ABSTRACT

Einstein, who abolished the absolute stationary coordinate system and proposed the theory of relativity, gave up on the static universe when he found distant galaxies receding. However, the hierarchical structure of the universe after the elimination of absolute time cannot be relative time symmetric, leaving many paradoxes and unsolved problems. This paper goes back to Galilean invariance before Newtonian mechanics and clarifies some paradoxes and unsolved problems of relativity in the time hierarchy of the receding universe by means of the "equivalence principle of the light's momentum under asymmetry relative rate transformation". And, Answers 16-year-old Einstein's question, "If you look in the mirror while moving at the speed of light".

Keywords

Relative time, hierarchical structure of the Universe, observational reference system, Doppler effect in light, asymmetry, time hierarchy, unsolved problems, flyby anomaly, hierarchy problem, scale effects, light's momentum, equivalence principle, perihelion shift, galaxies, dark matter

1. Introduction

Newtonian mechanics formulated Galilean invariance as a Galilean transformation based on the assumption of an absolutely stationary coordinate system, but due to an inconsistency with electromagnetism, the absolutely stationary coordinate system was abolished and Einstein's (1905)[1] special theory of relativity (SR) was born, based on Lorentz transformation. However, Einstein (1947)[2] said, "If we do not want to give up the physical interpretation of coordinates (something that is itself possible) in general, it is better to allow such a contradiction ---- but of course, in subsequent studies of the theory, it must be removed." The motion of an object whose sum of working forces (combined force) is zero is called inertial motion, and the inertial system is a concept that combines the inertial motion with the time and space measured by a clock and a ruler that moves with the object. Therefore, between inertial systems with different time rates, the square- cube law must be taken into account, but this is not defined in the Lorentz transformation. The global inertial system, which is the stage of the Lorentz transformation, is also not defined in line with the receding universe. This paper reviews the global inertial system of the receding universe and the relative rate transformation in the hierarchical structure of the universe, taking these into account, and explains "Hierarchical dynamics with relative rates".

2. Relative motion defined in relative time

Relative motion with absolute stationary reference has ±.

Relative velocity (v±) =
motion velocity of the object (vb) - motion velocity of the observer (va).　(1)

Eliminating this, relative motion with the speed of light as the observation criterion also has ±.

Motion velocity (v±)² =
| wave speed of the object (w±)² - speed of light of the observer (c)² |,
for w₊ > c, v₊: recession velocity, for w₋ < c, v₋: advanced velocity.　　(2)

This will spontaneously generate a light-speed-invariant observation reference system, without manual coordinate transformation, by interference at the boundary between the wave speed and the motion velocity of the object.

c² = w² ± v².　　(3)

In other words, the invariant system is generated from asymmetry, which is what Galilean invariance says. Therefore, the time unbounded by absolute time is just a unit made from the number of oscillations in "a certain wave speed", and if that "certain wave speed" is the speed of light, then the wave speed is the speed of relative time. If the relative time remaining after eliminating the background of the absolute stationary coordinate system is bounded by a spacetime background that treats time and space equally, then background independence is lost.

Relative rate factor γ = c / w = c / (c² ± v²)¹ᐟ², τ = γ t, x = c t = w τ, t: elapsed time of observer, τ: elapsed time of object, x: distance.　　　(4)

If the wave speed of the object is slow (c > w₋), it takes an elapsed time (τ) to travel the distance (x).
If the wave speed of the object is fast (c < w₊), it does not take an elapsed time (τ) to travel the distance (x).

If the light speed (c) is fixed at a defined value and the lengths of space-time and objects are allowed to vary, either the relationship between speed, time and distance is broken by acceleration, or Lorentz symmetry is broken between the three bodies, as in Bell's spaceship paradox.

For example, the ratio of the advance of an atomic clock on a GPS satellite to that of a clock on the earth's surface is:
Speed of light at the earth's surface (c): 299,792,458 m/s.
Geocentric gravitational constant (GM): 3.986e + 14 m³/s².
Earth radius (r): 6,378,000 m.
Wave speed at infinity, w₈ = (c² + 2GM / r)¹ᐟ².
GPS satellite altitude (h): 20,200,000 m.
GPS satellite orbit velocity (v): 3,874₈ m/s.
GPS satellite wave speed, w₉ = (w₈² - 2GM /[r + h] - v²).
Difference in the advance of the light clock, w₉ / c = 1 + 4.45e-10.

3. Deceleration of time in the Universe and observational reference systems

When looking at distant galaxies from this observational reference system, the
 recession velocity converted from the cosmological redshift is observed. This
 is considered as a slowing down of the observer's time, and all time of the universe is assumed to be slowing down. Einstein (1905) eliminated the
 boundary conditions for a change in the speed of time and derived a Lorentz   symmetric SR with only a advanced velocity derived from simultaneous
 relativity. However, the "Reciprocity of time dilation" that occur in Lorentz
 symmetry only at the advanced velocity will occur at the recession velocity as
 seen by observers in distant galaxies due to the slowing down of all time in the universe. Also, the twin paradox is resolved by the fact his time in the universe is further slowed down by the advanced velocity of one of the brothers.

4. Asymmetry of the Doppler effect in light

Since the asymmetric velocities of motion are not known in advance from relative time alone, not only the primary Doppler effect but also the light-specific secondary Doppler effect must be prepared for asymmetric Doppler effects.

4-1. The light source has a slower speed of time than the observer (c > w₋)

The wave speed of the light source (w₋) is ,

γ₊ = 1 / γ₋ = c / w₊, w₋ = γ₊ c = (c² - v₋²)¹ᐟ²,
v₋ = c (1 - γ₊²)¹ᐟ²= (c² - w₋²)¹ᐟ².　　(5)

The secondary Doppler frequency (f₋) generated at the light source is ,

γ₋ = 1 / γ₊ = c / w₋, f₋ = f₀ / γ₋,
f±: secondary Doppler frequency, f₀: reference frequency.　　(6)

When light is emitted from the source, at the boundary, the secondary Doppler wavelength (λ₋) is,

λ₋ = γ₋ λ₀,
λ± : secondary Doppler wavelength, λ₀ : reference wavelength.　　(7)

If the light source in the direction of the angle (θ) viewed from the observer is moving at advanced velocity (v₋), the observed frequency (f) and the observed wavelength (λ) are,

f = f₋ / (1 - v₋ cos θ / c), λ = c / f,
θ: angle of the light source as seen from the observer.         (8)

4-2. Observers have a slower speed of time than the light source (c < w₊)

The wave speed of the light source (w₊) is ,

w₊ = γ₋ c = (c² + v₊²)¹ᐟ², v₊ = c (γ₋² - 1)¹ᐟ²= (w₊² - c²)¹ᐟ².　　(9)

The secondary Doppler frequency generated at the light source is ,

f₊ = f₀ / γ₊.　　(10)

The secondary Doppler wavelength (λ₊) when incident on the observer's boundary is,

λ₊ = γ₊ λ₀.　　　(11)

If the light source is moving with a recession velocity (v₊) in the direction of the angle (θ) seen by the observer, the observed frequency (f) and the observed wavelength (λ) are,

f = f₊ / (1 - v₊ cos θ / w₊), λ = c / f.             (12)

4-3. Unresolved flyby anomaly

For example, in the unsolved flyby anomaly, the wave speed propagated from a spacecraft flybying the Earth, as seen from a Deep Space Network (DSN) at a certain latitude (φ), is superlight speed (c < w ₊R) due to the second-order Doppler shift.

w₊ = (c² + [ω R cos φ]²)¹ᐟ².　　　(13)

Therefore, the difference between (Eq. 8) and (Eq. 12), the light speed difference in the optical path (c: w₊), appears as the primary Doppler frequency difference (⊿f₊) in the velocity increment at infinity (flyby anomaly: ⊿v₊∞).

⊿f₊ / f = (f₊ - f ) / f = ⊿v₊∞ cos θ / c.　　(14)

Einstein (1905)[1] relied on Einstein synchronisation because he did not know which was moving more, but did not know that he could reduce it to an easily verifiable physical property of light propagation. Therefore, this anomaly is, as V. Guruprasad (2015)[3] says, an SR problem pertaining to the symmetry of the Doppler effect in light.

5. Hierarchy problems in the time hierarchy of the universe

5-1. Scale effects, and the equivalence principle of the light's momentum

In Newtonian mechanics, density refers to the mass per unit volume and can be expressed by the conversion equation: matter density = mass/volume. If the matter density is extended to the momentum density of electromagnetic waves and converted to the light's momentum (p), the relationship with energy (E) is,

E = c |p|.　　(15)

Requesting Galilean equivalence principle (generality of free fall), we can translate the energy magnitude and increase/decrease of gravitational mass (m₉ = E / c²) in Newtonian mechanics and the scale change of inertial mass (m₁ = γ m₉) due to motion as unrelated.

|p| = m₉ c = m₁ w, E = m₉ c² = γ m₁ w².　　(16)

That is, it is one inertial system only for (rest mass: m₀ = m₁ = m₉) and (c = w₀), which is consistent with Einstein's equivalence principle. This is called the equivalence principle of the light's momentum. This difference between gravitational and inertial mass is equivalent to the difference in the speed of time, which causes a scale effect and produces a deviation from the inverse square law in the time hierarchy of the universe.

Absolute time ⇒ Inverse square law.
↓
Relative time ⇒ Square-cube law.

5-2. Equivalence principle of the light's momentum, and the perihelion shift error

The perihelion shifts of the planets in the solar system can be explained almost entirely by gravitational perturbations from other planets according to Newtonian mechanics. For example, of the 575″ perihelion shifts that Mercury undergoes in 100 years, the gravitational influence from other planets is calculated by Newtonian mechanics to be 532″ in total, which is more than 90% of the total, Will, Clifford (2011)[5] reported in 1915 that Einstein's geodesic equations and He says that using approximate solutions of the space-time metrology around the Sun, he was able to match observations with the relativistic contribution to Mercury's perihelion migration. The equation of equilibrium between centrifugal force and universal gravitation in Newtonian mechanics is,

GM☉ m₉ / r² - m₁ v₋² / r = 0, GM☉: heliocentric gravitational constant,
r: distance between planet and sun.　　(17)

Placing the observational reference system for the speed of light (c) at infinity and eliminating the mass of the planet from (Eq. 17) using the equivalence principle of the light's momentum in (Eq. 16), we obtain,

γ₋ v² = GM☉ / r, γ₋ = c / w₋ = ｃ / (c² - v₋²)¹ᐟ².　　(18)

Replacing (v₋ = 2π r / T) in (Eq. 18) by Kepler's third law,

4 π² γ₋ = GM☉ T² / r³, T: orbital period of the planet.　　(19)

From the left-hand side of (Eq. 19), the error (∆φ) of one revolution of the orbit is,

Δφ = 4 π² (γ₋ - 1).　　(20)

From (Eq. 20), the arc second (x") of Mercury's 415 orbits in 100 years is,

x″ = ∆φ (180 / π) × 3600 × 415.         (21)

If the orbital velocity (v) and orbital period (T) are known from observations, the exact solution for the perihelion migration of each planet can be obtained from the relative rate factor (γ) using Newtonian mechanics without the absolute stationary coordinate system.

Einstein created SR by requesting symmetry under Lorentz transformations and the principle of invariance of the speed of light. Furthermore, by requesting symmetry under general coordinate transformations and the Einstein equivalence principle, he constructed a theory of gravity called General Relativity (GR). This paper follows only the advantages of Galilei-Newton mechanics and Einstein's relativity with the equivalence principle of the light's momentum and under asymmetry relative rate transformation.

5-3. Galactic rotation curve problem and dark matter

We have seen above the deviation from the inverse square law of inertial motion due to differences in the speed of time between systems, but it becomes more pronounced in the relationship between galaxy groups and galaxies. A galaxy group is the smallest group of galaxies. The number of galaxies does not exceed 50, typically with a diameter of 1-2 Mpc and a mass of about 10¹³ solar masses. The dispersion velocity (v₊) of the galaxies here is about 150 km/s. From the equivalent principle of the light's momentum from the observational reference system (c) inside the galaxy, the total mass (Mⓖ) of the galaxy above about 10¹¹ solar masses, replaced by a recession factor (k₊) in the reciprocal dimension (m-¹) of the distance, leads to the force The balance equation is ,

w₊² - c² = v₊² = GMⓖ k₊.　　(22)

This is the image of a galaxy rolling at a dispersive velocity (v₊), and (Eq. 22) can be rewritten in terms of mass (M[r]) within position (r) from the centre of the galaxy,

v(r)² = GM(r) / r + GM(r) k₊ = GM(r) (1 / r + k₊).　　(23)

The first term on the right-hand side of (Eq. 23) is the Newtonian potential (1 / r). The second term is the asymptotically increasing dark matter (GM[r] → GMⓖ). The combined result is a rotation (v[r] → v₊) in the galaxy.

6. Conclusion

The Galilean invariance is based on the principle that invariance occurs at each level of the hierarchical structure of the Universe. It is not a case of symmetry between systems, and the scale effects caused by differences in the rate of relative time cannot be ignored. There is no phenomenon in which the rate of relative time, which is not bound by absolute time or Lorentz symmetry, changes and the speed of light does not change; M. Senovilla et al [8] say that dark energy is not necessary because the universe is not expanding at an accelerated rate in the first place. What appears to be an acceleration is actually a deceleration of time itself, and is only an apparent phenomenon caused by the observation of supernova explosions in the past, when the passage of time was faster than it is now. If this is the case, then the seemingly permanent increase in the number of astronomical units relative to the metre, and the increase in the lunar eccentricity of this slight expansion of the eccentricity, which is not consistent with the dynamical model, are also related to this. The paradoxes and unsolved problems in the hierarchical structure of the Universe described in this paper, as well as the hierarchy problem in particle theory, may also be further addressed by "relativity in the time hierarchy of the receding Universe". In recent years, time crystals, which were said to be impossible in an equilibrium state, have been confirmed by tests in a non-equilibrium state, and are no longer unsolved problems. In the same way, wouldn't it be a shortcut to the theory of quantum gravity if we abolish symmetric relativity and reexamine it to Temporal Light?

Authorship Contribution Statement

Competing Interests

The author declares there are no competing interests.

Acknowledgments

The author would like to thank the members of the Q&A site who have worked with him on the problems of relativity for many years.

References

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[2] Einstein, Albert. (1947). "jiden nooto" Tokyo tosho(1978)p73～81 (fnorio.com)

[3] V. Guruprasad (2015), "Observational evidence for travelling wave modes bearing distance proportional shifts", EPL, 110 (5): 54001, arXiv:1507.08222, Bibcode:2015EL....11054001G, doi:10.1209/0295-5075/110/54001

[5] Will, Clifford M. "On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics." Proceedings of the National Academy of Sciences 108.15 (2011): 5938-5945. https://www.pnas.org/doi/full/10.1073/pnas.1103127108

[6] Nobili, Anna M., and Clifford M. Will. "The real value of Mercury's perihelion advance." Nature 320.6057 (1986): 39-41. https://www.nature.com/articles/320039a0

[8] Marc Mars, José M. M. Senovilla, Raül Vera. "Accelerating expansion and change of signature". arXiv:0712.1462