On the Galilean and Lorentz Transformations

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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

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

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

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 motorcycle 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 S' , 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

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' . Eq.(3) does not hold. The proof is given by theorems 1 and 2.

Theorem 1: Eq.s (1) and (2) for kinematic relationship among distances, times, and velocities of a rigid body moving in two referential frames hold if either x' ct' and x  ct or x' = 0 and x  vt.

Proof: Given distance x, time t and velocity v, three simple kinematic equations of a moving body are expressed as

x  vt, (4)

t  x / v , (5)

v  x / t . (6)

Eq.s (4) ~ (6) hold because anyone of them can be derived from one of two others. For example, Eq. (4) can be given from Eq. (5) multiplied by velocity v or from Eq. (6) multiplied by time t:

vt  vx / v  x or vt  x / t t  x .

Eq. (5) also can be obtained from Eq.s (4) divided by velocity v

x (vt)   t . v v

This is called "self-consistency" principle. Similarly, in Galilean transformation,

x  x'vt' (7)

t  t' , (8) we can derive Eq. (8) from Eq.(7):

x  u't'vt' (u'v)t' ut  (u'v)t'

ut  ut' t  t' where x  ut and u  u'v (addition of velocities). We also can get Eq. (7) from Eq.(8) multiplied by velocity u.

According to this “self-consistency” principle, Eq.(1) can be derived from Eq. (2) multiplied by speed of light c:

t'vx' / c2 ct'cvx' / c2 ct'vx' / c ct'v(x' / c) ct( x)  c    1 (v / c)2 1 (v / c)2 1 (v / c)2 1 (v / c)2 (9) ct'vt' x'vt'   1 (v / c)2 1 (v / c)2 where x  ct and x' ct' . In the case of x' 0 and x  vt  0, Eq. (1) can be obtained from Eq.(2) multiplied by velocity v

vt' vt( x)  . (10) 2 1 (v / c)

However, In the case of x' 0 , Eq.(1) cannot be obtained from Eq.(2) multiplied by velocity u  (u'v) /(1 u'v ) : c2

v u'v vx' u'v v u'v u(t' 2 x') [(u'v) /(1 2 )]t' 2  t' 2  2 x' 2  ut( x)  c  c c  1u'v / c c 1u'v / c 1 (v / c)2 1 (v / c)2 1 (v / c)2

u'v v u'v t' 2  u't' 2  u'v u'v u'v 1u'v / c 2 1u'v / c  2 t' 2  2 t' x'vt'  c  1u'v / c c 1u'v / c  (11) 1 (v / c)2 1 (v / c)2 1 (v / c)2

u'v u'v u'v u'v because of (u'v) /(1 2 )  u' and 2  2   v where u  2 is Einstein's addition of velocities ( Eq. c c 1u'v / c 1u'v / c

(3)), 0  u  u'v  c , v  0 , and x' u't' where 0  u' c . Eq.(1) also cannot be derived from multiplying 1u'v / c2 both sides of Eq. (2) by velocity u' :

u't' u'v x' x' u'2 v x' x' u'2 vt' c2 c2 u' c2 x'vt' u't( x)     1 (v / c)2 1 (v / c)2 1 (v / c)2 1 (v / c)2

because of u'2 / c2  1 or u'2 v / c2  v when 0  u' c . Given the Lorentz transformation, Eq.s (1), and (2), then x  ct and x' ct' can be directly proved: Let x  ut and x' u't' . Eq. (1) is rewritten as

u't'vt' ut  . (12) 1 (v / c)2

Dividing both sides of Eq.(12) by u, we have

u' v u' v x' u' (t' v x') u't'vt' u (t' u' t') u (t' u' u' ) u u'2 t     . (13) u 1 (v / c)2 1 (v / c)2 1 (v / c)2 1 (v / c)2

Eq. (13) can become into Eq.(2) if and only if u  u' c . End

Theorem 2: Given Eq.s (1) and (2), Lorentz transformation holds if and only if x' ct' and x  ct .

Proof: (a) According to theorem 1, for x' u't' ct', x' u't' 0 , Eq.s (1) and (2) do not hold, therefore, Lorentz transformation does not hold.

(b) In the case of x' u't' 0 and t' 0 , it is clear that Lorentz transformation does not hold because it does not obey

x  vt x'vt' the Lorentz invariance, or x' u't' 0 conflicts with x'  where x  1 v / c2 1 (v / c)2

vt'   0 . 1 (v / c)2

x'vt' ct'vt' ct'(1 v ) x'(1 v ) x    c  c  ' x' (14) 1 (v / c)2 1 (v / c)2 1 (v / c)2 1 (v / c)2

t' v x' v x' v v c2 t' c c t' c t' t'(1 c ) t      't' . (15) 1 (v / c)2 1 (v / c)2 1 (v / c)2 1 (v / c)2

1 v / c Here both t and x have a common factor' . Therefore, both sides of Eq.s (14) and (15) divided by 1 (v / c)2 factor ' yield

' x' x  x' x /' x 1 (v / c)2 1 (v / c)2 (1 v / c)(1 v / c) (1 v / c) x'  x  x  x (1 v / c) (1 v / c)2 (1 v / c)2 (1 v / c) (1 v / c)(1 v / c) 1 v / c x  (v / c)x ct  (v / c)ct (16)  x  x   x   (1 v / c)(1 v / c) 1 (v / c)2 1 (v / c)2 1 (v / c)2 x  vt  , 1 (v / c)2

't' t  t' t /' t 1 (v / c)2 1 (v / c)2 (1 v / c)(1 v / c) (1 v / c) t'  t  t  t  (1 v / c) (1 v / c)2 (1 v / c)2 (1 v / c) v x (1 v / c)(1 v / c) 1 v / c t  tv/ c t  t  t   t   c c (1 v / c)(1 v / c) 1 (v / c)2 1 (v / c)2 1 (v / c)2 (17) v t  2 x  c . 1 (v / c)2

Similarly, t' and x' also have a common factor  (1 v / c) / 1 (v / c)2 . The reversal process is that both sides of

Eq.s (16) and (17) divided by  yields

 x  x' x  x' /  1 (v / c)2 (1 v / c)(1 v / c) (1 v / c) (1 v / c)(1 v / c) x  x'  x'  x'  x' (18) 1 v / c (1 v / c)2 (1 v / c) (1 v / c)(1 v / c) 1 v / c x'(v / c)x' x'(v / c)ct' x'vt'  x'    . (1 (v / c)2 (1 (v / c)2 (1 (v / c)2 (1 (v / c)2

 t  t' t  t' /  1 (v / c)2 (1 v / c)(1 v / c) (1 v / c) (1 v / c)(1 v / c) t  t'  t'  t'  t' (19) 1 v / c (1 v / c)2 (1 v / c) (1 v / c)(1 v / c)

v 1 v / c t'(v / c)t' t'(v / c)(x' / c) t' 2 x'  t'    c . 2 2 2 v 2 (1 (v / c) (1 (v / c) (1  (v / c) (1 ( c )

Thus, Eq.s (16) and (18) allow x and x' to be directly transformed to each other and Eq.s (17) and (19) directly exchange between t and t' . End

Theorem 2 indicates that Lorentz transformation can directly be realized in mathematics without needing light spherical (Minkowski) equations

x2  y2  z2  c2t2  0, x'2 y'2 z'2 c2t'2  0 . (20)

In fact, Einstein‟s derivation[3] of Lorentz transformation was also based on x  ct and x' ct' . However, in the

2 2 2 2 light spherical (Minkowski) equations, x  c t  (y  z )  ct and x' c2t'2 (y'2 z'2 )  ct' unless y  y' 0 and z  z' 0 . It is necessary to indicate that, the light spherical equations cannot separately realize the

Lorentz exchanges between x and and between t and but realize transformation from a light spherical equation to another via Lorentz transformation equations. Thus, theorem 2 indicates that Eq. (3) is either meaningless for addition of velocities in the cases of u' c and u  c or u' 0 or does not hold in case of 0  u' c . In other words,

Lorentz transformation cannot lead to addition of velocities. From Eq.s (14) and (15), we can find

x x'' x' ct' ct' u     u'   c . However, as seen above, Galilean transformation can give velocity t t'' t' t' t' addition. 19

Corollary: From theorems 1 and 2, we can easily extend Lorentz transformation to a general form. For anyV  v ,

v t' x' x'vt' 2 x  and t  V (21) 1 (v /V )2 1 (v /V )2

v t  x x  vt 2 x' and t' V (22) 1 (v /V )2 1 (v /V )2

hold if and only if x' Vt 'and x  Vt . Eq.s (21) ~ (22) imply that Lorentz transformation also occurs in the case of velocity V < c and V = c is a special case.

Physical mechanisms of the Galilean and Lorentz transformations

An important concept that is one of cores in Einstein‟s special theory of relativity is that Lorentz transformation can be reduced to Galilean transformation when velocity (v) of referential frame S' moving relative to frame S is much small such that v / c  0 . This concept is actually incorrect because suppose that a body moves in referential frame with light speed c, from Eq.s(1) and (2), we have

t'(v / c2 )x' t' v x' t'0 x' t   c c  c  t  t' (23) 1 v / c2 1 v / c2 1 02

x'vt' x   x  x'vt'  ct  ct'vt' ct  (c  v)t' c  c v (24) 1 v / c2

where and v  0 . Eq.(24) indicates that Lorentz transformation cannot be reduced to Galilean transformation no matter how small velocity of frame moving relative to frame S is. In effect, they have completely different physical mechanisms that have been ignored so far. In other words, the two transformations deal with completely different physical relationships of a moving body to a moving frame .

According to the Galileo‟s ship, the physical mechanism for Galilean transformation is that a body moving with

a velocity ( u' ) in frame S' that moves with a constant velocity (v) relative to stationary frame S physically contacts

frame such that frame carries the body(Figure1). In such a system frame S is stationary relative to a moving

frame and frame is also stationary relative to a moving body. There are many examples for this system. The

most well-known example is a man walking on a train moving along a railway. The railway or embankment is

stationary relative to a moving train and the train is stationary relative to the man who is walking on the deck in the

direction of train moving. Suppose a stationary frame S and a moving frame are coincident at time of 0 and a

Figure 1: Galilean system Galilean system consists of a stationary frame S, a moving frame S' , and a moving body. The body contacts frame and so is carried by frame S' . There are two cases A and B. A: Frame moves with velocity v along coordinate x of stationary frame S. A body moves along coordinate x' of frame . Frame S is stationary relative to moving frame but the latter is also stationary relative to the moving body. Motion of body P on frame has the same time rule with that on frame S because motion of body P on frame has the same time rule with motion of frame on frame S, that is, t  t'. But since body P is carried by frame , the distance x of body P moving away from origin O is larger than the distance x' of body P moving away from origin O' , i.e., x  ut  x' u't' . leads to u  u' . B: A body moves on a surface formed by coordinates x' and z' of frame that moves relative to frame S at rest. Relative to frame , frame S is stationary and relative to body P, frame is stationary. Here the velocity of body P on coordinate x' is a component velocity of the body moving on the surface of coordinates x' and z' : 2 2 u' ux'  w  uz'

body is at origin O'of frame . Let frame and the body simultaneously start to move. Frame moves with

velocity (v) along coordinate x of frame S at rest and the body moves with velocity ( u' ) on coordinate x' of frame

. Relative to the moving frame , frame S is stationary and hence origin O of frame S is fixed. Thus, the

coordinate rules for distance and time of frame moving away from origin O along coordinate x are not changed

with motion of frame . As a result, for an observer at origin O' of frame , the distance xv of frame moving

away from origin O is a linear function of time t' :

xv  vt' . (25)

Relative to the moving body, frame S' is stationary and hence origin O'is fixed. Thus the coordinate rules for distance and time t' of the body moving away from origin O'also do not vary with motion of the body. Therefore, for the body, the distance of the body moving away from origin O' is also a linear function of time t' :

x' u't' . (26)

Likewise, relative to the moving body, frame S is stationary and origin O is fixed. So, similarly, for an observer at origin O of frame S, the distance of the body moving away from origin O is a linear function of time t:

x  ut . (27)

Since the body is physically carried by frame , the distance of the body moving away from origin O is sum of the distances of frame moving away from origin O and of the body moving away from origin O', that is,

x  x'xv  x'vt' (28)

Rearrangement of Eq.(28) gives

x  u't'vt' ut  (u'v)t'. (29)

If u  u'v , we must have t  t' . Thus, Galilean transformation is derived from the “physical contact and carry” mechanism and Galilean addition of velocities, u  u'v , is based on the same time in these two frames of reference. So in Galilean system, we must have u  u'but .

In Maxwell theory of electromagnetism, electromagnetic wave freely propagates in vacuum and is not carried by any object or frame. Therefore the physical mechanism for Lorentz transformation is that a particle moving with a constant velocity ( u' ) in frame that moves with a constant velocity v (v < ) in frame S does not physically contact frame S' such that frame does not carry the particle (Figure 2). In other words, when motion of the particle is relative to frame S, frame may be ignored and free for the particle. Therefore, velocity (u) of the particle moving relative to frame S does not contain velocity (v) of frame moving relative to frame S, that is, u  u' . Relative to frame S, frame is moving with velocity v. Relative to the moving particle, frame is also

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 moving with velocity v. Thus the coordinate rules for distance x' and time t' of the particle moving away from the origin O' of frame S' are changed with motion of frame because the moving particle possesses of a moving origin of frame with time t (or coordinate x). As a result, the distance of the particle moving away from origin O'of frame is linear function of time (change in coordinate rule of distance is the same with that in coordinate rule

of time ). On the other hand, because frame S is at rest, at different points of time t, for example, t1 and t 2 ,the

Figure 2: Lorentz system

Lorentz system consists of a stationary frame S, a moving frame S' , and a moving body P. The body does not contact frame and so is not carried by frame S' . There are two cases A and B. A: Frame moves with velocity v along coordinate x of stationary frame S. Body P moves along coordinate x' of frame . Frame S is stationary relative to moving frame but the latter is also moving relative to the moving body. When body P moves relative to frame S, frame can be ignored and free for the body. This means that velocity (u) of the body moving relative to frame S does not contain velocity (v) of frame moving relative to frame S, so u  u'. In addition, frame S is stationary relative to the moving body but frame moves toward the body, so distance ( x' ) of the body moving away from origin O' is contracted. Thus distance (x) of the body moving away from origin O is larger than that ( ) of the body moving away from origin , that is , x  ut  x' u't' . leads to t  t', which is contrary to that in Galilean system. In particular, if the body moves with light speed c, then u  u' c . B: A body flies in the empty three-dimension space of frame that moves along coordinate x of frame S. Since body P is a free particle, it does not contact frame and hence is not carried by frame . Note that velocity (V) of particle moving on coordinate x' is a component of velocity , that is w' p 2 2 2 . V  u  u' u'x'  w' p u' y' u'z' particle P fixes origin O of frame S and hence the coordinate rule for distance of the particle moving away from origin O does not vary with time t, therefore, the distance of the particle moving away from the fixed origin of frame

S is a linear function of time t. But, as seen in Figure 2, due to the fact that frame is moving toward the particle while frame S is being at rest, the distance x of the body moving away from origin O of frame S is larger than distance x' of the particle moving away from the origin O' of frame , that is, x  ut  x' u't' . From u  u', we get t  t'. On the basis of this mechanism, we have got derivation of Lorentz transformation elsewhere [49].

Since a ray of light freely propagates with the refraction coefficient of 1 in vacuum, it is not carried by the earth.

Thus the velocity of light should be always c wherever and whenever we measure it. This is why all the results of

Michelson -Morley‟s and Trouton-Noble‟s experiments [5,6,27] were negative in different directions. Due to the fact that a light ray obeys the “no contact and no carry” mechanism, using Galilean transformation to transform

Maxwell‟s equations must get a wrong result.

“Partial contact and partial carry” mechanism and its transformation

Velocity of light moving in the rest water is c/n that is smaller than c where n, refraction coefficient, is larger than 1, indicating that the water drags light such that its speed becomes slower in water than in vacuum. So when a ray of light runs in the flowing water, the light ray is partially carried by the flowing water. What is “partial carry”?

Let us image that a man driving a motorcycle runs on the containers and flies across gaps between containers on a train in the direction of a train moving along a railway (Figure 3). When the motorcycle runs on a container, the motorcycle is carried by the train; when it flies in a gap, it does not touch the train and is also not carried by the

train. So the whole process of the man driving

motorcycle to run on the containers and to fly

across the gaps of the train is a “partial contact

and partial carry” process. Light running in the

flowing water is in between the case in which

the man walks on the train moving along a

Figure 3: „‟Partial contact and carry” system. railway and the case in which a light ray flies in A train carries many containers and moves with velocity v relative to embankment. A man drives a motorcycle with velocity u >> v on containers vacuum. Thus, as mentioned above, Galilean and flies across gaps between containers in the same direction of train moving along railway. On containers, the motorcycle is carried and train is stationary transformation and Lorentz transformation can relative to the motorcycle but in the gaps, the motorcycle is free and not carried by the train. The whole process of motorcycle moving on the train is so-called “partial contact and carry”. be only appropriate to two extreme cases: “complete physical contact and complete carry” and “no contact and no carry”, respectively, and hence they each cannot be applied to such a general case: when a body is running in the flowing media, the body is partially carried by the media. So both the Galilean and Lorentz transformations cannot be used to explain the result of the Fizeau experiment [9]: In Galilean system, velocity of a

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 light ray moving relative to the pipe is u  c / n  v which is larger than c / n  v(11/ n2 ) , while in Lorentz system, u  u' c / n is smaller than c / n  v(11/ n2 ) . To date, physicists have not really distinguished Lorentz transformation from Galilean transformation in physical mechanism [44-47]. For example, most physicists believe that

Lorentz transformation is reduced to Galilean transformation when the velocity is very small. Therefore, Lorentz transformation is naturally used to describe the relativity of light running in the flowing water. Einstein used his addition of velocities [10] based on Lorentz transformation to explain the result of the Fizeau experiment [9].

Therefore, so far no transformation has been proposed for the mechanism of “partial contact and partial carry”. We here utilize refraction coefficient n to combine Lorentz transformation with Galilean transformation and get a general transformation formula:

1 x'vt' 1 (30) x  2  (1 2 )(x'vt') n 1 (v / w)2 n

1 t'vx' / w2 1 t  2  (1 2 )t' (31) n 1 (v / w)2 n where w  V / n in a stationary frame in the mixed system. V is the velocity of body moving in the Lorentz system and 1/ n2 is used to measure proportion of carry-free component. So when n 2 =1, Eq.s(30) and (31) are reduced to the Lorentz transformation, when n 2 is large enough so that carry-free component is ignored and is approximately zero, then Eq.s(30) and (31) are reduced to Galilean transformation. Eq.s(30) and (31) are derived as follows

1  1  x  xL  xG  x  1 x (32) n2  n2 

1  1  t  tL  tG  t  1 t (33) n2  n2 

where x L and t L are the Lorentz components in distance and time of the body in stationary frame in this mixed

system while xG and tG are the Galilean components in this mixed system. , , , and are unknown, but x, 25

1 1 time t: one is Lorentz system, the other is Galilean system, take x and t from Lorentz system and n2 n 2

 1   1  1 x and 1 t from Galilean system, and mix them to construct a new system. In the mixed system, we  n2   n2  have Lorentz transformation components

1 1 x'vt' 1 1 t'vx' / w2 and 2 x  2 2 t  2 . n n 1 (v / w)2 n n 1 (v / w)2

Note that in Lorentz system, velocity w of the body is the same in frames S' and S, that is, x  wt and x' wt ' . By

Eq.s (14) and (16), (15) and (17), we have

1 1 x  vt 1 1 t  vx/ w and . 2 x' 2 2 t' 2 n n 1 (v / w)2 n n 1 (v / w)2

Galilean transformation components are

1 1 x  1 1 (x'vt')  1 1 x  1 1 x' 1 1 vt' 1 1 x  n2   n2   n2   n2   n2   n2 

 1 1 vt' 1 1 x' 1 1 (x  vt')  1 1 x' 1 1 x' 1 1 (x  vt)  n2   n2   n2   n2   n2   n2  and

(1 1 )t  (1 1 )t' . n2 n2

Since x' 1 x'(1 1 )x'and t' 1 t'(1 1 )t' , we get the transformations from x to x' : n2 n2 n2 n2

1 x  vt  1  x' 2  1 2 (x  vt) (34) n 1 (v / w)2  n  and from t to t' :

1 t  vx/ w2  1  t' 2  1 2 t . (35) n 1 (v / w)2  n 

Let us derive the result of Fizeau experiment from Eq.s(30) and (31). Here w =c/n and v is velocity of the water flowing in the pipe. The flowing water drags (partially carry) a light ray, so this is a typical “partial carry” case in which the light ray has the same speed in Lorentz state and in Galilean state, for the light ray moving in the direction of the water flowing, we have

1 x'vt' 1 (36) x  2  (1 2 )(x'vt') n 1 (vn/ c)2 n

v t' x' 1 (c / n)2 1 t  2  (1 2 )t' . (37) n 1 (vn/ c)2 n

2 Since nv<< c, nv / c  0 . Thus, Eq.s (36) and (37) become respectively

1 1 x  (x'vt')  (1 )(x'vt')  x'vt' (38) n2 n2

1 1 1  c n2  1 v 2 . (39) t  2 t'vx' /(c / n)  (1 2 )t' 2 t'v t' 2   t' 2 t' t' t' n n n  n c  n nc

Then, we have velocity u of the light ray moving relative to the pipe:

2 c c c v c v v x x'vt' t'vt'  v   v1  n  v  2  nc c  1  u    n  n  n nc  n   v1  , (40) v v v v v v 2 2 t t' nc t' t'1 nc  1 nc 1 nc 1 nc  1 nc  n  n  which is just Einstein‟s result. Thus in the mixed system partial addition of velocities is

x'vt' V t'vt' V  v u'v u   n  n  v v v v , (41) t' nu' t' t' nV t' 1 nV 1 2 n u'

V x'vt' (  v)t' V  v =1, we have u   n   V  u', that is, no addition of velocities. Since we do not v v v t' nV t' (1 nV )t' 1 V require v /V  0 in the case of which v is much smaller than V, Eq.(41) does not have self-inconsistence occurring in

v Eq.(23) or Eq.(24). In Galilean system, since n is large enough so that  0 , we have n2u'

x'vt' x'vt' u't'vt' u     u'v , (42) t' v t' t' t' n 2u' a linear addition of velocities. Note that u' V / n , so u' V .

Conclusions

Lorentz transformation cannot be reduced to Galilean transformation even when the velocity ( v ) of the referential frame moving relative to a stationary frame is very low such that v / c  0 because their physical bases are complete different. In Lorentz transformation equations x  (x'vt') / 1 (v /V )2 and t  (t'vx' /V 2 ) / 1 (v /V )2 , V= c hold if and only if x  ct and x' ct' , therefore, the addition of velocities cannot be derived from Lorentz transformation.

Galilean transformation exists in such a referential frame system in which a moving body physically contacts referential frame S' moving relative to stationary frame S and is carried by frame S' . In this referential frame system, the time for motion of the body in moving frame is the exact same with that in stationary frame S, that is, t' t but velocity (u) of the body moving relative to stationary frame S is sum of velocity ( u' ) of the body moving relative to the moving frame and that (v) of frame moving relative to frame S, i.e., u  u'v or u  u' .

Lorentz transformation exists in such a referential frame system in which a moving body does physically not contact referential frame moving relative to stationary frame S and hence is not carried by frame . In this referential frame system, the motion of the body is not impacted by frame , so the velocity of the body moving relative to stationary frame S is the exact same with that of the body moving relative to moving frame , that is, 28

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  u' , but the time for motion of the body in the stationary frame is different from that in the moving frame, i.e., t  t' because the coordinates x' and t' for motion of the body are changed as frame S' moves while coordinates x and t for motion of this body are not changed with frame moves.

Velocity addition occurs if and only if a moving body is carried by a frame moving in a stationary frame.

Besides Galilean transformation and Lorentz transformation systems that are two extreme systems, there is a general system in which a moving body partially contacts a referential frame moving relative to a stationary frame and is partially carried by the moving frame. The transformation equations can be given by combination of Galilean transformation equations with Lorentz transformation equations. A typical example of this system is the Fizeau‟s experimental system where a light ray is partially carried (dragged) by the water flowing in the pipe. Thus, in vacuum, light is not carried by any frame so that light speed is the same in all (moving and stationary) frames but in the flowing water light is partially carried by water such that it has non-linear addition of speeds in the stationary frame.

Acknowledgments

We here specially thank Prof. Palash B. Pal in Saha Institute of Nuclear Physics, India for many important and constructive suggestions for this paper.

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