Can Physics Benefit from a New Concept of Time?

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. In Euclidean relativity (ER), we replace Einstein time (coordinate time of an observer) with Euclidean time (proper time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light. For each observed object, Euclidean time flows in a 4D direction equal to its direction of motion. The projection of the object’s 4D vector “flow of time” to an observer’s direction of motion yields its motion in his Einstein time. Because of the projection, Einstein time provides less information than Euclidean time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR. Yet 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. We conclude: Physics based on Euclidean time penetrates to a deeper level and makes less assumptions.

Full text

Can Physics Benefit from a New Concept of Time? 1
Markolf H. Niemz 1,* and Siegfried W. Stein 2 2
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1 Heidelberg University, Theodor-Kutzer-Ufer 1–3, 68167 Mannheim, Germany
2 no affiliation
* Correspondence: markolf.niemz@medma.uni-heidelberg.de
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. In Euclidean rela-
tivity (ER), we replace Einstein time (coordinate time of an observer) with Euclidean time (proper
time of each object). In Euclidean spacetime (ES), all energy is moving at the speed of light. For each
observed object, Euclidean time flows in a 4D direction equal to its direction of motion. The projec-
tion of the object’s 4D vector “flow of time” to an observer’s direction of motion yields its motion in
his Einstein time. Because of the projection, Einstein time provides less information than Euclidean
time. ER gives us the same Lorentz factor as in SR and the same gravitational time dilation as in GR.
Yet ER outperforms SR in explaining time’s arrow and . 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. We conclude:
Physics based on Euclidean time penetrates to a deeper level and makes less assumptions.
Keywords: cosmology; Hubble constant; gravitation; wave–particle duality; entanglement 22
23
Important Remarks 24
We kindly ask all readers including editors and reviewers to read these preliminary 25
remarks. They help you to avoid those traps that previous reviewers already stepped into. 26
Most readers seem to believe that our theory is just another attempt to identify an issue in 27
Einstein’s theory of special relativity (SR) [1]. Since SR has been experimentally confirmed 28
many times over, our theory is considered a waste of time. What they don’t see: The issue 29
is in Einstein’s concept of time! It affects all of physics including SR, general relativity (GR) 30
[2], and quantum mechanics. We do not dispute any predictions made by SR or GR. Quite 31
the opposite is true: The Lorentz factor is recovered in our theory, and we explain why SR 32
and GR work so well despite the issue in Einstein’s concept of time. 33
Yet it is because of this issue in today’s concept of time that GR is not compatible with 34
quantum mechanics. We make three changes to the foundations of physics—new concepts 35
of time, distance, and energy—that make relativity compatible with quantum mechanics. 36
Isn’t that reason enough to give our theory of Euclidean relativity (ER) a chance? We must 37
ask this question because one editor informed us that some journals do not consider refu- 38
tations of SR. Sorry, but why is that? Do they fear that their reputation could be damaged? 39
Have SR and GR turned into a dogma that must not be questioned anymore? A theory is 40
scientific only if it is falsifiable [3]. Neither SR nor GR nor ER is ever set in stone! 41
Our recent submission to a renowned journal was rejected by one peer reviewer who 42
argued: “Modern physics is the discovery that a notion of universal time plays no role in 43
the phenomena we observe.” Nonsense! Today’s physics works well without an absolute 44
time, but only if other—more speculative—concepts are added. This one reviewer didn’t 45
consider all the benefits of our theory. If he did, he would have noticed that we get along 46
without cosmic inflation, expansion of space, dark energy, and non-locality. Isn’t that rea- 47
son enough to promote scientific discussion of our theory in a journal? 48
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Six pieces of advice: (1) Imagine that sticking to Einstein’s egocentric concept of time could 49
be the same mistake as sticking to the geocentric model. Learn in our paper about all the benefits 50
of replacing Einstein’s concept of time. (2) Don’t be prejudiced against a theory which claims 51
to solve 15 mysteries at once. It isn’t unusual that new concepts give many answers at once. 52
(3) Don’t take SR and GR for granted while evaluating ER. Previous reviewers made a severe, 53
systematic error by doing so. ER is different. In ER, everything is moving at the speed of 54
light. (4) Evaluate ER reasonably. The Lorentz factor is recovered in ER. So, this argument 55
doesn’t favor SR. Yet ER does solve mysteries that SR and GR haven’t solved in 100+ years. 56
(5) Be patient and fair. Don’t expect that we address all of physics in one paper. SR and GR 57
have been tested for 100+ years. We must wait for ER to prove itself, too. (6) Let illustrations 58
assist you to conceive of 4D. Geometric derivations are equivalent to equations. 59
Also consider: (7) It is very unlikely that a new concept of time is mistaken if it solves 60
15 fundamental mysteries at once. (8) Science should discuss an original stance on funda- 61
mental issues rather than dismissing it on a knee-jerk reaction. (9) Today’s physics cannot 62
be complete since GR is not compatible with quantum mechanics. (10) If you are an expert 63
in inflation theory, dark energy, or quantum gravity, bear in mind that you are biased. ER 64
doesn’t need any of these theories/concepts, and that might be hard to accept. 65
To sum it all up: Predictions made by SR and GR are correct, but ER penetrates to a 66
deeper level. We do apologize for having published several preprint versions. It was really 67
tricky to figure out why SR and GR make correct predictions despite the issue in Einstein’s 68
concept of time. Sect. 2 is about disclosing this issue. Sect. 3 gives us an intuitive approach 69
to Euclidean time. In Sect. 4, we derive the Lorentz factor and gravitational time dilation. 70
In Sect. 5, we solve 15 mysteries and declare four concepts of today’s physics redundant. 71
In our Conclusions, Occam’s razor knocks out Einstein’s concept of time. 72
1. Introduction 73
Today’s concepts of space and time were coined by Albert Einstein. His theory of SR 74
[1] is based on a flat spacetime with an indefinite distance function. SR is often interpreted 75
in Minkowski spacetime (MS) because Minkowski’s geometric interpretation [4] was very 76
successful in explaining relativistic effects. Predicting the lifetime of muons [5] is one ex- 77
ample that demonstrates the power of SR. General relativity (GR) [2] is based on a curved 78
spacetime with a pseudo-Riemannian metric. GR is supported, for example, by the deflec- 79
tion of starlight during a solar eclipse [6] and by the high accuracy of GPS. Quantum field 80
theory [7] unifies classical field theory, SR, and quantum mechanics, but not GR. 81
We call our theory “Euclidean relativity” and build it on these three postulates: (1) In 82
Euclidean spacetime (ES), all energy is moving at the speed of light. (2) The laws of physics 83
have the same form in each observer’s “reality” (orthogonal projections of ES to his proper 84
3D space and to his proper flow of time). (3) All energy is “wavematter” (electromagnetic 85
wave packet and matter in one). Our first postulate is stronger than Einstein’s second pos- 86
tulate. The speed of light  is both absolute and universal. Everything is moving through 87
ES at the speed . Moving through MS at the speed  is a pointless idea: Objects “at rest” 88
in 3D space would move in time at one second per one second. Our second postulate matches 89
Einstein’s first postulate, except that it isn’t limited to inertial frames, but to an observer’s 90
reality. Our third postulate makes relativity compatible with quantum mechanics. 91
We aren’t the first physicists to investigate ER: In the early 1990s, Montanus already 92
described ES [8]. He also formulated electrodynamics and gravitational lensing in ES [9]. 93
Almeida compared trajectories in MS with trajectories in ES [10]. Gersten demonstrated 94
that the Lorentz transformation is equivalent to an SO(4) rotation [11]. van Linden studied 95
energy and momentum in ES [12]. Pereira claimed a “hypergeometrical universe”, where 96
matter is made from deformed space [13]. Yet none of these models identifies the issue in 97
Einstein’s concept of time. And they all run into geometric paradoxes discussed in Sect. 4 98
because they don’t project ES to an observer’s reality. Only Machotka added a “bounded- 99
ness postulate” to avoid paradoxes [14], but it sounds rather contrived. We overcome such 100
paradoxes by limiting our second postulate to an observer’s reality. 101
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It is instructive to compare our theory with Newton’s physics and Einstein’s physics. 102
In Newton’s physics, all objects are moving through 3D space as a function of an inde- 103
pendent time. The speed of matter is   . In Einstein’s physics, all objects are moving 104
through 4D spacetime given by 3D space and time, where time is linked to, but different 105
from space (time is measured in seconds). The speed of matter is   . In our theory, 106
all objects are moving through 4D ES given by four symmetric distances (all distances are 107
measured in light seconds), where time is only a subordinate quantity. The 4D speed of 108
everything is   . Newton’s physics inspired Kant’s philosophy [15]. Our theory will 109
have a huge impact on modern physics and philosophy. Replacing the concept of time is 110
probably the biggest adjustment since the formulation of quantum mechanics. 111
2. An Issue in Einstein’s Concept of Time 112
Today’s concept of time traces back to Albert Einstein. We thus call it “Einstein time” 113
. § 1 of SR [1] is an instruction of how to synchronize two clocks at the positions P and Q. 114
At “P time” , an observer sends a light pulse from P towards Q. At “Q time” , it is 115
reflected at Q towards P. At “P time” 
, it is back at P. Both clocks synchronize if 116
117
    
  . (1) 118
119
In § 3 of SR [1], Einstein derives the Lorentz transformation for two systems moving 120
relative to each other at a constant speed. The coordinates     of an event in a sys- 121
tem K are transformed to the coordinates 
󰆒 
󰆒 
󰆒 󰆒 of that event in a system K’ by 122
123

󰆒  󰇛 󰇜 , (2a) 124
125

󰆒    , (2b) 126
127

󰆒    , (2c) 128
129
󰆒  󰇛   󰇜 , (2d) 130
131
where the system K’ is moving relative to K in the axis  and at the constant speed . 132
The factor   󰇛  
󰇜 is the Lorentz factor. 133
Eqs. (1) and (2a-d) are correct for one observer R in K describing his reality. Because 134
of the relativity postulate, we can write down a similar set of equations for one observer 135
B in K’ describing his reality. So, all theories that are consistent with SR (such as electro- 136
dynamics) will be valid for either observer. SR works well for each observer describing 137
his reality, but Einstein time has an issue. It arranges all events in the universe in a 1D line 138
on my watch, yet neither cosmology nor quantum mechanics care about my watch. Ein- 139
stein time is egocentric: It considers the watch of an observer (“ego”) the center of time, just 140
as the geocentric model considers Earth (“geo”) the center of the solar system. This analogy 141
(and the pun “ego/geo”) should give food for thought to all skeptics. 142
In order to find an alternative concept of time, we now take a closer look at the effect 143
of time dilation. In § 4 of SR [1], Einstein derives that there is a dilation in Einstein time: 144
The clock of an observer B in K’ is slow with respect to the clock of an observer R in K by 145
the factor . Time dilation has been experimentally confirmed. So, any alternative concept 146
must recover it and the same . Now watch out as the next sentences are our entrance to 147
ER: Most physicists aren’t aware that there are two variables in which this time dilation 148
can show up for the same (!) observer R. Einstein and Minkowski assumed that the clock 149
of B is slow with respect to R in 󰆒 (“proper time” of B). As we explain next, it can also be 150
slow with respect to R in  (“coordinate time” of R). 151
Fig. 1 top illustrates a Minkowski diagram of two identical rockets—except for their 152
color—with a proper length of 0.5 Ls (light seconds). They started at the origin and move 153
relative to each other in the axis  at a speed of . We choose these very high values 154
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to visualize relativistic effects. We show that moment when the red rocket has moved 1 s 155
in . Observer R is in the rear end of the red rocket r. His/her view is the red frame with 156
the coordinates  and . Observer B is in the rear end of the blue rocket b. His/her view 157
is the blue frame with the coordinates 
󰆒 and 󰆒. Only for visualization do we draw our 158
rockets in 2D although their width is in the dimensions   or 
󰆒 
󰆒 (not displayed in 159
Fig. 1). For R, the blue rocket contracts to 0.4 Ls because of length contraction. For B, the rear 160
end of the blue rocket has moved only 0.8 s in 󰆒 because of time dilation. 161
162
Fig. 1 Minkowski diagram, ES diagram, and 3D projection for two identical rockets. Top: The Min- 163
kowski diagram depicts the reality of just one observer (here of R who synchronizes all clocks inside 164
both rockets). Our diagram doesn’t depict the reality of B who would also synchronize these clocks. 165
Center: The ES diagram can be projected to either reality. Bottom: Projection to the 3D space of R 166
It is well known that simultaneity isn’t absolute in SR. In Fig. 1 top, R synchronized 167
all clocks inside r and b according to § 2 of SR [1]:   . In this diagram, clocks inside 168
b display a different time for B: 󰆒  and 󰆒 . Clocks that are synchronized for 169
R aren’t synchronized for B. Yet we must assume that B would also synchronize all clocks 170
inside r and b. To depict the reality of B, we must draw a second Minkowski diagram (not 171
shown here) where clocks inside r aren’t synchronized for R. Since we need two diagrams, 172
we can’t take the measurements of R and B seriously at once. In SR, there is no “at once for 173
both”. Each observer claims just for himself that all clocks are synchronized. 174
In experimental physics, we are used to take measurements of all observers seriously 175
at once. We can do so if we claim: Each observer measures clocks inside his own rocket as syn- 176
chronous, while he measures all moving clocks as asynchronous. We get to this “Euclidean time” 177
by replacing the asymmetric axes  and  with symmetric distances  and , and by 178
rotating rocket b thereafter. We then end up with an ES diagram (Fig. 1 center) in which 179
the two values “0.8” and “0.5” show up in  (which belongs to R). 180
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In MS,  is the coordinate time of R, and 󰆒 is the proper time of B. In ES, R uses the 181
same variable  for measuring the time of R and for measuring the time of B. In either 182
case (MS and ES), the clock of B is slow with respect to R. In MS, it is slow with respect 183
to R in 󰆒 (which belongs to B). In ES, it is slow with respect to R in  related to  (which 184
belongs to R). Common sense tells us that two identical clocks run the same whether or 185
not they move relative to each other. This is true in ES: Only by observing a moving clock 186
(by projecting 
󰆒 to ) does this clock become slow with respect to R. 187
3. Introducing Euclidean Time and Euclidean Spacetime 188
MS comes with an indefinite (not positive semidefinite) distance function, which is 189
usually written as 190
191
󰇛󰇜   󰇛 󰇜 
 
 
 , (3a) 192
193
where  is the proper time of an object and  is the coordinate time of an observer. We 194
can rearrange the terms in Eq. (3a), so that we end up with a Euclidean metric 195
196
󰇛󰇜  
 
 
 
 , (3b) 197
198
where   for    and . The roles of Einstein time  (coordinate time of 199
an observer) and Euclidean time  (proper time of each object) have switched: All invar- 200
iants are now based on , whereas the fourth dimension in all vectors is now based on . 201
The switch affects all time-dependent equations of physics and must not be confused with 202
the “Wick rotation” [16], which replaces  by , but keeps  as the invariant. 203
Euclidean time isn’t egocentric (centered in the observer), but universal (centered in 204
each object of the universe). Our word “object” includes observers, while “observed ob- 205
ject” excludes observers. Because of the symmetry in Eq. (3b), we are free to label the four 206
axes. We assume: Each object moves only in its axis . According to our first postulate, 207
it does so at the speed . Euclidean time is distance covered in ES, divided by . 208
209
    (Euclidean time). (4) 210
211
Eq. (4) tells us that Euclidean time is not a fundamental quantity, but only a subordi- 212
nate quantity derived from covered distance. Distance and speed are more significant than 213
time! So, we suggest to define new units for distance, speed, and time: All distances should 214
be specified in “light seconds”,  in its own new unit to be given by the community, and 215
 in “light seconds per this new unit”. We do prefer the term “Euclidean spacetime” over 216
“Euclidean space” because covered distance relates to Euclidean time. 217
For each object, we define a 4D vector “flow of time” 218
219
    (Flow of time), (5) 220
221
where  is the Cartesian ES velocity of the object and  is a unit 4D vector specifying 222
the current direction of motion of the object. The Cartesian ES velocity  has four com- 223
ponents  . From Eq. (3b), we get 224
225

  
 
 
    . (6) 226
227
From Eqs. (5) and (6), we can convince ourselves that there is indeed     . 228
For each observed object, Euclidean time flows in a 4D direction equal to its direction of 229
motion. The projection of the object’s 4D vector “flow of time” to an observer’s direction 230
of motion yields its motion in his Einstein time. Because of this projection, Einstein time 231
provides less information than Euclidean time! Einstein time hides that there is a unique 232
4D vector “flow of time” for each object. 233
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ES is an open 4D manifold with a Euclidean metric. We can describe ES either in four 234
hyperspherical coordinates (   ), where each  is a hyperspherical angle and  235
is radial distance from an origin,—or in four symmetric, Cartesian coordinates (   ), 236
where each  is axial distance from an origin. Be aware that in ES distance isn’t covered 237
as a function of independent time. Only by covering distance is Euclidean time passing 238
by. The drawback of ER is that we must get used to a different meaning of time: Euclidean 239
time isn’t a quantity enabling motion, but a quantity resulting from motion. 240
Hyperspherical coordinates are good for grasping the big picture in cosmology. We 241
claim: The Big Bang injected a huge amount of energy into ES all at once at what we take 242
as “origin O”. It also provided an overall radial momentum: Shortly after the Big Bang, all 243
energy moved radially away from O. Today, some energy is moving transversally because 244
of energy conversion events, such as plasma recombination and supernovae. 245
Cartesian ES coordinates serve as a “master reference frame”: An observer’s reality is 246
only created by projecting the coordinates  orthogonally to his proper 3D space and to his proper 247
flow of time. The symmetry of all  supports the idea of natural units. “Space” and “time” 248
in everyday life are just different interpretations of distance covered in ES. There is 249
250
     , (7a) 251
252
      , (7b) 253
254
        , (7c) 255
256
        . (7d) 257
258
In our ES diagrams, we often choose Cartesian coordinates in which an object starts 259
moving from some origin P other than O. We assume: Each observer moves only in his axis 260
; each observed object moves only in its axis 
󰆒. Below our ES diagrams, we project ES to an 261
observer’s proper 3D space. Here we are free to label the three axes that we project to. In 262
most cases, we assume: There is relative motion only in  and . Our ES diagrams then 263
display  and , while our 3D projections display . The projections to    and 264
to  are orthogonal. We don’t replace the concept of space because    are equal 265
to   . We replace the concept of time because there is    only for clocks moving 266
in the axis . If a clock moves in a direction 
󰆒 other than , its distance covered in ES 267
is projected to an observer’s axis . Such a clock is slow in his Einstein time, but Euclidean 268
time flows at the same rate for him (in ) and for the observed clock (in 
󰆒). 269
4. Geometric Effects in Euclidean Spacetime 270
We consider the same two rockets as in Fig. 1. Observer R (or B) in the rear end of the 271
red rocket r (or else blue rocket b) uses     (or else 
󰆒 
󰆒 
󰆒 
󰆒) as coordinates. 272
   span the 3D space of R, and 
󰆒 
󰆒 
󰆒 span the 3D space of B.  relates to the 273
Einstein time of R, and 
󰆒 relates to the Einstein time of B. The rockets move relative to 274
each other in either 3D space at the constant speed  (Fig. 2 bottom). As just explained, 275
all 3D motion is in  (or else 
󰆒). Our ES diagrams (Fig. 2 top) must fulfill these require- 276
ments: (1) According to our first postulate, both rockets must move at the speed . (2) Our 277
second postulate must be fulfilled. (3) Both rockets started at the same point P. There is 278
only one way of how to draw our ES diagrams: We must rotate the two reference frames 279
with respect to each other. Only a rotation guarantees full symmetry, so that the laws of 280
physics have the same form in the 3D spaces of R and of B. 281
We now verify two effects in ES: (1) Since B moves relative to R, the proper 3D space 282
of B is rotated with respect to the proper 3D space of R causing length contraction. (2) Since 283
B moves relative to R, the time of B and the time of R flow in different directions causing 284
time dilation. We define  (or  ) as length of the rocket  as measured by the observer 285
R (or else B). In a first step, we project the blue rocket in Fig. 2 top left to the axis . 286
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      󰇛 󰇜 󰇛󰇜   , (8) 287
288
      (Length contraction), (9) 289
290
where   󰇛  
󰇜 is the same Lorentz factor as in SR. The blue rocket appears 291
contracted to observer R by the factor . 292
293
Fig. 2 ES diagrams and 3D projections for two identical rockets. All axes are in Ls (light seconds). 294
Top left and top right: In the ES diagrams, both rockets are moving at the speed , but in different 295
directions. Bottom left: Projection to the 3D space of R. The relative speed is . The blue rocket 296
contracts to . Bottom right: Projection to the 3D space of B. The red rocket contracts to  297
We now ask: Which distances will R observe in his axis ? For the answer, we men- 298
tally continue the rotation of the blue rocket in Fig. 2 top left until it is pointing vertically 299
down (  ) and serves as R’s ruler in the axis . In the projection to the 3D space of 300
R, this ruler contracts to zero: The axis  “is suppressed” (disappears) for R. In a second 301
step, we project the blue rocket in Fig. 2 top left to the axis . 302
303
      󰇛 
󰆒󰇜  󰇛 󰇜   , (10) 304
305
     
󰆒 , (11) 306
307
where  (or 
󰆒) is the distance that B has moved in  (or else 
󰆒). With 
󰆒  308
(full symmetry in ES) and the substitutions    and   , we get 309
310
   (Einstein time dilation), (12) 311
312
where  (or ) is the distance that R (or else B) has moved in the Einstein time  of R. 313
Eq. (12) tells us that the clock of B is slow with respect to R in the variable , and not in 󰆒. 314
There is no Euclidean time dilation because  is absolute ( ). 315
Despite the Euclidean metric in ES, the Lorentz factor  is recovered in Eqs. (9) and 316
(12). This is no surprise because Weyl showed that the Lorentz group is generated by 4D 317
rotations [17]. Gersten [11] demonstrated that the Lorentz transformation is equivalent to 318
an SO(4) rotation in a “mixed space”   󰆒. While this is mathematically correct, 319
such a “mixed space” doesn’t make sense physically. Yet it is a hint that Einstein time has 320
an issue! In ER,  of an observed object (and not 󰆒) is taken as the fourth coordinate of 321
that object. The SO(4) rotation now takes place in     (Fig. 2). The Lorentz factor 322
 is recovered in an observer’s reality by projecting ES to    and to . 323
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And the Lorentz transformation? It is recovered, together with the Lorentz factor , 324
whenever the observer ignores the richness of  and holds on to . Since his selected con- 325
cept of time ( or ) has no effect on how clocks are running, it also has no effect on the 326
physics involved. SR and all theories based on SR work equally well in either concept 327
of time. Yet if the observer selects , he won’t be able to grasp the big picture in cosmology 328
and quantum mechanics (see Sect. 5). So, the issue in Einstein time is real! 329
In order to understand how an acceleration in 3D space manifests itself in ES, let us 330
assume that the blue rocket b in Fig. 3 accelerates in the axis . According to Eq. (6), the 331
speed  of b must then increase at the expense of its speed . So, b is rotating and moving 332
along a curved trajectory in Cartesian ES coordinates. Any acceleration of an object in 3D space 333
relates to a 4D rotation and a curved trajectory in Cartesian ES coordinates. 334
335
Fig. 3 ES diagram and 3D projection for two identical rockets. Top: In the ES diagram, the red rocket 336
moves in the steady axis . The blue rocket accelerates in the axis . Bottom: Projection to the 3D 337
space of R. The red rocket is “at rest”. The blue rocket accelerates against the red rocket 338
Up next, we demonstrate that the ES geometry can also improve our understanding 339
of gravitation. Let us imagine that Earth is located to the right of the blue rocket in Fig. 3 340
bottom. We assume that the blue rocket is accelerating in the gravitational field of Earth. 341
Eq. (6), which we applied for drawing Fig. 3, tells us: If an object accelerates in the axis  342
of an observer, it automatically decelerates in his axis  (in his flow of time). 343
Gravitational waves [18] support the idea of GR that gravitation would be a property 344
of spacetime, but they might be predicted by ER, too. Particle physics is still considering 345
gravitation a force that has not yet been unified with the other three forces of physics. We 346
claim: Curved trajectories in Cartesian ES coordinates replace curved spacetime in GR. Eq. (6) is 347
the key equation which relates any motion in    to a motion in . To support our 348
claim, we now use Cartesian ES coordinates to calculate the Einstein time dilation in the 349
gravitational field of Earth. Let A and B be two identical clocks far away from Earth. They 350
are synchronized, next to each other, and move in the axis  at the speed . Clock B is 351
then allowed to approach Earth in the axis  of A. The kinetic energy of B (mass ) is 352
353


    , (13) 354
355
where  is the speed of B in the axis  of A,  is the gravitational constant,  is the 356
mass of Earth, and  is the distance of B to Earth’s center. By applying Eq. (6), we get 357
358

 
      
    , (14) 359
360

     󰇛󰇜 , (15) 361
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where  is the speed of B in the axis  of A. With    and    362
(there is no steady axis 
󰆒, but B keeps moving at the speed ), we get 363
364
   󰇛  󰇛󰇜󰇜 , (16a) 365
366
    (Gravitational Einstein time dilation), (16b) 367
368
where  is the distance that B has moved in the Einstein time  of A, while A itself has 369
moved the distance . The dilation factor   󰇛  󰇛󰇜󰇜 is the same as in 370
GR [2]. The value of  won’t change if B suddenly stops its motion relative to Earth. 371
If clock B returns to A, the time displayed by B will be behind the time displayed by A. In 372
GR, this effect is due to a curved spacetime. Applying Eq. (6) in Eq. (14) indicates: In ER, 373
this effect is due to a curved trajectory of B which is projected to the axis  of A. 374
We finish this section by discussing three instructive paradoxes (Fig. 4). They demon- 375
strate the benefit of our concept “distance” and of the projections from ES to an observer’s 376
reality. Problem 1: A rocket moves along a guide wire. In ES, rocket and wire move at the 377
speed . We assume that the wire moves in some axis . As the rocket moves along the 378
wire, its speed in  must be slower than . Wouldn’t the wire eventually be outside the 379
rocket? Problem 2: A mirror passes a rocket. An observer in the rocket’s tip sends a light 380
pulse to the mirror and tries to detect the reflection. In ES, all objects move at the speed , 381
but in different directions. We assume that the observer moves in some axis . How can 382
he ever detect the reflection? Problem 3: Earth revolves around the sun. We assume that 383
the sun moves in some axis . As Earth covers distance in   , its speed in  must 384
be slower than . Wouldn’t the sun escape from the orbital plane of Earth? 385
386
Fig. 4 Graphical solutions to three geometric paradoxes. Left: A rocket moves along a guide wire. 387
In 3D space, the guide wire remains within the rocket. Center: An observer in a rocket’s tip tries to 388
detect the reflection of a light pulse. Between two snapshots (0–1 or 1–2), rocket, mirror, and light 389
pulse move 0.5 Ls in ES. In 3D space, the light pulse is reflected back to the observer. Right: Earth 390
revolves around the sun. In 3D space, the sun remains in the orbital plane of Earth 391
The questions in the last paragraph seem to imply that there are geometric paradoxes 392
in ER, but there aren’t. The fallacy in all problems lies in the assumption that there would 393
be four observable (spatial) dimensions. Yet just three distances of ES are observable! We 394
solve all problems by projecting 4D ES orthogonally to 3D space (Fig. 4). Then the axis  395
is suppressed. The projection tells us what an observer’s reality is like because “suppressing ” 396
is equivalent to “length contraction makes  disappear”. Suppressed distance is felt as time. 397
We easily verify in 3D space: The guide wire remains within the rocket; the light pulse is 398
reflected back to the observer; the sun remains in the orbital plane of Earth. Other models 399
[8–13] run into paradoxes because they don’t project ES to an observer’s reality. 400
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5. Solving 15 Fundamental Mysteries of Physics 401
Why should we know about ER and the master frame ES if SR and GR work so well 402
for each observer? In this section, we demonstrate that ER outperforms SR and GR in the 403
understanding of time, time’s arrow, , cosmology, and quantum mechanics. 404
5.1. Solving the Mystery of Time 405
Euclidean time is distance covered in ES, divided by . Time originates from each object 406
rather than from my watch. Because time can flow in countless 4D directions, the metaphor 407
of “time running in a straight 1D line” is limited in scope. By contrast, there is no definition 408
of Einstein time other than “what I read on my watch” (attributed to Einstein himself). 409
5.2. Solving the Mystery of Time’s Arrow 410
“Time’s arrow” is a synonym for time moving only forward. The arrow emerges from 411
the fact that the distance covered in ES and  always have a positive value. 412
5.3. Solving the Mystery of  413
In SR, where forces are absent, the total energy  of an object is given by 414
415
        , (17) 416
417
where  is its kinetic energy in 3D space and  is its “energy at rest”. SR doesn’t 418
tell us why there is a  in the energy of objects that in SR never move at the speed . ER 419
gives us this missing clue and is thus superior to SR:  is an object’s kinetic energy in 420
the axes    of an observer,  is its kinetic energy in his axis , and  is the sum 421
of these energies—and likewise its kinetic energy in its axis 
󰆒. The  in Eq. (17) tells us that 422
everything is moving through ES at the speed . In SR, we are also familiar with 423
424
      
   , (18) 425
426
where  is the total momentum of an object and  is its momentum in 3D space. ER is 427
again superior to SR: After dividing Eq. (18) by , it becomes the vector addition of an object’s 428
momentum  in the axes    of an observer and its momentum  in his axis . For 429
observer R inside the red rocket (Fig. 1 center or Fig. 2), there is    and 
  430
for the red rocket, but    and 
  for the blue rocket. 431
5.4. Solving the Mystery of Relativistic Effects 432
In SR, length contraction and time dilation can be derived from the Lorentz transfor- 433
mation, but the physical cause of these relativistic effects remains in the dark. ER discloses 434
that they stem from projecting the master frame ES to an observer’s reality. 435
5.5. Solving the Mystery of Gravitational Time Dilation 436
Eq. (16b) tells us: The Einstein time of an object in a gravitational field passes by more 437
slowly with respect to an observer who is very far away from the center of this field. The 438
object’s curved trajectory in Cartesian ES coordinates is projected to the observer’s proper 439
3D space (here the object accelerates) and to his proper flow of time (here it decelerates). 440
5.6. Solving the Mystery of the Cosmic Microwave Background (CMB) 441
Now we are ready for our new model of cosmology based on ER. There is no need to 442
create ES. It exists just like numbers. Because of some reason that we don’t know, there 443
was a Big Bang. In today’s model of cosmology, it makes no sense to ask where it occurred: 444
Because space inflated from a singularity, it occurred everywhere. In ES, the Big Bang can 445
be localized at what we take as origin O. The Big Bang injected a huge amount of energy 446
into ES all at once. It also provided an overall radial momentum. 447
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Right after the Big Bang, the concentration of energy was extremely high in ES. In the 448
projection to any reality, a very hot and dense plasma was created. While this plasma was 449
expanding, it cooled down. During plasma recombination, electromagnetic radiation was 450
emitted that we observe as CMB today [19]. At temperatures of roughly 3,000 K, hydrogen 451
atoms formed and the universe became transparent for the CMB. In today’s model of cos- 452
mology, this stage was reached about 380,000 years “after” the Big Bang. In ER, these are 453
380,000 light years “away from” the Big Bang. The value “380,000” still needs to be recal- 454
culated because we claim that there was no cosmic inflation (see Sect. 5.9). 455
Fig. 5 left shows the ES diagram for observers on Earth (here Earth is moving in ). 456
Most energy is moving radially: It keeps the radial momentum provided by the Big Bang. 457
The CMB is moving transversally to the axis : It can’t move in  as it already moves 458
in  at the speed of light. All energy is confined to an expanding 4D hypersphere; most energy 459
is confined to its 3D hypersurface. We now explain three striking observations regarding the 460
CMB: (1) It is nearly isotropic because it was created equally in   . Cosmic inflation 461
is not needed! (2) The temperature of the CMB is very low because of a very high recession 462
speed 
󰆒 (see Sect. 5.10) and thus a very high Doppler redshift. (3) We observe the CMB 463
today because it started moving at a speed 󰆒  in a very dense medium. 464
465
Fig. 5 ES diagrams and 3D projections (not to scale) for solving the three mysteries 5.6, 5.7, and 5.10. 466
The displayed circular arcs are part of a 3D hypersurface, which is expanding in ES at the speed . 467
Left: The CMB is isotropic because it was created equally in    (  not shown here). The 468
CMB has a very low temperature because of a very high 
󰆒. We observe the CMB today because it 469
started moving at a speed 󰆒 . Right: A supernova S’ occurred when the radius 󰆒 was smaller 470
than today’s radius . Team B measures S’ in a distance  . Earth moved the same , but in 471
, when the light of S’ arrives. A supernova S occurring today (same ) recedes slower than S’ 472
5.7. Solving the Mystery of Hubble’s Law 473
Fig. 5 left shows a galaxy G, which is moving away from the origin O and from Earth. 474
The recession speed  relates to the distance  as  relates to the radius . 475
476
       (Hubble’s law), (19) 477
478
where   is the Hubble constant,  is in km/s, and  is in Mpc. There it is! Eq. (19) 479
is Hubble’s law [20]: The farther a galaxy, the faster it is moving away from Earth. 480
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5.8. Solving the Mystery of the Flat Universe 481
ES is projected orthogonally to an observer’s proper 3D space. So, this 3D space has 482
no curvature in the fourth dimension. Each observer experiences a flat 3D universe. 483
5.9. Solving the Mystery of Cosmic Inflation 484
Many physicists believe that an inflation of space in the early Universe [21,22] would 485
explain the isotropic CMB, the flatness of the observed universe, and large-scale structures 486
(inflated from quantum fluctuations). We showed in Sects. 5.7 and 5.8 that ES can explain 487
the first two of these observations. It also explains the third observation if we only assume 488
that the impacts of early quantum fluctuations have been expanding at the speed of light. 489
Cosmic inflation is a redundant concept. 490
5.10. Solving the Mystery of Competing Hubble Constants 491
There are several methods of calculating the Hubble constant , but unfortunately 492
the results vary from one method to another. Here we consider measurements of the CMB 493
made with the Planck space telescope [23]. We compare them with calibrated distance ladder 494
techniques (distance and redshift of celestial objects) using the Hubble space telescope [24]. 495
We now explain why the values of  obtained by the two teams don’t match within the 496
specified error margins. According to team A [23], there is   . 497
According to team B [24], there is   . 498
Team B made efforts to minimize the error margin by optimizing the distance meas- 499
urements. Yet, as we will prove now, misinterpreting the redshift measurements causes a 500
systematic error in team B’s calculation of . We assume that 67.66 km/s/Mpc would be 501
today’s value of . We simulate a supernova at a distance of . If this 502
supernova occurred today (S in Fig. 5 right), Eq. (19) would give us the recession speed 503
504
       , (20) 505
506
        , (21) 507
508
where the redshift parameter  tells us how any wavelength  of the supernova’s light 509
is either passively stretched by an expanding space (team B)—or how  is redshifted by 510
the Doppler effect of objects that are actively receding in ES (our model). 511
In this and the next paragraph, we demonstrate that team B measures a higher value 512
󰆒, and thus calculates a higher value 
󰆒, and thus calculates a higher value 
󰆒 (which 513
is not the same as ). In Fig. 5 right, there is one circle called “past”, where the supernova 514
S’ occurred that team B is measuring, and a second circle called “present”, where its light 515
arrives on Earth. Today, this supernova has turned into a neutron star. Because everything 516
is moving at the speed , Earth moved the same distance , but in the axis , when the 517
light of S’ arrives. Hence, team B is receiving data from an ancient time 󰆒 
󰆒 when 518
there was a smaller radius 󰆒 and a larger Hubble constant 
󰆒. 519
520

󰆒   󰆒    󰇛  󰇜    , (22) 521
522

󰆒  . (23) 523
524
Because of this higher value and of Eq. (19), all data measured and calculated by team 525
B relate to a higher 3D speed 
󰆒 for the same . Because of 󰆒 
󰆒, 526
this is going to happen: Team B measures a redshift of 󰆒 , which is indeed higher 527
than 0.0903. Because of this higher value of 󰆒, team B will calculate 
󰆒 528
from 󰆒 
󰆒 and thus 
󰆒 from Eq. (19). Hence, team B will con- 529
clude that 74.37 km/s/Mpc would be today’s value . In truth, team B ends up with a 530
value 
󰆒 of the past because it isn’t aware of Eq. (22) and of the ES geometry. 531
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For a shorter distance of   , Eq. (22) tells us that team B’s Hubble constant 532

󰆒 deviates from team A’s Hubble constant  by only 0.009 percent. Yet when plotting 533

󰆒 versus  for various distances (we chose 50 Mpc, 100 Mpc, 150 Mpc, ..., and 450 Mpc 534
as we didn’t have the raw distance data used by [24]), the resulting slope (team B’s Hubble 535
constant) is 8 to 9 percent higher than team A’s Hubble constant. We kindly ask team B to 536
improve its calculation by eliminating the systematic error in the redshift measurement. 537
It must adjust the calculated speed 
󰆒 to today’s speed  by converting Eq. (22) to 538
539

󰆒   󰇛  󰇜    󰇛 󰇜 , (24) 540
541
    
󰆒󰇛 
󰆒󰇜 . (25) 542
543
We conclude: The redshift is caused by the Doppler effect of objects that are actively receding 544
in ES. Matching the two competing values of  (team B’s published value is indeed 8 to 545
9 percent higher than team A’s value) is probably the strongest proof of our theory. Team 546
A’s value is correct:  . If the 3D hypersurface in Fig. 5 has always 547
been expanding at the speed , the total time having elapsed since the Big Bang would be 548
equal to , which is 14.5 billion years rather than 13.8 billion years [25]. The adjusted 549
age would explain the existence of stars as old as 14.5 billion years [26]. 550
Of course, team B is well aware of the fact that the supernova’s light was emitted in 551
the past. Yet in the Lambda-CDM model, all that counts is the timespan  during which 552
light is traveling from the supernova to Earth. Along the way, its wavelength is passively 553
stretched by expanding space. So, the total redshift is only developing during the journey 554
to Earth. We can put it this way: The redshift parameter 󰆒 starts from zero and increases 555
continuously during the journey to Earth. The fact that the supernova occurred long ago 556
in the past at a time  is irrelevant for team B’s calculation. 557
In ER, the moment  (when a supernova occurs) is significant, but the timespan  558
(during which light is traveling to Earth) is irrelevant. The wavelength of the supernova’s 559
light is initially redshifted by the Doppler effect. During its journey to Earth, the parame- 560
ter 󰆒 remains constant. Here we can put it this way: The redshift parameter 󰆒 is tied up 561
at the moment  “in a package” and sent to Earth, where it is measured. In the Lambda- 562
CDM model, space itself is expanding. In ER, a 3D hypersurface (actively receding energy, 563
not space!) is expanding in ES. Expansion of space is a redundant concept. 564
5.11. Solving the Mystery of Dark Energy 565
The CDM model of cosmology assumes an expanding space to explain the distance- 566
dependent recession of celestial objects. Meanwhile, it has been extended to the Lambda- 567
CDM model, where Lambda is the cosmological constant. Cosmologists are now favoring 568
an accelerating expansion [27,28] over a uniform expansion. This is because the calculated 569
recession speeds deviate from values predicted by Eq. (19) if  is taken as an averaged 570
constant. The deviations increase with distance  and are compensated by assuming an 571
accelerating expansion of space. Such an acceleration would stretch the wavelength even 572
more and thus increase the recession speeds according to Eq. (21). 573
Our model gives a much simpler explanation for the deviations from Hubble’s law: 574
Because of     , the Hubble constant  is a function 󰇛󰇜. 
󰆒 from 575
every past is higher than today’s value. The older the redshift data are, the more will 
󰆒 576
deviate from today’s value , and the more will 
󰆒 deviate from . The small white 577
circle in Fig. 5 right helps us understand these deviations: If a new supernova S occurred 578
today at the same distance    as the shown supernova S’ in the past, S would 579
recede slower (27,064 km/s) than S’ (29,748 km/s) just because of the different values of 580
 and 
󰆒. As long as the ES geometry is unknown, higher redshifts are attributed to an 581
accelerating expansion of space. Now that we know about the ES geometry, we can attrib- 582
ute higher redshifts to data from deeper pasts. 583
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We conclude that any expansion of space—uniform as well as accelerating—is only 584
virtual. There is no accelerating expansion of the Universe even if a Nobel Prize was given 585
“for the discovery of the accelerating expansion of the Universe through observations of 586
distant supernovae” [29]. This phrasing actually contains two misconceptions: (1) In the 587
Lambda-CDM model, the term “Universe” implies space, but space isn’t expanding at all. 588
(2) There is a uniform expansion of a 3D hypersurface (actively receding energy), but no 589
“accelerating expansion” whatsoever. 590
The term “dark energy” [30] was coined to come up with a cause for an accelerating 591
expansion of space. We just explained that there is no expansion of space. So, dark energy 592
is an artifact of Einstein time. Dark energy is a redundant concept. It has never been ob- 593
served anyway. Radial momentum provided by the Big Bang drives galaxies away from 594
the origin O. They are driven by themselves rather than by dark energy. 595
Tab. 1 summarizes huge differences in the meaning of Big Bang, Universe/universe, 596
space, and time. In the Lambda-CDM model, the Big Bang was the beginning of the Uni- 597
verse. In our model, the Big Bang was the injection of energy into ES. In the Lambda-CDM 598
model, Universe (capitalized) is all space, all time, and all energy. In our model, universe 599
is the proper 3D space of one observer. In the Lambda-CDM model, spacetime is curved. 600
In our model, trajectories of objects are curved in Cartesian ES coordinates. There is also 601
a significant difference regarding the underlying theory of relativity: GR isn’t compatible 602
with quantum mechanics; ER is compatible with quantum mechanics. 603
604
Tab. 1 Comparing the Lambda-CDM model with our model of cosmology 605
5.12. Solving the Mystery of the Wave–Particle Duality 606
We can’t tell which solved mystery is the most important one. Yet the wave–particle 607
duality has certainly kept physicists busy since it was first discussed by Niels Bohr and 608
Werner Heisenberg [31]. The Maxwell equations tell us that electromagnetic waves are 609
oscillations of an electromagnetic field that move through 3D space at the speed of light 610
. In some experiments, objects behave like “waves” (electromagnetic wave packets). But 611
in other experiments, the same objects behave like particles. In today’s physics, an object 612
can’t be both at once because waves distribute energy in space over time, while the energy 613
of particles is localized in space at a given time. This is why we added our third postulate: 614
All energy is “wavematter” (electromagnetic wave packet and matter in one). By combin- 615
ing our concepts of distance and wavematter, we now demonstrate: Waves and particles are 616
actually the same thing (energy), but seen from two perspectives. 617
Fig. 6 illustrates in Cartesian ES coordinates what our new concept of wavematter is 618
all about. If I observe a wavematter (we call it the “external view”), this wavematter comes 619
in four orthogonal dimensions: It propagates in my axis  at some speed   , and it 620
oscillates in my axes  (electric field) and  (magnetic field); propagating and oscillat- 621
ing are functions of Euclidean time  (related to my fourth axis ). So, I can observe how 622
this wavematter is propagating and oscillating: I deem it wave. 623
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624
Fig. 6 Concept of wavematter. Artwork illustrating how one object can be deemed wave or matter. 625
Wavematter comes in four orthogonal dimensions: propagation, electric field, magnetic field, and 626
Euclidean time. Each wavematter deems itself matter at rest (internal or in-flight view). If it is ob- 627
served by some other wavematter (external view), it is deemed wave 628
From its own perspective (we call it the “internal view” or the “in-flight view”), each 629
wavematter propagates in its axis 
󰆒 at the speed . Yet because of length contraction at 630
the speed , the axis 
󰆒 is suppressed for this wavematter. For this reason, its own prop- 631
agating disappears for itself: It deems itself matter at rest. It still observes the other objects 632
propagating and oscillating in its proper 3D space as it keeps on feeling Euclidean time, 633
while it is invisibly propagating in its axis 
󰆒. We conclude that there is an external view 634
and an internal view of each wavematter. In today’s physics, there is no reference frame 635
moving at the speed  and thus no internal view of a photon. Be aware that “wavematter” 636
isn’t just another word for the duality, but a generalized concept of energy disclosing why 637
there is a wave–particle duality in an observer’s proper 3D space. 638
As an example, we now investigate the symmetry in three wavematters , , 639
and . We assume that they are all moving away from the same point P in ES, but in 640
different directions (Fig. 7 top left).     are Cartesian coordinates in which  641
moves only in . Hence,  is that axis which  deems time multiplied by , and 642
   span ’s 3D space (Fig. 7 bottom left). As the axis  disappears because of 643
length contraction,  deems itself matter at rest ().  moves orthogonally to 644
. 
󰆒 
󰆒 
󰆒 
󰆒 are Cartesian coordinates in which  moves only in 
󰆒 (Fig. 7 top 645
right). In this case, 
󰆒 is that axis which  deems time multiplied by , and 
󰆒 
󰆒 
󰆒 646
span ’s 3D space (Fig. 7 bottom right). As the axis 
󰆒 disappears because of length 647
contraction,  also deems itself matter at rest (). 648
649
Fig. 7 ES diagrams and 3D projections for three wavematters. Top left: ES in coordinates where 650
 moves in . Top right: ES in coordinates where  moves in 
󰆒. Bottom left: Projection 651
to ’s 3D space.  deems itself matter at rest () and  wave (). Bottom right: Pro- 652
jection to ’s 3D space.  deems itself matter at rest () and  wave () 653
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Yet how do  and  move in each other’s view? We must fulfill our first two 654
postulates and the requirement that they both started at the same point P. There is only 655
one way of how to draw our ES diagrams: We must rotate the two reference frames with 656
respect to each other. Only a rotation guarantees full symmetry, so that the laws of physics 657
have the same form in the 3D spaces of  and of . We can put it this way: ’s 658
4D motion “swings completely” (rotates by an angle of ) into ’s 3D space, so that 659
 deems  wave (). Regarding , we split its 4D motion into a motion par- 660
allel to ’s motion (internal view) and a motion orthogonal to ’s motion (external 661
view). So,  can deem  either matter () or wave (). 662
The secret to understanding our new concepts “distance” and “wavematter” is all in 663
Fig. 7. Here we see how they go hand in hand: We claim the symmetry of all four Cartesian 664
coordinates in ES and, on top of that, the symmetry of waves and matter. What I deem wave, 665
deems itself matter. Just as distance is spatial and temporal distance in one, so is wavematter 666
wave and matter in one. Here is a compelling reason for this unique claim of our theory: 667
Einstein taught that energy is equivalent to mass. Full symmetry of waves and matter is a 668
consequence of this equivalence. As the axis  disappears because of length contraction, 669
the energy in a propagating wave “condenses” to mass in matter at rest. 670
In a double-slit experiment, an observer detects coherent waves which pass through 671
a double-slit and produce some pattern of interference on a screen. He observes wavemat- 672
ters from ES whose 4D motion “swings completely” (rotates by an angle of ) into his 673
proper 3D space. He deems all these wavematters waves because he isn’t tracking through 674
which slit each wavematter is passing. If he did, the interference pattern would disappear 675
immediately. So, he is a typical external observer. 676
The photoelectric effect is quite different. Of course, one can externally witness how 677
one photon releases one electron from a metal surface. But the physical effect itself (“Do I 678
have enough energy to release one electron?”) is all up to the photon’s view. Only if the 679
photon’s energy exceeds the binding energy of an electron is this electron released. So, we 680
must interpret the photoelectric effect from the internal view of each wavematter. Here its 681
view is crucial! It behaves like a particle, which is commonly called “photon”. 682
The wave–particle duality is also observed in matter, such as electrons [32]. Accord- 683
ing to our third postulate, electrons are wavematter, too. From the internal view (if I track 684
them), electrons are particles: “Which slit will I go through?” From the external view (if I 685
don’t track them), electrons behave more like waves. Because I automatically track slow 686
objects, I deem all macroscopic wavematters matter: Their speed in my 3D space is rather 687
low compared with the speed of light thus favoring the internal view. This justifies draw- 688
ing solid rockets and celestial objects in most of our ES diagrams. 689
5.13. Solving the Mystery of Quantum Entanglement 690
The term “entanglement” [33] was coined by Erwin Schrödinger when he published 691
his comment on the Einstein–Podolsky–Rosen paradox [34]. The three authors argued that 692
quantum mechanics wouldn’t provide a complete description of reality. John Bell proved 693
that quantum mechanics is incompatible with local hidden-variable theories [35]. Schrö- 694
dinger’s word creation didn’t solve the paradox, but demonstrates up to the present day 695
the difficulties that we have in comprehending quantum mechanics. Several experiments 696
have meanwhile confirmed that entangled particles violate the concept of locality [36–38]. 697
Ever since has quantum entanglement been considered a non-local effect. 698
We will now “untangle” quantum entanglement without the concept of non-locality. 699
All we need to do is discuss quantum entanglement in ES. Fig. 8 displays two wavematters 700
that were created at once at the same point P and move away from each other in opposite 701
directions at the speed . We claim that these wavematters are entangled. We assume that 702
they are moving in the axes  and , respectively. If they are observed by a third 703
wavematter that is moving in a direction other than , they are deemed two objects. 704
This third wavematter can’t understand how the entangled wavematters are able to com- 705
municate with each other in no time. This is again the external view. 706
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707
Fig. 8 Quantum entanglement in ES. Artwork illustrating internal view and external view. For each 708
displayed wavematter, the axis  disappears because of length contraction. It deems its twin and 709
itself one object (internal view). For a third wavematter that is moving in a direction other than , 710
the axis  doesn’t disappear. It deems the displayed wavematters two objects (external view) 711
And here comes the internal (in-flight) view in ES: For each entangled wavematter in 712
Fig. 8, the axis  disappears because of length contraction at the speed . That is to say: 713
In the projection to their common 3D space spanned by   , either wavematter deems 714
itself at the very same position as its twin. From either perspective, they are one object that has 715
never been separated. This is why they communicate with each other in no time! Entangle- 716
ment is another strong evidence that everything is moving through ES at the speed . Our 717
solution to entanglement isn’t limited to photons. Electrons or atoms can be entangled as 718
well. They move at a speed    in my 3D space, but in their axis  they also move 719
at the speed . We conclude: Even non-locality is a redundant concept. 720
5.14. Solving the Mystery of Spontaneity 721
In spontaneous emission, a photon is emitted by an excited atom. Prior to the emission, 722
the photon’s energy was moving with the atom. After the emission, this energy is moving 723
by itself. Today’s physics can’t explain how this energy is boosted to the speed  in no 724
time. In ES, both atom and photon are moving at the speed . So, there is no need to boost 725
any energy to the speed . All it takes is energy from ES whose 4D motion “swings com- 726
pletely” (rotates by an angle of ) into an observer’s proper 3D space—and this energy 727
speeds off at once. In absorption, a photon is spontaneously absorbed by an atom. Today’s 728
physics can’t explain how the photon’s energy is slowed down to the atom’s speed in no 729
time. In ES, both photon and atom are moving at the speed . So, there is no need to slow 730
down any energy. Similar arguments apply to pair production and annihilation. We consider 731
spontaneity another clue that everything is moving through ES at the speed . 732
5.15. Solving the Mystery of the Baryon Asymmetry 733
According to the Lambda-CDM model, almost all matter in the Universe was created 734
shortly after the Big Bang. Only then was the temperature high enough to enable the pair 735
production of baryons and antibaryons. Yet the density was also very high so that baryons 736
and antibaryons should have annihilated each other again. Since we do observe a lot more 737
baryons than antibaryons today (also known as the “baryon asymmetry”), it is assumed 738
that more baryons than antibaryons must have been produced in the early Universe [39]. 739
However, an asymmetry in pair production has never been observed. 740
Our theory offers a unique solution to the baryon asymmetry: Since each wavematter 741
deems itself matter, there was matter in 3D space right after the Big Bang. Pair production 742
isn’t needed to create matter, and an asymmetry in pair production isn’t needed to explain 743
the baryon asymmetry. There is much less antimatter than matter because antimatter is created 744
only in pair production. One may ask why wavematter doesn’t deem itself antimatter, but 745
this question is missing the point. Energy has two faces: wave and matter. “Antimatter” 746
is matter, too, but with the opposite electric charge. 747
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6. Conclusions 748
To this day, all attempts to unify GR and quantum mechanics have failed miserably. 749
In Sects. 5.1 through 5.15, ER solves mysteries which SR and GR either haven’t solved in 750
100+ years—or that have meanwhile been solved, but only by applying concepts (cosmic 751
inflation, expansion of space, dark energy, non-locality) that we proved to be redundant. 752
Now we let Occam’s razor, a powerful tool in science, do its job: Because ER outperforms 753
SR and GR, Occam’s razor knocks out Einstein time and these four redundant concepts. 754
The weakness in today’s cosmology is: It uses two concepts of time (universal Euclidean 755
time in all Big Bang models, Einstein time in all measurements). The weakness in today’s 756
quantum mechanics is: An observer’s Einstein time can’t describe things that are beyond 757
any observation. Egocentric Einstein time hides the big picture. 758
Since SR and GR have been experimentally confirmed many times over, they are con- 759
sidered two of the greatest achievements of physics. We proved that their concept of time 760
is flawed. Albert Einstein, one of the most brilliant physicists ever, wasn’t aware of ER. It 761
was a wise decision to award him with the Nobel Prize for his theory of the photoelectric 762
effect [40] rather than for SR or GR. We campaign for ER as it penetrates to a deeper level. 763
For the first time ever, mankind understands the nature of time: Time isn’t a fundamental 764
quantity, but distance covered in ES, divided by the speed of light. Imagine: The human 765
brain is able to grasp the idea that our energy is moving through ES at the speed of light. 766
With that said, conflicts of mankind become all so small. 767
ER solves 15 mysteries at once: (1) time, (2) time’s arrow, (3) , (4) relativistic ef- 768
fects, (5) gravitational time dilation, (6) CMB, (7) Hubble’s law, (8) flat universe, (9) cosmic 769
inflation, (10) competing Hubble constants, (11) dark energy, (12) wave–particle duality, 770
(13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 771
are 15 confirmations of ER. It isn’t unusual that new concepts give many answers at once. 772
So, the answer to our title question is: Yes, physics benefits from Euclidean time. Einstein 773
sacrificed absolute space and time. We sacrifice the absoluteness of waves and matter, but 774
add a new absolute time derived from covered distance. Quantum leaps can’t be planned. 775
They just happen like the spontaneous emission of a photon. ☺ 776
We introduced new concepts of time, distance, and energy: (1) There is absolute time. 777
(2) Spatial and temporal distance aren’t two, but one [41]. (3) Wave and matter aren’t two, 778
but one. We explained these concepts and confirmed how powerful they are. We can even 779
tell the source of their power: symmetry and beauty. Once you have cherished this beauty, 780
you will never let it go again. Yet to cherish it, you first need to give yourself a little push—781
accepting that an observer’s reality is only created by projecting ES to his proper 3D space 782
and to his proper flow of time. Questions like “Why would reality only be a projection?” 783
must not be asked in physics. The magic of “reality being a projection” compares to the 784
magic of “reality being a probability function”. The latter is well accepted. 785
It looks like philosopher Plato was right with his Allegory of the Cave [42]: Mankind 786
experiences a projection that is blurred because of quantum mechanics! We would be mis- 787
taken if we thought that the concepts of nature were on the same level as all the tangible 788
realities perceived by us. Our advice: Think of a problem in physics and try to solve it in 789
ER. We predict that ER covers gravitational waves, too. Our new concepts lay the ground- 790
work for ER. Anyone is welcome to join us. Hopefully, physics will be improved. 791
Acknowledgements: We wish to thank Matthias Bartelmann, Dennis Dieks, Dirk Rischke, and Jür- 792
gen Struckmeier for reading and commenting on earlier versions of this paper. 793
Author Contributions: Markolf has a Ph.D. in physics and is a full professor at Heidelberg Univer- 794
sity, Germany. He studied in Frankfurt, Heidelberg, at UC San Diego, and Harvard. He found the 795
issue in Einstein time, interpreted reality as “projection from a master frame”, and contributed the 796
concepts “distance” and “wavematter” that make ER compatible with quantum mechanics. He also 797
drafted this paper. Siegfried taught physics and math at the Waldorf School in Darmstadt, Germany. 798
He contributed most of the ES diagrams and solved the mystery of the competing Hubble constants. 799
Funding: No funds, grants, or other support was received. 800
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Conflict of Interest: The authors have no competing interests to declare. 801
Data Availability Statement: All data that support this study are included. 802
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Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 May 2023