Dynamic Medium of Reference: a New Theory of Gravitation Olivier Pignard

Dynamic medium of reference: A new theory of gravitation Olivier Pignard

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Olivier Pignard. Dynamic medium of reference: A new theory of gravitation. Physics Essays, Cenveo Publisher Services, 2019, 32 (4), pp.422-438. 10.4006/0836-1398-32.4.422]. hal-03130502

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Dynamic medium of reference: A new theory of gravitation Olivier Pignarda) 16 Boulevard du Docteur Cathelin, 91160 Longjumeau, France (Received 25 May 2019; accepted 22 August 2019; published online 1 October 2019) Abstract: The object of this article is to present a new theory based on the introduction of a nonmaterial medium which makes it possible to obtain a Preferred Frame of Reference (in the context of special relativity) or a Reference (in the context of general relativity), that is to say a dynamic medium of reference. The theory of the dynamic medium of reference is an extension of Lorentz–Poincare’s theory in the domain of gravitation in which instruments (clocks, rulers) are perturbed by gravitation and where only the measure of the speed of light always gives the same result. The presence of a massive body creates a centripetal flux of the medium, which has three fundamental effects: the dilatation of the period of material clocks, the contraction of the length of material rulers, and the slowdown of light. Thanks to the centripetal flux of the medium and these three effects, it is possible to find the correct expression of the deflection of a ray of light and the Shapiro delay. The dynamic medium of reference allows to establish a gravitational transformation and to find the fundamental equations of movement for light and matter. Hence, the theory of the dynamic medium of reference allows to find the main results of general relativity, but with important differences: The simultaneity is absolute, existence of the Preferred Frame of Reference, the physical reality is the universal present moment and not a global space-time (block-universe), light is slowed down by a gravitational field. VC 2019 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-32.4.422]

Resume: L’objet de cet article est de presenter une nouvelle theorie basee sur l’introduction d’un milieu non-materiel qui permet d’obtenir un Referentiel Privilegie (dans le contexte de la relativite restreinte) ou une Reference (dans le contexte de la relativitegenerale), c’est-a-dire un milieu dynamique de reference. La theorie du milieu dynamique de reference est une extension de la theorie de Lorentz–Poincare au domaine de la gravitation dans laquelle les instruments (horloges, re`gles) sont perturbees par la gravitation et seule la mesure de la vitesse de la lumie`re donne le même resultat. La presence d’un corps massif cree un flux centripe`te du milieu ce qui gene`re trois effets: La dilatation de la periode des horloges materielles; - La contraction de la longueur des re`gles materielles; et - Le ralentissement de la lumie`re. Grâce au flux centripe`te du milieu et a ces trois effets, il est possible de trouver l’expression correcte de la deflection d’un rayon de lumie`re et de l’effet Shapiro. Le milieu dynamique de reference permet d’etablir une transformation gravitationnelle et de trouver les equations fondamentales du mouvement pour la lumie`re et la matie`re. Ainsi, la theorie du milieu dynamique de reference permet de trouver les principaux resultats de la relativite generale, mais avec des differences importantes: La simultaneite est absolue, le Referentiel Privilegie existe, la realite physique est le moment present universel et non pas un espace-temps global (univers bloc), la lumie`re est ralentie par un champ gravitationnel.

Key words: General Relativity; Preferred Frame of Reference; Dynamic Medium of Reference; Lorentz–Poincare’s Theory; Simultaneity; Speed of Light.

I. INTRODUCTION 15 pages on Lorentz’s theory in his book5 showing that this theory is perfectly acceptable. Up to now, general relativity is the best existing theory The present article is the presentation of a theory that is of gravitation and it provides very accurate values to many an extension of Lorentz–Poincare’s theory in the domain of experiments and observations.1–3 gravitation. Lorentz–Poincare’s theory competes with Einstein’s spe- The proposed theory is based on the following concepts: cial relativity. Although contested by some, Lorentz’s theory is perfectly coherent. It has been defended by Henri • Simultaneity is an absolute notion; 4 Poincare and Michel Lambert dedicates a whole chapter of • Existence of a dynamic medium of reference; • Existence of a reference time or privileged time, that is to say a universal present moment rather than a block- a)olivier_paciﬁ[email protected] universe containing all past, present, and future events;

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• Centripetal flux of the medium created by a massive body; The last two points explain that a ray of light is deflected and by a massive body. • The contraction of rulers and the dilatation of the periods of the clocks are physical effects due to the movement of The presence of a massive body creates a flux of the rulers and clocks with regard to the medium or due to the medium (centripetal that is to say, directed toward the center movement of the medium created by a massive body. of gravity of the massive body) of speed rffiffiffiffiffiffiffiffiffiffi 2GM Several renowned scientists have defended or defend Vflux ¼ ; (1) most of these concepts. r Of course, Lorentz and Poincare maintained their inter- and acceleration pretation confronting Einstein’s special relativity. John Bell also maintained the idea of returning to a prerelativistic the- GM c ¼ ; (2) ory, close to Lorentz’s theory and the fact that the medium flux r2 of propagation of light had been rejected based on erroneous arguments.6,7 where r refers to the distance to the center of gravity of the Finally, in his book “Time Reborn,”8 Lee Smolin claims massive body. the following arguments: In the theory of the dynamic medium of reference, the factor “The fact that it is always some moment in our perception, and that we experience that moment as 2GM 1=2 KðrÞ¼ 1 (3) one of a flow of moments, is not an illusion. It is c2r the best clue we have to fundamental reality. 0 plays a role quite equivalent to the factor This means giving up the relativity of simultaneity ! 1=2 and embracing its opposite: that there is a V2 preferred global notion of time.” cðVÞ¼ 1 2 : (4) c0 All these arguments bring support to the theory which will be developed in the present paper. • The material rulers are contracted by the factor c(V) due to After the present introduction, Section II describes the their movement with regard to the dynamic medium of ref- fundamental characteristics of this medium. Section III pro- erence and they are contracted by the factor K(r) due to a vides three applications of the proposed theory correspond- gravitational field (centripetal movement of the medium ing to well-known tests of general relativity. Section IV created by a massive body); gives the fundamental equations of movement in the theory • The material clocks have their period dilated by a factor of the dynamic medium of reference. Finally, the conclusion c(V) due to their movement with regard to the dynamic highlights the differences between this new theory and medium of reference and they have their period dilated by Einstein’s relativity. the factor K(r) due to a gravitational field (centripetal movement of the medium created by a massive body). II. THEORY OF THE DYNAMIC MEDIUM OF REFERENCE However, there is a great difference between the two cases: A. Presentation of the dynamic medium of reference • The proposed theory introduces a dynamic nonmaterial In the case of the movement of a material ruler or clock, it medium which is present in the whole Universe. The charac- is the movement of the ruler or the clock with regard to the teristics of this medium are: Preferred Frame of Reference that creates the effect. The medium is not distorted. • This medium enables one to deduce a Preferred Frame of • In the case of a gravitational field created by a massive Reference or rather a REFERENCE in the whole Universe body, the medium undergoes a centripetal flux increasing and at all scales, when one gets closer to the massive body. This flux, as a • This REFERENCE enables one to obtain a privileged result, physically alters the length of the material rulers time. The present moment is universal, that is to say the and the period of the material clocks. Another important same in the whole Universe, point: as the medium is being physically altered by the • This medium is also the medium of propagation of light, presence of the massive body, this alteration plays the role • This medium verifies the principle of reciprocal action: of the curvature of space-time of general relativity. ᭺ The medium is distorted by matter and energy like the space-time of general relativity, In the case of a gravitational field, for a fix referential ᭺ The warping of this medium determines the with regard to the massive body, the Preferred Frame of trajectories of the particles (material particles and Reference movespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at the same speed as a laboratory in free light particles). fall VPFR ¼ 2GM=r. 424 Physics Essays 32, 4 (2019) 1=2 The material clocks and the material rulers are then physi- L0 2GM 2 2 1=2 L ¼ with KðrÞ¼ 1 2 : (7) cally affected with the factor cðVPFRÞ¼ð1 ðVPFR=c0ÞÞ KðrÞ c0 r which gives the factor of general relativity 2GM 1=2 D. Speed of light in the presence of a gravitational KðrÞ¼ 1 : (5) field c2r 0 1. Speed of light for a radial trajectory From the two previous points, we can deduce the funda- This parallel between special relativity and gravitation is mental result: very important and justifies the claim that, for all dynamic phenomena in a gravitational field, the part due to the tempo- The speed of light varies according to the distance ral effect is exactly the same as the part due to the spatial from the center of gravity of a massive body. effect. Indeed, very far from the massive body, the speed of This is due to the fact that in special relativity, the factor light is used for the temporal part (clocks) is the same as the one used in the spatial part (rulers). L0 c0 ¼ : (8) Remark: The centripetal flux of the medium is linked to T0 the fundamental notion of the free fall referential of general relativity. This concept of free fall referential is so essential that it On the other hand, at the distance r from the center of is systematically used by Clifford Will in his book “Was Ein- gravity of the massive body, when light follows a radial tra- stein right?”2 to explain the following phenomena in the jectory, the speed of light actually has the following expres- presence of a gravitational field: sion with regard to a referential linked to the massive body: • Shifting of the clocks (Chapter 3), L L =KðrÞ c 2GM c ¼ ¼ 0 ¼ 0 ¼ c 1 : (9) • 2 0 2 Shifting toward the red (redshift) of the light (Chapter 3), T T0:KðrÞ K ðrÞ c0 r • Deflection of the light (Chapter 4), • Shapiro delay (Chapter 6). In the frame of radial trajectories, we can deduce an index of refraction of the medium due to gravitation Also, in their book “RelativiteGenerale,” Julien Grain 9 c 1 and Aurelien Barrau often use the free fall referential: nðrÞ¼ 0 ¼ : (10) cðrÞ 2GM 1 “In a free fall referential, the correct laws must be c2:r the ones satisfying special relativity.” 0

“From a geometric point of view, geodetics are the Light is therefore slower close to a massive body shorter lines between events of space-time. Physi- than very far from it. cally, they are the trajectories of the test-masses in This notable result allows one to find a good approxima- free fall.” tion of the Shapiro delay due to a massive body. Another major point is that the measure of the speed of light results in the same value whatever distance we are B. Material clock in a gravitational field from a massive body. As a result of what we have seen in Section II A, a mate- Indeed, if we own a ruler of length L and a clock of rial clock, of period T0 when situated very far from all gravi- period T, in such a way that the light covers the length of the tational fields, undergoes a dilatation of its period of a factor ruler in a period of the clock, whatever the strength of the K(r) when situated at the distance r from the center of grav- gravitational field we measure ity of a massive body, in such a way that its period is equal L c ¼ : (11) to T 2GM 1=2 T ¼ T0 KðrÞ with KðrÞ¼ 1 2 : (6) c0 r In fact, whether we are very far from all massive bodies (absence of gravitational field) or very close to the surface of a massive body, the light covers the length of the same ruler C. Material ruler in a gravitational field during the period of the same clock. The measure provides an identical result of the speed of Likewise, a material ruler, of length L0 when situated light, so apparently that light has a constant speed. very far from all gravitational fields, undergoes a contraction 2. Speed of light in a general case of a factor K(r) when it is situated at the distance r from the center of gravity of a massive body, in such a way that its The expression of the speed of light found in Section length is equal to II D 1 is only valid for a radial trajectory. Physics Essays 32, 4 (2019) 425

For any kind of trajectory of light, the expression of E. The three fundamental effects of the theory of the speed is more complex and we are going to establish it. dynamic medium of reference The speed vector of light must be decomposed along The three fundamental effects due to gravitation pre- two components which do not follow the same rule because dicted by the theory of the dynamic medium of reference are the rulers are contracted by the factor K(r) only when they the following: are disposed along a radial of the massive body. The radial component can be written • (1a) dilatation of the period of the clocks: T ¼ T0 K, • (1b) contraction of the length of the rulers: L ¼ L0=K, L== L0== =KðrÞ c0== k • (2) distortion of the dynamic medium of reference:pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The grav- c== ðrÞ¼ ¼ ¼ 2 ¼ c0== : 1 : T T0 KðrÞ K ðrÞ r itation creates centripetal flux of speed Vflux ¼ 2GM=r. (12) The two first effects have been noted (1a) and (1b) The ortho radial component can be written because: rffiffiffiffiffiffiffiffiffiffiffi • L? L0? c0? k They are quite equivalent to the dilatation of durations and c?ðrÞ¼ ¼ ¼ ¼ c0? 1 ; (13) T T0 KðrÞ KðrÞ r contraction of lengths of special relativity, • For the light, the two cumulated effects imply the effect where (1c): diminution of the speed of light in a gravitational field according to the rule = k 1 2 2GM KðrÞ¼ 1 and k ¼ : (14) c0 r c2 cðrÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 0 K 1 þðK2 1Þ cos2b

Moreover, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The third effect (2) is specific to the domain of gravitation. 2 2 c0 ¼ c0== þ c0?: (15) Thus, it is an additional effect compared to special relativity (which corresponds to a nondistorted medium).

! ! We name b ¼ðu== ; c Þ the angle between the vector F. New postulates of the theory of the dynamic ! ! medium of reference u== and the speed vector of light c . ! The vector c0 represents the speed vector of light if there In the context of the theory of the dynamic medium of was not any massive body. reference, the two Einstein’s postulates become: We can write () (1) The measure of the speed of light is constant in all c ðrÞ¼cðrÞ cos b referentials and in all directions, including in the pres- == which gives us: c ðrÞ¼cðrÞ sin b ence of a gravitational field. ()? This implies that all instruments, in particular, the c ¼ K2c ðrÞ¼K2 cðrÞ cos b material clocks and the material rulers undergo physi- 0== == : (16) cal effects due to their movement or the presence of a c0? ¼ K c?ðrÞ¼K cðrÞ sin b gravitational field in order to ensure a constant measure of the speed of light. We also have (2) All the inertial/Galilean referentials are equivalent for the expression of physical laws and there is a Preferred qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Frame of Reference or medium of reference in which 2 2 2 2 c0 ¼ c0== þ c0? ¼ K cðrÞ ðK: cos bÞ þ sin b: the light really propagates at the speed c0 and the mate- (17) rial clocks and rulers do not undergo any physical effect of dilatation of their period or contraction of their length. This Preferred Frame of Reference or Thus, we obtain the modulus of the speed vector of light medium of reference makes it possible to define an absolute simultaneity and a privileged time every- c cðrÞ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 where in the Universe. K ðK cos bÞ2 þ sin2b Important remark: In this theory, it is not question to which can also be written come back to an absolute time as in Newton’s theory where c0 all the clocks of the Universe beat in unison whatever their cðrÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi or even: K 1 þðK2 1Þ cos2b speed and whatever the gravitational field. rffiffiffiffiffiffiffiffiffiffiffi 1=2 In contrast, this theory proposes that all the material k k 2 clocks (of same building) in all the parts of the Universe beat cðrÞ¼c0 1 1 þ cos b : (18) r r k at exactly the same rhythm if and only if they are immobile 426 Physics Essays 32, 4 (2019) with regard to the dynamic medium of reference. All these According to Lorentz–Poincare’s theory, this material clocks provide the universal time of reference. clock undergoes a physical dilatation of its period according to the formula ! 1=2 2 1=2 G. Discussion on the principle of equivalence V=PFR 2GM T ¼ c T with c ¼ 1 ¼ 1 : 0 c2 c2 r This part presents three objectives: 0 0 (19) • to show that the principle of equivalence such as stated by Einstein in 1907 and 1911 can be used in the frame of Lorentz–Poincare’s theory in order to extend it to the T0 indicates the period that the clock would have if it domain of gravitation, was immobile with regard to the Preferred Frame of • to provide a new formulation of the principle of equiva- Reference. lence in the frame of the theory of the dynamic medium of The equivalence principle10 allows us to claim that this reference and to show that it is sufficient to postulate the clock undergoes the same effect being in the presence of a existence of the dynamic medium of reference to demon- massive body of mass M creating a gravitational field of strate the principle of equivalence, and acceleration C ¼ðGM=r2Þ, that is to say, that its period • to show that the formulation of the principle of equiva- undergoes a physical dilatation according to the formula lence in the frame of the theory of the dynamic medium of 1=2 reference allows to establish that the inertia force due to a 2GM T ¼ KðrÞT0 with KðrÞ¼ 1 2 : (20) variation of the speed vector of a material object is funda- c0 r mentally of the same nature that the gravitational force.

T0 indicates the period that the clock would have in the absence of gravitational field. 1. Use of the principle of equivalence in the frame of 10 Lorentz–Poincare’s theory The equivalence principle allows us to state that a material ruler undergoes exactly the same effect in the two 10 The equivalence principle allows us to state that a following cases: material clock undergoes exactly the same effect in the two following cases: • The ruler moves with an acceleration C and with the speed V with regard to the Preferred Frame of Reference. • The clock moves with an acceleration C and with the speed According to Lorentz–Poincare’s theory, the material ruler V with regard to the Preferred Frame of Reference. undergoes a physical, real contraction of its length, According to Lorentz–Poincare’s theory, the material • The ruler is fixed with regard to a massive body of mass M clock undergoes a physical, real dilatation of its period, creating a gravitational field of acceleration C. According • The clock is fixed with regard to a massive body of mass to the proposed interpretation, which is an extension of M creating a gravitational field of acceleration C. Accord- Lorentz–Poincare’s theory in the domain of gravitation, ing to the proposed interpretation, which is an extension of the material ruler undergoes a physical, real contraction of Lorentz–Poincare’s theory in the domain of gravitation, its length. the material clock undergoes a physical, real dilatation of its period. We can state the following important result: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For a ruler moving with the speed V=PFR ¼ 2GM=r First, we arep goingffiffiffiffiffiffiffiffiffiffi to show that an object moving at the with regard to the Preferred Frame of Reference PFR, this 2 speed V=R ¼ 2A=r with regard to a referential R (r indi- ruler undergoes the acceleration C=PFR ¼ðGM=r Þ. cates the distance from the origin O of the referential R According to Lorentz–Poincare’s theory, this material to the position of the object) undergoes an acceleration ruler undergoes a physical contraction of its length according 2 C=R ¼ðA=r Þ (the signs are given with regard to a unit to the formula ! vector ur going from the origin O toward the position of the ! 1=2 object). 2 1=2 pffiffiffiffiffiffiffiffiffiffi V=PFR 2GM = L ¼ L0=c with c ¼ 1 ¼ 1 : Starting from the speed V ¼ 2A r, the acceleration c2 c2:r can be written 0 0 ! (21) pffiffiffiffiffiffi rffiffiffiffiffiffi dV dV dr dV 1 2A 2A A C V : L indicates the length that the ruler would have if it ¼ ¼ ¼ ¼ 3=2 ¼ 2 0 dt dr dt dr 2 r r r was immobile with regard to the Preferred Frame of Reference. The equivalence principle10 allows us to claim that this So we can state the following important result: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ruler undergoes the same effect being in the presence of a For a clock moving with the speed V=PFR ¼ 2GM=r massive body of mass M creating a gravitational field of with regard to the Preferred Frame of Reference PFR, this acceleration C ¼ðGM=r2Þ, that is to say, that its length 2 clock undergoes the acceleration C=PFR ¼ðGM=r Þ. undergoes a physical contraction according to the formula Physics Essays 32, 4 (2019) 427 2GM 1=2 A. Redshift of light L ¼ L0=KðrÞ with KðrÞ¼ 1 2 : (22) c0:r The redshift or Einstein’s effect is the frequential shifting of light or an electromagnetic signal due to a gravitational field. L0 indicates the length that the ruler would have in the This effect can be considered as a “corollary” of the absence of gravitational field. slowdown of the clocks due to a gravitational field. Conclusion: The principle of equivalence such as stated Indeed, if we consider two clocks, one located at a dis- by Einstein in 1907 and 1911 is sufficient to establish the tance r1 and the other at a distance r2 from the center of grav- laws concerning the clocks and the rulers in a gravitational ity of a spherical body of mass M, the ratio between the field and so to establish the speed of light in a gravitational periods of the two clocks is given by the following formula: field and finally to extend the Lorentz–Poincare’s theory to the domain of gravitation. 1=2 T2 Kðr2Þ 2GM ¼ with KðrÞ¼ 1 2 : (23) T1 Kðr1Þ c0:r 2. New formulation of the principle of equivalence in the theory of the dynamic medium of reference Subsequently, it is sufficient to convert the periods at the In the frame of the theory of the dynamic medium of ref- points of emission and reception of light (or of the electro- erence, the principle of equivalence is formulated in terms of magnetic signal) in frequencies. A signal emitted at the fre- speed and not acceleration and the two following statements quency F at the distance r of the center of the spherical are naturally equivalent: 1 1 body is received at the distance r2 at the frequency F2 • The effects on material clocks and rulers are due to their according to the formula: movement with regard to the medium of reference, 1=2 • The effects on material clocks and rulers in the presence of F1 T2 Kðr2Þ 2GM ¼ ¼ with KðrÞ¼ 1 2 : (24) a massive body are due to the movement of the medium F2 T1 Kðr1Þ c0:r with regard to the massive body and so with regard to the clocks and the rulers. The proposed theory allows one to find the famous for- This formulation is a justification of the principle of mula of the gravitational redshift. equivalence because it is obvious that the relative movement Remark: If the distance r2 is very large (“infinite”), it of the clocks and the rulers with regard to the medium is follows that K(r2) ¼ 1 and F2 ¼ F0. Therefore, the previous equivalent to the relative movement of the medium with formula can be written: F1=F0 ¼ T0=T1 ¼ 1=Kðr1Þ or regard to the clocks and the rulers. F1 ¼ F0=Kðr1Þ. In every case, it is the relative movement (speed) of the Therefore, at the surface of the Sun, the atoms will emit material clock and ruler with regard to the medium (which is photons with a frequency F1 smaller by a factor K(r1) with the reference) which creates the effects (dilatation of the regard to F0 which would be the frequency of the photons period of the material clock and contraction of the length of emitted by the same atoms without the presence of a gravita- the material ruler). tional field.

B. Deflection of a ray of light 3. Inertia force 1. Snell-Descartes law In the theory of the dynamic medium of reference, the It has been known for many centuries that light is force of gravitation is due to the movement of the medium deflected when passing from one medium to another, which created by a massive body, which means that the force of is called refraction. gravitation is due to a variation of the speed vector of a mate- If the first medium possesses a refraction index n1, the rial body with regard to the medium. second medium an index n2, then the link between the inci- Thanks to the principle of equivalence established in the dence angle i and the refracted angle r is the famous Snell- frame of the dynamic medium of reference, it is possible to Descartes law which can be written: n1: sin ðiÞ¼n2: sin ðrÞ. state that, without gravitation field, all modification of the If we use the two expressions c1 ¼ c0=n1 and speed vector of a material object with regard to the medium c2 ¼ c0=n2, Snell-Descartes law can be written of reference creates a force: It is the inertia force. sin ðiÞ sin ðrÞ ¼ : (25) c c III. APPLICATION OF THE THEORY OF THE DYNAMIC 1 2 MEDIUM OF REFERENCE TO THREE TESTS OF If the speed c is smaller than the speed c , then the GENERAL RELATIVITY 2 1 angle of refraction is smaller than the incidence angle. This section details three applications of the theory of We can deduce that light is deflected in the direction the dynamic medium of reference corresponding to three where the speed of light is the smallest and we can thereby well-known tests of general relativity. establish the following rule: 428 Physics Essays 32, 4 (2019)

A massive body has the effect of slowing down light 3. Total angle of deflection of a ray of light when it gets closer to its center of gravity and of deflecting a. The part due to the slowdown of the light the ray of light oriented toward the massive body. From Eq. (18), we can obtain the approached expression of 2. Huygens–Fresnel’s approach the speed of light ÀÁ Using the Huygens–Fresnel’s approach, the elementary k cðrÞc 1 1 þ cos2b : (30) angle of deflection of the ray of light is given by the follow- 0 2r ing formula: da ¼ð@c=@sÞdt ¼ð@c=@sÞðds=c0Þ, where s represents the curvilinear abscissa covered by the ray of light, ds an ele- At an infinite distance of the massive body we have: mentary part of this curvilinear abscissa, and dt ¼ ds=c0. cðr1Þ¼c0. At a given point M on the path of the ray of light, we At the minimal distance of the massive body r ¼ rmin we ! ! call u the angle between the vector ds and the vector dr. have b p=2 which shows that We therefore have: dr ¼ cosu:ds and thus k ds ¼ dr= cosu and @c=@s ¼ð@c=@rÞ cosu. cðrminÞc0 1 : By taking into account this link between ds and dr,we 2rmin can write the elementary angle of deflection of the ray of light in the following way: c0 cðrminÞ By using the relation a ¼ 2 we immediately c0 1 @c 1 @c obtain the angle of deflection da ¼ ds ¼ dr (26) c0 @s c0 @r k 2GM a ¼ ¼ 2 : (31) for a ray of light moving away from the massive body, that rmin c0:rmin is to say, for which ð@c=@rÞ > 0. The angle of deflection of the ray of light due to the By using the relation a ¼ 2lnðc =cðr ÞÞ we obtain slower speed of light is therefore given by the formula 0 min k k 2GM rðmin þ1ð a ¼2ln 1 ¼ 2 : (32) 1 @c 1 @c 2rmin rmin c :rmin a ¼ a1 þ a2 with a1 ¼ dr and a2 ¼ dr: 0 c0 @r c0 @r 1 rmin

Here, rmin represents the distance between the trajectory of b. The part due to the distortion of the medium the light and the center of gravity of the massive body when The part due to the distortion of the medium is caused by the the light is closest to the massive body. fact that the medium of propagation of light undergoes a cen- Ð 1 @c 1 cð1Þcðr Þ tripetal flux in the presence of a massive body. We have: a ¼ þ1 dr¼ ½cðrÞþ1 ¼ min 2 r rmin min c0 @r c0 c0 This centripetal flux is even faster as we get closer to the c0 cðrminÞ ¼ because cð1Þ¼c0. massive body and its acceleration is expressed at the first c0 approximation by the following formula: For obvious reasons of symmetry, we have a1 ¼ a2 (which is confirmed by the calculus of a1) so finally we GM obtain c ¼ : r2 c cðr Þ a ¼ 2 0 min : (27) c 0 The medium of propagation of light undergoing a cen- Remark: tripetal flux, obviously this impacts a ray of light moving in A more accurate expression of the elementary angle of its medium of propagation and this bends its trajectory in the deflection of the ray of light is the following: direction of the center of gravity of the massive body. The fundamental law is that a ray of light maintains its 1 @c 1 @c da ¼ ds ¼ dr: (28) direction in a referential linked to its medium, that is to say, c @s c @r the Preferred Frame of Reference (we should say, the Reference since this referential is distorted seen by a refer- We therefore obtain the angle of deflection of the ray of ential linked to the massive body). light due to the slowing down of speed of light This implies that in a referential linked to the massive rð1 body, for an elementary moving of the ray of 1 @c a ¼ 2 dr ¼ 2ln½ðÞcðrÞ þ1 that is to say light dL ¼ c0:dt, the Preferred Frame of Reference moved rmin c @r in a centripetal way of the length dLPFR ¼ VPFR:dt with rmin V ¼ c :dt and c ¼ GM=r2. c PFR PFR PFR a ¼ 2ln 0 : (29) For an almost rectilinear ray of light that we choose par- cðrminÞ allel to the axis of the abscissa in a landmark where the Physics Essays 32, 4 (2019) 429 origin is placed at the center of gravity of the massive body, ð þ1ð þ1ð GMR dx GMR dx it undergoes an elementary deflection of an angle da given a ¼ da ¼ ¼ 2 3 2 2 2 3=2 c0 r c0 ðÞx þ y by the following formula: 1 1 tan ðdaÞ¼ðdLPFR: cos ðp=2 uÞ=dLÞ by taking (r, u) þ1ð the polar coordinates of the current point M and (x, y) its GMR dx : Cartesian coordinates shown by the scheme below (Fig. 1). 2 3=2 c0 ðÞx2 þ R2 The term cos ððp=2ÞuÞ comes from the fact that the 1 ray of light follows a trajectory almost parallel to the axis of the abscissa whereas the centripetal flux of the medium is on By putting u ¼ x=R we have the radial of the massive body at the current point M. The green rectangle symbolizes a free fall referential þ1ð þ1ð GMR R du GM du (for example, a laboratory or an elevator in free fall with a a ¼ 2 3=2 ¼ 2 3=2 centripetal movement toward the center of gravity of the c 2 c R 2 0 ðR uÞ þ R2 0 ðÞu þ 1 massive body). It also symbolizes a volume of the medium 1 1 þ1ð that undergoes a centripetal flux. 2GM du Let us pursue the calculus of the elementary angle of ¼ : c2R 2 3=2 deflection 0 ðÞu þ 1 0 p (34) dL cos u PFR 2 tanðdaÞ¼ dL We therefore use the well-known result p p Ð pffiffiffiffiffiffiffiffiffiffiffiffiffi V dt cos u V cos u þ1 du=ðu2 þ 1Þ3=2 ¼½u= u2 þ 1þ1 ¼ 1 which allows us PFR 2 PFR 2 0 0 ¼ ¼ to finally obtain the angle of deflection of the ray of light due c0:dt c0 to the distortion of the medium generating a centripetal flux p c dt cos u PFR 2 2GM ¼ : a ¼ : (35) c flux 2 0 c0R

2 tan ðdaÞ¼ðGM=r Þðsin ðuÞ=c0Þdt with dL ¼ c0 dt dx c. Total angle of deflection due to the slowing down of and sin ðuÞ¼y=r R=r where R refers to the distance from the speed of light and the flux of the medium the closest point of the trajectory of the ray of light to the center The total angle of deflection is the sum of the angle due to of gravity of the massive body. the slowing down of the speed of light and the angle due to Whence finally the expression of the elementary angle the centripetal flux of the medium GMR dx GMR dx 2GM 2GM 4GM da atan : (33) a ¼ a þ a ¼ þ ¼ : (36) c2 r3 c2 r3 total speed flux 2 2 2 0 0 c0rmin c0R c0rmin

By assuming that the ray of light comes from the infinite and goes to the infinite along the axis of abscissa, the total We find once again the famous formula of general angle of deflection is given by the following formula: relativity.

FIG. 1. Deﬂection of a ray of light by the Sun. 430 Physics Essays 32, 4 (2019)

Remark: By ignoring k with regard to r1 and r2, we finally obtain Still in the context of the radial case, by using the formula 2GM r2 n1: sin ðiÞ¼n2: sin ðrÞ or the formula sin ðiÞ=c1 ¼ sin ðrÞ=c2 Dt ¼ DtNewton þ 3 ln : (40) on successive elementary portions of the ray of light in a simu- c0 r1 lation, we obtain the exact value of the angle of deflection of the ray of light a ¼ 1.75 s of arc in the case of a ray of light approaching the Sun’s surface. Example: Case of a photon going from the surface of the Sun of radius RS and arriving at the surface of the Earth of C. Shapiro effect radius RT, by taking into account only the gravitational field due to the Sun (D represents the distance Sun–Earth taken 1. Radial Shapiro effect ST with regard to the centers of gravity) We study the Shapiro Effect on a radial of a massive body of mass M and center of gravity O. Since the trajectory of the light is radial, we will use the 2GM DST RT Dt ¼ DtNewton þ 3 ln : (41) result established in Section II D 1 showing that light propa- c0 RS gates more slowly near a massive body with a speed repre- sented as 2. Non radial Shapiro effect k 2GM c c 1 with k : (37) ¼ 0 ¼ 2 In this case, we consider the path of a ray of light going r c0 from point M1 of Cartesian coordinates (x1, y1) to point M2 of Cartesian coordinates (x2, y2) (Fig. 2). We consider the time taken by light to travel from point The time taken by light to go from point M1 to point M2 M1 (r1 ¼ OM1) to another point M2 (r2 ¼ OM2), the three is points O, M1, M2 being aligned in this order Mð2

ðr2 ðr2 ðr2 dx dr 1 dr 1 r Dt ¼ : (42) Dt ¼ ¼ ¼ dr cðrÞ k M1 cðrÞ c0 1 r c0 r k r1 r1 r1

ðr2 ðr2 ðr2 1 r k þ k 1 k dr The speed of light is approximately expressed as ¼ dr ¼ dr þ ; (38) c0 r k c0 c0 r k r1 r1 r1 k ÀÁ cðrÞc 1 1 þ cos2b which gives us 0 2r ÀÁ r r k k 2 Dt ¼ 2 1 þ ½ln ðr kÞ r2 that is to say ¼ c0 1 1 þ cos ðÞu þ a r1 2r c0 c0 ÀÁ r2 r1 k r2 k k 2 Dt ¼ þ ln : (39) c0 1 1 þ cos u : (43) c0 c0 r1 k 2r

We immediately recognize the ﬁrst term which is the By using the formula cos u ¼ x=r we have 2 3 duration in the Newtonian case DtNewton ¼ðr2 r1Þ=c0. cðrÞc0½1 ðk=2rÞðkx =2r Þ and therefore

FIG. 2. Slowdown of light due to the Sun. Physics Essays 32, 4 (2019) 431

Mð2 xð2 ÀÁ Finally, we have dx 1 k 2 xð2 Dt ¼ 1 þ 1 þ cos u dx k ÀÁ cðrÞ c0 2r Dt Dt þ 2f 0ðxÞg0ðxÞ dx M x Newton 1 1 2c0 x xð2 xð2 xð2 h1 i 1 k dx k x2 k : ¼ dx þ þ 3 dx: (44) DtNewton þ 2f ðx2Þ2:f ðx1Þgðx2Þþgðx1Þ c0 2c0 r 2c0 r 2c0 x1 x1 x1 Ð k x2 þ r2 x2 x1 x2 Dt DtNewton þ 2 ln þ Dt ¼ 1=c dx ¼ðx x Þ=c represents the 2c0 x1 þ r1 r2 r1 Newton 0 x1 2 1 0 duration of the travel of the ray of light in the Newtonian (45) case. This formula can"# also be written: ! If we define f ðxÞ¼ ln ðx þ rÞ then we have k ðr þ x Þ:ðr x Þ x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt Dt þ 2: ln 2 2 1 1 2 þ 1 . 2x Newton 2c y2 r r dr d x2 þ y2 1 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 2 1 df ðxÞ 1 þ 1þ 2 x2 þ y2 By using once again the relation cos u ¼ x=r we finally f 0ðxÞ¼ ¼ dx ¼ dx ¼ dx x þ r x þ r x þ r obtain x 1 þ 1 Dt DtNewton ¼ r ¼ : x þ r r k r1:r2 þ 2: ln 2 ðÞ1 cos u1 :ðÞ1 þ cos u2 2c0 y1 By writing gðxÞ¼x=r we have: 0 dr=dx 2 3 þ cos u cos u : (46) g ðxÞ¼ð1=rÞx r2 ¼ð1=rÞðx =r Þ. 1 2

TABLE I. Sum up of the results of the proposed theory and comparison with general relativity.

General relativity Theory of the dynamic medium of reference

A clock is affected by a gravitational ﬁeld because of the curvature A material clock is affected by the centripetal ﬂux of the dynamic of space-time itself medium of reference 2GM 1=2 2GM 1=2 T ¼ T1:KrðÞwith KrðÞ¼ 1 2 T ¼ T1:KrðÞwith KrðÞ¼ 1 2 c0:r c0:r

A ruler is affected by a gravitational ﬁeld because of the curvature A material ruler is affected by the centripetal ﬂux of the dynamic of space-time itself medium of reference 1=2 1=2 L1 2GM L1 2GM L ¼ with KrðÞ¼ 1 2 L ¼ with KrðÞ¼ 1 2 KrðÞ c0:r KrðÞ c0:r

Invariance of the speed of light The measure of the speed of light is constant, but light is really slowed down by a gravitational ﬁeld

c Radial case: c ¼ 0 K2ðÞr c General case: c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 K 1 þ ðÞK2 1 cos2b

Frequency shift for a photon emitted at the surface of a massive Frequency shift for a photon emitted at the surface of a massive body due to the curvature of space-time body due to the dynamic medium of reference F F F ¼ 1 F ¼ 1 KrðÞ KrðÞ

Deﬂection of light due to the curvature of space-time Deﬂection of light due to the dynamic medium of reference 4GM 4GM a ¼ 2 a ¼ 2 c0:r c0:r

Shapiro delay due to a longer path because of the curvature of Shapiro delay due to the slowdown of light by a gravitational ﬁeld space-time 2GM r2 2GM r2 Radial case: Dt ¼ DtNewton þ 3 ln Radial case: Dt ¼ DtNewton þ 3 ln c0 r1 c0 r1 2GM r1:r2 2GM r1:r2 General case: Dt DtNewton þ 3 ln 4 2 1 General case: Dt DtNewton þ 3 ln 4 2 1 c0 rmin c0 rmin 432 Physics Essays 32, 4 (2019)

Case where u1 p and u2 0 (that is to say, that is to say cos u1 1 and cos u2 1). By observing that y r we have 1 min dr dr 2 2 2 : k r1 r2 V== ¼ K V== N or V== N ¼ K V== or ¼ K dt N dt Dt DtNewton þ ln 4 2 1 : (47) c0 rmin (49)

LðrÞ=L1 • 1 D. Sum up of the obtained results and comparison Ortho radial speed: V ¼ V N ¼ V N 1=2 ? ? ? 1 k 1 TðrÞ=T1 ð rÞ with general relativity ¼ V?NK Table I sums up the main results of the Sections II and that is to say III. 1 d/ d/ V? ¼ K V?N or V?N ¼ KV? or r ¼ Kr ; IV. FUNDAMENTAL EQUATIONS OF MOVEMENT IN dt N dt THE THEORY OF THE DYNAMIC MEDIUM OF (50) REFERENCE 2GM 1=2 2GM The purpose of this section is to establish equations of where KðrÞ¼ 1 2 and k ¼ 2 . c0r c0 movement, for photons and material particles in the frame of Finally, we obtain the first fundamental equation the theory of the dynamic medium of reference. 2 2 2 In order to do this, we are going to apply the following VN ¼ V== N þ V?N method: 2 2 2 dr d/ 2 2 ¼ þ r ¼ K V== þ ðÞKV? • Establish equations of movement in a prerelativistic con- dt N dt N text (i.e., Newtonian), 2 2 dr d/ • Apply to these equations a transformation due to the gravi- ¼ K2 þ Kr : (51) tation taking into account the results of Section II concern- dt dt ing the measure of durations and lengths. As in the proposed theory, the gravitation is due to a flux of the medium directed toward the center of gravity of the A. Gravitational transformation in the context of the massive body, it is considered that the conservation of the theory of the dynamic medium of reference angular momentum is valid The purpose of this section is to establish a gravitational 2 d/ d/ transformation whose principle is to start from prerelativistic rN ¼ rN r ¼ rNV?N ¼ A ¼ constant: dt N dt N (Newtonian) equations and to apply to them the results of Section II concerning the measure of durations and lengths. We are going to establish the gravitational transforma- By using the expressions V?N ¼ K V? and rN ¼ K r, tion for a speed expressed in polar coordinates (r, /) since we obtain the second fundamental equation the flux of the medium is centripetal (i.e., directed toward the center of gravity of the massive body). 2 2 2 d/ rNV?N ¼ K r V? ¼ K r ¼ A: (52) In polar coordinates, the speed can be written dt ! V ¼ V u! þ V u! N == N == ?N ? B. Photon dr d/ with V== N ¼ and V?N ¼ r : In the context of the theory of the dynamic medium of dt N dt N reference, the approach is the following: • start from the equation c ¼ c , Thus, the modulus of the speed can be written photon 0 • write this equation in polar coordinates in the prerelativist 2 2 (Newtonian) context, dr d/ 2 2 2 • VN ¼ V== N þ V?N ¼ þ r : (48) transform this last equation using the gravitational dt N dt N transformation. Taking into account the results of Section II concerning the measure of durations and lengths we have 2 2 The equation cphoton ¼ c0 can be written in polar coordi- 2 2 2 2 nates: VN ¼ðdr=dtÞN þðrðd/=dtÞÞN ¼ c0. • k 1=2 1 By using the gravitational transformation we obtain the LðrÞ=L1 r • Radial speed: V ¼ V ¼ V fundamental formula == == N TðrÞ=T == N 1=2 1 k 1 2 2 ÀÁ r 2 dr d/ 2 k 2 K þ Kr ¼ c0: (53) ¼ V== N 1 r ¼ V== NK dt dt Physics Essays 32, 4 (2019) 433

In the frame of general relativity, this equation can be Finally, we have obtained thanks to the Lagrangian. For a null geodesic (case of the photon), the ﬁrst integral dr 2 2GM ¼ 6K c0 ¼ 6 1 2 c0: (59) of the geodesic equations is dt c0r "# 2 2 2 2 2 dr d/ L ¼ K c0 K Kr ¼ 0; (54) We could have directly found this formula by using dt dt 2 2 2 ðdr=dtÞN ¼ c0 and ðdr=dtÞN ¼ K ðdr=dtÞ. where L indicates the Lagrangian (see Ref. 11, Chap. 9). C. Material particle It is possible to interpret the previous result in the following way: In the context of the theory of the dynamic medium of reference, the approach is the following: • In the medium of propagation (medium of reference) the ! ! • speed of a photon is cphoton ¼ c0 . Start from the equation • In a referential linked to the massive body (with for origin ! ! ! GM ! the center of gravity of the massive body) the speed of the C particule ¼ C fluxwith C f lux ¼ u== ; (60) 2 2 2 2 r2 photon is such: ðK ðdr=dtÞÞ þðKrðd/=dtÞÞ ¼ c0. • Write this equation in polar coordinates in a prerelativist ! ! We are going to carry on the calculations in order to (Newtonian) context, that is to say ð C particuleÞN ¼ðC fluxÞN obtain an equation in dr=d/. an equation in which the acceleration of the ﬂux in a By using the second fundamental equation (52) in prerelativistic context has the approximate expression ! 2 ! Eq. (53) we obtain ð C fluxÞN ðGM=rNÞ u== , • Integrate this equation in order to obtain an equation with 2 2 dr A speed, K2 þ ¼ c2: (55) dt Kr 0 • Transform this last equation using the gravitational transformation.

By using the formula dr=dt ¼ðdr=d/Þðd/=dtÞ¼ ðA=K2r2Þðdr=d/Þ we obtain In polar coordinates the acceleration can be written: 2 2 2 2 2 ! ! ! A dr A 1 dr 1 c CN ¼ C== N u== þ C?N u? þ ¼ c2 or þ ¼ 0 r2 d/ Kr 0 r2 d/ Kr A 2 2 2 with C== N ¼ðd r=dt ÞN rNðd/=dtÞN and (56) 2 2 C?N ¼ rNðd /= dt ÞN þ2ðdr=dtÞNðd/=dtÞN. ! ! 2 ! The equation ð C particuleÞ ¼ðC fluxÞ ðGM=r Þ u== 2 N N N If we put u ¼ 1=r we obtain: dr=d/ ¼ð1=u Þðdu=d/Þ can be written and so we get 2 2 2 2 d r d/ GM du c0 r : (61) þ K2u2 ¼ dt2 N dt r2 d/ A N N N du 2 2GM c 2 The movement of the particle being with a radial accel- or þ u2 u3 ¼ 0 : d/ c2 A eration, the conservation of the angular momentum (deduced 2 (57) from C?N ¼ 0) is valid: rNðd/=dtÞN ¼ A ¼ constant. Equation (61) can be written as 2 2 If we derive this expression with respect to / and then d r A GM 2 3 2 : (62) divide it by 2ðdu=d/Þ 6¼ 0 we obtain dt N rN rN 2 d u 3GM 2 2 þ u ¼ 2 u : (58) d/ c After multiplying this last equation by 2ðdr=dtÞN we obtain This equation makes it possible to determine the deﬂec- tion of rays of light by the Sun and also by cluster of galaxies dr d2r A2 dr GM dr (gravitational lens, gravitational mirage, Einstein’s ring). 2 2 2 3 2 2 : dt N dt N rN dt N rN dt N Case of the radial trajectory: In this case, we have the very simple expression After integrating this last equation, we obtain 2 2 2 2 2 2 ðdr=dtÞN þðA =rNÞð2GM=rNÞþC where C is an 2 dr 2 2 dr integration constant taking into account the initial conditions K ¼ c or also K ¼ 6c0: dt 0 dt in position and speed of the particle. 434 Physics Essays 32, 4 (2019) 2 2 2 Using the relation rNðd/=dtÞN ¼ A we can rewrite the A dr A 2GM þ ¼ þ C2 or furthermore previous equation in this way r2 d/ Kr r 2 2 2 2 2 1 dr 1 2GM C dr d/ 2GM ÀÁ 2 2 2 2 þ ¼ 2 þ 2 : (68) þ r þ C Vflux N þ C : (63) r d/ Kr A r A dt N dt N rN

Putting u ¼ 1=r we obtain dr=d/ ¼ð1=u2Þðdu=d/Þ By using the gravitational transformation, we obtain the which gives us following fundamental formula: du 2 2GM C2 2 2 K2u2 u or dr d/ 2GM þ ¼ 2 þ 2 K2 þ Kr ¼ þ C2 ¼ V2 þ C2: d/ A A dt dt r flux du 2 2GM 2GM C2 (64) þu2 u3 ¼ uþ : d/ c2 A2 A2 In the context of general relativity, this equation can be (69) obtained thanks to the Lagrangian. For a nonnull geodesic, the first integral of the geodesic equations is If we derive this expression with respect to / and then 2 divide it by 2ðdu=d/Þ 6¼ 0 we obtain L ¼ c0 which can be written d2u GM 3GM "# u u2: (70) 2 2 þ ¼ 2 þ 2 dr 2 d/ 2GM d/ A c c2 L ¼ K2 K2 þ Kr C2 ¼ 0; 0 dt dt r (65) This equation allows one to determine the trajectory of the planets of the solar system and in particular, the preces- where L indicates the Lagrangian (see Ref. 11, Chap. 9). sion of Mercury’s perihelion. It is possible to interpret the previous result in the following way: Case of a radial trajectory of a material particle with • In the dynamic medium of reference, the speed of a mate- ! ! a null speed at an infinite range rial particle is V ¼ V , particule 0 In this case, we have the very simple expression • In a referential linked to the massive body (with for origin the rffiffiffiffiffiffiffiffiffiffi center of gravity of the massive body) the speed of a particle 2 2 2 dr 2GM dr 2GM is such: ðK2ðdr=dtÞÞ þðKrðd/=dtÞÞ ¼ð2GM=rÞþC2. K2 ¼ or also K2 ¼ 6 ¼ 6V : dt r dt r flux

Remark: In the case of a material particle with a null speed at an infinite range (r ¼1), we have dr/dt ¼ 0, Finally, we have r.d//dt ¼ A/r ¼ 0, K ¼ 1 which implies C ¼ 0. rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi So, the equation can be written dr 2 2GM 2GM 2GM ¼ 6K ¼ 6 1 2 : (71) dt r c0r r 2 dr 2 d/ 2GM K2 þ Kr ¼ ¼ V2 : (66) dt dt r flux We could have directly found this formula by using 2 We considerpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the expression of the speed of the flux dr 2GM dr 2 dr ! ! and ¼ K : V flux ¼ ð2GM=rÞ u== , which was only an approxima- dt N r dt N dt tion in a prerelativistic context, is accurate in the context of the theory of the dynamic medium of reference. V. CONCLUSION We are going to carry on the calculations in order to The present article proposes a new theory with the fol- obtain an equation in dr=d/. lowing features: Using the second fundamental equation (52) in Eq. (64) we obtain • A material ruler is physically contracted due to its motion with regard to the medium of reference; 2 2 dr A 2GM • A material clock has its period physically dilated due to its K2 þ ¼ þ C2: (67) dt Kr r motion with regard to the medium of reference; • A material ruler is physically contracted due to the movement of the medium created by a massive body; Using the formula ðdr=dtÞ¼ðdr=d/Þðd/=dtÞ¼ • A material clock has its period physically dilated due to ðA=K2r2Þðdr=d/Þ we obtain the movement of the medium created by a massive body; Physics Essays 32, 4 (2019) 435

• Light is slowed down by the gravitational field (centripetal past no longer exists, the future does not exist yet and is flux of the medium) created by a massive body. open. One can choose one or the other theory. According to the previous features, the theory of the dynamic medium of reference allows to obtain the following results: ACKNOWLEDGMENTS I would like to thank Mr. Roger Balian, member of the • deviation of light by the gravitational field (centripetal flux of the medium) created by a massive body, French Science Academy, for his pertinent remarks and advice concerning my document. I would also like to thank • Shapiro delay due to the gravitational field (centripetal flux of the medium) created by a massive body, Mr. Michel Barreteau, researcher at Thales Research & Technology, for reading and correcting this article. • gravitational transformation which allows to obtain the equations of movement for photons and material particles.

The concepts, the starting hypothesis and postulates, the APPENDIX A: A POSSIBLE DESCRIPTION OF THE physical phenomenon explaining gravitation and above all DYNAMIC MEDIUM OF REFERENCE the interpretation differ greatly between the theory of the This part gives a possible description of the dynamic dynamic medium of reference and general relativity. medium of reference. In the context of general relativity, gravitation is a geo- The theory of the dynamic medium of reference is based metric modification of space-time. A massive body curves on Le Sage theory, but it adds many deep changes and the space-time. evolutions. In the context of the theory of the dynamic medium of Numerous scientists have studied Le Sage theory. Just reference, gravitation is a modification of the medium. A to mention a few of them: massive body affects the medium by generating flux directed Newton, Huygens, Leibniz, Euler, Laplace, Lord Kelvin, toward the center of gravity of the massive body. Maxwell, Lorentz, Hilbert, Darwin, Poincare, and Feynman. Table II gives the main differences between the two Henri Poincare has studied this theory and written a syn- theories. thesis in Science et Methode. 4 The theory of the dynamic medium of reference is an Poincare sums up the principle of Le Sage theory like extension of Lorentz–Poincare’s theory in the domain of this: gravitation where instruments (rulers and clocks) are dis- “It is proper to establish a parallel between these consid- torted by gravitation and it is only the measure of the speed erations and a theory proposed a long time ago in order to of light which always gives the same result, the light being explain the universal gravitation. Let’s suppose that, in the really slowed down by a gravitational field. interplanetary spaces, very tiny particles move in all direc- Most well-known scientists working on general relativity tions, with very high speeds. A single body in the space will think that physical reality is global space-time, called block- not be affected, apparently, by the impact of these cor- universe, which contains all past, present and future events. puscles, since these impacts are equally divided in all direc- All events exist eternally. tions. But, if two bodies A and B are in the space, the body B In the context of the theory of the dynamic medium of will play the role of a screen and will intercept a part of these reference, the only physical reality is the universal present corpuscles which would have hit A. Then, the impacts moment, where travels in the past are impossible because the received by A in the opposite direction of the one of B, will

TABLE II. Main differences between general relativity and the theory of the dynamic medium of reference.

General relativity Theory of the dynamic medium of reference

Simultaneity relative to the observer12,13 Absolute simultaneity Independent of the observer Existence of a block-universe13 The only physical reality is the universal present moment, the past (containing all past, present, and future events) does not exist any longer, the future does not exist yet and is open The aether does not exist Existence of the medium of propagation of light No Preferred Frame of Reference Existence of the Preferred Frame of Reference6,7 Invariance of the speed of light12 The measure of the speed of light is constant, but light is really slowed down by a gravitational field Gravitation ¼ geometric modification of space-time Gravitation ¼ modification of the medium (flux directed toward the center of gravity of the massive body) A gravitational wave is a small deformation of space time A gravitational wave is a small deformation of the medium propagat- propagating at the speed of light ing at the speed of light The speed of propagation of gravitation is the speed of light The speed of propagation of gravitation is much greater than the speed of light 436 Physics Essays 32, 4 (2019) not have compensation any longer, or will be imperfectly PNG ! V rffiffiffiffiffiffiffiffiffiffi compensated, and they will push A toward B. Such is Le G=R ! i¼1 ! 2GM! Sage theory.” CG=R ¼ ¼ Vflux ¼ ur : NG r It is possible to demonstrate rather easily that the “push” is inversely proportional to the square distance between the The demonstration of the link between the acceleration two bodies (like the Newton law). and the speed of the flux is the following: One can also demonstrate that, if the corpuscles are very If we derivate the acceleration we obtain tiny, the push is approximately proportional to the number of dV dV dr dV 1 d 2 nucleons and so the mass of the body, and not the apparent cflux ¼ ¼ ¼ :V ¼ ðV Þ so we have: surface of the body. Ðr dt dr dt dr 2 dr 2 Moreover, only a tiny fraction of corpuscles hits the Vflux ¼ 2cfluxdr. þ1 atoms of the body, which explains that the push (the gravita- 2 Using the expression cflux ¼ðGM=r Þ we obtain: tional force) is so weak. Ðr 2 2 r Vflux ¼ ð2GM=r Þdr ¼½2GM=rþ1 ¼ 2GM=r. The main evolutions of the theory of the dynamic þ1 medium of reference versus Le Sage theory are the Finally, wep obtainffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the expression of the speed of the ! ! following: flux: Vflux ¼ ð2GM=rÞur . • one must not use the notion of impact with the corpuscles, Remark: The speed of the flux is zero at an infinite dis- • one must consider that the corpuscles are nonmaterial and tance of the massive body creating the gravitational field constitute a medium, (Vflux ¼ 0 for r !þ1). • the total energy of an entity is the sum of its kinetic energy of translation and of its kinetic energy of rotation about Definition of the Preferred Frame of Reference based on itself, and the entities constituting the medium: Let us consider a Galilean referential R (a laboratory) • fundamental law: conservation of the total energy of an entity: the total energy of one entity remains constant: and an elementary volume linked to this referential. In this very small volume, imagine that we can count the E ¼ E þ E ¼ constant. total translation rotation entities in it (gravitons) and we can also know the speed vec- ! Afterward, I will call these entities gravitons (but these tor of each graviton VG=R . gravitons have nothing to do with the graviton of spin 2 of Knowing this, it is possible to compute the vectoral quantum mechanics). average of the speed vectors of the gravitons: If the total energy of the gravitons remains constant, ! PNG ! CG=R ¼ VG=R =NG. then the gravitons do not give energy to the atoms of the i¼1 Earth and so do not raise the temperature of the Earth. It is postulated that the gravitons which interact with the atoms of the Earth lose some of their kinetic energy of translation which turns into kinetic energy of rotation. The gravitons which interact with the atoms of the Earth lose a part of their translation speed and win some rotation speed. So the Earth would be a huge “transformer” of “standard gravitons” in “gravitons-spin.” This physical phenomenon has no effect on Earth (at least not the elevation of temperature), but it has an effect on the medium. The medium undergoes a centripetal flux due to the presence of the Earth. Indeed, let us consider a reference frame at the surface of the Earth and an elementary volume linked to it. If one measures the speed vectors of all the gravitons in this elementary volume, the average of the speed vectors gives a resulting speed vector which is centripetal (because the gravitons-spin coming from the ground have a smallest translation speed than the standard gravitons coming from the sky). One can demonstrate that the acceleration of the flux has ! 2 ! the following expression cflux ¼ðGM=r Þur from which one can deduce that the centripetal speed of the flux at a distance r from the center of gravity of the Earth of mass M is equal to (measured in a reference frame R linked to the Earth): FIG. 3. Distortion of the gravitons field due to the Earth. Physics Essays 32, 4 (2019) 437

This resultant vector means that, at the center of this It is essential to be aware that all these difficulties are given elementary volume, the Preferred Frame of Reference only in Einstein’s vision, i.e., the special relativity (and gen- ! eral relativity). moves at the speed CG=R versus the referential R and that the ! If one chooses the version of Lorentz and Poincare, all referential R (the laboratory) moves at the speed CG=R ver- these difficulties disappear. sus the Preferred Frame of Reference (defined by the In The Ghost in the Atom,14 John Bell says about the medium), i.e., versus the medium. EPR paradox (page 48): “I would say, that the cheapest reso- Figure 3 shows the distortion of the gravitons field cre- lution is something like going back to relativity as it was ated by the Earth. before Einstein, when people like Lorentz and Poincare The distortion of the gravitons field (the flux of the thought that there was an aether—a preferred frame of refer- medium): ence—but that our measuring instruments were distorted by • is generated by the presence of the matter of the Earth, motion in such a way that we could not detect motion • is always centripetal, i.e., radial, oriented toward the center through the aether. Now, in that way one can imagine that of gravity of the Earth, there is a preferred frame of reference, and in this preferred • is maximum at the surface of the Earth and decreases frame of reference things do go faster than light.” when going away from the Earth, In the same book,14 David Bohm says about locality: “I • has a constant modulus on every sphere whose center is would be quite ready to relinquish locality; I think it is an the one of the Earth, arbitrary assumption.” and Basil Hiley says about non local- • gives the impression to follow the Earth in its movement ity: “If you have an absolute spacetime,oranabsolute (whatever its speed) because it remains identical to itself. time, in the background then you don’t get into causal loops. So, the causal paradoxes won’t arise.” APPENDIX B: SPEED OF PROPAGATION OF The entanglement of two particles (let us say, two pho- GRAVITATION tons) could be realized by the proposed gravitons which move much faster than light. To determine the speed of gravitons, one reasoning is to 1. Gravitational waves consider that the connection between two particles remains In the theory of general relativity, the gravitational even if the two particles are situated at two opposite extremi- waves are a small perturbation of space-time which propa- ties of our Universe and that our more accurate instruments gates at the speed of light. cannot detect a duration between the measurement of the first particle and the measurement of the second particle, i.e., the In the theory of the dynamic medium of reference, the 44 gravitational waves are a small perturbation of the medium duration is smaller than the Planck time tPlanck ¼ 5.4 10 s. which propagates at the speed of light. So, we obtain the following speed for the gravitons and the speed of gravitation: 2. Speed of propagation of gravitation DUniverse 69 Vgravitons ¼ ¼ 3 10 m=s: tPlanck In the theory of the dynamic medium of reference, the speed of gravitation (the physical effect which produces that It is only in Einstein’s version (the theory of relativity) an object falls toward a massive body) is the one of the grav- that nothing can move faster than light. itons introduced in the previous paragraph and is supposed to be much greater than the speed of light. 4. Proposed experiment to measure the speed of In the theory of the dynamic medium of reference, propagation of gravitation which is an extension of Lorentz–Poincare’s theory in the The only way to be really sure of the speed of propaga- domain of gravitation, nothing prevents the existence of enti- tion of gravitation is a direct measure, that is to say, the ties moving much faster than light. effect of the modification of a first material system on a sec- It is only in Einstein’s relativity that nothing can over- ond material system including the modification of the distri- take the speed of light. bution of matter, the modification of the distance between the two systems at a high speed. 3. Entanglement of two particles The cosmos seems to be the best laboratory to measure In The Ghost in the Atom,14 there is the following dis- the speed of propagation of gravitation. cussion about the EPR experiment and locality (page 39): For example, we can consider a gigantic triangle (sys- “If locality is abandoned, it is possible to re-create a tem 1, system 2, Earth) and we suppose that system 1 under- description of the microworld closely similar to that of the goes a titanic and violent change (explosion, supernova, …). everyday world, with objects having a concrete independent We suppose that the distances between the two systems existence in well-defined states and possessing complete sets and between the systems and the Earth are perfectly known. of physical attributes. No need for fuzziness now. We call « optical image » of an event, the light emitted The trade-off is, of course, that nonlocal effects bring or reflected by the object concerned by the event at the time their own crop of difficulties; specifically, the ability for sig- of the event. This optical image propagates at the speed of nals to travel backward into the past. This would open the light and can be received at different times by different way to all sorts of causal paradoxes.” observers. 438 Physics Essays 32, 4 (2019)

By knowing the time of arrival on Earth of the optical Obviously, the bigger the distance dS1S2 will be, the image of the violent phenomena which occurs in system 1 as more definitive it will be to decide between general relativity well as the time of arrival on Earth of the optical image of and the theory of the dynamic medium of reference. the gravitational effects which affect system 2, it is possible Other proposition of experiment with swift massive to deduce the speed of propagation of gravitation. bodies: We call d the distance between system 1 and the S1E We can also envisage the measure of the gravitational Earth, d the distance between system 2 and the Earth, S2E effect of a massive body, moving at a speed as close as possi- d the distance between the two systems and V the speed S1S2 G ble to the speed of light, on a second massive body. of propagation of gravitation. The first massive body would start at a long distance We call t the time when the violent event occurs in sys- 1 from the second body, where its gravitational effect is negli- tem 1, t0 the time of arrival on Earth of the optical image of 1 gible, and then would pass very close to the second body at a this event, t the beginning of the gravitational effects on 2 very high speed. system 2 and finally t 0 the time of arrival on Earth of the 2 If the measure is possible, we could see if the gravita- optical image of this last event which occurs in system 2. tional effect propagates at the speed of light or much faster. We have

0 dS1E dS1S2 0 dS2E 1 t1 ¼ t1 þ t2 ¼ t1 þ and t2 ¼ t2 þ : C. Will, Theory and Experiment in Gravitational Physics (Cambridge Uni- c VG c versity Press, Cambridge, 1993). 2C. Will, Was Einstein Right? Putting General Relativity to the Test (Basic We can deduce Books, New York, 1993). 3J.-C. Boudenot, Electromagn etisme et Gravitation Relativistes (Ellipses, 0 0 dS2E dS1E t2 t1 ¼ t2 þ t1 þ Etuz, France, 1989). c c 4H. Poincare, Science et Methode (Flammarion, Paris, France, 1908). 5 d d d M. Lambert, Relativite Restreinte Et Electromagnetisme (Ellipses, Etuz, ¼ t þ S1S2 þ S2E t þ S1E France, 2000). 1 1 6 VG c c J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge d d d University Press, Cambridge, 1987). ¼ S1S2 þ S2E S1E : 7J. Gribbin, Schrodinger’s€ Kittens and the Search for Reality (Phoenix VG c International Publications, Phoenix, AZ, 1996). Then we deduce the speed of propagation of gravitation 8L. Smolin, Time Reborn (Houghton Mifﬂin Harcourt, San Diego, CA, 2013). 0 Dd 0 0 0 9A. Barrau and J. Grain, RelativiteG en erale (Dunod, Paris, France, 2011). VG ¼ dS1S2= Dt with Dt ¼ t2-t1 and Dd¼ dS2E –dS1E. c 10A. Einstein, Ann. Phys. 35, 898 (1911). Particular cases 11M. Holson, G. Efstathiou, and A. Lasenby, RelativiteG en erale (De Boeck, Paris, France, 2010). 0 0 12 (1) VG ¼ c, then we have: t2 t1 ¼ðdS1S2=cÞþðDd=cÞ.It A. Einstein, La Theorie de la Relativite Restreinte et Gen erale (Dunod, Paris, France, 2004). is the case of general relativity. 13 0 0 B. Greene, The Fabric of the Cosmos (Penguin Books, 2008). o = 14 (2) VG c,thenwehave:t2 t1 Dd c. It is the case of P. Davies and J. Brown, The Ghost in the Atom (Cambridge University the theory of the dynamic medium of reference. Press, Cambridge, 2000).