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On the Galilean and Lorentz Transformations American Open Advanced Physics Journal Vol. 1, No. 2, November 2013, PP: 11 - 32 Available online at http://rekpub.com/Journals.php Research article On The Galilean and Lorentz Transformations Yuan-De Tan Department of Physics, Hunan Normal University, Changsha, Hunan, 410081, P.R. China E-mail: [email protected] Abstract Mathematical proofs show that Lorentz transformation holds if and only if x ct and x' ct' where c is light speed. However, unlike Galilean transformation, Lorentz transformation cannot lead to addition of velocities that can be well explained by a physical mechanism that motion of a body is independent of a frame moving relative to a stationary frame. We extend Lorentz transformation to a general situation of V >> v where V and v are velocities of a moving body and a moving frame, respectively. Galilean transformation can be well derived from the physical mechanism that a moving body is completely carried by a frame moving relative a stationary frame. We also prove that Lorentz transformation cannot be reduced to Galilean transformation no matter how small velocity of a moving frame is. The Lorentz and Galilean transformation systems are two extreme cases and a general case is “partial contact and carry” based on which we propose a new transformation. The new transformation is reduced to Lorentz transformation when a moving body is absolutely independent of the moving frame or to Galilean transformation when the body is completely carried by the frame. Interestingly but unsurprisingly, the new transformation naturally leads to the result of Fizeau‟s experiment. Keywords: Galilean transformation, Lorentz transformation, relativity theory, referential frames, addition of velocities, light speed PACS Nos: 01.55.+b, 03.30.+p 11 Copyright © rekpub.com, all rights reserved. American Open Advanced Physics Journal Vol. 1, No. 2, November 2013, PP: 11 - 32 Available online at http://rekpub.com/Journals.php Introduction Galilean transformation was intuitively given by Galileo based on his description of uniform motion and an assumption that time is absolute, which is at the core of Galilean transformation [1]. To our knowledge, indeed, no one gives its theoretical derivation because it may be self-evident for most physicists. As basic knowledge of classic physics, it is well known that Galilean transformation satisfies invariance of space and time intervals and invariance of acceleration [1,2], which makes invariance of Newton‟s laws hold well. Therefore, Galilean transformation is foundation of Newton mechanics. However, Galilean transformation is subjected to a big difficulty in application to the particle motion. For example, when Galilean transformation is used to convert Maxwell equations between two frames, it generates asymmetric forms between referential frames: very simple in a stationary frame of reference but quite complicated in a moving frame [3,4]. Again, invariance of light speed c that Michelson and Morley found [5,6] conflicts with addition of velocities in Galilean transformation. To explain the null results of Michelson and Morley‟s experiments under the assumption that ether is luminiferous media for light propagation, in 1904 Lorentz developed a new transformation between two frames of reference [7], now called Lorentz transformation. Based on Lorentz transformation, in 1905 Einstein proposed his special theory of relativity [8] where Einstein used light speed as a speed limit of universe (the second postulation) to explain invariance of light speed and used his addition of velocities to explain the result of Fizeau‟s experiment of light in the flowing water [9] and refuted effect of water dragging light. The special theory of relativity has a great attraction due to its time dilation and length contraction. However, in theory, although there are many approaches [10-26] to derive the Lorentz transformation, except for Einstein‟s derivation [10] based on x ct and x' ct' , none of them can theoretically prove v / c in the Lorentz factor. In particular, Einstein‟s addition of velocities does not hold because we will prove that Lorentz transformation cannot lead to velocity addition. In experiments a lot of tests [27-41] were conducted for Einstein‟s special theory of relativity, but most of these experiments focused on testing for invariance and independence of light speed, isotropy of universe, and luminiferous ether [42,43]. The results of these experiments, as indicated by Gezari [43], are not robust to support this theory because they are not specific for special relativity. In addition, many textbooks and books of physics [44-48] specially emphasize that Newton theory requires an absolute stationary frame and absolute time, whereas in special theory of relativity space and time are relative. On the other hand, they all indicate that special theory of relativity is reduced to Newton mechanics in the case of low velocities because 12 Copyright © rekpub.com, all rights reserved. American Open Advanced Physics Journal Vol. 1, No. 2, November 2013, PP: 11 - 32 Available online at http://rekpub.com/Journals.php Lorentz transformation is deemed to be reduced to Galilean transformation when the velocity of frame S' moving relative to a stationary frame is much lower than light speed c, that is, v / c 0 [44-48]. These indicate that in the physical mechanism, the distinction between Galilean and Lorentz transformations have not been made so far. The physical relationship between a body or particle and a moving frame is not described in all books of physics. In this paper, we first prove in mathematics that, unlike Galilean transformation, Lorentz transformation cannot give velocity addition. We then give the physical mechanisms to derive Galilean transformation and to explain why Lorentz transformation has the same velocity in two referential frames and Galilean transformation has the same time in two referential frames, why Lorentz transformation instead of Galilean transformation can be used to convert Maxwell‟s electromagnetic wave and why light speed is invariant in vacuum. Finally we develop a general transformation for motion of a body in a flowing media and theoretically derive the result of Fizeau‟s experiment [9] based on the new transformation. We also prove in mathematics that even at a very low velocity, Lorentz transformation cannot be reduced to Galilean transformation and show that Galilean transformation definitely works only in mechanical movement with low velocities in most cases while Lorentz transformation works in movements of a body independent of two frames with either high or low velocities. Methods We used the basic kinematic equations to give self-consistency principle and based on this principle, we proved two theorems with regard to Lorentz transformation. These two theorems demonstrate that Lorentz transformation does not lead to Einstein addition of velocities. To interpret Galilean and Lorentz transformations we proposed "complete carry" and "no carry" mechanisms. "Complete carry" logically deduces invariance of time while "no carry" gives invariance of light speed. This is very well consistent with no addition of velocities in Lorentz transformation. It is easy to find that "no carry" and "complete carry" are not related to velocity of the moving frame, which is completely agreeable with mathematical demonstration that Lorentz transformation cannot be reduced to Galilean transformation no matter how small velocity of the moving frame relative to light speed is. "No carry", like light in vacuum, and "complete carry", like man walking on a moving train, are two extreme cases. Thus, logically, there should be "partial carry" in between "no carry" and "complete carry", that is, there should be a general transformation due to "partial carry". The typical example for "partial carry" is light moving in water 13 Copyright © rekpub.com, all rights reserved. American Open Advanced Physics Journal Vol. 1, No. 2, November 2013, PP: 11 - 32 Available online at http://rekpub.com/Journals.php flowing in a pipe. In order to understand "partial carry", we gives a clear example in Figure 3 where a motorcycleS' flies over gaps between containers and runs on containers on a moving train. For flying over gaps, the motorcycle motion relative to the train and railway can be described by Lorentz transformation because the motorcycle is not carried by the train, for running on containers, the motorcycle motion relative to the train and railway can be described by Galilean transformation because the motorcycle is carried by the train. When intervals of gaps and length of containers become infinitesimal, what we see just likes light moving in the flowing water. Therefore, in the "partial carry" system, Galilean linear velocity addition and Lorentz invariant velocity form an Einstein-like nonlinear addition of velocities. This nonlinear addition of velocities is demonstrated by the result of Fizeau experiment[9]. Results and Discussion On addition of velocities Einstein‟s addition of velocities [10] which is derived from Lorentz transformation is one of important components of special theory of relativity. But, as mentioned above, unlike Galilean transformation, Lorentz transformation does not have a property of addition of velocities. To prove this point, let us write Lorentz transformation formula as x'vt' x (1) 1 (v / c)2 t'vx' / c 2 t (2) 1 (v / c)2 where x and x' , t and t' are distances and times of motion of a rigid body in referential frames S and , respectively. Here z z' and y y' are ignored for the convenience. v is velocity of frame moving relative to frame S at rest and c is speed of light in vacuum. From Eq.s (1) and (2), Einstein‟s addition of velocities in the special theory of relativity is 14 Copyright © rekpub.com, all rights reserved. American Open Advanced Physics Journal Vol. 1, No. 2, November 2013, PP: 11 - 32 Available online at http://rekpub.com/Journals.php u'v u (3) 1 u'v / c2 where u x / t (x 0) /(t 0) x / t and u' x'/ t' (x'0) /(t'0) x'/ t' .
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