally confirmed many times over, they are considered two of the greatest achievements of physics. We proved that their concept of time is flawed. Albert Einstein, one of the most brilliant physicists ever, wasn’t aware of ER. It was a wise decision to award him with the Nobel Prize for his theory of the photoelectric effect [40] rather than for SR or GR. We campaign for ER as it penetrates to a deeper level. For the first time ever, mankind understands the nature of time: Time isn’t a fundamental quantity, but distance covered in ES, divided by the speed of light. Imagine: The human brain is able to grasp the idea that our energy is moving through ES at the speed of light. With that said, conflicts of mankind become all so small.

ER solves 15 mysteries at once: (1) time, (2) time’s arrow, (3)

{m c}^{2}

, (4) relativistic effects, (5) gravitational time dilation, (6) CMB, (7) Hubble’s law, (8) flat universe, (9) cosmic inflation, (10) competing Hubble constants, (11) dark energy, (12) wave–particle duality, (13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions are 15 confirmations of ER. It isn’t unusual that new concepts give many answers at once. For quantum leaps in understanding, we must question existing concepts. It certainly was to our advantage that we weren’t dazzled by the success of SR and GR. Einstein sacrificed absolute space and time. We sacrifice the absoluteness of waves and matter, but we restore absolute time and pair it with an absolute hypersphere. Quantum leaps can’t be planned. They just happen like the spontaneous emission of a photon. J

We introduced new concepts of time, distance, and energy: (1) There is absolute time. (2) Spatial and temporal distance aren’t two, but one [41]. (3) Wave and matter aren’t two, but one. We explained these concepts and confirmed how powerful they are. We can even tell the source of their power: symmetry and beauty. Once you have cherished this beauty, you will never let it go again. Yet to cherish it, you first need to give yourself a little push—accepting that an observer’s reality is only created by projecting ES to his proper 3D space and to his proper flow of time. Questions like “Why would reality only be a projection?” must not be asked in physics. The magic of “reality being a projection” compares to the magic of “reality being a probability function”. The latter is well accepted.

It looks like philosopher Plato was right with his Allegory of the Cave [42]: Mankind experiences a projection that is blurred because of quantum mechanics! We would be mistaken if we thought that the concepts of nature were on the same level as all the tangible realities perceived by us. Our advice: Think of a problem in physics and try to solve it in ER. We predict that ER covers gravitational waves, too. Our new concepts lay the groundwork for ER. Anyone is welcome to join us. Hopefully, physics will be improved.

Conflict of Interest: The authors have no competing interests to declare.

Author Contributions

Markolf has a Ph.D. in physics and is a full professor at Heidelberg University, Germany. He studied in Frankfurt, Heidelberg, at UC San Diego, and Harvard. He found the issue in Einstein time, interpreted reality as “projection from a master frame”, and contributed the concepts “distance” and “wavematter” that make ER compatible with quantum mechanics. He also drafted this paper. Siegfried taught physics and math at the Waldorf School in Darmstadt, Germany. He contributed most of the ES diagrams and solved the mystery of the competing Hubble constants.

Data Availability Statement

All data that support this study are included.

Acknowledgements

We wish to thank Matthias Bartelmann, Dennis Dieks, Dirk Rischke, and Jürgen Struckmeier for reading and commenting on earlier versions of this paper.

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Figure 1. Minkowski diagram, ES diagram, and 3D projection for two identical rockets. Top: The Minkowski diagram depicts the reality of just one observer (here of R who synchronizes all clocks inside both rockets). Our diagram doesn’t depict the reality of B who would also synchronize these clocks. Center: The ES diagram can be projected to either reality. Bottom: Projection to the 3D space of R.

Figure 2. ES diagrams and 3D projections for two identical rockets. All axes are in Ls (light seconds). Top left and top right: In the ES diagrams, both rockets are moving at the speed

c

Figure 2. ES diagrams and 3D projections for two identical rockets. All axes are in Ls (light seconds). Top left and top right: In the ES diagrams, both rockets are moving at the speed

c

Figure 3. ES diagram and 3D projection for two identical rockets. Top: In the ES diagram, the red rocket moves in the steady axis

d_{4}

Figure 3. ES diagram and 3D projection for two identical rockets. Top: In the ES diagram, the red rocket moves in the steady axis

d_{4}

Figure 4. Graphical solutions to three geometric paradoxes. Left: A rocket moves along a guide wire. In 3D space, the guide wire remains within the rocket. Center: An observer in a rocket’s tip tries to detect the reflection of a light pulse. Between two snapshots (0–1 or 1–2), rocket, mirror, and light pulse move 0.5 Ls in ES. In 3D space, the light pulse is reflected back to the observer. Right: Earth revolves around the sun. In 3D space, the sun remains in the orbital plane of Earth.

Figure 5. ES diagrams and 3D projections (not to scale) for solving the three mysteries 5.6, 5.7, and 5.10. The displayed circular arcs are part of a 3D hypersurface, which is expanding in ES at the speed

c

c

Figure 6. Concept of wavematter. Artwork illustrating how one object can be deemed wave or matter. Wavematter comes in four orthogonal dimensions: propagation, electric field, magnetic field, and Euclidean time. Each wavematter deems itself matter at rest (internal or in-flight view). If it is observed by some other wavematter (external view), it is deemed wave.

Figure 7. ES diagrams and 3D projections for three wavematters. Top left: ES in coordinates where

{W M}_{1}

Figure 7. ES diagrams and 3D projections for three wavematters. Top left: ES in coordinates where

{W M}_{1}

Figure 8. Quantum entanglement in ES. Artwork illustrating internal view and external view. For each displayed wavematter, the axis

d_{4}

Figure 8. Quantum entanglement in ES. Artwork illustrating internal view and external view. For each displayed wavematter, the axis

d_{4}

Table 1. Comparing the Lambda-CDM model with our model of cosmology.

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An Issue in Einstein’s Concept of Time

Abstract

1. Introduction

2. An Issue in Einstein’s Concept of Time

3. Introducing Euclidean Time and Euclidean Spacetime

4. Geometric Effects in Euclidean Spacetime

5. Solving 15 Fundamental Mysteries of Physics

5.1. Solving the Mystery of Time

5.2. Solving the Mystery of Time’s Arrow

5.3. Solving the Mystery of m c 2

5.4. Solving the Mystery of Relativistic Effects

5.5. Solving the Mystery of Gravitational Time Dilation

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

5.7. Solving the Mystery of Hubble’s Law

5.8. Solving the Mystery of the Flat Universe

5.9. Solving the Mystery of Cosmic Inflation

5.10. Solving the Mystery of Competing Hubble Constants

5.11. Solving the Mystery of Dark Energy

5.12. Solving the Mystery of the Wave–Particle Duality

5.13. Solving the Mystery of Quantum Entanglement

5.14. Solving the Mystery of Spontaneity

5.15. Solving the Mystery of the Baryon Asymmetry

6. Conclusions

Conflict of Interest: The authors have no competing interests to declare.

Author Contributions

Data Availability Statement

Acknowledgements

References

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5.3. Solving the Mystery of $m c^{2}$