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 Lorentz transformation 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 aberration, 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. American Journal of Mathematics and Statistics 2012, 2(5): 153-163 155 r 2. Aberration moving along any arbitrary direction with velocity V with The speed of light 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.