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Error in Modern Astronomy Error in Modern Astronomy Eric Su [email protected] https://sites.google.com/view/physics-news/home (Dated: August 1, 2019) The conservation of the interference pattern from double slit interference proves that the wave- length is conserved in all inertial reference frames. However, there is a popular belief in modern astronomy that the wavelength can be changed by the choice of reference frame. This erroneous belief results in the problematic prediction of the radial speed of galaxy. The reflection symmetry shows that the elapsed time is conserved in all inertial reference frames. From both conservation properties, the velocity of the light is proved to be different in a different reference frame. This different velocity was confirmed by lunar laser ranging test at NASA in 2009. The relative motion between the light source and the light detector bears great similarity to the magnetic force on a moving charge. The motion changes the interference pattern but not the wavelength in the rest frame of the star. This is known as the blueshift or the redshift in astronomy. The speed of light in the rest frame of the grism determines how the spectrum is shifted. Wide Field Camera 3 in Hubble Space Telescope provides an excellent example on how the speed of light can change the spectrum. I. INTRODUCTION location of the light band is y1. The separation between the parallel slits is d1. The double slit interference can be formulated with the If d1 << y1 and d1 << D1, the constructive interfer- principle of superposition[1]. The interference pattern is ence can be described by the equation of phase shift[1] a function of the incident wavelength. The pattern is for the constructive phase difference as conserved in all inertial reference frames. This conser- p 2 2 vation determines how the wavelength may change in a D1 + y1 y1 = m ∗ λ1 ∗ (1) different inertial reference frame. d1 The spectrum from a diffraction grating can be for- mulated from the double slit interference. It exhibits the λ1 is the wavelength in F1. m is a positive integer. same conservation property. The spectrum is also a func- q 2 2 tion of the wavelengths. λ1 = λ1x + λ1y (2) However, the spectrum changes if there is relative mo- tion between the light source and the diffraction grating. The derivation of equation (1) is located next to the con- In the rest frame of the light source, the wavelength is in- clusion section. tact but the spectrum becomes different. Therefore, the spectrum is actually a function of both the wavelength and the relative motion. The correct equation for double slit interference will be derived to include the relative motion. This is similar to the magnetic force on a moving charge. The relative motion must be incorporated to account for the spectral shift. II. PROOF FIG. 1. Double Slit Interference A. Double Slit Interference Let another reference frame F2 move at a velocity of (- v,0) relative to F1. The interference pattern is conserved A light emitter emits coherent light along the x- in all inertial reference frames because the emitter is sta- direction through a plate with two parallel slits to reach tionary relative to F1. In F2, the conservation of phase a projection screen. Both the plate and the screen are shift is represented by aligned with the y-z plane. The emitter, the plate, and pD2 + y2 the screen are all stationary relative to a reference frame y = m ∗ λ ∗ 2 2 (3) 2 2 d F1. 2 A series of alternating light and dark bands appear on the projection screen along the y-direction. Let the q 2 2 distance between the plate and the screen be D1. The λ2 = λ2x + λ2y (4) 2 The choice of inertial reference frame along the x- C. Doppler Effect direction has no effect on the measurement along the y-direction. Let an observer P1 be stationary relative to F1. The emitter is stationary relative to P1. The frequency of the y2 = y1 (5) light is f1 for P1 Let another observer P2 be stationary relative to F2. d2 = d1 (6) The emitter moves toward P2. The frequency of the light is f2 for P2. According to the Doppler effect[2], λ2y = λ1y (7) f > f (19) From equations (1,3,5,6), 2 1 q q The speed of light for P1 is 2 2 2 2 λ2 D2 + y2 = λ1 D1 + y1 (8) c1 = f1 ∗ λ1 (20) The choice of inertial reference frame along the x- direction may alter the measurement along the x- The speed of light for P2 is direction. Let γ be the proportional factor between the c2 = f2 ∗ λ2 (21) original measurement in F1 and the new measurement in F2. From equations (18,19,20,21), D2 = γ ∗ D1 (9) c2 > c1 (22) The speed of light increases if the light emitter moves λ2x = γ ∗ λ1x (10) toward the observer. From equations (4,7,10), q D. Conservation of Length 2 2 2 λ2 = γ ∗ λ1x + λ1y (11) From equations (9,17), From equation (8,9,11), 2 2 2 2 2 2 2 2 2 D2 = D1 (23) (γ λ1x +λ1y)((γ∗D1) +y1) = (λ1x +λ1y)(D1 +y1) (12) The length is conserved in all inertial reference frames. γ4λ2 D2 + γ2(λ2 D2 + y2λ2 ) + λ2 y2 (13) Therefore, length contraction from Lorentz transforma- 1x 1 1y 1 1 1x 1y 1 tion is proven to be incorrect. 2 2 2 2 = (λ1x + λ1y)(D1 + y1) (14) E. Elapsed Time (γ4 − 1)λ2 D2 + (γ2 − 1)(λ2 D2 + y2λ2 ) = 0 (15) 1x 1 1y 1 1 1x Let t2 be the time of F2. Let the projection screen be at the location of (r,0) in F2. The projection moves at 2 2 2 2 2 2 2 2 the velocity of (v,0) in F2. (γ − 1)(λ1xD1(γ + 1) + λ1yD1 + y1λ1x) = 0 (16) dr v = (24) γ2 − 1 = 0 (17) dt2 The choice of inertial reference frame along the x- Let t1 be the time of F1. Let the projection screen be direction does not alter the measurement along the x- at the origin in F1. The origin of F2 is located at (-r,0) direction. in F1 and moves at the velocity of (-v,0) in F1. d(−r) −v = (25) B. Conservation of Wavelength dt1 From equations (24,25), From equations (2,11,17), dt1 = dt2 (26) λ2 = λ1 (18) The elapsed time is conserved in all inertial reference The wavelength is conserved in all inertial reference frames. Therefore, time dilation from Lorentz transfor- frames. mation is proven to be incorrect. 3 F. Velocity of Light G. Isotropy of Light Let ~c1 = (c1x; c1y) be the velocity of light in F1. Let The speed of light in F2 depends on the choice of ref- T1 be the elapsed time for the light to travel from the slit erence frame and also the orientation. q plate to the projection screen in F1. 2 2 c1 ∗ T1 = D1 + y1 (37) c ∗ T = y (27) 1y 1 1 q 2 2 c2 ∗ T2 = (D2 + v ∗ T2) + y2 (38) p 2 2 c1x ∗ T1 = D1 (28) Place a rod of length D1 + y1 along the x-direction in F1. Let the light pass through the whole length of this Let ~c2 = (c2x; c2y) be the velocity of light in F2. Let rod. T2 be the elapsed time for the light to travel from the slit T3 is the elapsed time for the light to travel from end plate to the projection screen in F2. to end of the rod in F1. q c ∗ T = D2 + y2 (39) c2y ∗ T2 = y2 (29) 1 3 1 1 T4 is the elapsed time for the light to travel from end to end of the rod in F2. Let c0 be the speed of light in c ∗ T = D + v ∗ T (30) 2x 2 1 2 the x-direction in F2. q 2 2 The screen has moved an extra distance of v ∗ T2 during c0 ∗ T4 = D1 + y1 + v ∗ T4 (40) T2. From equations (5,27,29), p 2 2 D1 + y1 T4 = (41) c0 − v c1y ∗ T1 = c2y ∗ T2 (31) From equations (37,39), From equations (28,30), T3 = T1 (42) Therefore, c2x ∗ T2 = c1x ∗ T1 + v ∗ T2 (32) T4 = T2 (43) From equation (26), the elapsed time is conserved in all inertial reference frames. From equations (38,40), p 2 2 c2 ∗ T2 (D2 + v ∗ T2) + y T = T (33) = 2 (44) 1 2 c ∗ T p 2 2 0 4 D1 + y1 + v ∗ T4 From equations (32,33), From equation (43,44), p 2 2 c2 (D2 + v ∗ T2) + y c2x = c1x + v (34) = 2 (45) c p 2 2 0 D1 + y1 + v ∗ T2 From equations (31,33), From equation (5,45), p 2 2 c = c (35) (D2 + v ∗ T2) + y 2y 1y c = c 2 (46) 2 0 p 2 2 D2 + y2 + v ∗ T2 From equations (34,35), Let the diffraction angle θ2 be y2 ~c2 = ~c1 + (v; 0) (36) tan(θ2) = (47) D2 The speed of light in F2 is greater than the speed of light From equations (46,47), in F1 if v > 0. q v∗T2 2 2 (1 + ) + tan(θ2) The velocity of light is ~c1 and is constant in the rest D2 c2 = c0 < c0 (48) frame of the emitter. It changes to ~c2 only in another p 2 v∗T2 1 + tan(θ2) + inertial reference frame. D2 In 2009, the speed of laser was measured with lunar c2 is equal to c0 only if θ2 is zero.
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