Adv. Studies Theor. Phys., Vol. 6, 2012, no. 3, 139 – 145
Three Lorentz Transformations Considering Two Rotations Mukul Chandra Das* Singhania University, Rajasthan, India
[email protected] Rampada Misra Department of electronics, Vidyasagar University West Bengal, India Abstract A formulation for combination of three Lorentz transformations considering two rotations has been developed in this paper. This procedure physically means the composition of three linear velocities in different directions resulting in a single velocity. The same process has been applied to a particle or body possessing three simultaneous rotational motions having mutual effects. This leads to the assumption that a body may possess three simultaneous superimposed spins. Key words: superimposed rotational motions, superimposed spins.
1. Introduction One of the basic questions in Lorentz transformation is velocity addition. According to Einstein’s relativistic theorem, we have the well known formula of co-linear velocity addition [1], u (u v) /(1 u v / c 2 ) . Møller extended this approach to develop the velocity addition formula for combining two velocities in different directions in planar motion [2]. Chandrau Iyer and G.M Probhu describe the LRL transformation for composition of two Lorentz boosts in different directions in plane [3]. This means the composition of two velocities u and v resulting in a single velocity w . In this paper we first develop the LRLRL transformations yielding a *Communication address : Satmile High School , P.O. Satmile 721452, W .B., India
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single resultant Lorentz boost through a proper way. We extend this to compose three superimposed motions in different directions into a resultant velocity and following this constructive method we show that a body or particle may possess three simultaneous superimposed spins.
2. LRLRL Transformations In this section we want to derive the matrix for a conventional Lorenz transformation L xu followed by a planar rotation of the XY plane ( R ) and then followed by another conventional Lorenz transformation L xv which is again followed by a planar rotation of the XY plane ( R ) and then by another conventional Lorenz transformation L xw . For clarity, we may visualize six inertial frames S , S1 , S 2 , S3 , S 4 and S 5 in three dimensional space. Frames S and S1 have both their co-ordinate axes aligned and S1 is moving at a velocity u along X 1 axis as observed by S . The inertial frame S1 has another co-ordinate reference frame S2 , where X 2 axis of S2 , are rotated by an angle counter clockwise with respect to S1 on X 1Y1 plane. Frames S 2 and S 3 have both their co-ordinate axes aligned and S 3 is moving at a velocity v along X 3 axis as observed by S 2 . The inertial frame S 3 has another coordinate reference frame S 4 where X 4 axis of S 4 are rotated by an angle counter clockwise with respect to S 3 on X 3Y3 plane. Frames S 4 and S 5 have both their co-ordinate axes aligned and S 5 is moving at a velocity w along X 5 axis as observed by S 4 . Then the transformation of co-ordinates of an event from frame S to frame S 5 is given by 4 4 order matrix M i j which is equal to the matrix product. L xw R xz ( ) L xv R xy ( ) L xu .The matrix
Lxu , R xy ( ) , L xv , R xz ( )
and Lxw, are as specified in (1) , (2), (3), (4) and (5) respectively
Lorentz transformations
0 0 x1 u 1 0 0 y1 0 0 1 z1 u u 0 0 t1 2 c 0 0 x3 v 1 0 0 y3 0 0 1 z3 v v 0 0 t3 2 c 0 w x5 1 0 y5 0 0 z5 w w 0 t5 c2
141
u u x 0 y (1), 0 z u t v v x 2 0 y2 (3), 0 z2 v t 2 0 w w x 4 0 0 y4 1 0 z4 0 w t 4
x 2 cos y 2 sin z2 0 t2 0
sin cos 0 0
0 0 1 0
0 x1 0 y1 0 z1 1 t 1
(2)
x 4 cos y4 0 z 4 sin t4 0
0 sin 1 0 0 cos 0 0
0 x3 0 y 3 0 z 3 1 t 3
(4)
(5)
The inverse of this operation is given by N Matrices M i j and N i j turn out as M
w 0 0 w w c2
0 0 1 0 0 1
cos sin 0 0
sin cos 0
0 0 1
0
0
0
0
cos 0 sin 0 w 0 u 0 0 0 0 1 u2 u c
w 0 0
i j
i j
L xw R xz ( ) L xv R xy ( ) L xu
0 sin 1 0 0 cos
w
0
0
0 0 0 1
0
0 0 1 0 0 1 0
u 0 0
u
L x ( u ) R xy ( ) L x ( v ) R xz ( ) L x ( w)
u
=
v 0 0 v v c2
0 0 1 0 0 1 0
0
v 0 0
v
v
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uv cos u v w ( c 2 cos cos vw cos uw 2) c2 c u sin ( uv sin u v c2 cos sin ) u v w (v cos c2 uvw cos u c2 w cos cos )
Similarly N
u 0 0 u u c2
i j
vw sin c2 sin cos )
w sin
cos
0
v sin sin
cos
v w(
v w (v sin c2 w sin cos )
w
w
c
sin 2
u v w (v cos uvw cos w 2 c u cos cos ) u u sin u v (v sin u sin cos ) uv cos u v w( c2 uw cos cos c2 vw cos 1) c2
L x ( u ) R xy ( ) L x ( v ) R xz ( ) L x ( w) =
0 0 u u cos sin 1 0 0 sin cos 0 1 0 0 0 0 0 0 u 0 0 cos 0 sin 0 w 1 0 0 1 0 0 0 0 sin 0 cos 0 w 0 0 1 2 w 0 0 c
0 0 v 0 0 0 0 1 0 v 0 1 2v c 0 w w 0 0 1 0 0 w
0 0 v v 1 0 0 0 1 0 0 0 v
(6)
Lorentz transformations
u v w (cos cos vw cos uw c2 c2 uv cos ) c2 v w (sin cos vw sin ) c2 w sin u v w (v cos 2 c uvw cos w c2 u cos cos )
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u sin
u v (sin cos
cos
uv sin ) c2
v sin sin
0
cos
u u sin c2
u v (v sin c2 u cos sin )
u v w (v cos uvw cos u c2 w cos cos )
v w (v sin w sin cos ) w w sin uv cos u v w( c2 uw cos cos c2 vw cos 1) 2 c
(7)
3. Resultant velocity due to composition of three velocities For composition of three velocities we may visualize six inertial frames as discussed in section-2 where, the co-ordinate of an event at any point on S 5 with respect to S would be written as H Ni jK (8) x where, H y , z t
x5 y K 5 z5 t 5
Equation (8) gives the relations as
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x N 11 x 5 N 12 y 5 N 13 z 5 N 14 t 5 y N 21 x 5 N 22 y 5 N 23 z 5 N 24 t 5
(9)
z N 31 x 5 N 32 y 5 N 33 z 5 N 34 t 5 t N 41 x 5 N 42 y 5 N 43 z 5 N 44 t 5 From spherical coordinate system one can also write dx 5 w sin(900 ) cos w cos cos dt 5 dy 5 w sin(900 ) sin w sin cos dt5
(10)
dz 5 w cos(900 ) w sin dt 5
Following equations (9) and (10) we could find out the resultant velocity ' G ' of the event of S 5 as observed by S , whose magnitude also can be evaluated by considering the motion of the origin of S 5 ( x5 0, y 5 0, z 5 0) as G (
dx 2 dy 2 dz 2 ) ( ) ( ) dt dt dt
(11)
This may be written as
G
P 2 Q 2 R 2 (12) T P N 11 w cos cos N 12 w sin cos N 13 w sin N 14 Q N 21 w cos cos N 22 w sin cos N 23 w sin N 24
where,
R N 31 w cos cos N 32 w sin cos N 33 w sin N 34 T N 41 w cos cos N 42 w sin cos N 43 w sin N 44
When 900 then the resultant velocity G would be given by
u 2v 2 w 2 u 2v 2 v 2 w 2 w 2u 2 12 G (u v w 2 2 2 ) c4 c c c 2
2
2
(13)
Lorentz transformations
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4. Resultant velocity due to three rotational motions From the above discussion it is seen that u , v , and w are the velocities in three different directions where, angle between u and v is , between v and w is and that between w and u is which depends on and . In that sense for composition or addition of three simultaneous superimposed rotational motions we can consider the velocity addition formula (12). For clarity we may visualize six inertial frames as discussed in section-2, but S1 , S3 and S5 don’t move with rectilinear motion. They move only with angular velocity 1 , 2 and 3 respectively, about X 1 , X 3 and X 5 axes as observed by S and others (i.e. S , S 2 and S 4 ) be same as discussed in section-2, When origins of frames are same with respect to S, then the resultant velocity of an event of S5 will be same as G in (12)
as observed by S where, u 1 r , v 2 r and w 3 r become separately
relativistic velocities and r is the position vector of the event in S 5 . It is to be mentioned that if a body or particle be present at the origin of S 5 then it will possess three simultaneous superimposed spins as observed by S .
5. Conclusions The above derivations and discussions reveal the field of relativistic effects on a body due to three motions. The expression for resultant velocity shows that it is dependent upon all the linear velocities arising from the rotations (viz. equation (12))
References [1] Robert Resnick, Introduction to Special Relativity, 2009, p.80 ISBN 978 8126511006 [2] Møller C., The theory of Relativity, Oxford University press, Oxford, 1952, 51-52 [3] Chandru Iyer, and G.M. Prabhu, Composition of two Lorenz boosts through spatial and Space-time rotations, Journal of Physical and Natural Sciences Vol. 1, (2007), no.2 Received: September, 2011
