Article title: On the Relativity of the Authors: XD Dongfang[1] Affiliations: Wutong Mountain National Forest Park[1] Orcid ids: 0000-0002-3644-5170[1] Contact e-mail: [email protected] License information: This work has been published open access under Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at https://www.scienceopen.com/. Preprint statement: This article is a preprint and has not been peer-reviewed, under consideration and submitted to ScienceOpen Preprints for open peer review. DOI: 10.14293/S2199-1006.1.SOR-.PPBT1S7.v1 Preprint first posted online: 21 December 2020 Keywords: Unitary principle; Lorentz transformation; complete space-time transformation; constant speed of light in isolated reference frame; relative variable light speed On the Relativity of the Speed of Light

X. D. Dongfang Wutong Mountain National Forest Park, Shenzhen, China

Einstein’s assumption that the speed of light is constant is a fundamental principle of modern with great influence. However, the nature of the principle of constant speed of light is rarely described in detail in the relevant literatures, which leads to a deep misunderstanding among some readers of special relativity. Here we introduce the unitary principle, which has a wide application prospect in the logic self consistency test of mathematics, natural science and social science. Based on this, we propose the complete space-time transformation including the Lorentz transformation, clarify the definition of relative velocity of light and the conclusion that the relative velocity of light is variable, and further prove that the relative variable light speed is compatible with Einstein’s constant speed of light. The specific conclusion is that the propagation speed of light in vacuum relative to the observer’s inertial reference frame is always constant c, but the propagation speed of light relative to any other inertial reference frame which has relative motion with the observer is not equal to the constant c; observing in all inertial frame of reference, the relative velocity of light propagating in the same direction in vacuum is 0, while that of light propagating in the opposite direction is 2c. The essence of Einstein’s constant speed of light is that the speed of light in an isolated reference frame is constant, but the relative speed of light in vacuum is variable. The assumption of constant speed of light in an isolated frame of reference and the inference of relative variable light speed can be derived from each other.

I. INTRODUCTION not change because of the different metrics. The results of the transformation from different mathematical form- Because of relativity, the speed of light in vacuum has s of natural laws expressed in different metrics into the a very special position in modern physics. Although the same metric must be the same as the inherent form under hypothesis of invariance of light speed[1, 2] and the prin- this metric, 1 = 1, which means the transformation is u- ciple of relativity[3], which are the basis of special rela- nitary. This principle is called the unitary principle. It tivity, are regarded as the two basic principles that can can be widely used to test the self consistency of logic in stand the test of experiment[4–10] and the demonstra- mathematics, physics and even philosophy. By using the tion of logical completeness[11–13], the essences of the unitary principle to test the physical hypothesis and in- hypothesis of constant speed of light and the inferences of ference that can not be proved, the controversial problem Lorentz transformation are not always clear to all physics can be transformed into the problem of definite solution readers. This is because the relevant experimental results of mathematical physics, so as to find the real solution of can be interpreted differently[14, 15]. Of course, the so- the problem under the condition of definite solution and called superluminal and slow light speed phenomena[16– distinguish it from the formal solution, and the problem 24] are not variable light speed phenomena in the usual will have a clear answer. sense, while various superluminal and variable light speed The unitary principle test of quantum mechanics has theories[25–30] are only limited to formal reasoning. promoted the discovery of many important problems and conclusions from modern physics to com quantum The conclusions of theoretical physics often come from theory[31–34]. In this paper, we use the unitary princi- mathematical deduction. We know that from different ple to test the concept of space-time in the sense of spe- angles, the same mathematical physics problem can have cial relativity, and put forward the complete space-time completely different formal solutions, and each formal transformation that includes the Lorentz transformation solution can have different interpretations. Only the re- and thus conforms to the meaning of special relativity. al solution of a mathematical physics problem conforms We can find that the definition of relative speed of light to the natural law, and the real solution of mathemat- exists, which is no longer constant. Furthermore, we can ical physics problem should be unique. If we discuss prove that the relative variable light speed is compatible the transformation of time and space parameters from with Einstein’s constant speed of light. the perspective of the existence and uniqueness theorem for the solution of mathematical physics problems, the essences of the hypothesis of invariable light speed and the inference of Lorentz transformation will be clear. At II. COMPLETE TRANSFORMATION OF TIME this time, it is very important to put forward the rel- AND SPACE PARAMETERS evant mathematical and physical problems. There is a universal principle to test the self consistency of natural Let the relative speed of two inertial reference frames science theory: different metrics can be used to describe Σ and Σ′ be v, and the unitary principle test of the logic natural laws, and there is a certain transformation rela- self consistency of the hypothesis of constant light speed tionship between them. The natural laws themselves do in vacuum contains the following four metrics: (a) the 2 speed of light measured on reference frame Σ relative to object relative to its own frame and relative to other reference frame Σ; (b) the speed of light measured on inertial frames, but the clock used can only be the clock of reference frame Σ′ relative to reference frame Σ′; (c) the that reference frame. The meaning of time here accords speed of light measured on reference frame Σ relative to with Einstein’s definition, that is, when the reading t = reference frame Σ′; (d) the speed of light measured on ref- t′ = 0 of the calibrated synchronous clocks resting in erence frame Σ′ relative to reference frame Σ. According the reference frame Σ and Σ′ respectively, the coordinate to the unitary principle, the result of the transformation origin points of the two inertial frames coincide exactly. of light speed between different metric must be the same There is a class of basic facts that have a potential im- as the inherent speed of light under this metric, and the pact on physical theory. For example, when the moon is transformation is unitary. on the line between the sun and the earth, measuring the It should be pointed out that the self consistent log- distance between the sun and the earth on the earth can ic must satisfy the unitary principle, and the logic that be divided into two parts: the distance between the sun violates the unitary principle must not be self consistent and the moon and the distance between the moon and and implies many contradictions, while the physical logic the earth; When the moon is not on the line between the with local self consistency may not be the natural law. sun and the earth, measuring the position vector of the Logical self consistency is only a necessary condition but sun relative to the earth on the earth can be decomposed not a sufficient condition for the theory of natural sci- into two parts: the position vector of the sun relative to ence. With regard to the space-time coordinates of the the moon and the position vector of the moon relative to so-called event P , the concept of special relativity de- the earth. This kind of fact is abstracted as an axiom: scription is as follows: the position vector of an object relative to other inertial reference frames in all inertial reference frame obeys the (i) The time-space coordinate (x, y, z, t) of event P operational rule of vector superposition. We call it the measured on reference frame Σ relative to reference principle of the relative position vector superposition. frame Σ; (ii) The time-space coordinate (x′, y′, z′, t′) of event P ¦ ¦c measured on the reference frame Σ′ relative to the y yc reference frame Σ′;

There are actually two other concepts: P r, ȡ (iii) The spatiotemporal coordinate (ξ, ζ, η, τ) of event P measured on reference frame Σ relative to refer- rc, ȡc ence frame Σ′; y, y c ,9 , 9 c (iv) The spatiotemporal coordinate (ξ′, ζ′, η′, τ ′) of event P measured in the reference frame Σ′ rela- o oc [, xc x, x c tive to the reference frame Σ. z, z c ,K , K c It can be seen that for two inertial reference frames, an event has four sets of space-time concepts. The special x, [ c theory of relativity based on the Lorentz transformation only describes the first two sets of space-time concepts z zc (x, y, z, t) and (x′, y′, z′, t′). On reference frame Σ, measuring the time and space FIG. 1. Four sets of spatiotemporal concepts of the same event relative to two inertial reference frames parameters of event P relative to reference frame Σ and to reference frame Σ′, the clock used is the clock on ref- erence frame Σ, and the concept of time belongs to ref- The above problems (iii) and (iv) can now be solved. erence frame Σ, so τ = t. Similarly, on reference frame As shown in Fig. 1, according to the agreement of spe- ′ cial relativity, the two inertial reference frames Σ and Σ , measuring the time and space parameters of event ′ P relative to reference frame Σ′ and relative reference Σ have a common x axis. When the moment of refer- frame Σ, the clock used is clock on reference frame Σ′, ence frame Σ is t, the distance between the coordinate ′ and the concept of time belongs to the reference frame origins of the two coordinate frames is oo = vt. Since Σ′, so τ ′ = t′. the distance between an object measured in all inertial reference frame relative to other inertial frames can be τ = t, τ ′ = t′ (1) divided, then x = oo′ = vt + ξ, therefore ξ = x − vt. Because ζ = y, η = z, then the transformation relation- The above equations show the characteristics of isolated ship between the spatiotemporal coordinates (x, y, z, t) reference frame of time, that is, in all inertial reference relative to reference frame Σ and the spatiotemporal co- frame, the observer can measure the motion law of an ordinates (ξ, ζ, η, τ) relative to reference frame Σ′ of event 3

P measured in reference frame Σ is as follows v. The form of the equations is as follows,   ξ = x − vt, ζ = y, η = z, τ = t (2)  ′ x − vt  x = √  ′ ξ  1 − v2c−2  x = √ ′   − 2 −2 Similarly, observed in the reference frame Σ , the trans-  ′ ′  1 v c y = y = ζ = ζ ′ ′ ′ formation relationship between the spatiotemporal co- ′ ′ ⇔ ξ = x + vt ′ ′ ′ ′ ′  z = z = η = η  ′ ′ ordinate (x , y , z , t ) of event P at time t relative to   τ = t, τ = t ′  − −2  √ the reference frame Σ and the spatiotemporal coordi-  ′ t vc x  ′ ′ ′ ′ ′  t = √ ξ = 1 − v2c−2 (ξ + vt) nate (ξ , ζ , η , τ ) of the relative reference frame Σ is as 1 − v2c−2 follows (8) The corresponding inverse transformation relation is ξ′ = x′ + vt′, ζ′ = y′, η′ = z′, τ ′ = t′ (3)  ′ ′   x + vt ′  x = √  ξ The meaning of Einstein’s constant speed of light hy-  − 2 −2  x = √  1 v c  − 2 −2 pothesis is that the speed of the observed light relative  ′ ′ ′  1 v c y = y = ζ = ζ − to the reference frame in any inertial frame is c, which ′ ′ ⇔ ξ = x vt | | | ′ ′|  z = z = η = η  ′ ′ is expressed as dr/dt = c, dr /dt = c. Therefore,   t = τ, τ = t the invariance of light speed in special relativity belongs  ′ −2 ′  √  t + vc x  − ′ ′ to the constant speed of light in an isolated reference t = √ ξ = 1 − v2c 2 (ξ − vt ) 1 − v2c−2 frame. Einstein derived Lorentz transformation accord- (9) ing to the principle of constant light speed in isolated The above transformation relations (8) and (9) conform reference frame and relativity principle to the special relativity meaning, and are called complete space-time transformation. According to the complete x − vt t − vc−2x x′ = √ , y′ = y, z′ = z, t′ = √ (4) space-time transformation, the complete transformation 1 − v2c−2 1 − v2c−2 relations between the intrinsic quantity and the relative quantity of the corresponding physical quantity such as The inverse transformation is momentum and energy between different inertial refer- ′ ′ ′ −2 ′ ence frames can be derived, which is omitted here. We x + vt ′ ′ t + vc x x = √ , y = y , z = z , t = √ (5) also need to give priority to discovering the deeper spa- − 2 −2 − 2 −2 1 v c 1 v c tiotemporal parameter transformations Obviously, Lorentz transformation in the sense of special relativity describes the relationship between space-time parameters of isolated reference frame between two iner- III. VARIABILITY OF RELATIVE SPEED OF tial reference frames. LIGHT Now we can determine the relations among various de- fined spatial parameters. By substituting the first for- The Lorentz transformation is essentially a speed pre- mula of the set of relations (2) into the first formula of serving transformation, which stipulates that the speed Lorentz transformation (4), or the first formula of the set of light in the vacuum of two isolated reference frames of relations (3) into the first formula of Lorentz inverse is constant, thus defining the transformation relationship transformation (5), new transformation and correspond- of the space-time parameters of the two inertial reference ing inverse transformation are obtained respectively frames. Since the speed of the experimental observation light relative to the laboratory reference frame in the vac- ξ ξ′ x′ = √ , x = √ (6) uum is constant c, Einstein proposed the hypothesis that 1 − v2c−2 1 − v2c−2 the speed of light does not change. In turning, he de- rived the Lorentz transformation and established special ′ ′ ′ After substituting the equivalent form x = ξ −vt of the relativity[35]. Now that we have a complete space-time first formula of (3) into the first formula of formula (6), transformation, we have a clear definition of the rela- or substituting the equivalent form x = ξ + vt of the first tive speed of light. According to the complete space-time formula of (2) into the second formula of formula (6), we transformation, the relative speed of light is variable, and get the following results the relative variable light speed is compatible with the √ √ constant speed of light in an isolated reference frame. ξ = 1 − v2c−2 (ξ′ − vt′) , ξ′ = 1 − v2c−2 (ξ + vt) It is assumed that when the coordinate origin o of the (7) inertial reference frame Σ coincides with the coordinate These transformations are different from both Lorentz origin o′ of the inertial reference frame Σ′, the light source transformation and Galileo transformation. at the common origin emits a to the positive di- Equations (2), (3), (4) and (5) describe the transforma- rection of the x axis. According to the complete space- tion relations among four groups of space-time parame- time transformation, the two inertial frames of reference ters of two inertial reference frames with relative velocity contain four definitions of light speed, namely: 4

a) The isolated speed of the photon relative to the obtained. In the calculation, the isolated light speed √ ( ) reference frame Σ measured on reference frame Σ, ′ ( ′ ) ′ 2 ( ′ ) √ ( ) dr dx 2 dy dz 2 ( ) 2 ( ) dt′ = dt′ + dt′ + dt′ = c of the reference dr dx 2 dy dz 2 = + + = c ; ′ dt dt dt dt frame Σ is used again, so the results of the relative light dρ′ velocity ′ of relative to the reference frame Σ b) The relative light speed of the photon relative to dτ ′ measured in the reference frame Σ satisfy the relation- ′ dρ Σ measured on reference frame Σ, dτ =? ; ship √( ) ( ) ( ) c) The isolated speed of the photons relative to ref- ′ ′ 2 ′ 2 ′ 2 ′ ′ dρ dξ dς dη erence frame√ Σ measured on reference frame Σ , = + + ( ) dτ ′ dτ ′ dτ ′ dτ ′ ′ ( ′ ) ′ 2 ( ′ ) dr dx 2 dy dz 2 √ dt′ = dt′ + dt′ + dt′ = c ; ( ) ( ) ( ) dx′ 2 dy′ 2 dz′ 2 = + v + + d) The relative light speed of the photon relative to Σ dt′ dt′ dt′ ′ √ (12) ′ dρ ( ) ( ) ( ) measured on reference frame Σ , ′ =?. 2 2 2 dτ dx′ dy′ dz′ dx′ = + + + 2 v + v2 dt′ dt′ dt′ dt′ The constant speed of light in an isolated frame of refer- √ ence is well known, and the results are written directly dx′ above. = c2 + 2v + v2 ≠ c dt′ Now let’s calculate the two relative speeds of light dρ dτ ′ For the light propagating along the common x axis of dρ and dτ ′ . According to the transformation formula (2), the two inertial reference frames, the velocity of the light dξ d(x−vt) dx dζ dy dη dz ′ = = −v, = and = are obtained. measured on the reference frame′ Σ relative to the refer- dτ dt dt dτ dt √ dτ dt( ) ′ dx′ ± dy dz′ ( ) 2 ( ) ence frame Σ is dt′ = c, dt′ = 0 and dt′ = 0, while dr dx 2 dy dz 2 The isolated light speed dt = dt + dt + dt = the speed of the light measured on the reference frame Σ′ relative to the reference frame Σ is c on the reference frame Σ will be used in the calculation. Therefore, the results of the relative light velocity dρ of ′ dτ dρ ′ = |c ± v| (13) photons relative to the reference frame Σ measured on dτ ′ the reference frame Σ satisfy the relationship √( ) ( ) ( ) If the reference frame Σ is also a photon, v = ±c. There- 2 2 2 dρ dξ dς dη fore, the relative velocity of light propagating in the same = + + dτ dτ dτ dτ direction is 0 while that of light propagating in the op- √ posite direction is 2c measured on the reference frame ( )2 ( )2 ( )2 dx dy dz Σ. = − v + + dt dt dt Formulas (10) and (13) show that when the velocity (10) √ between two inertial reference frames is not zero, the two ( )2 ( )2 ( )2 dx dy dz dx relative light velocities are related to the relative veloc- = + + − 2 v + v2 dt dt dt dt ities of inertial reference frames. Obviously, the relative √ velocity of light propagating in the same direction is 0 dx = c2 − 2v + v2 ≠ c and that of light propagating in the opposite direction dt is 2c when observed in any inertial reference. Under the If the light propagates along the common x axis of the complete space-time transformation, the relative speed two inertial reference frames, that is, the velocity of light of light is variable, and the Einstein’s constant speed of relative to the reference frame Σ measured on the refer- light in isolated frame of reference is a necessary condi- dx ± dy dz tion for the relative variable speed of light. ence frame Σ is dt = c, dt = 0 and dt = 0, then the speed of light relative to the reference frame Σ′ measured on the reference frame Σ is, IV. COMPATIBILITY BETWEEN RELATIVE

dρ LIGHT SPEED AND SOLITARY LIGHT SPEED = |c ∓ v| (11) dτ Now we know that Einstein’s constant speed of light If the reference frame Σ′ is also a photon, v = ±c. There- hypothesis means that the speed of light in an isolated fore, the relative velocity of light propagating in the same frame of reference is constant, while the relative speed of direction is 0 while that of light propagating in the op- light is variable. The unitary principle requires that the posite direction is 2c measured on the reference frame result of the transformation of the relative speed of light Σ. in vacuum into the isolated speed of light in vacuum must Similarly, according to the transformation formula (3), ′ ′ ′ ′ ′ ′ be the constant c, otherwise it will constitute a negative dξ d(x +vt ) dx′ dζ dy dη dz′ ̸ dt′ = dt′ = dt′ + v, dτ ′ = dt′ and dτ ′ = dt′ are result of 1 = 1. Therefore, the constant Einstein speed 5 of light must also be able to be derived from the variable frame. The constant speed of light in isolated frame of relative speed of light. The proof is as follows. reference is compatible with the relative variable speed dx dξ dξ dy dς dς Relationships dt = dt + v = dτ + v, dt = dt = dτ of light, and they do not constitute a seemingly oppo- dz dη dη site negative basis on the surface. All these calculations and dt = dt = dτ are obtained from the complete in- verse space-time transformation (9). First, substituting are elementary, but elementary does not mean that the dr corresponding problems and conclusions are of little im- √theses relationships into the calculation formula dt = ( ) ( )2 ( ) portance. Often, the discovery of primary problems is dx 2 dy dz 2 dt + dt + dt of the speed in isolated refer- more difficult than the discovery of higher problems, and ∑ the impact of primary problems is more profound. ence frame( ) of , and( then) using relative light speed 2 2 ( ) 2 dρ dξ dς 2 dη 2 − dx 2 dτ = dτ + dτ + dτ = c 2v dt +v given in equation (10), we have V. CONCLUSIONS AND COMMENTS √( ) ( ) ( ) 2 2 2 dr dξ dς dη This paper introduces the unitary principle, which has = + v + + dt dτ dτ dτ a wide application prospect. Then, the space-time trans- √ formation of special relativity is tested by the unitary ( )2 ( )2 ( )2 dξ dς dη dξ principle, and the complete space-time transformation is = + + + 2v + v2 dτ dτ dτ dτ proposed. It is shown that the relative variable speed (14) √ of light is compatible with Einstein’s hypothesis of con- dx dξ = c2 − 2v + 2v + 2v2 stant speed of light. Einstein’s invariable speed of light dt dτ belongs to the constant speed of light in isolated refer- √ ( ) dξ dξ ence frame, that is, the speed of light in vacuum relative = c2 − 2v + v + 2v + 2v2 = c to that reference frame measured on any isolated inertial dτ dτ reference frame is c, which is the real meaning of constan- ′ ′ ′ dx′ dξ dξ dy dς′ t light speed. In fact, the concept of ”event” in special Similarly, relationships ′ = ′ −v = ′ −v, ′ = ′ = ′ ′ dt dt dτ dt dt relativity defines the characteristics of isolated reference dς′ dz′ dη dη dτ ′ and dt′ = dt′ = dτ ′ are obtained from the complete frame of space and time concepts used to describe phys- inverse space-time transformation (8). By substituting ical laws. The mapping relationship between space-time theses relationships( ) ( into) the( into) the calculation formula concepts of each isolated reference frame is determined ′ ′ 2 ′ 2 ′ 2 dr dx dy dz by the Lorentz transformation. From the laboratory ref- dt′ = dt′ + dt′ + dt′ of the speed in isolated ∑′ erence system, if the light moves in the opposite direction reference frame ( of ) , and( using) ( again) the relative light with other reference systems, the vacuum light speed rel- ′ 2 ′ 2 ′ 2 ′ 2 ′ dρ dξ dς dη 2 dx 2 speed ′ = ′ + ′ + ′ = c +2v ′ +v ative to that reference frame is greater than c; if the light dτ dτ dτ dτ dt moves in the same direction with other reference system- given by formula (12), the following result is obtained, s, the vacuum light speed relative to that reference frame √ ( ) ( ) ( ) is less than c. ′ ′ 2 ′ 2 ′ 2 dr dξ dς dη = − v + + The condition of an inertial reference frame is very dt′ dτ ′ dτ ′ dτ ′ √ special because there is no real inertial reference frame ( ) ( ) ( ) in nature. Even if the inertial reference frame conditions dξ′ 2 dς′ 2 dη′ 2 dξ′ = + + − 2v + v2 are satisfied, if the speed of a particle relative to other dτ ′ dτ ′ dτ ′ dτ ′ √ (15) inertial reference systems measured in the laboratory ref- dx′ dξ′ erence frame is greater than the speed of light in vacuum, = c2 + 2v − 2v + 2v2 the isolated speed of a particle measured on that inertial dt′ dτ ′ √ ( ) reference frame will always be less than the speed of light dξ′ dξ′ in vacuum. This is the essence of the speed limit of spe- = c2 + 2v − v − 2v + 2v2 = c dτ ′ dτ ′ cial relativity. The Lorentz transformation determines that the velocity of the isolated reference frame of mat- It can be seen that according to relative light speed in ter motion is less than that of light in vacuum. However, the complete space-time transformation, constant light this speed limit theory does not apply to relative velocity. speed in isolated reference frame can be derived. It can The observer cannot stand on photons, and the relative be said that the relative variable light speed is also a velocity of light to light in the vacuum observed in the necessary condition for constant light speed in isolated laboratory is actually distributed between 0 and 2c. It reference frame. should be stated again that the length of an object mea- The calculation results of (10)-(15) above show that sured in the same inertial frame is separable, that is, the there are definitions of constant solitary light speed and length of straight lines observed in all inertial reference variable relative light speed under the complete space- frame can be added or subtracted. In a broad sense, the time transformation. Einstein’s invariance of light speed position vectors between moving objects observed in the belongs to the constant speed of light in isolated reference same inertial frame obey the rule of vector superposition. 6

Because of this, if the relative variable speed of light is The complete space-time transformation based on the negated, the constant speed of light in Einstein’s isolated unitary principle solves the doubts caused by the invari- reference frame is also denied. However, there may be no able speed of light. However, the relationship between causal relationship between the relative variable speed of it and the laws of nature needs to be further revealed. light and the constant speed of light in an isolated refer- The unitary principle can be widely used in logic self ence frame, although the relative variable speed of light consistency test, so as to effectively solve those tangled and the constant speed of light in isolated reference frame logical problems, and will play a more important role in can be derived from each other in the sense of complete the future new scientific theory. space-time transformation.

[1] Nerlich, G. Special relativity is not based on causality. and slow light propagation in a room-temperature solid, The British Journal for the Philosophy of Science 33, American Association for the Advancement of Science, 361-388 (1982). 301, 200-202 (2003). [2] Garrison, J., Mitchell, M., Chiao, R. & Bolda, E. Super- [20] Bortman-Arbiv, D., Wilson-Gordon, A. & Friedmann, luminal signals: causal loop paradoxes revisited. Physics H. Phase control of group velocity: From subluminal to Letters A 245, 19-25 (1998). superluminal light propagation. Physical Review A 63, [3] Einstein, A. Relativity: The Special and the General 43818 (2001). Theory-100th Anniversary Edition. (Princeton Universi- [21] Kang, H., Hernandez, G. & Zhu, Y. Superluminal and ty Press, 2019). slow light propagation in cold atoms. Physical Review A [4] Michelson, A. A. ART. XXI. The relative motion of the 70, 11801 (2004). Earth and the Luminiferous ether. American Journal of [22] Bo, F., Zhang, G. & Xu, J. Transition between superlu- Science (1880-1910) 22, 120 (1881). minal and subluminal light propagation in photorefrac- [5] M¨uller,H., Herrmann, S., Braxmaier, C., Schiller, S. & tive Bi 12 SiO 20 crystals. Phys. Rev. A 64, 053809 Peters, A. Modern Michelson-Morley Experiment using (2001). Cryogenic Optical Resonators. Physical Review Letters [23] Sun, H. et al. Light propagation from subluminal to su- 91, 020401 (2003). perluminal in a three-level Λ-type system. Physics Letters [6] M¨uller,H et al. Tests of Relativity by Complementary A 335, 68-75 (2005). Rotating Michelson-Morley Experiments. Physical Re- [24] Salart, D., Baas, A., Branciard, C., Gisin, N. & Zbinden, view Letters 99, 50401 (2007). H. Testing the speed of ‘spooky action at a distance’. [7] Haugan, M. P. and Will, C. M. Modern tests of special Nature 454, 861-864 (2008). relativity. Phys. Today 40, 69 (1987). [25] Recami, E. How to recover causality in special relativity [8] Zhang, Y. Z. Special relativity and its experimental foun- for tachyons. Foundations of Physics (Historical Archive) dations. (World Scientific, 1997). 8, 329-340 (1978). [9] Antonini, P., Okhapkin, M., G kl? E. & Schiller, S. Test [26] Schwartz, C. Some improvements in the theory of faster- of constancy of speed of light with rotating cryogenic than-light particles. Physical Review D 25, 356-364 optical resonators. Physical Review A 71, 50101 (2005). (1982). [10] Magueijo, J. New varying speed of light theories. Reports [27] Gonzalez-Mestres, L. Cosmological implications of a pos- on Progress in Physics 66, 2025 (2003). sible class of particles able to travel faster than light. [11] Mansouri, R. and Sexl, R. U. A test theory of special rel- Nucl. Phys. B, Proc. Suppl. 48 (1996). ativity: I. Simultaneity and clock synchronization. Gen- [28] Everett, A. Tachyons, broken Lorentz invariance, and a eral relativity and Gravitation 8, 497 (1977). penetrable light barrier. Physical Review D 13, 785-794 [12] Mansouri, R. & Sexl, R. U. A test theory of special rel- (1976). ativity: III. Second-order tests. and [29] Pant, M., Bisht, P., Negi, O. & Rajput, B. Dirac spinors Gravitation 8, 809-814 (1977). and tachyon quantization. Canadian Journal of Physics [13] Will, C. M. Theory and experiment in gravitational 78, 303-315 (2000). physics. (Cambridge university press, 2018). [30] Milonni, P., Furuya, K. & Chiao, R. Quantum theory [14] Cl´ement, G. Does the Fizeau experiment really test spe- of superluminal pulse propagation. Phys. Rev 52, 1525 cial relativity? American Journal of Physics 48, 1059 (1995). (1980). [31] Chen, R. New Exact Solution of Dirac-Coulomb Equa- [15] Wang, L-J., Liu, N., Lin, Q. & Zhu, S. Superluminal tion with Exact Boundary Condition International Jour- propagation of light pulses: A result of interference. nal of Theoretical Physics 47 (2008) 881-890. Physical Review E 68, 66606 (2003). [32] Chen, R. Real Solution of Dirac Equation. arX- [16] Alv¨ager,T., Kreisler, M. Quest for faster-than-light par- iv:0908.4320 (physics.gen-ph, 2009). ticles. Physical Review 171, 1357-1361 (1968). [33] Dongfang, X. D. Com Quantum Theory, ResearchGate, [17] Parker, L. Faster-Than-Light Intertial Frames and the update will be persistent. Tachyons. Physical Review 188, 2287-2292 (1969). [34] Dongang, X. D. The Morbid Equation of Quantum Num- [18] Wang, L-J., Kuzmich, A. & Dogariu, A. Gain-assisted bers, ResearchGate, 2019. superluminal light propagation. Nature 406, 277-279 [35] Einstein, A. On the electrodynamics of moving bodies. (2000). Annalen der physik 17, 891-921 (1905). [19] Bigelow, M., Lepeshkin, N. & Boyd, R. Superluminal