Vavel experiment: the relative constancy of the speed of light

Vavel experiment: the relative constancy of the speed of light Author: Vavel Sanchez, JM e-mail: lobra333@gmail.com Madrid 01 August 2019, Spain Word key: Michelson-Morley, Maxwell, Fizeau, Special relativity, Speed of light, Vavel experiment, Interference spectrum of light NOTE: This paper is a partial translation to English of the paper: “Variación de la Velocidad de la Luz en el Vacio: Experimento VAVEL” in the Academia,edu web Index 1 Precisions about the Experiences that Support the Constancy of the Speed of light in Vacuum. Need for a New Experiment .............................................................................. 3 1.1 Introduction...................................................................................................................... 3 1.2 Analysis of the Michelson-Morley experiment ................................................................ 4 1.2.1 Explanation of the Michelson-Morley experiment in Classical Mechanics for the observer “Os” outside the Earth. ..................................................................................... 5 1.2.2 Explanation of the Michelson-Morley experiment in Relativistic Mechanics for the observer “Os” outside the Earth. ................................................................................... 10 1.3 Maxwell's equations developed for an observer in relative motion ............................. 13 1.4 Explanation of the Fizeau experiment of light dragging by the medium, in Classical Mechanics....................................................................................................................... 15 1.5 Conclusions..................................................................................................................... 18 2 Vavel (Variación de la Velocidad de la Luz) experiment ................................................ 19 2.1 Technical data of equipment.......................................................................................... 20 2.2 Results of experiment .................................................................................................... 22 2.2.1 Method used in the experiment research...................................................................... 22 2.2.2 Experiment's development ............................................................................................ 22 2.3 Analysis of the obtained results ..................................................................................... 25 3 Conclusions..................................................................................................................... 29 4 Last objection ................................................................................................................. 30 5 Figure list ........................................................................................................................ 30 6 Picture list ....................................................................................................................... 31 1 Precisions about the Experiences that Support the Constancy of the Speed of light in Vacuum. Need for a New Experiment 1.1 Introduction The demonstration that the speed of light is constant is, mistakenly, attributed to the MichelsonMorley experiment. Nothing further from reality because, as will be demonstrated in the following points, this experiment finds its perfect explanation in classical mechanics, without resorting to the contractions of space and time proposed by relativistic mechanics. A first objection to the application of the equations of relativistic mechanics to the Michelson-Morley experiment is that, in said experiment, there is no relative movement between the elements that make it up, all of them (light bulb, mirrors and observer) are of the same "inertial frame", the Earth, therefore, there is no relative velocity and, consequently, the equations of relativistic mechanics will hardly be applicable. The Michelson-Morley experiment proves just the opposite of what he was trying to prove, that is, "the non-existence of the ether" as a necessary fluid for the transmission of light in vacuum. The application of the relativistic theory to this experiment implies, in reality, the implicit acceptance that the ether exists, because only in the case that it existed it would make sense since then it is true that the inertial frame “Earth” moves in the ether, which is fixed and the light travels through the ether so there would be relative motion and the Lorentz equations would apply. But if the ether does not exist, then the whole set is in the same inertial frame and there would be no relative moving element. Nor do other experiments of the same style, such as Kennedy-Torndike, demonstrate this constancy of the speed of light in vacuum, since the explanation is similar to that of the first experiment: none of the elements of the experiment are in relative motion respect to others, not being, therefore, relative speed between them. Another of the theoretical fundaments of the constancy of the speed of light are Maxwell's equations for electromagnetic waves. As will also be demonstrated, if the equations for magnetic and electric fields are in relative motion respect to the observer, the apparent constancy of the speed of the electromagnetic waves in vacuum, disappears. The same applies to the Fizeau experiment of light dragged by the medium. The difference found between the theoretical calculation and the experimental value is due to an erroneous theoretical analysis of the fundamentals of the experiment, as we will show later. The only really valid position in relation to the constancy of the speed of light is that of those who maintain that said constancy is established as a principle and, therefore, is the capacity of the theory (which develops from it) to explain the physical phenomena that occurs in nature, the one that establishes the validity or not of it and, as a consequence, the validity of the principles on which it is based. I have nothing to object to this argument since it is irrefutable and is the basis of all physical theory. However, given the importance of this principle, and that it is still an exception in nature (it states that the speed of light in vacuum is the maximum that an object can reach and that it is independent of the speed of the focus) I think it is of the utmost importance for the world of theoretical physics to be able to demonstrate whether this is so or not. How can it be demonstrated or experimented? With an experiment like the Michelson-Morley, modified so that the elements of the experiment are in relative motion with each other. We can observe if the interference spectrum stripes are fixed in these conditions or not. This is the VAVEL Experiment that we have carried out and which results are shown in next section-2 1.2 Analysis of the Michelson-Morley experiment First, we will establish the different observers of the experiment. A first observer will be the one in the same inertial frame as the experiment (the Earth) that we will call Oe, and a second observer will be the one outside the Earth, the Sun for instance, respect to which the Earth moves with velocity V and which we will call Os. See Figure-1 in the next section. The first observer Oe cannot apply the equations of the relative Movement, because as we mentioned in the previous section, there is no relative speed between the elements that make up the experiment (Focus, mirrors and observer). Therefore, the only thing that the observer can do is measure the distances, which are equal for both mirrors, measure the speed of light that is the same in all cases, measure the number of waves in each incident and reflected ray, that they will always be the same, and conclude that for him the whole experiment is logical and his explanation does not represent any problem. Everything is the same, the time it takes for the incident beam to reach the mirror A, the beam reflected in the mirror A until he observes it, the time it takes for the incident beam to reach the mirror B and the time it takes for the reflected beam in B until he watches it again. Consequently, the interference spectrum remains constant and immutable all the time, regardless of the direction in which we place the mirrors. The second observer, the observer Os who is in the Sun, sees that there is a displacement of the elements of the experiment while it is taking place, so that the distances and times that he can measure will be different than those of the observer Oe. We are going to present the observations of this observer Os, from the point of view of classical mechanics and from the point of view of relativistic mechanics and compare the conclusions to which the observer Os would arrive in each case. 1.2.1 Explanation of the Michelson-Morley experiment in Classical Mechanics for the observer “Os” outside the Earth. I suppose that the reader knows well the V Michelson-Morley experiment, so I omit Desplazamiento its description. If we look at Figure-1, the F A mirror A is in the direction of the Earth's displacement and the mirror B in the perpendicular direction. As established in the experiment, the distance of both mirrors to the focus (F) is the same, that is, the distances FA and FB are equal and the value is L, for the observer Oe. B At the moment that the light beam leaves Figure 1.- Michelson-Morley experiment the focus F, the mirror A is at a distance L from F, but the distance that the light beam must travel to reach the mirror A is greater. When the light reaches the space point A, the mirror is no longer there because it has moved due to the movement of the Earth. We will calculate what is the distance traveled by the ray to reach the mirror A. Starting from the principle that the speed of light is affected by the speed of the focus and, therefore, by the speed of displacement of the Earth, we will have the following data: • Speed of light = c + V • Distance = L For the observer Os, while the beam travels the distance L, the mirror moves a distance of value: V L , and when the beam travels this new increase, the mirror has moved again a smaller c +V distance, of value: V  L 2 c +V = L V , following this development in series, the c +V (c + V ) 2 V distance "da1" finally travelled by the beam before reaching the mirror is: d a1 = L + L  V V2 V3 + L L + + ............. = L   c +V (c + V ) 2 (c + V ) 3 1 V 1− c +V since da1 is the sum of the terms of an indefinite and decreasing geometric progression of ratio V / (c + V), so we will finally have: da1 = L  1 + V  [1-1] c For the observer Os, in the return trip of the light beam from the mirror A to the detector F the distance is less than L, because the detector F will approach to the ray as it advances to it, due to the movement from the earth. On the other hand, the speed of the light beam will also be different from the one it was carrying on the outward trip, the starting data for this new calculation are: • Speed of light = c - V • Distance = L While the beam travels the distance L, the detector F has approached the beam a distance of value: L  V , so they would have met before, but while the beam travels the distance L c −V minus the previously calculated segment, the point F will have approached the beam a distance: V 2 c − V V = L  V − L  V , and in the meantime, point F will have c −V c −V c − V  2 L− L approached a distance:  V V2  V V V3 V2 + L  = L − L + L L − L  c −V c −V (c − V )2  c − V (c − V )2 (c − V )3  Continuing with the approximations, the new distance travelled by the beam to find the detector will be: d a2 V3 V2 V = L − L + L − L + ............... c −V c − V  2 c − V  3 which represents the sum of the terms of an indefinite and decreasing geometric progression of ratio –V / (c-V) which value is: da2 = L  c −V V 1 = L = L  1 −  [1-2] V c c 1+ c −V from equations [1-1] and [1-2] we will obtain the times it takes for the ray of light to travel the distances, these times being the following: • • one-way time: ta1 = return time: ta 2 = V  c =L c +V c L  1 + V  c =L c −V c L  1 − That is, for the purposes of the observer Os, both times are equal and with a value such as if the speed of light was not affected by the speed of the Earth's displacement. We are going to analyse what happens to the ray that comes perpendicular to the Earth's movement towards the mirror B (See Figure 2). When the ray leaves F, it will be affected by the F F c Vluz Earth's velocity as well as the mirror B, so that the ray of light actually travels through the hypotenuse of the triangle FBB' in Figure-2 and the mirror B becomes the position B'. If the speed of the Earth V was not affected by the light beam, it would follow a perpendicular path, the FB, and when it would reach space point B the mirror would no longer be there, not being able to reflect on it and the B B' experiment would not take place. The distance travelled by the ray of light to reach the mirror B Figure 2.-Beam perpendicular to the motion of the Earth ’will be: db1 = L2 +  BB' 2 ; but on the other hand: BB' = L  V (by similarity of triangles) and substituting in the square root, we have: c V V db1 = L2 + L2    2 = L  1 +   2 c c The speed of the light ray is influenced by the speed of the Earth's displacement, so the velocity    module will be equal to the vector module of c + V = Vluz ,then the value of the module is:  Vluz = c 2 + V 2 ; the values of the distance travelled by the light beam, as well as the value of the speed module, are equal both for the light beam incident in the mirror B, and for the reflected beam returning to the detector F. The time for both rays in their trip will be: tb1 = tb 2 = db1 = Vluz V2 c2 = L c c2 + V 2 L  1+ That is, for the observer Os the time taken by the ray of light that moves perpendicular to the Earth's trajectory, is the same for the emission beam as for the reflected beam and is the same as for the beam that moves in the direction of the trajectory of the Earth. All of them coinciding with an equivalent value as to be constant the speed of light and coinciding with the results of the experiment. We are going to finish this study with the analysis of the number of waves in each ray of light, both in the ray that comes out of the focus, and in the ray reflected in the respective mirrors. This will give us an idea of the coherence of the light rays and the interference spectrum they form. The fundamental idea is that a variation in the speed of light changes the wavelength of the light, but not the frequency that is a property of the beam in origin and therefore unalterable. We make it so because, otherwise, the experiment would not have taken place, as the light rays would lost coherence giving rise to a non-static interference spectrum. By that it is a plausible hypothesis within the experience. Let's start with the ray of light that goes to the mirror A. The wavelength of the emitted ray will be: a1 = c +V 0 : where ω0 is the frequency of light. The number of waves that there will be, is equal to: na1 = d a1 a1 = V  c = L  = L 0 c +V 0 c L  1 + 0 Where λ0 is the wavelength of the light beam for constant velocity and equal to “c”. According with this equation, the number of waves in the emitted ray is the same as in a ray which speed would not have been altered by the speed of the Earth. Let's see what happens with the reflected ray. The wavelength of this ray will be: a 2 = And the number of waves is: na 2 = da2 a 2 = c −V 0 V  c = L  = L 0 c −V 0 c L  1 − 0 Then the number of waves for the incident ray and reflected ray is the same and equal to those that would have a beam of light which speed had not been affected (static conditions). Let's see what happens with the incident ray and reflected ray in the mirror B. As it has been seen before, having the same speed and being equal the distances travelled by both rays, the wavelength and the number of waves in the incident and reflected ray shall be the same value. Let's calculate its value: b1 = b 2 = nb1 = nb 2 = c2 + V 2 0 d b1 b1 , and the number of waves is: L  1+ = V2 c2 c2 +V 2 = L c 0 = L 0 0 Then the number of waves is independent of the ray path, perpendicular or parallel to the trajectory of the Earth. Also, the number of waves is the same value as for static conditions of velocity, maintaining static the interference spectrum at all times. All the previous demonstration is easy to repeat with identical results when the rays maintaining any angle with the Earth's trajectory. According to previously stated, Classical Mechanics can give a coherent explanation to the Michelson-Morley experiment without needing to establish constant speed of light and, therefore, the contractions of space and time. On the other hand, the explanations given by the Classical Mechanics make coincident the observation of observer Os with the observation made by the static observer Oe and both with the results of the experiment. 1.2.2 Explanation of the Michelson-Morley experiment in Relativistic Mechanics for the observer “Os” outside the Earth. The only difference with stated above for Classical Mechanics is that the speed of light is constant in any inertial frame. Returning to Figure-1 the equations of the distances travelled by the incident ray and reflected ray in each of the mirrors, which the observer Os would measure, are the following for the incident beam: d a1 L V V3 V2 V L c = L + V +  V + ..... = L + L  + L  2 + L  3 + ...... = L  c c c c c c 1 = L V c −V 1− c and for the reflected beam the distance that the beam must travel to reach the detector will be: da 2 L V 1 c V3 V2 L V = L − V + c  V − .... = L − L  + L  2 − L  3 + ..... = L  = L V c c c +V c c c 1+ c The time it will take each ray to travel the previous distances will be: d a1 L = c c −V • Incident ray: ta1 = • Reflected ray: t a 2 = d a2 L = c c +V And for round trip: L L L 1 2c c  ta = + = = L 2 2 2 c −V c −V c +V V V2 1− 2 1− 2 c c [1-3] 2 The first thing that is observed is that the travel time of the incident ray is greater than the travel time of the reflected ray. We are going to analyse now what happens to the ray that moves perpendicular to the Earth's trajectory going to the mirror B. If we look at Figure-2 of the previous section, the first thing that is observed is that, if the incident ray, from the focus F, is not affected by the movement of the Earth, it would never find mirror B, since when it reaches the point where it was, it will have moved to point B'. Forgetting this inconvenience, it is logical to think that the incident beam has to travel the distance FB' to reach the mirror and that the reflected beam, overlooking the same problem mentioned above, will travel the distance B'F to reach the detector. The distances and times will be as follows: d b1 = d b 2 = FB ' = L2 +  BB') 2 , but BB' = FB ' V , since the condition is that the speed of light c is constant and equal to “c”, therefore, the time that the beam takes to travel the distance FB’ is the time that the mirror takes for going from B to B’. Substituting and clearing, we will have: L db1 = db 2 = tb1 = tb 2 V2 1− 2 c , and the time it will take to travel these distances will be: L d c , and for round trip: = b1 = c V2 1− 2 c L c [1-4] tb = V2 1− 2 c 2 If we compare equation [1-3] and [1-4] the first thing we see is that both times are different, but as the experiment tells us they must be the same, if we substitute in the [1-3] equation, the expression of the [1-4], we will have: 𝛥𝑡𝑎 = 𝛥𝑡𝑏 2 √1−𝑉2 𝑐 [1-5] that is the relativistic equation of the time contraction in the direction of relative motion. If we multiply both members of equation [1-5] by the speed of light in vacuum, we will have: da = db V2 1− 2 c [1-6] ; since 𝑐 × 𝛥𝑡𝑎 = 𝑑𝑎 𝑎𝑛𝑑 𝑐 × 𝛥𝑡𝑏 = 𝑑𝑏 that are the distances travelled by the incident ray and reflected ray in the mirror A and B respectively. As can be seen, equation [1-6] is the relativistic equation of the space contraction in the direction of relative motion. We will now study the necessary condition that the number of waves in each ray is the same, so that the interference spectrum remains unchanged. If the number of waves is N, the wavelength of each ray will be the distance travelled divided by N, therefore, applying it to equation [1-6], we will have: db da b N ; and as the speed of light is considered constant, the = a = = 2 N V V2 1− 2 1− 2 c c V2 V2 =  a =  1 − 2 = b  1 − 2 frequency of each ray will be: c c a b c c which is the frequency of a ray of light reflected in a mirror, which moves with velocity V in the same direction of the ray, when the frequency of the incident ray is ωb. The latter introduces a contradiction, since, if the ray reflected in the mirror A and the ray reflected in the mirror B do not have the same frequency, they could not produce static interference spectrum, since the rays would not be coherent. However, the spectrum is static. The imposed condition of the constancy of the speed of light in any inertial frame complicates the measurements of the observer Os, who must introduce some corrections to the measured times and lengths, overlook some objections, such as finding the mirror B, or not leaving perpendicular to the Earth's trajectory (change of angles) and finally having difficulties with the frequency of the rays. All this to match their measurements with the measurements of the static observer and the results of the experiment. 1.3 Maxwell's equations developed for an observer in relative motion Y' Y ξ O O' v X ≡ X' β Z Z Figure 3.- Electric and magnetic field in relative motion The equations of electromagnetic waves developed by Maxwell can no longer be used as a mathematical demonstration of the constancy of the speed of light in vacuum. The reason for this statement is that Maxwell's equations are developed for the electric and magnetic fields fixed with respect to the observer, that is, without being relative movement between the observer and the fields, therefore, it is logical that the velocity of the electromagnetic waves is equal to the speed of light in vacuum and constant. We will develop these equations of the electromagnetic field for an observer respect to which the source of electric and magnetic field moves. The observer O is static respect to the magnetic and electric field ξ and β respectively. If this observer applies Maxwell's equations will have: ξx = 0; ξy = ξ; ξz = 0; βx = 0; βy = 0; βz = β If the fields are in vacuum and there are not electric or magnetic charges, that is, ρ = 0 and j = 0, we will have:     = 0; = 0; = 0; =0 y z z y [1-7]     ;− =  0 0 =− t x t x There is another observer O’ respect to which the inertial frame of the observer O moves with constant velocity V, in the direction of the positive axis of the X / X’, as indicated in Figure-3. For this observer O’ Maxwell's equations will be the same as before, but making the change of independent variables, since for him X is X’ and t is t’, linked by the following transformation equations: X ’= X + V x t Z ’= Z Y ’= Y t ’= t That is a Galilean transformation of axes change. Making the change of variables in the differential equations [1-7] above, we will have:  x  t  x  t  x'  t '    + = 1 + 0 = x' x t ' x x' t ' x'   x'  t '  V + = + = t ' x' t t ' t x'   x'  t '   0 = 1 + = + = x' t ' x' x t ' x x'   x'  t '  V + = + = t ' x' t t ' t x' = Finally, Maxwell's equations for this observer O’ are:     = 0; = 0; = 0; =0 y' z ' z ' y'       = − V + ; − =  0 0   V +  x' x' t ' x' x' t ' [1-8] To solve these differential equations, we will derive the last two with respect to x’ and t’: 2 2  2 1 − 9  − = − V x' t ' x'2 x'2  2 2 2 =−  V − 2 1 − 10 x' t ' x' t ' t ' 2 2    2     1 − 11   + = − V   0 0 2 x' t '  x'2  x' 𝜕2 𝛽 𝜕𝑥′𝜕𝑡′ = −𝜇0 𝜀0 × ( 𝜕2 𝜉 𝜕𝑥′𝜕𝑡′ ×𝑉+ 𝜕2 𝜉 𝜕𝑡′2 ) [1-12] Substituting in equation [1-9] the equations [1-11] and [1-12], it will be: 2  2  2  2  2 2   = 0 0V + 0 0V + 0 0V + 0 0 2 x'2 x'2 x' t ' x' t ' t ' Or:  2  2 1   2  2  2 + 2V V − = 0 ; or using another annotation type: +  0 0  x' x' t ' t ' 2   2 1 V −  0 0    2  Dx ' + 2VD x ' Dt ' + Dt2'  = 0 ; solving this differential equation:    1   − 2V  4V 2 − 4V 2 − 0 0   D= ; whose solutions are: 2 D1 = −V + 1  0 0 ; and D2 = −V −   1  Dt ' −  − V +   0 0        = 1  x'−V −  1  0 0 ; and we can write the differential equation as:     Dx '   Dt ' −  − V − 1     0 0       Dx '  = 0 ; whose solution is:         t ' +  2  x'−V + 1  0 0  0 0     1   t ' ; which is the equation of two waves     moving along the X / X' axis, each one in a direction of said axis and whose travel speeds for the observer O' are V - c and V + c, the first one for the waves that travel in the negative direction of the axis X and the second one for the waves that move in the positive direction of the axis X. Therefore, the resolution of Maxwell's equations for an observer respect to which the sources of the electric and magnetic fields are in movement, results in electromagnetic waves affected by such movement. 1.4 Explanation of the Fizeau experiment of light dragging by the medium, in Classical Mechanics Another of the experiments that the Theory of Relativity has taken as an example and basis, is the Fizeau experiment of the drag of light by the medium in which it moves, carried out in 1851. We will demonstrate that Relativity is not necessary to explain what was observed with this experiment and that only with Classical theory it has explanation. First, we will describe the experiment conducted by Fizeau and the scheme of the equipment used to perform it. The scheme of the equipment used is the one shown in Figure 4. Figure 4.- Fizeau interferometer scheme Description of the interferometer and the experiment: A beam of light from the source S' is reflected by a beam splitter G and is collimated in a parallel beam by the lens L. After passing through the slits O1 and O2, two rays of light travel through pipes A1 and A2. Inside these pipes water circulates in two opposite directions, as indicated by the arrows. The rays are reflected in the mirror m located in the focus of the lens L’, so that each ray always propagates through the branch of the pipe through which water circulates in the same direction. Thus, the beam that travels through the upper branch on the outward path is reflected in m and travels through the lower branch on the way back and vice versa. After making the round trip in the water pipes, the rays are joined in S where there are bands of interference that can be visualized by means of the display that appears in the image. By varying the speed of water circulating through the branches, the interference pattern can be analysed to determine the speed of light traveling in each branch of the pipe and analyse how the speed of water (propagation medium) influences the speed of Light. The data and results of the experiment are as follows: • Wavelength of the light used (λ): 5.3 x 10-7 m. • Speed of light in vacuum (c): 299,792,458 m / s • Water refractive index (n): 1.33 • Water speed in each branch (v): 7m/s • Length of each branch (L): 1.5 m Under these conditions, Fizeau observed that the displacement corresponded to 0.23 waves. If the theoretical calculations of the displacement that should occur in the interference spectrum according to classical mechanics are made, we will have: c = 565.646.147.169.811 waves / s. • Frequency of the used light (ω) = • Speed of light in the co-current of water: (c1) =  c + v = 225,407,870,157895 m / s n • Speed of light in the counter-current of water: (c2) = c − v = 225,407,856,157895 m / n s • Travel time of ray in co-current to crossing the branches (t1) = 2L = c1 13,309,206985x10-12 s. • Travel time of ray in counter-current to crossing the branches (t2) = 2L = c2 13,309,2078117x10-12 s • The time difference is (t2 - t1) = 8.2663 x 10-16 seconds • The number of displaced waves will be = (t2 - t1) x ω = 0.46758 waves This is where the problem appears, as we can see, the theoretical calculation gives us a displacement double of the observed. For this reason, the Fresnel factor is introduced, which then explains the Relativity Theory and whose formulation is F = 1 − 1 n2 that corrects the water speed, and reducing the drag by said factor F, although not with total accuracy, but with an acceptable approximation, since the value of the correction factor is F = 0.4346 and not 0.5 as it should have been. We will demonstrate that this is not necessary and that the difference between the theoretical and experimental results is due to a lack of deepening of the study of light interference. To simplify we go from two Simple Harmonic Movements with the following equations and that interfere in a point in space: 1 = A1 sen( K1 r1 − t ) 2 = A2 sen( K 2 r2 − t ) These equations represent two waves of equal frequency, but with a phase of K1r1 = φ1 for the first and K2r2 = φ2 for the second. If we take φ1 as a reference base, the gap between the waves is worth:  =  2 − 1 = K 2 r2 − K1 r1 , and we get that:  2 = 1 +  = 1 + (K 2 r2 − K1 r1 ) The interference wave equation comes out of solving the following system of equations: A cos  = A1 cos 1 + A2 cos  2 A sen  = A1 sen  1 + A2 sen  2 We will solve the equations system and calculate the value of the φ phase of the interference wave. For this we make A1 = A2, since in Fizeau's experiment the amplitude of both waves was the same as was the frequency ω. If we divide the second equation by the first, we get: tg  = A1 sen1 + A2 sen 2 sen 1 + sen  2 = ; since A1 = A2, if we apply the Theorems of A1 cos 1 + A2 cos  2 cos 1 + cos  2 Trigonometric Functions Addition, we have:   + 2    +  2   1 −  2  2sen  1   sen  1  cos sen 1 + sen  2 2  2   2    + 2    = tg  1 tg  = = =  cos 1 + cos  2 2   1 +  2   1 +  2   1 −  2   2 cos    cos  cos   2   2   2  And  = 1 +  2 2 Where φ is the phase of the interference wave resulting, and φ1 + φ2 is the sum of phases corresponding each one to the two waves, so that the gap of the interference wave will be: δi = φ - φ1 and whose value is:  i =  − 1 = 1 +  2 2 − 1 =  2 − 1 2    r   (t − t ) − r K r K r   1 1 2 1 =  2 − 1 = = = 2 2 c  2 2 2c 2 +v  −v n n  where it is seen that the actual gap of the interference wave is half the gap of the interfering waves. Therefore, the speed of a beam of light moving in a moving medium has a drag equal to the speed of the medium (of water in the case of Fizeau) and the Fresnel factor is not necessary. 1.5 Conclusions From the above, the conclusion is that the Michelson-Morley experiment has a perfect explanation within Classical Mechanics, we would even say that a better explanation than within Relativistic Mechanics. Therefore, from this experiment, the need for the speed of light to be constant in any inertial frame, for any observer and independent of the focus speed, is not deduced. The Maxwell equations also do not condition that the speed of electromagnetic waves in vacuum is constant and independent of the observer and/or the speed of the fields focus. Similarly, Fizeau's experiment can also be explained by Classical Mechanics without the need to resort to the Relativist. Considering that the constancy of the speed of light in vacuum is a strong constraint imposed and that many physicists would be more comfortable without that imposition (Quantum Mechanics versus Relativistic Mechanics) and that the constancy of the speed of light is the “cornerstone” on which the entire theoretical building of current physics is based, it seems logical to propose and carry out an experiment that can directly prove this constancy in order to discard doubts about it and lay down definitively the most important basis of Physics. My proposal is to carry out an experiment type like the Michelson-Morley but modified, so that at least one mirror is in relative movement in relation to the other parts of the experiment. The results of an experiment of this nature would be the verdict on the constancy or not of the speed of light, since the predictions are totally different made according to Classical Mechanics and Relativistic Mechanics, so there could be no doubt about which of they are right or wrong. 2 Vavel (Variación de la Velocidad de la Luz) experiment This experiment was carried out in 1995. The topic of VAVEL’s experiment is to move one of the mirrors of Michelson-Morley’s experiment and check if such movement changes anyhow the beam reflected. No model is known that predicts what happens with the reflected beam when the mirror is moving from the observer and from the light source. According with the actual theory nothing should change, so the angles, parameters and other characteristics of the reflected beam shouldn’t change in spite of mirror’s movement. In order to build the experiment and test the above mentioned, the interference spectrum of two homogeneous light beams is going to be studied. One of them shall be reflected on the static mirror and the other one shall be reflected on the mirror that is in relative movement referred to the observer, the first mirror and the light source. The light spectrum shall be obtained for different mirror relative speed in order to analyse the effect of speed on the reflected light beam and quantified such effect. As we have previously concluded, the experiment is basically composed of two cylindrical mirrors one in front of the other as we see in Figure-5. MIRROR-2 Observer MIRROR-1 Figure 5.- Basic design of the VAVEL experiment A monochrome light beam is reflected on mirror nº 2 and part of the reflected beam, is also reflected in mirror nº 1. This beam reflected and the beam part reflected in mirror nº 2 only, interfere with each other in the shady area in Figure-5, giving rise to an interference spectrum in that area. Next we make mirror nº1 spin and we analyse what happens to the interference spectrum. The simplified experiment's model is the one of the reflexions of a light beam on a mobile mirror, whose displacement velocity direction is parallel to the mirror. In order to design the experiment and analyse the results of the experiment, it was necessary to develop a mathematical model that, depending on the geometric parameters of the experiment, could predict what were the angles of incidence and reflection, could know how the movement of the mirror affected the reflected beam and could finally obtain, the expected interference spectrum. This mathematical model is the program LUZ. 2.1 Technical data of equipment The equipment of the experiment is comprised of two mirrors, a static one and a moving one, which consist on metal disks 10 mm. thick and with 250 mm. radius, whose outer surfaces are polished in a way that they work as mirrors. The light focus is a He-Ne laser tube with 5 mw power and high beam stability. Picture 1.- VAVEL equipment The interference spectrum detector is made up by a cylindrical plane-concave glass lens with a focal length of 12.7 mm. This lens amplifies the interference spectrum. Behind the lens there is a linear detector silicon array. The number of sensible elements (pixels) is 1024 and 25 microns spacing. A laser probe is a detecting device with software for sending the data to an oscilloscope. We have also a 20 Mhz analogic oscilloscope. An electrical motor, which can either rotate to the right or to the left, rotates the moving mirror. Rotation's velocity can be regulated in both cases. The equipment is completed with the necessary security, regulation and control elements. (See Picture-1). The maximum peripheral velocity of the moving mirror is 157 m/s. The reading velocity of silicon array detector is 8 microseconds for pixel, this is to say that it can read the whole spectrum in 8,2 milliseconds. When the moving mirror turns, its radius increases because of the dynamic stresses. We will also find some variations in the spectrum because the moving mirror is not a perfect circle, (being the maximum difference between two radius of 28 microns), and because there are vibrations when the mirror turns. So that we have looked for a mirror that can rotate to the right or to the left, because the mirror's deformation, the radius increase by dynamic stresses and the vibration's effects are the same, regardless what the spinning direction is. If we observe any variation between the spectrums when the mirror turns to right or to left at the same velocity, it will be caused by the velocity of the mirror and not by any other fact. 2.2 Results of experiment 2.2.1 Method used in the experiment research The experiment was set out in the following way: a) Firstly, we obtain a reference spectrum with both mirrors static to which we can refer the other interference spectrums. This spectrum taken with the still mirrors will be the reference during the rest of the experiment. b) We will make the mirror spin with a determined velocity and we will get the interference spectrum for that velocity. c) Next we make the mirror spin in the opposite direction and the same velocity as "b" point, so we will get another spectrum. d) We check the differences between both spectrums with the same velocity but opposite spinning direction. e) We go on making "b", "c" and "d" points for spinning velocities increasing it every time until we get the highest one allowed by the mirror. f) We introduce all the dimensional data and spinning velocities in LUZ program and obtain the predicted results. g) We compare the experimental data with the mathematical predictions 2.2.2 Experiment's development The experiment has been carried out separating the mirrors 2.1 mm. from each other. The distance from the exit of the Laser beam tube to the Y-axis, which links the mirror's centres was 447 mm. The Laser beams strike the static mirror with incidence angles between 85.54 and 90 degrees. The incidence beams strike the moving mirror with incidence angles between 81.72 and 90 degrees. The lens to amplifier the image is placed 243 mm. away from the above-mentioned Y-axis and it has a focal distance of 12.7 mm. The silicon array Picture 2.- Refence spectrum (V = 0) detector is placed behind the lens 476 mm. from it. It makes the image 36.48 times bigger. Considering that the distance between pixels in the silicon array is 25 microns, the final resolution of the spectrum's image will be 25/36.48 = 0.685 microns. In this condition we obtained the interference spectrum for a still mirror of the equipment, the results are shown in Picture-2, where we can see a spectrum made up by five complete waves whose wavelength is approximately 140 microns. We calculate it in the following way: • Length of Silicon array: 25 x 1024 = 25,600 microns • Length of detected spectrum: 25,600/36.48 = 701.75 • Number of waves in the spectrum: 5 • Wavelength: 701.75/5 = 140 microns. microns. Next we made the moving mirror spin with a velocity of +1,607 r.p.m., understanding the plus sign when the mirror is spinning the same direction that the light beam velocity and the peripheral velocity vector of the mirror, in the contact point. So we get the spectrum in Picture-3. Afterwards the mirror shall spin with velocity 1,611 r.p.m., understanding the minus sign when the mirror is spinning the opposite direction to the light beam velocity and the peripheral velocity vector of the mirror, in the Picture 3.- Interference spectrum for V=+1607 rpm contact point. So, we got the interference spectrum in Picture-4. Considering that the spinning velocities are practically the same value, the radius increase, as effect of the dynamic stresses, is the same. As the temperature is also the same, both spectrums should be the same as well. However, we can see that, having both the same wave shape, they are displaced one in relation to the other. We went on getting spectrums in the same Picture 4.- Interference spectrum for V=-1611 rpm way for different velocities with the same value but different direction in order to get rid of the effect that the increase of the radius causes. In the Picture-5 we have the spectrum for a mirror spinning with velocity +2,766 r.p.m. In the Picture-6 we have the spectrum for velocity -2,764 r.p.m. which is noticeably similar to the previous one, but the interference spectrums do not coincide being displaced relating to each other. Picture 5.- Interference spectrum for V=+2766 rpm In the same way we made the moving mirror spin with velocities +3,378 r.p.m. and -3,372 r.p.m. whose interference spectrums are shown in Picture-7, left for the positive velocity and right for the negative one. Both spectrums do not coincide either, even having been obtained in the same geometric conditions just changing the spinning direction. The fact that the spectrums obtained for practically the same velocities in both Picture 6.- Interference spectrum for V=-2764 rpm Picture 7.- Interference spectrum for V=+3378 rpm (left) and V=-3372 rpm (right) directions don't coincide means that the mirror velocity affects to reflected beams. The fact that the spectra for very similar rotation speeds, in one direction and on the contrary, do not coincide does not have an immediate explanation in the Model of Relativistic Mechanics because the spectrums should be practically the same in both cases. Next, we will try to apply the different mathematical models to the experiment to see how they explain it 2.3 Analysis of the obtained results We are going to apply the Model developed with the LUZ program, in which the rotation of the mirror affects to the reflected light beam in two ways: one varying the reflection angles and the other varying the speed of the light beam. The fact that the speed of light varies could have three consequences: that such variation affects the frequency or affects the wavelength or both. According to the results of the experiment, it is clear that the frequency cannot be affected since, in this case, the rays reflected in the moving mirror would not be coherent with the rays reflected in the fixed mirror and therefore there could not be interference spectrum. Given that said spectrum exists, the only thing that can be affected by the change in speed is the wavelength. Applying the mathematical model, we obtained the theoretical interference spectrum that can be seen below. Comparing these spectra with the pictures of the experiment, we can observe a large coincidence between the results of the Mathematical Model and the results obtained in the experiment. In order to rule out that the variation of the spectra at equal speeds, but with opposite directions of rotation were due to the effect of the rotation on the reflection angles, but not to the variation of the wavelength, other calculations were made in in which the effect on the wavelength was suppressed, or what is the same, the reflected and incident lightning speed was made the same and constant, that is, we applied a Model of Relativistic Mechanics. In the spectra calculated with this Model, we could see that the effect of the variation of the reflection angles is so small that it cannot be seen in the graphic representation of the spectra, being only appreciable in the numerical values of the calculation. Therefore, the only thing that can explain the variation of the interference spectra when the mirror rotates at equal speeds, but in the opposite direction, is that the rotation affects the wavelength of the reflected ray and, since the frequency remains constant while the spectrum still exists, it is necessary that the velocity of the reflected ray is different than that of the incident ray. Finally, based on the pairs of photographs of equal speed, measurements have been made on them that allow us to make a calculation, simply approximating, of how much the spectrum shift is worth by changing the direction of rotation, obtaining the following results. Since the spectrum occupies 8.4 divisions of the oscilloscope screen, this implies that each division of the screen is equivalent to 83.54 microns of the actual measured spectrum, since the sensitive length of the detector is 25,600 microns and the magnification due to the lens is 36.48 times, therefore, the actual length of the measured spectrum is 701.75 microns. Always taking as a reference the spectrum corresponding to positive velocity, we have: CASO W = 1.609 rpm v = + 1.607 rpm Displacement=25.06 μm. v = - 1.611 rpm Displacement to the right of the -1611 rpm spectrum in relation to the other: 0.3 divisions. Real displacement length in the experiment: 25.06 microns 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 16,4 Figure 6.- Interference spectrum in Classical Mechanics Model for V=+1607 rpm 16,5 16,6 16,7 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 16,4 16,5 16,6 16,7 Figure 7.- Interference spectrum in Classical Mechanical Model for V=-1611 rpm Displacement measured in the Mathematical Model through numerical values: 32 microns Deviation between model and experiment: (1-25,06/32) = 21,7% CASO W = 2.765 rpm v = + 2.766 rpm Displacement=50.12 μm. v = - 2.764 rpm Displacement to the right of the -2,764 rpm spectrum in relation to the other: 0.6 divisions. Real displacement length in the experiment: 50,12 microns 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 16,4 16,5 16,6 16,7 16,5 16,6 16,7 Figure 8.- Interference spectrum in Classical Mechanics Model for V=+2766 rpm 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 16,4 Figure 9.- Interference spectrum in Classical Mechanics for V=-2764 rpm Displacement measured in the Mathematical Model through numerical values: 54 microns Deviation between model and experiment: (1-50,12/54) = 7,18% CASO W = 3.375 rpm v = +3.378 rpm. Displacement=66,83 μm. v = - 3.372 rpm Displacement to right of the -3372 rpm spectrum in relation to the other: 0.8 divisions. Real displacement length in the experiment: 66,83 microns 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 16,4 16,5 16,6 16,7 16,4 16,5 16,6 16,7 Figure 10.- Interference spectrum in Classical Mechanics for V= +3378 rpm 1,2 1 0,8 0,6 0,4 0,2 0 15,8 15,9 16 16,1 16,2 16,3 Figure 11.- Interference spectrum in Classical Mechanics for V= -3372 rpm Displacement measured in the Mathematical Model through numerical values: 67 microns Deviation between model and experiment: (1-66,83/67) = 0,25% Considering that the mistakes made in the measurement on the photographs can be significant, it is important to note the great coincidence between theoretical and experimental values. 3 Conclusions As we have demonstrated before, both theoretically and experimentally, maintaining the constancy of the speed of light in vacuum, for any inertial frame and for any observer, is not possible. We believe it is important that the VAVEL experiment can be reproduced in other laboratories in order to check if the results obtained by us are reproducible and true and if they are true, as scientists, accept the consequences and modify the current physical models for it. 4 Last objection Many people will ask if the speed of Light is not constant, what happens to the famous equation of E = mxc2? I have developed a physical model considering the new situation. The model is based on the fact that our reality moves in a four-dimensional reference system (three spatial axes and one temporal axis), and therefore there is a spatial velocity of expansion and temporal velocity of expansion, that is, we move in the temporal axis as we move in the space axis. With these bases we have developed a very similar equation that is: E = mxf2, where E is the energy, m is the mass measured by a static observer and f is the speed of displacement of our reality in the time axis. Therefore, new models will have to be developed opening a very interesting field. 5 Figure list Figure 1.- Michelson-Morley experiment ..................................................................................... 5 Figure 2.-Beam perpendicular to the motion of the Earth ........................................................... 7 Figure 3.- Electric and magnetic field in relative motion ............................................................ 13 Figure 4.- Fizeau interferometer scheme .................................................................................... 16 Figure 5.- Basic design of the VAVEL experiment ....................................................................... 20 Figure 6.- Interference spectrum in Classical Mechanics Model for V=+1607 rpm .................... 26 Figure 7.- Interference spectrum in Classical Mechanical Model for V=-1611 rpm ................... 27 Figure 8.- Interference spectrum in Classical Mechanics Model for V=+2766 rpm .................... 28 Figure 9.- Interference spectrum in Classical Mechanics for V=-2764 rpm ................................ 28 Figure 10.- Interference spectrum in Classical Mechanicss for V= +3378 rpm ........................... 29 Figure 11.- Interference spectrum in Classical Mechanics for V= -3372 rpm ............................. 29 6 Picture list Picture 1.- VAVEL equipment ...................................................................................................... 21 Picture 2.- Refeence spectrum (V = 0) ........................................................................................ 22 Picture 3.- Interference spectrum for V=+1607 rpm .................................................................. 23 Picture 4.- Interference spectrum for V=-1611 rpm ................................................................... 23 Picture 5.- Interference spectrum for V=+2766 rpm .................................................................. 24 Picture 6.- Interference spectrum for V=-2764 rpm ................................................................... 24 Picture 7.- Interference spectrum for V=+3378 rpm (left) and V=-3372 rpm (right) .................. 24