The Platonic Theory of Relativity

Introduction 

Around the year 375 BC the Greek philosopher Plato, in his work Republic, suggested that the world we see around us is a mere projection of the actual reality, similar to 2-dimensional shadows cast on a wall by a 3-dimensional world. Almost 2100 years later, in the year 1687, the English scientist Isaac Newton published his works Philosophiae Naturalis Principia Mathematica in which he described the theory of Classical Mechanics. Classical mechanics is still taught and used today and accurately describes nearly everything that moves or stands still in the world we see around us, with one important exception:things that move extremely fast. This was discovered exactly 200 years after the publication of Philosophiae Naturalis Principia Mathematica, when the Americal Scientists Albert Michelson and Edward Morley delivered strong experimental evidence against the, then-prevalent, ether theory [1]. Their findings led to the Special Theory of Relativity and its famous postulate that the speed of light in vacuum is the same for all observers, regardless their relative motion, and that elecromagnetic radiation is the only wave form that does not require a medium of propagation. The Platonic Theory of Relativity suggests an alternative interpretation of the invariability of the speed of light. It assumes that the 3-dimensional world around us is indeed a projection of a 4-dimensinal reality, exactly like Plato suggested 2400 years ago. By using Classical Mechanics only, and without requiring the elimination of ether as a medium of propagation, the Platonic Theory of Relativity explains observations currently only addressed by Special Relativity, including Lorentz length contraction, time dilation, mass increase, relativistic simultaneity as well as a theoretical value for the Hubble constant.

 

Hypothesis 

The Simple Theory of Relativity considers our 3-dimensional (3D) universe as the outer shell of a 4-dimensional (4D) sphere, which is expanding at the speed of light c with respect to its centre, into a space with 4 orthogonal spatial dimensions. In Figure 1 this 4D sphere is shown in blue, with the extension of our universe in green. An observer located at point A is consequently moving with velocity c into the w dimension. However, the observer can only perceive the three orthogonal spatial dimensions x, y and z (indicated by the orange dotted line), corresponding to the three directions perpendicular to his direction of motion into the 4th spatial dimension w. 

Derivation of Hubble’s constant 

Consider a far-away galaxy at point B. This galaxy moves with velocity c under an angle 𝜑 with the direction of the motion of the observer. The observer only perceives the x, y and z dimensions, and therefore sees the galaxy receding with a velocity v = c sin(𝜑). Similarly, the distance between A and B is perceived as D = R sin(𝜑). The Hubble constant, which relates the receding velocity of celestial objects to their distance, can now be written as 

 

      H = v/D = c/R      (1) 

 

where R is the radius of the 4D sphere. Setting the start of the expansion at 13.8 billion years ago, the estimated age of the universe, we find H = 70.8 km/s/Mpc, in alignment with the latest measurements [2]. Note that when 𝜑 approaches 90 degrees, the galaxy will recede from the observer with a velocity close to the speed of light. The orthogonal projection of the 4D sphere on the 3D xyz-space can consequently be recognised as the traditional Hubble sphere. Also note that the circumference of the 4D sphere amounts to 2𝜋R, so that the perceived diameter of our 3D universe amounts to 2𝜋 times 13.8 = 87 billion light years, close to current estimates. 

 

Doppler shift and the invariability of the speed of light 

We now consider electromagnetic radiation emitted by the galaxy at point B. We hypothesise that electromagnetic radiation, just like all other known wave forms, requires a medium of propagation (the ether) and obeys the 4D electromagnetic wave equations (c² ∇² - 𝜕²/𝜕t²) E(w,x,y,z,t) = 0 and (c² ∇² - 𝜕²/𝜕t²) B(w,x,y,z,t) = 0, with E and B 4D electromagnetic field vectors and ∇² =  𝜕²/𝜕w² + 𝜕²/𝜕x² + 𝜕²/𝜕y² + 𝜕²/𝜕z². With the same arguments as before, the observer can only see the orthogonal projection of this 4D wave on his own 3D environment. In order to illustrate the resulting dynamics, we performed a numerical simulation of an electromagnetic point source with fixed frequency f creating a wave in a medium of propagation, while simultaneously moving at the wave's phase velocity. Figure 2 shows how the point source leaves the center of the picture and moves in the w direction (corresponding to 𝜑 = 0 degrees in Fig. 1). 

The wave amplitude in the medium of propagation is shown in shades of grey. The horizontal axis is labelled ‘xyz’ and is the analogy of the 3D space indicated by the horizontal axis in Fig. 1, while the curve at the bottom is the orthogonal projection of the wave on the horizontal axis, i.e., the integral of all local amplitudes in the vertical direction for each horizontal position. The projection curve takes the shape of a symmetric, modulated sine function with the same frequency as its source. The analogy in Figure 1 would be a 3-dimensional spherical electromagnetic wave, located at the position of the observer. 

Figure 3 shows what happens when the radiation source travels under an angle of 30 degrees towards the top right. In Fig. 1 this would correspond to 𝜑 = 30 degrees and hence the galaxy receding at a velocity of c/2 from the observer. 

We see that the projected wave now gets an asymmetrical shape, corresponding to a red shift (increased wavelength) seen by the observer. It is straightforward to demonstrate that the fractional change in wavelength indeed corresponds to z = v/c (neglecting relativistic time effects at the radiation source). Not that the phase velocity of the projected wave perceived by the observer is identical in Figures 2 and 3, as it is solely determined by the medium of propagation. For reasons of symmetry, the same applies when the observer moves with respect to the radiation source. As a result, it is easily understood that if a 3D light wave is an orthogonal projection of its 4D parent, its perceived phase velocity remains constant, irrespective the relative velocity between observer and source. This explains the invariability of the speed of light, even in the presence of a medium of propagation. 

 

Lorentz contraction 

The reduction of length in the direction of motion by a factor 𝛾 = 1/√(1-v²/c²), known as Lorentz contraction, can also be easily explained by our model. Imagine an observer next to a space ship that is launched in any direction in our x-y-z universe (see Figure 4). 

No matter in what direction the space ship is launched, it will always be perpendicular to the w direction. Assume it is launched in the z-direction. Prior to the launch, even though standing still with respect to the observer, the space ship moves with a velocity c with respect to the centre of the 4D sphere. As a result, according to Newtonian mechanics, its acceleration will be perpendicular to its velocity, so that the space ship will execute a circular motion in the w-z plane, while its orbital velocity in the 4D space will remain constant at c. After the acceleration stops, the space ship will have tilted into the w dimension over an angle 𝜑 with respect to the velocity of the observer, and continue to move in a straight line, always with velocity c. As before, the observer sees the space ship receding at a velocity v = c sin(𝜑). Since the observer cannot discern the w dimension, it will only see the orthogonal projection of the tilted space ship, which will be a 3D space ship with length L = L_o cos(𝜑), with L_o the space ship’s rest length. Since sin(𝜑) = v/c, it is quickly found that L = L_o /𝛾, with 𝛾 = 1/√(1-v²/c²), the well-known relativistic factor from Special Relativity. 

 

Time Dilation 

Velocity time dilation, or time slowing down when objects move, can be explained in a similar fashion. It is hypothesised that atomic interactions are driven by the stream of ether particles, or ‘ether wind’, inherent to the aforementioned motion of all matter through a 4D space. As a result, time passed for any object is proportional to distance travelled in 4D space. In addition, the perceived 3D space, perpendicular to an observer's direction of motion in 4D space, is also considered the space in which the observer manifests himself. We call it the space of existence or SoE. With these two hypotheses velocity time dilation is explained in Fig. 5. 

Two observers, a red one and a green one, recede from each other with velocity v. Their trajectories in 4D space are shown at 4 successive, equidistant, Newtonian time points, with the dotted lines indicating their SoE’s. At t = 3 the SoE of the red observer crosses the SoE of the green observer at t = 2 (yellow dotted lines). As a result, the red observer can see a projection of the green observer at t = 2. This effect only works one way, because the green observer at t = 2 is not inside the SoE of the red observer at t = 3. Instead, at t = 2 he will see the red observer at t = 4/3, and - for symmetry reasons - only when arriving at t = 3 will he see the red observer at t = 2. This clarifies the well-known but hard to imagine symmetry of velocity time dilation: both observers are simply looking into each other’s past. From Fig. 5 it is straightforward to derive that this time dilation is indeed proportional to the same factor 𝛾 mentioned before, in agreement with the Special Theory of Relativity. 

 

Conclusion 

We proposed a new physical model to describe the invariability of the speed of light, and several phenomena known from the Special Theory of Relativity, by using Newtonian mechanics only. This approach eliminates several inconsistencies in contemporary physics, including the dualistic model for electromagnetic radiation (photons versus waves) and the absence of a medium of propagation for electromagnetic waves, by reintroducing the ether as medium of propagation for electromagnetic waves as well as the driving force behind the concept of time. Additional aspects, such as relativistic mass increase, gravity and gravitational time dilation, as well as the hypothesis that the ether may be the source of the widely ununderstood concept of dark matter and dark energy may be explained by our model as well and will be subject of future studies. 

 

References 

[1] Michelson, Albert A.; Morley, Edward W. (1887). "On the Relative Motion of the Earth and the Luminiferous Ether" . American Journal of Science. 34 (203): 333–345.

[2] https://map.gsfc.nasa.gov/universe/uni_expansion.htm

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