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An Issue in Einstein’s 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. 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 . Euclidean time is distance covered in ES, divided by .

For each object, Euclidean time flows in a unique 4D direction equal to its current direction of mo-

tion. 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 . 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: cosmology; Hubble constant, gravitation; wave–particle duality; entanglement 23

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Important Remarks 25

We kindly ask all readers including editors and reviewers to read these preliminary 26

remarks. They help you to avoid those traps that previous reviewers already stepped into. 27

Most readers seem to believe that our theory is just another attempt to identify an issue in 28

Einstein’s theory of special relativity (SR) [1]. Since SR has been experimentally confirmed 29

many times over, our theory is considered a waste of time. What they don’t see: The issue 30

is in Einstein’s concept of time! It affects all of physics including SR, general relativity (GR) 31

[2], and quantum mechanics. We do not dispute any predictions made by SR or GR. Quite 32

the opposite is true: The Lorentz factor is recovered in our theory, and we explain why SR 33

and GR work so well despite the issue in Einstein’s concept of time. Cosmology and quan- 34

tum mechanics work well only if concepts are added (cosmic inflation, expansion of space, 35

dark energy, non-locality) that we prove to be redundant. 36

It is because of this issue in Einstein’s concept of time that GR is not compatible with 37

quantum mechanics. We make three changes to the foundations of physics—new concepts 38

of time, distance, and energy—that make relativity compatible with quantum mechanics. 39

Isn’t that reason enough to give our theory of Euclidean relativity (ER) a chance? We must 40

ask this question because one editor informed us that some journals do not consider refu- 41

tations of SR. Sorry, but why is that? Have SR and GR turned into a dogma that must not 42

be questioned anymore? According to Karl Popper, a theory is scientific only if it is falsi- 43

fiable [3]. Neither SR nor GR nor ER nor any concept of time is ever set in stone! 44

Five pieces of advice: (1) Be willing to question Einstein’s concept of time. Do not hold on 45

to it just because you have always worked with it so far. There is progress in science only 46

if we are open for new concepts. (2) Do not take SR and GR for granted when evaluating ER. 47

Neither must we take the geocentric model for granted when evaluating the heliocentric 48

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model. Previous reviewers made a systematic error when evaluating ER with concepts of 49

SR and GR. ER is different. In ER, everything is moving at the speed of light. (3) Evaluate 50

ER reasonably. SR comes with Einstein’s concept of time. ER comes with Euclidean time. 51

The Lorentz factor is recovered in ER. So, this argument doesn’t favor SR over ER. Yet ER 52

solves mysteries which SR and GR haven’t solved in 100+ years. (4) Be patient and fair. Do 53

not expect us to address all of physics in this paper. SR and GR have been tested for 100+ 54

years. We must hang on for ER to prove itself, too. (5) Let illustrations inspire you. Artwork 55

assists us to conceive of 4D. Geometric derivations are equivalent to equations. 56

To sum it all up: Predictions made by SR and GR are correct, but ER penetrates to a 57

deeper level. We do apologize for having published several preprint versions. It was really 58

tricky to figure out why SR and GR make correct predictions despite the issue in Einstein’s 59

concept of time. Sect. 2 is about disclosing this issue. Sect. 3 gives us an intuitive approach 60

to Euclidean time. In Sect. 4, we derive the Lorentz factor and gravitational time dilation. 61

In Sect. 5, we solve 15 mysteries and declare four concepts of today’s physics redundant. 62

In our Conclusions, Occam’s razor knocks out Einstein’s concept of time. 63

1. Introduction 64

Today’s concepts of space and time were coined by Albert Einstein. His theory of SR 65

[1] is based on a flat spacetime with an indefinite (not positive-semidefinite) distance func- 66

tion. SR is often interpreted in Minkowski spacetime (MS) because Hermann Minkowski’s 67

geometric interpretation [4] was very successful in explaining relativistic effects. Predict- 68

ing the lifetime of muons [5] is one example that demonstrates the power of SR. General 69

relativity (GR) [2] includes gravitation and is based on a curved spacetime with a pseudo- 70

Riemannian metric. GR is supported, for example, by the deflection of starlight during a 71

solar eclipse [6] and by the high accuracy of GPS. Quantum field theory [7] unifies classical 72

field theory, SR, and quantum mechanics, but not GR. 73

We call our theory “Euclidean relativity” and build it on these three postulates: (1) In 74

Euclidean spacetime (ES), all energy is moving at the speed of light. (2) The laws of physics 75

have the same form in each observer’s “reality” (orthogonal projections of ES to his proper 76

3D space and to his proper flow of time). (3) All energy is “wavematter” (electromagnetic 77

wave packet and matter in one). Our first postulate is stronger than the second SR postu- 78

late. The speed of light is both absolute and universal. Everything is moving through 79

ES at the speed . Moving through MS at the speed of light is a pointless concept as objects 80

at rest in 3D space would then move in time at “one second per one second”. Our second 81

postulate is the same as the first SR postulate, except that there is no limitation to inertial 82

frames and that we distinguish ES from an observer’s reality. Our third postulate makes 83

relativity compatible with quantum mechanics. 84

For all followers of our theory: We loosened our first postulate. Only shortly after the 85

Big Bang was all energy moving radially in ES. Today, some energy is moving transversally 86

because of energy conversion events, such as plasma recombination and supernovae. We 87

were misled by other models, but now our statement is consistent: All energy is confined 88

to an expanding 4D hypersphere; most energy is confined to its 3D hypersurface (see Sect. 89

5.6). We hold on to our three new concepts of time, distance, and energy. 90

We aren’t the first physicists to investigate ER: In the early 1990s, Montanus already 91

described ES [8]. He also formulated electrodynamics and gravitational lensing in ES [9]. 92

Almeida compared trajectories in MS with trajectories in ES [10]. Gersten demonstrated 93

that the Lorentz transformation is equivalent to an SO(4) rotation [11]. van Linden studied 94

energy and momentum in ES [12]. Pereira claimed a “hypergeometrical universe”, where 95

matter is made from deformed space [13]. Yet none of these models identifies the issue in 96

Einstein’s concept of time. And they all run into geometric paradoxes discussed in Sect. 4 97

because they don’t project ES to an observer’s reality. Only Machotka added a “bounded- 98

ness postulate” to avoid paradoxes [14], but it sounds rather contrived. We overcome such 99

paradoxes by limiting reality in our second postulate. An observer’s reality is only created by 100

projecting ES orthogonally to his proper 3D space and to his proper flow of time. 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”, which replaces by , but keeps as the invariant. 203

Euclidean time isn’t egocentric (centered in the observer), but universal (centered in 204

each observed object). Because of the symmetry in Eq. (3b), we are free to label the four 205

axes. We assume that each object moves only in its axis . According to our first postu- 206

late, it does so at the speed . Euclidean time is distance covered in ES, divided by . 207

208

(Euclidean time). (4) 209

210

Eq. (4) tells us that Euclidean time is only a subordinate quantity derived from cov- 211

ered distance. Time isn’t fundamental to physics as already claimed by other authors [16]. 212

Distance and speed are more significant than time. So, we suggest to choose new units for 213

speed and time. should be specified in its own new unit to be given. should be spec- 214

ified in “light seconds per this new unit”. We claim that the coordinates in Eq. (3b) are 215

projected to an observer’s proper 3D space and to his proper flow of time, which is equal 216

to his Einstein time. For each object, we define a 4D vector “flow of time” 217

218

(Flow of time), (5) 219

220

where is the Cartesian ES velocity of the object and is a unit 4D vector pointing to 221

the current direction of motion of the object. The Cartesian ES velocity has four com- 222

ponents . From Eq. (3b), we get 223

224

. (6) 225

226

From Eqs. (5) and (6), we calculate that there is indeed . For each ob- 227

ject, Euclidean time flows in a unique 4D direction equal to its current direction of motion. 228

Clocks project this 4D flow to a 1D flow of time. So, each clock displays Einstein time. Be 229

aware that 1 s in Einstein time is equal to 1 s in Euclidean time. The only difference is that 230

flows in 1D, whereas flows in 4D. Einstein time hides that there is a unique 4D vector 231

“flow of time” for each object. Einstein time makes us believe that time would flow in one 232

direction for all objects in the universe! 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. and all are “spatial and temporal dis- 237

tance in one”. Distance isn’t covered as a function of independent time. Only by covering 238

distance is Euclidean time passing by. All distances are measured in light seconds (Ls) 239

by odometers. There is no need to calibrate the odometers because light seconds in ES are 240

absolute. The symmetry of all supports the idea of natural units. “Space” and “time” 241

in everyday life are only interpretations of distance. 242

Hyperspherical coordinates are good for grasping the big picture in cosmology. We 243

claim: The Big Bang injected a huge amount of energy into ES all at once at what we take 244

as “origin O”. It also provided an overall radial momentum: Shortly after the Big Bang, all 245

energy moved radially away from O. Today, some energy is moving transversally because 246

of energy conversion events, such as plasma recombination and supernovae. 247

Cartesian ES coordinates serve as a “master reference frame”: Each observer’s reality 248

is created by projecting the coordinates orthogonally to his proper 3D space and to his 249

proper flow of time. The coordinates can be calculated from 250

251

, (7a) 252

253

, (7b) 254

255

, (7c) 256

257

. (7d) 258

259

In our ES diagrams, we often choose Cartesian coordinates in which an object starts 260

moving from some origin P other than O. Because of the ES symmetry, we are free to label 261

all four axes. As mentioned before, each object shall move only in its axis . Below our 262

ES diagrams, we project ES to an observer’s proper 3D space. Here we are free to label the 263

axis that we project to. We assume: There is relative motion only in and . So, our ES 264

diagrams display and , while our 3D projections display . 265

The projections of to and to are orthogonal. We don’t re- 266

place the concept of space because are equal to . We replace the concept 267

of time because there is only for clocks moving in . If a clock moves in a direction 268

other than , its distance covered in ES is projected to the axis of an observer. 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

once the observer ignores the richness of and holds on to . Since his selected concept 325

of time ( or ) has no effect on how clocks are running, it also has no effect on the physics 326

involved. SR and all theories based on SR work equally well in either concept of time. 327

Yet if the observer selects , he won’t be able to grasp the big picture in cosmology and 328

quantum mechanics (see Sect. 5). This is why 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]. Be aware that depends on , but not on . If clock B returns to A, the time 371

displayed by B will be behind the time displayed by A. In GR, this effect is due to a curved 372

spacetime. Applying Eq. (6) in Eq. (14) indicates: In ER, this effect is due to a curved tra- 373

jectory 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. Unlike our precise definition of 408

Euclidean time, there is no definition of Einstein time other than “what I can read on my 409

watch” (attributed to Albert Einstein himself). 410

5.2. Solving the Mystery of Time’s Arrow 411

“Time’s arrow” is a synonym for time moving only forward. The arrow emerges from 412

our definition of Euclidean time: The distance covered in ES and always have a positive 413

value. There is no “negative distance” and thus no “going back in time”. 414

5.3. Solving the Mystery of 415

In SR, where forces are absent, the total energy of an object is given by 416

417

, (17) 418

419

where is an object’s kinetic energy in 3D space and is its “energy at rest”. SR 420

doesn’t tell us why there is a in the energy of objects that in SR never move at the speed 421

of light. ER gives us this missing clue and is thus superior to SR: is the kinetic energy 422

of moving through the fourth dimension. The in Eq. (17) is strong evidence that everything 423

is moving through ES at the speed , while it is at rest in its proper 3D space. There is also 424

425

, (18) 426

427

where and are the momenta of an object in ES and in 3D space. Dividing Eq. (18) 428

by gives us the vector addition of an object’s momentum in its proper 3D space and its 429

momentum of moving through the fourth dimension. 430

5.4. Solving the Mystery of Relativistic Effects 431

In SR, length contraction and time dilation can be derived from the Lorentz transfor- 432

mation, but the physical cause of these relativistic effects remains in the dark. ER discloses 433

that they stem from projecting the master frame ES to an observer’s reality. 434

5.5. Solving the Mystery of Gravitational Time Dilation 435

Eq. (16b) tells us: The Einstein time of an object in a gravitational field passes by more 436

slowly with respect to an observer who is very far away from the center of this field. The 437

object’s curved trajectory in Cartesian ES coordinates is projected to the observer’s proper 438

3D space (here the object accelerates) and to his proper flow of time (here it decelerates). 439

Curved trajectories in Cartesian ES coordinates replace curved spacetime in GR. 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

11 of 19

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: It can’t move in the axis because 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

12 of 19

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

13 of 19

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

14 of 19

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

15 of 19

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

16 of 19

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

17 of 19

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

18 of 19

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

We also conclude: ER is compatible with quantum mechanics. Egocentric Einstein time pre- 755

vents physicists from grasping the big picture in cosmology and quantum mechanics. We glimpse 756

“behind the curtain” if we only replace Einstein time with Euclidean time. 757

Since SR and GR have been experimentally confirmed many times over, they are con- 758

sidered two of the greatest achievements of physics. We proved that their concept of time 759

is flawed. Albert Einstein, one of the most brilliant physicists ever, wasn’t aware of ER. It 760

was a wise decision to award him with the Nobel Prize for his theory of the photoelectric 761

effect [40] rather than for SR or GR. We campaign for ER as it penetrates to a deeper level. 762

For the first time ever, mankind understands the nature of time: Time isn’t a fundamental 763

quantity, but distance covered in ES, divided by the speed of light. Imagine: The human 764

brain is able to grasp the idea that our energy is moving through ES at the speed of light. 765

With that said, conflicts of mankind become all so small. 766

ER solves 15 mysteries at once: (1) time, (2) time’s arrow, (3) , (4) relativistic ef- 767

fects, (5) gravitational time dilation, (6) CMB, (7) Hubble’s law, (8) flat universe, (9) cosmic 768

inflation, (10) competing Hubble constants, (11) dark energy, (12) wave–particle duality, 769

(13) quantum entanglement, (14) spontaneity, (15) baryon asymmetry. These 15 solutions 770

are 15 confirmations of ER. It isn’t unusual that new concepts give many answers at once. 771

For quantum leaps in understanding, we must question existing concepts. It certainly was 772

to our advantage that we weren’t dazzled by the success of SR and GR. Einstein sacrificed 773

absolute space and time. We sacrifice the absoluteness of waves and matter, but we restore 774

absolute time and pair it with an absolute hypersphere. 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

References 803

1. Einstein, A.: Zur Elektrodynamik bewegter Körper. Ann. Phys. 17, 891 (1905) 804

2. Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49, 769 (1916) 805

3. Popper, K.: Logik der Forschung. Mohr, Tübingen (1989) 806

4. Minkowski, H.: Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Math. Ann. 68, 472 807

(1910) 808

5. Rossi, B., Hall, D.B.: Variation of the rate of decay of mesotrons with momentum. Phys. Rev. 59, 223 (1941) 809

6. Dyson, F.W., Eddington, A.S., Davidson, C.: A determination of the deflection of light by the sun’s gravitational field, from 810

observations made at the total eclipse of May 29, 1919. Phil. Trans. R. Soc. London A 220, 291 (1920) 811

7. Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Westview Press, Boulder (1995) 812

8. Montanus, J.M.C.: Special relativity in an absolute Euclidean space-time. Phys. Essays 4, 350 (1991) 813

9. Montanus, J.M.C.: Proper-time formulation of relativistic dynamics. Found. Phys. 31, 1357 (2001) 814

10. Almeida, J.B.: An alternative to Minkowski space-time. arXiv:gr-qc/0104029 (2001) 815

11. Gersten, A.: Euclidean special relativity. Found. Phys. 33, 1237 (2003) 816

12. van Linden, R.F.J.: Dimensions in special relativity theory. Galilean Electrodynamics 18, 12 (2007) 817

13. Pereira, M.: The hypergeometrical universe. World Scientific News. http://www.worldscientificnews.com/wp-content/up- 818

loads/2017/07/WSN-82-2017-1-96-1.pdf (2017). Accessed 14 February 2023 819

14. Machotka, R.: Euclidean model of space and time. J. Mod. Phys. 9, 1215 (2018) 820

15. Kant, I.: Kritik der reinen Vernunft. Hartknoch, Riga (1781) 821

16. Rovelli, C.: The Order of Time. Allen Lane, London (2018) 822

17. Weyl, H.: Gruppentheorie und Quantenmechanik, chap. III, § 8c. Hirzel, Leipzig (1928) 823

18. LIGO Scientific Collaboration, Virgo Collaboration: Observation of gravitational waves from a binary black hole merger. 824

arXiv:1602.03837 (2016) 825

19. Penzias, A.A., Wilson, R.W.: A measurement of excess antenna temperature at 4080 Mc/s. Astrophys. J. 142, 419 (1965) 826

20. Hubble, E.: A relation between distance and radial velocity among extra-galactic nebulae. Proc. Nat. Acad. Sci. 15, 168 (1929) 827

21. Linde, A.: Inflation and Quantum Cosmology. Academic Press, Boston (1990) 828

22. Guth, A.H.: The Inflationary Universe. Perseus Books, Reading (1997) 829

23. Planck Collaboration: Planck 2018 results. VI. Cosmological parameters. arXiv:1807.06209 (2021) 830

24. Riess, A.G., Casertano, S., Yuan, W., et al.: Milky Way Cepheid standards for measuring cosmic distances and application to 831

Gaia DR2: Implications for the Hubble constant. arXiv:1804.10655 (2018) 832

25. Choi, S.K., Hasselfield, M., Ho, S.-P.P., et al.: The Atacama Cosmology Telescope: A measurement of the cosmic microwave 833

background power spectra at 98 and 150 GHz. arXiv:2007.07289 (2020) 834

26. Bond, H.E., Nelan, E.P., VandenBerg, D.A., et al.: HD 140283: A star in the solar neighborhood that formed shortly after the 835

Big Bang. arXiv:1302.3180 (2013) 836

27. Perlmutter, S., Aldering, G., Goldhaber, G., et al.: Measurements of Ω and Λ from 42 high-redshift supernovae. arXiv:astro- 837

ph/9812133 (1998) 838

28. Riess, A.G., Filippenko, A.V., Challis, P., et al.: Observational evidence from supernovae for an accelerating universe and a 839

cosmological constant. arXiv:astro-ph/9805201 (1998) 840

29. The Nobel Prize. https://www.nobelprize.org/prizes/physics/2011/summary/ (2011). Accessed 14 February 2023 841

30. Turner, M.S.: Dark matter and dark energy in the universe. arXiv:astro-ph/9811454 (1998) 842

31. Heisenberg, W.: Der Teil und das Ganze. Piper, Munich (1969) 843

32. Jönsson, C.: Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Z. Phys. 161, 454 (1961) 844

33. Schrödinger, E.: Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften 23, 807 (1935) 845

34. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. 846

Rev. 47, 777 (1935) 847

35. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964) 848

36. Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938 (1972) 849

37. Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 850

1804 (1982) 851

38. Bouwmeester, D., Pan, J.-W., Mattle, K., et al.: Experimental quantum teleportation. Nature 390, 575 (1997) 852

39. Canetti, L., Drewes, M., Shaposhnikov, M.: Matter and antimatter in the universe. arXiv:1204.4186 (2012) 853

40. Einstein, A.: Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Ann. Phys. 854

17, 132 (1905) 855

41. Niemz, M.H.: Seeing Our World Through Different Eyes. Wipf and Stock, Eugene (2020). Niemz, M.H.: Die Welt mit anderen 856

Augen sehen. Gütersloher Verlagshaus, Gütersloh (2020) 857

42. Plato: Politeia, 514a 858