American Journal of Mathematics and Statistics 2012, 2(5): 153-163 DOI: 10.5923/j.ajms.20120205.08

Applications of different Types of Lorenz Transformations

Atikur Rahman Baizid1,*, Md. Shah Alam2

1Department of Business Administration, Leading University, Sylhet, 3100, Bangladesh 2Department of Physics, Shahjalal University of Science and Technology, Sylhet, 3114, Bangladesh

Abstract The is well known. In this paper, we have presented various types of applications of different Lorenz Transformations according to the nature of movement of one inertial frame relative to the other inertial frame such as relativistic , relativistic Doppler’s effect and reflection of light ray by a moving mirror. When one frame moves along X- axis with respect to the rest frame then we can find these applications for special Lorentz transformation. When the motion of the moving frame is not along X-axis relative to the rest frame but the motion is along any arbitrary direction then we can find these formulae for most general Lorentz transformation. We can generate these formulae for different types of most general Lorentz transformations using mixed number, quaternion and geometric product. Ke ywo rds Special Lorentz Transformation, Most General, Mixed Nu mbe r, Quaternion, Geometric product, Re lativistic Aberration, Re lativ istic Doppler’s Effect, Reflection of Light Ray.

1. Introduction 1.2. Most Gener al Lorentz Transfor mati on When the motion of the moving frame is along any 1.1. Special Lorentz Transformation r arbitrary direction instead of the X-a xis , i.e. , the velocity V ¢ Let us consider two inertial frames of reference S and S , has three components Vx , Vy and Vz , then the relation ¢ where the frame S is at rest and the frame S is moving along between the space and time co-ordinates of S and S¢ is the X-axis with velocity v with respect to the S frame. The called the most general Lorentz transformation[2] which can space and time co-ordinates of S and S¢ are (x, y, z, t) and (x¢, ¢ ¢ ¢ be written as y , z , t ) respectively. The relation between the co-ordinates r r ¢ rì(r.V ) ü of S and S is called the special Lorentz transformation [1] r¢ = r + g - - g r r V í 2 ( 1) t ý which can be written as î V þ (3) x - vt r r x¢ = , y¢ = y, z¢ = z (1) t¢ = g {t - r . V} 1- v 2 c 2 and the inverse most general Lorentz transformation[2] t - vx c 2 which can be written as t¢ = r r 2 2 rì(r ¢. V ) ü 1- v c r = r¢ + g - + ¢g r r V í 2 ( 1) t ý and the inverse special Lorentz transformation[1] can be V ¢ (4) î þ written as r r x¢ + vt¢ t = g (t¢ + r ¢. V ) x = , y = y¢, z = z¢ (2) 1 2 2 - 1- v c æ V 2 ö 2 Where , g = ç1- ÷ and c =1 ¢ vx¢ ç 2 ÷ t + 2 è c ø t = c 2 2 1- v c 1.3. Mixe d Number Lorentz Transfor mation

In the case of the most general Lorentz transformation, the * Corresponding author: r [email protected] (Atikur Rahman Baizid) velocity V of S¢ with respect to S is not along the Published online at http://journal.sapub.org/ajms r Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved X-axis; i.e., the velocity V has three components, Vx, Vy,

154 Atikur Rahman Baizid et al.: Applications of different Types of Lorenz Transformations

r r¢ and Vz. Let in this case r and r be the space parts in S x¢ = g {x - tvx - (yvz - zv y )} ¢ and S frames, respectively. Then using the mixed ¢ r r r r r r y = g {y - tv y - (zvx - xvz )} product A Ä B = A.B + iA´ B , the mixed number (11) z¢ = g {z - tv - (xv - yv )} Lorentz transformations [3– 6] can be written as z y x r r t¢ = g (t - r.V ) t¢ = g (t + xvx + yvy + zvz ) r r r r r (5) r ¢ = g (r - tV - ir ´V ) and ¢ ¢ ¢ ¢ and the inverse mixed number Lorentz transformation[3– 6], x = g {x + t vx + (y vz - z v y )} can be written as = g ¢ + ¢ + ¢ - ¢ r r y {y t v y (z vx x vz )} t = g (t¢ + r¢.V ) (12) (6) = g ¢ + ¢ + ¢ - ¢ r r r r r z {z t vz (x v y y vx )} r = g (r¢ + t¢V + ir¢´V ) t = g (t¢ - x¢vx - y¢v y - z¢vz ) Let, rx = x, ry = y, rz = z, rx¢ = x, ry¢ = y, 1.5. Geometric Product Lorentz Transformation r¢ = z¢ . If Vx, Vy, and Vz denote the components of the z r velocity of the system S' relative to S then equation (5) In this case the velocity V of S' with respect to S has and (6) can be written as also three components, Vx, Vy, and Vz as the Most general r r¢ x¢ = g {x - tvx - i(yvz - zv y )} Lorentz transformation. Let in this case r and r be the space parts in S and S¢ frames respectively. Then y¢ = g {y - tv y - i(zvx - xvz )} (7) using the geometric product of two vectors ¢ = g - - - r r r r r r z {z tvz i(xvy yvx )} AB = A.B + A´ B the geometric product Lorentz ¢ = g - - - transformation [11, 12] can be written as t (t xvx yvy zvz ) r ¢ r and t = g (t - r.V ) r r r r r (13) x = g {x¢ + t¢vx + i(y¢vz - z¢vy )} r ¢ = g (r - tV - r ´V ) and the inverse geometric product Lorentz transformation, y = g {y¢ + t¢vy + i(z¢vx - x¢vz )} (8) [11, 12] can be written as ¢ ¢ ¢ ¢ r r z = g {z + t vz + i(x vy - y vx )} t = g (t¢ + r ¢. V ) r r r r r (14) t = g (t¢ + x¢vx + y¢vy + z¢vz ) r = g (r ¢ + t¢V + r ¢ ´V )

Let, rx = x, ry = y, rz = z, rx¢ = x, ry¢ = y, 1.4. Quater nion Lorentz Transfor mation r rz¢ = z¢ .If Vx, Vy, and Vz denote the components of the In this case the velocity of ¢ with respect to V S S velocity of the system S' relative to S then equation (13) has also three components, V , V and V as the most general x y, z r r and (14) can be written as Lorentz transformation. Let in this case r and r ¢ be the x¢ = g {x - tvx - (yvz - zv y )} space parts in S and S¢ frames respectively. Then r r r r r r y¢ = g {y - tv y - (zvx - xvz )} using the quaternion product AB = -A.B + A´ B the (15) ¢ quaternion Lorentz transformation [7-10] can be written as z = g {z - tvz - (xvy - yvx )} r r t¢ = g (t + r.V ) t¢ = g (t - xvx - yvy - zvz ) r r r r r (9) r ¢ = g (r - tV - r ´V ) and and the inverse quaternion Lorentz transformation[7-10] can x = g {x¢ + t¢vx + (y¢vz - z¢vy )} be written as (16) r r y = g {y¢ + t¢v + (z¢v - x¢v )} t = g (t¢ - r ¢. V ) y x z r r r r r (10) z = g {z¢ + t¢v + (x¢v - y¢v )} r = g (r ¢ + t¢V + r ¢ ´V ) z y x t = g (t¢ + x¢vx + y¢v y + z¢vz ) Let, rx = x, ry = y, rz = z, rx¢ = x, ry¢ = y, Now we are going to derive the formulae for re lativ istic ¢ ¢ .If V , V and V denote the components of the rz = z x y, z aberration, relativistic Doppler’s effect and reflection of light velocity of the system S' relative to S then equation (9) ray by a moving mirror of different types of lorentz and (10) can be written as transformations respectively.

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r 2. Aberration moving along any arbitrary direction with velocity V with The is independent of the medium of respect to S frame as shown in figure -2. Let the angles transmission but the direction of light rays depends on the made by the light ray in X-Y plane from the star P at any motion of the source emitting light and observer [10] instant in two systems atO andO¢ bej (j =< POX ) and j¢ j¢ =< ¢ ¢ respectively. Here the angle j is 2.1. Relativistic Aberration of Special Lorentz ( PO X ) Transfor mati on same in Fig. 1 and 2. But the angle j¢ is different Fixed The earth moves round the sun in its orbit. Consider sun is Moving to be the system S and the earth is in the system S¢ is moving with velocity V relative to the system S along positive direction of common X-axis. Let the star P observed fro m the observers O and O¢ in system S and S¢ where the frame S¢ is moving along X-a xis with velocity V with respect to S frame. Let the angles made by the light ray in X-Y plane from the star P at any instant in two systems at O and O¢ be j (j =< POX ) and j¢ j¢ =< ¢ ¢ respectively [Figure 1]. ( PO X ) Fi gure 2. Direct ion of light rays observed from a fixed frame S and a moving frame S¢ ( the frame S¢ is moving along any arbit rary direct ion wit h velocit y V wit h respect to S frame)

It also can be shown that for two dimension case, the Relativistic aberration of most general Lorentz transformation is clear which has been described [10] by the following relat ionship ì 2 ü v y vx v y (g -1) 2 + í1+ (g -1) 2 ýtanj + v yg secj v î v þ (19) tanj¢ = ì 2 ü v v Fi gure 1. Direct ion of light rays observed from a fixed frame S and a vx y x í1+ (g -1) 2 ý + (g -1) tanj + v secj v 2 x moving frame S¢ (the frame S¢ is moving along common X-ax is î þ v with velocity V with respect to S frame) From equation (19), we have j andj¢ are not same in the two systems. i.e. the direction of light rays depends on the It can be shown that [13] motion of the source emitting light and observer. Equation tanj 1- b 2 (19) gives the relativistic formula for aberration of most tanj¢ = general Lorentz transformation. 1- b secj where 2.3. Relativistic Aberration of Mixed Number Lorentz b = v (17) Transfor mati on c Consider the same situation as described in Figure 2. Let, and rx = x, ry = y, rz = z, rx¢ = x, tanj¢ 1- b 2 tanj = (18) ry¢ = y, rz¢ = z¢ ,then using equation (5) and (6) we can 1- b secj¢ easily find out the aberration formula for two dimension case It is clear from equation (17) and (18),j andj¢ are not of the mixed number Lorentz transformation[10] same in the two systems; they depend upon the motion of tanj + v y secj source and observer. Hence equation (17) describes the tanj¢ = (20) relativistic aberration of special Lorentz transformation. 1+ vx secj From equation (20), we have j andj¢ are not same in the 2.2. Relativistic Aberration of Most General Lorentz two systems. i.e. the direction of light rays depends on the Transfor mati on motion of the source emitting light and observer. Equation Consider the light from a star P observed by the observers (20) gives the relativistic formula for aberration of mixed O and O¢ in system S and S¢ where the frame S¢ is number Lorentz transformation.

156 Atikur Rahman Baizid et al.: Applications of different Types of Lorenz Transformations

2.4. Relativistic Aberration of Quaterni on Lorentz u¢y {u y - vy - (uz vx - vzux )} Transfor mati on = u¢ {u - v - (v u - v u )} Consider the same situation as described in Figure 2. Let, x x x z y y z If we consider two dimension case, then we can write r = x, r = y, r = z, r¢ = x, r¢ = y, r¢ = z¢ x y z x y z u¢ u - v then using equation (9) and (10) we can easily find out the y y y = (25) aberration formula for two dimension case of the quaternion u¢x ux - vx Lorentz transformation [14] In this case the star light travelling in x-y plane with tanj + v y secj tanj¢ = (21) velocity c, has component ccos(p +j) and + j 1 vx sec c cos(p +j¢) along positive direction of X-axis in system ¢ 2.5. Relativistic Aberration of Geometric Product S and S respectively. Also those csin(p +j) and Lorentz Transfor mati on csin(p +j¢) are along positive direction of Y-a xis in Consider the same situation as described in figure 2. Let, system S and S¢ respectively. Thus we have = = = ¢ = ¢ = ¢ = ¢ rx x, ry y, rz z, rx x, ry y, rz z ux = ccos(p +j) = -ccosj Equation (13) can be written as u y = csin(p +j) = -csinj x¢ = g {x - tvx - (yvz - zv y )} (26) u¢x = ccos(p +j¢) = -ccosj¢ y¢ = g {y - tv - (zv - xv )} y x z u¢ = csin(p +j¢) = -csinj¢ (22) y z¢ = g {z - tvz - (xvy - yvx )} From Equation (25) and (26) we have t¢ = g (t - xvx - yvy - zvz ) -csinj¢ -csinj - v y Differentiating both sides of equation (22) we get = - ccosj¢ - ccosj - vx dx¢ = g {dx - vx dt - (vz dy - v y dz)} csinj + v ¢ y dy¢ = g {dy - v y dt - (vx dz - vz dx)} or, tanj = ccosj + vx dt¢ = g (dt - v dx - v dy - v dz x y z Considering c =1 we have from the above equation we have sinj + v y tanj¢ = dx¢ g {dx - vx dt - (vz dy - v y dz )} = cosj + vx dt¢ g (dt - vx dx - v y dy - vz dz ) tanj + v y secj or, tanj¢ = (27) ì æ dy öü + j ídx - v - çv - v dz ÷ý 1 vx sec dt x è z dt y dt ø î þ From equation (27), we have j andj¢ are not same in the or, u¢x = dx dy dz 1- vx - v y - vz two systems. i.e. the direction of light rays depends on the dt dt dt motion of the source emitting light and observer. Equation {u - v - (v u - v u )} (27) gives the relativistic formula for aberration of geometric ¢ x x z y y z or, ux = (23) product Lorentz transformation. 1- vxux - v yu y - vzuz and

dy¢ g {dy - vy dt - (vxdz - vz dx)} 3. Relativistic Doppler’s Effect = dt¢ g (dt - vxdx - vydy - vz dz The Doppler’s effect [15] is an apparent change in frequency of a wave that results from the motion of source or dy - v - (v dz - v dx ) observer or both. The relat ivistic Doppler’s effect [16] is the { dt y x dt z dt } change in frequency and wavelength of light, caused by the or, u¢y = 1- v dx - v dy - v dz relative motion of the source and observer such as in the x dt y dt z dt regular Doppler’s effect, when taking into account effects of special . The relativistic Doppler’s effect {u - v - (u v - v u )} or, u¢ = y y z x z x (24) is different fro m the true (non – relativistic) Doppler’s effect y as the equations include the effect of the special 1- vxux - v yu y - vzuz relativity. They described the total difference in the Dividing equation (23) and (24) we get frequencies and possess the Lorentz symmetry.

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We can analyse the Doppler’s effect [17] of light by If n and n ¢ be the actual and observed frequencies of considering a light source as a clock that ticks n 0 times per light pulses, respectively .This equation (31) is the formula second and emits a wave of light with each tick. In the case of the relativistic Doppler’s effect of Specia l Lorentz of observer receding from the light source, the observer transformation for light. travels the distance Vt away from the source between the ticks, which mean that the light wave from a given tick takes Vt longer to reach him than the previous one. Hence the c total time between the arrivals of successive waves is V 1+ Vt T = t + = t c and the observed frequency is c 0 V 1- c V 1- Fi gure 3. The frame S¢ is moving along common X-axis with velocity c n =n V wit h resp ect to S frame 0 V 1+ c 3.2. Relati vistic Doppler’s Effect of most Gener al In the case of observer approaching the light source, the Lorentz Transfor mati on observed frequency is Consider two frames of references S and S¢ the V r 1+ velocity V of S¢ with respect to S is not along X-a xis r n =n c (28) 0 V i.e. the velocity V has three components Vx, Vy and Vz. Let 1- the transmitter and receiver be situated at origins O and c O¢ of frames S and S¢ respectively. Let t wo light 3.1. Relati vistic Doppler’s Effect of Special Lorentz signals or pulses be transmitted at time t = 0 and t = T , Transfor mati on T being the true period of light pulses. Let Dt¢ be the Consider two frames of references S and S¢ where interval between the receptions of the pulses by the receiver in the frame S¢ . Since the observer and receiver is at rest in S¢ is moving along common X-a xis with velocity V with frame S¢ , Dt¢ is the proper time interval T ¢ between respect to S frame. Let the transmitter and receiver be ¢ situated at origins O and O¢ of frames S and S¢ these pulses. Since the observer continues to be at O all D ¢ respectively. Let two light signals or pulses be transmitted at the time, the distance r covered by him in the frame ¢ during the reception of the two pulses is zero. time t = 0 and t = T , T being the true period of light S pulses. Let Dt¢ be the interval between the receptions o f the pulses by the receiver in the frame S¢ . Since the observer and receiver is at rest in fra me S¢ , Dt¢ is the proper time interval T ¢ between these pulses. Since the observer continues to be at O¢ all the time, the distance Dr¢ covered by him in the frame S¢ during the reception of the two pulses is zero. Using the inverse Lorentz transformation we can write, VT ¢ T ¢ Dx = & Dt = (29) 2 2 (1-V ) (1-V ) But Dt = T + Dx where c =1 (30) Fi gure 4. The frame S¢ is moving along any arbit rary direct ion with r Using the equations (29) and (30) we can write [13] velocity V with respect to S frame r 1-V n ¢ =n r (31) Using the space part and time part of inverse most general 1+V Lorentz transformation [2] any one can get the formula of

158 Atikur Rahman Baizid et al.: Applications of different Types of Lorenz Transformations

r r relativistic Doppler’s effect [18] in the most general Lorentz Dr = g T ¢V (36) transformation which is r This equation (36) rshows the second pulse has to travel 1-V this extra distance Dr then the first pulse in the frame S to n ¢ =n r (32) ¢ ¢ 1+V be able to reach at origin O in the moving frame S . Using the time part of the geometric product Lorentz This formula of the re lativ istic Dopple r’s effect coincides transformation of equation (13) with the formula of Special Lorentz transformation r Dt = g (Dt¢ + Dr¢ . V ) (37) r 3.3. Relativistic Doppler’s Effect of Mi xe d Nu mber Since Dr = 0 and Dt¢ = T ¢ then equation (37) can be Lorentz Transfor mati on written as Consider the same situation described in Figure 4. Using Dt = g T ¢ (38) the similar way of most general Lorentz transformation [2] This relation includes both the actual time period T of the r we can find the formula of Relativistic Doppler’s Effect of Dr mixed number Lorentz transformation [18] which is given by pulses and the time taken by the second pulse to cover r c 1-V r n ¢ =n the extra distance Dr in the frame S , i.e. r (33) r 1+V Dr Dt = T + (39) 3.4. Relativistic Doppler’s Effect of Quaternion Lorentz c Using c =1 the equation (39) can be written as Transfor mati on r Dt = T + Dr (40) Consider two frames of references S and S¢ as Using (36), (38) and (40) we can write ¢ r described in Figure 4. If n and n be the actual and T ¢g = T +VT ¢g observed frequencies of light pulses, respectively, then the r 2 formula for the relat ivistic Doppler’s Effect of quaternion 1-V = ¢ ( ) Lorentz transformation [14] can be found as or, T T 2 (41) r (1-V ) ¢ n = g n (1-V ) (34) Using equation (23), the equation (41) can be written as r 3.5. Relativistic Doppler’s Effect of Geometric Product 1-V T = T ¢ r (42) Lorentz Transfor mati on 1+V Consider two frames of references S and S¢ . The If n and n ¢ be the actual and observed frequencies of r velocity V of S¢ with respect to S is not along X-a xis 1 1 light pulses, respectively, we have n = and n ¢ = r ¢ i.e. the velocity V has three components Vx, Vy and Vz. Let T T then equation (42) gives the transmitter and receiver be situated at origins O and r O¢ of frames S and S¢ respectively. Let two light 1-V n ¢ =n r (43) signals or pulses be transmitted at time t = 0 and t = T , 1+V T being the true period of light pulses. Let Dt¢ be the this equation (43) is the formula of the re lativistic Doppler’s interval between the receptions of the pulses by the receiver effect of geometric product Lorentz transformation for light. in the frame S¢ . Since the observer and receiver is at rest in frame S¢ , Dt¢ is the proper time interval T ¢ between 4. Reflection of Light by a moving these pulses. Since the observer continues to be at O¢ all Mirror the time, the distance Dr¢ covered by him in the frame 4.1. Reflection of light by a moving mirror in Special S¢ during the reception of the two pulses is zero. Lorentz Tr ans for mati on Using the space part of the inverse geometric product Lorentz transformation of equation (13) Consider two frames of references S and S¢ where r r r Dr = g (Dr¢ + Dt¢V + iDr¢ ´V ) (35) S¢ is moving along common X-a xis with velocity V with r Since Dr = 0 and Dt¢ = T¢ then equation (35) can be respect to S frame. Consider a mirror M to be fixed in the written as Y¢ - Z¢ plane of system S¢ . Consider the mirror is r r Dr = g Dt¢V perfectly reflecting, moving along the direction of its normal or, relative to S¢ . Let a ray of light in X ¢ - Y ¢ be incident at

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angle f¢ at O¢ in system S¢ .As mirror M is stationary 4.2. Reflection of Light by a moving Mirror in Most General Lorentz Transformation in system S¢ ordinary laws of reflection hold good and so f¢ Consider two frames of references S and S¢ . The the angle of reflection will be and the reflected ray lies in r X ¢ - Y ¢ plane. Let the angles of incidence and reflection velocity V of S¢ with respect to S is not along X-a xis r are measured in system S bef1 &f2 , we get [2] i.e. the velocity V has two components Vx, and Vy . 2 Consider a mirror M to be fixed in the Y¢ - Z¢ plane of sinf1 sinf2 1- b = (44) system S¢ . Consider the mirror is perfectly reflecting, cosf + b (cosf - b ) 1 2 moving along the direction of its normal relative to S¢ . 1 1 V Using g = = ,c =1 & b = Let a ray of light in X ¢ - Y ¢ be incident at angle f¢ at O¢ 2 - b 2 c V 1 in system S¢ . 1- 2 c As mirror M is stationary in s ystem S¢ ordinary laws of we get fro m (44) reflection hold good and so the angle of reflection will be sinf sinf 1 = 2 (45) f¢ and the reflection ray lies in X ¢ - Y ¢ plane . cosf1 +V g (cosf2 -V ) Let the angles of incidence and reflection are measured in This equation (45) is the law of reflection of light by a system S be f1 &f2 , we get [3] moving mirror [2] in the case of special Lorentz transformation

Fi gure 5. The reflect ion of light by a moving mirror where M is fixed in t he Y¢ - Z¢ plane of sy stem S¢

Let the incident ray in system S and S¢ be represented by y and y ¢ where

A é ì x cos(p + f1 )+ y sin (p + f1 )üù y = expê2p i n ít - ýú (46) g ë î c þû and A¢ é ì x¢cos(p + f¢)+ y¢sin (p + f¢)üù y ¢= expê2p i n ¢ít¢ - ýú (47) g ë î c þû As phase is a Lorenz invariant quantity, using equations (46) and (47) we must have

160 Atikur Rahman Baizid et al.: Applications of different Types of Lorenz Transformations

ì x cos(p + f )+ y sin (p + f )ü ì x¢cos(p + f¢)+ y¢sin (p + f¢)ü n ít - 1 1 ý =n ¢ít¢ - ý î c þ î c þ

ì x cosf + y sinf ü ì x¢cosf¢ + y¢sinf¢ü or, n ít + 1 1 ý =n ¢ít¢ + ý (48) î c þ î c þ

Using c =1 we get from equation (48)

n{t + x cosf1 + y sinf}=n ¢{t¢ + x¢cosf¢ + y¢sinf¢} (49)

Using equations (3), (4) and (49) and considering the velocity V has two components Vx and Vy , we can get the formula [19] of the reflection of light ray by a moving mirror in most general Lorentz transformation

2 2 2 g Vy V + sinf1V + (g -1)VxVy cosf1 + (g -1)Vy sinf1 2 2 2 g Vx V + cosf1V + (g -1)VxVy sinf1 + (g -1)Vx cosf1 - g V V 2 - sinf V 2 + (g -1)V V cosf - (g -1)V 2 sinf = y 2 x y 2 y 2 g V V 2 + cosf V 2 - (g -1)V V sinf + (g -1)V 2 cosf x 2 x y 2 x 2

r Fi gure 6. The frame S¢ is moving along any arbit rary direct ion with velocit y V with respect to S frame where a mirror M is fixed in the Y¢ - Z¢ plane of system S¢

equation (7) and (49) we have considering the velocity V has 4.3. Reflection of Light by a moving Mirror in mixed two components V and V number Lorentz Transformation x y ¢ Consider two frames of reference S and S¢ . The n{t + xcosf1 + ysinf}=n gt - (50) r n ¢g - + - f¢ + - f¢ velocity V of S¢ with respect to S is not along X-a xis {xVx yVy (x tVx )cos (y tVy )sin } r i.e. the velocity V has three components Vx, Vy and Vz. = = Consider same situation as described in the Figure 6.Using [QVz 0 & z 0]

American Journal of Mathematics and Statistics 2012, 2(5): 153-163 161

Equating the coefficient of t of equation (50) we get, Vy + sinf1 tanf¢= (59) n =n ¢g - f¢ - f¢ (51) (1 Vx cos Vy sin ) Vx + cosf1 Similarly, for reflected ray the relation between angle of Equating the coefficient of x of equation (50) we get, reflection in system S and S¢ can be obtained as n cosf 1 =n ¢g (cosf¢ - Vx ) -Vy - sinf2 tanf¢= (60) or, Vx + cosf2 From equation (59) and (60), we get, n ¢g (1-Vx cosf¢ -Vy sinf¢)cosf1 =n ¢g (cosf¢ -Vx ) Vy + sin f1 -Vy - sin f2 = (61) ¢ (cosf - Vx ) Vx + cosf1 Vx + cosf2 or, cosf 1 = (52) (1 - Vx cosf¢ - V y sinf¢) This equation (61) is the law of reflection of light ray by a moving mirror in mixed number Lorentz transformation. Equating the coefficient of y of equation (50) we get, n sinf =n ¢g (sinf¢ - V ) 4.4. Reflection of Light by a moving Mirror in 1 y Quaternion Lorentz Transformation or, Consider two frames of reference S and S¢ . The n ¢g - f¢ - f¢ f =n ¢g f¢ - r (1 Vx cos Vy sin )sin 1 (sin Vy ) velocity V of S¢ with respect to S is not along X-a xis r i.e. the velocity has three components V , V and V . (sinf¢ - V y ) V x y z or, sinf = (53) Consider same situation as described in the Figure 6. 1 ¢ ¢ (1 - Vx cosf - V y sinf ) Using equation (11) and (49) we have considering the

Now from equation (52) and (53) we get, velocity V has two components Vx and Vy we can get by ¢ sinf - V y the similar process of mixed number Lorentz transformation, tanf = (54) 1 cosf¢ - V the same formula for the reflection of light ray by a moving x mirror in quaternion Lorentz transformation Using equation (8) and (49) we have considering the Vy + sin f1 -Vy - sin f2 velocity V has two components Vx and Vy = (62) Vx + cosf1 Vx + cosf2 ng {t¢ + x¢V + y¢V + (x¢ + t¢V )cosf + (y¢ + t¢V )sinf } x y x 1 y 1 (55) =n ¢{t¢ + x¢cosf¢ + y¢sinf¢} 4.5. Reflection of Light by a moving Mirror in Geometric Product Lorentz Transfor mati on Equating the coefficient of t¢ of equation (55) we get, Consider two frames of reference S and S¢ . The ng (1+Vx cosf1 +Vy sinf1 ) =n ¢ (56) r velocity V of S¢ with respect to S is not along X-a xis Equating the coefficient of x¢ of equation (55) we get, r i.e. the velocity V has three components V , V and V . ¢ ¢ x y z ng (Vx + cosf1 ) =n cosf Consider same situation as described in the Figure 6. or, Using equation (15) and (49), we have considering the velocity V has two components Vx and Vy then after a little ng (V + cosf ) =ng 1+V cosf +V sinf cosf¢ x 1 ( x 1 y 1 ) bit lengthy calculation according to the similar process of mixed number Lorentz transformation we can get the same V + cosf or, cosf¢ = x 1 (57) formula for the reflection of light ray by a moving mirror in geometric product Lorentz transformation 1+Vx cosf1 +Vy sinf1

Equating the coefficient of y¢ of equation (55) we get, Vy + sin f1 -Vy - sin f2 = (63) ¢ ¢ ng (Vy + sinf1 ) =n sinf Vx + cosf1 Vx + cosf2 or, ng (V + sinf ) =ng (1+V cosf +V sinf )sinf¢ y 1 x 1 y 1 5. Comparison of the Study Vy + y sinf1 or, sinf¢= (58) + f + f 5.1. Comparison of Relativistic Aberrations of Special, 1 Vx cos 1 Vy sin 1 Most General, Mi xe d Number, Quater nion and Now from equation (57) and (58) we get, Geometric Product Lorenz Transformations

162 Atikur Rahman Baizid et al.: Applications of different Types of Lorenz Transformations

Names of Lorentz transformations Relativistic aberration formula

2 Special tanj 1- b tanj¢ = Lorent z transformat ion 1- b secj

ì 2 ü v y vx v y (g -1) 2 + í1+ (g -1) 2 ýtanj + v g secj v v y Most general î þ tanj¢ = Lorent z transformat ion ì 2 ü v v vx y x í1+ (g -1) 2 ý + (g -1) tanj + vx secj î v þ v 2

j + j Mixed Number tan v y sec tanj¢ = Lorent z transformat ion 1+ vx secj

j + j Quaternion tan v y sec j¢ = Lorent z transformat ion tan 1+ vx secj

j + j Geometric product Lorent z tan v y sec j¢ = t ransformat ion tan 1+ vx secj

5.2. Comparison of Relativistic D o pple r’s Effect of Special, Mos t General, Mixed Number, Quaternion and Geometric Product Lorenz Transfor mati ons

Names of Lorentz transformations Relativistic Doppler’s effect formula

1-V Special Lorentz transformation n ¢ =n 1+V

r 1-V Most general Lorent z transformat ion n ¢ =n r 1+V

r 1-V Mixed number Lorent z transformat ion n ¢ =n r 1+V

r Quaternion Lorent z transformat ion n ¢ = gn (1-V )

r 1-V Geometric product Lorent z transformat ion n ¢ =n r 1+V

5.3. Comparison of the Formulae of the Reflection of Light by a Moving Mirror in S pecial, Most Gener al, Mi xe d Number, Quaternion and Geometric Product Lorenz Transfor mati ons

American Journal of Mathematics and Statistics 2012, 2(5): 153-163 163

Names of Lorentz transformations The formulae of reflect ion of light by a moving mirror

sinf1 sinf2 Special Lorentz transformation = cosf1 +V g (cosf2 -V )

2 2 2 g Vy V + sinf1V + (g -1)VxVy cosf1 + (g -1)Vy sinf1 g V V 2 + cosfV 2 + (g -1)V V sinf + (g -1)V 2 cosf Most general Lorent z x 1 x y 1 x 1 t ransformat ion - g V V 2 - sinf V 2 + (g -1)V V cosf - (g -1)V 2 sinf = y 2 x y 2 y 2 2 2 2 g Vx V + cosf2V - (g -1)VxVy sinf2 + (g -1)Vx cosf2 + f - - f Mixed number Lorent z Vy sin 1 Vy sin 2 = t ransformat ion Vx + cosf1 Vx + cosf2

+ f - - f Quaternion product Lorent z Vy sin 1 Vy sin 2 = t ransformat ion Vx + cosf1 Vx + cosf2

Vy + sin f1 -Vy - sin f2 Geometric Lorent z transformat ion = Vx + cosf1 Vx + cosf2

6. Conclusions [7] Alam, M.S. 2003. Different types of product of vectors, NewsBull.Cal.Math.Soc., 26(1,2 & 3) 21-24. The Relativistic aberration, Doppler’s effect and the [8] Datta, B.K., De Sabbata V. and Ronchetti L. 1998. reflection of light ray by a moving mirror formulae of special, Quantization of gravity in real space time, Il Nuovo Cimento, most general, mixed number, quaternion and geometric 113B. product Lorenz transformations have been derived clearly. We have found that the formulae for Relativistic aberration [9] Datta, B.K., Datta R. 1998. Einstein field equations in spinor formalism, Foundations of Physics letters, 11,1.. and for the reflection of light ray by a moving mirror of mixed number, quaternion and geometric product Lorenz [10] Md. Shah Alam and Md. Didar Chowdhury, Relativistic transformations are simpler than the formula of most general Aberration of Mixed number Lorenz Transformation, 2006, Lorenz transformation . Journal of the national Science foundation of Sri Lanka34 (3) [11] http://mathworld.wolfram.com/Quaternion.html [12] http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000 /quaternions1.html REFERENCES [13] Satya Prokash. Relativistic M echanics, Pragati Prakashan-2000, India. [1] Resnick, R. 1994. Introduction to , Wiley Eastern limited. [14] Md. Shah Alam and Sabar Bauk, Quaternion Lorenz Transformation, Physics Essays,2011, American Institute of [2] Moller, C. 1972. The Theory of Relativity, Oxford University Physics, Canada. press, London. [15] A. Beiser, 1964.The Fundamentals of [3] Alam, M.S. 2000. Study of Mixed Number. Proc. Pakistan Physics.Addison-Welsey Publishing,Inc. Acad. of Sci. 37(1): 119-122. [16] A. Beiser, 1995. Concepts of Modern Physics. Mc-Graw-Hill, [4] Alam, M.S. 2001. Mixed product of vectors, Journal of Inc.Fifth Edition. Theoretics, 3(4). http://www.journaloftheoretics.com/ [17] Md. Shah Alam and Md. Didar Chowdhury, Study of [5] Alam, M.S. 2003. Comparative study of mixed product and Relativistic Doppler’s Effect, Proc.Pakistan quaternion product, Indian J. Physics A 77: 47-49. Acad.Sci.44(2):109-115.2007. [6] Kyrala, A. 1967. Theoretical Physics, W.B. Saunders [18] Md. Shah Alam and Md. Didar Chowdhury, Study of Company. Philadelphia & London, Toppan Company reflection of light by a moving mirror,2009,J.Physics.83(2) Limited. Tokyo, Japan. 233-