Atikur Rahman Baizid 1, Md. Shah Alam 2
1Department of Business Administration, Leading University, Sylhet, 3100, Bangladesh
2Department of Physics, Shahjalal University of Science and Technology, Sylhet, 3114, Bangladesh
Correspondence to: Atikur Rahman Baizid , Department of Business Administration, Leading University, Sylhet, 3100, Bangladesh.
| Email: |  |
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
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.
Keywords:
Special Lorentz Transformation, Most General, Mixed Number, Quaternion, Geometric Product, Relativistic Aberration, Relativistic Doppler’s Effect, Reflection of Light Ray
1. Introduction
1.1. Special Lorentz Transformation
Let us consider two inertial frames of reference S and S′ , where the frame S is at rest and the frame S′ is moving along the X-axis with velocity v with respect to the S frame. The space and time co-ordinates of S and S′ are (x, y, z, t) and (x′, y′, z′, t′) respectively. The relation between the co-ordinates of S and S′ is called the special Lorentz transformation[1] which can be written as  | (1) |
and the inverse special Lorentz transformation[1] can be written as  | (2) |
1.2. Most General Lorentz Transformation
When the motion of the moving frame is along any arbitrary direction instead of the X-axis , i.e. , the velocity
has three components Vx , Vy and Vz , then the relation between the space and time co-ordinates of S and 
is called the most general Lorentz transformation[2] which can be written as  | (3) |
and the inverse most general Lorentz transformation[2] which can be written as | (4) |
Where , 
and 
1.3. Mixed Number Lorentz Transformation
In the case of the most general Lorentz transformation, the velocity 
of 
with respect to 
is not along the X-axis; i.e., the velocity 
has three components, Vx, Vy, and Vz. Let in this case 
and 
be the space parts in 
and 
frames, respectively. Then using the mixed product 
, the mixed number Lorentz transformations[3– 6] can be written as | (5) |
and the inverse mixed number Lorentz transformation[3– 6], can be written as | (6) |
Let, 
. If Vx, Vy, and Vz denote the components of the velocity of the system S' relative to 
then equation (5) and (6) can be written as | (7) |
and  | (8) |
1.4. Quaternion Lorentz Transformation
In this case the velocity 
of 
with respect to 
has also three components, Vx, Vy, and Vz as the most general Lorentz transformation. Let in this case 
and 
be the space parts in 
and 
frames respectively. Then using the quaternion product
the quaternion Lorentz transformation[7-10] can be written as  | (9) |
and the inverse quaternion Lorentz transformation[7-10] can be written as | (10) |
Let, 
.If Vx, Vy, and Vz denote the components of the velocity of the system S' relative to 
then equation (9) and (10) can be written as | (11) |
and | (12) |
1.5. Geometric Product Lorentz Transformation
In this case the velocity 
of S' with respect to 
has also three components, Vx, Vy, and Vz as the Most general Lorentz transformation. Let in this case 
and 
be the space parts in 
and 
frames respectively. Then using the geometric product of two vectors 
the geometric product Lorentz transformation[11, 12] can be written as | (13) |
and the inverse geometric product Lorentz transformation,[11, 12] can be written as | (14) |
Let, 
.If Vx, Vy, and Vz denote the components of the velocity of the system S' relative to 
then equation (13) and (14) can be written as | (15) |
and | (16) |
Now we are going to derive the formulae for relativistic aberration, relativistic Doppler’s effect and reflection of light ray by a moving mirror of different types of lorentz transformations respectively.
2. Aberration
The speed of light is independent of the medium of transmission but the direction of light rays depends on the motion of the source emitting light and observer[10]
2.1. Relativistic Aberration of Special Lorentz Transformation
The earth moves round the sun in its orbit. Consider sun is to be the system 
and the earth is in the system 
is moving with velocity V relative to the system 
along positive direction of common X-axis. Let the star P observed from the observers 
and 
in system 
and 
where the frame 
is moving along X-axis with velocity V with respect to 
frame. Let the angles made by the light ray in X-Y plane from the star P at any instant in two systems at 
and 
be 
and 
respectively[Figure 1]. | Figure 1. Direction of light rays observed from a fixed frame  and a moving frame  (the frame  is moving along common X-axis with velocity V with respect to frame) |
It can be shown that[13]
where  | (17) |
and | (18) |
It is clear from equation (17) and (18), 
and
are not same in the two systems; they depend upon the motion of source and observer. Hence equation (17) describes the relativistic aberration of special Lorentz transformation.
2.2. Relativistic Aberration of Most General Lorentz Transformation
Consider the light from a star P observed by the observers 
and 
in system 
and 
where the frame 
is moving along any arbitrary direction with velocity 
with respect to 
frame as shown in figure -2. Let the angles made by the light ray in X-Y plane from the star P at any instant in two systems at
and
be
and
respectively. Here the angle 
is same in Fig. 1 and 2. But the angle 
is different Fixed Moving | Figure 2. Direction of light rays observed from a fixed frame  and a moving frame  ( the frame  is moving along any arbitrary direction with velocity V with respect to  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 relationship | (19) |
From equation (19), we have 
and
are not same in the two systems. i.e. the direction of light rays depends on the motion of the source emitting light and observer. Equation (19) gives the relativistic formula for aberration of most general Lorentz transformation.
2.3. Relativistic Aberration of Mixed Number Lorentz Transformation
Consider the same situation as described in Figure 2. Let, 
,then using equation (5) and (6) we can easily find out the aberration formula for two dimension case of the mixed number Lorentz transformation[10] | (20) |
From equation (20), we have 
and
are not same in the two systems. i.e. the direction of light rays depends on the motion of the source emitting light and observer. Equation (20) gives the relativistic formula for aberration of mixed number Lorentz transformation.
2.4. Relativistic Aberration of Quaternion Lorentz Transformation
Consider the same situation as described in Figure 2. Let, 
then using equation (9) and (10) we can easily find out the aberration formula for two dimension case of the quaternion Lorentz transformation[14] | (21) |
2.5. Relativistic Aberration of Geometric Product Lorentz Transformation
Consider the same situation as described in figure 2. Let, 
Equation (13) can be written as  | (22) |
Differentiating both sides of equation (22) we get 
from the above equation we have  | (23) |
and  | (24) |
Dividing equation (23) and (24) we get 
If we consider two dimension case, then we can write | (25) |
In this case the star light travelling in x-y plane with velocity c, has component
and 
along positive direction of X-axis in system 
and 
respectively. Also those
and 
are along positive direction of Y-axis in system 
and 
respectively. Thus we have  | (26) |
From Equation (25) and (26) we have 
Considering
we have | (27) |
From equation (27), we have 
and
are not same in the two systems. i.e. the direction of light rays depends on the motion of the source emitting light and observer. Equation (27) gives the relativistic formula for aberration of geometric product Lorentz transformation.
3. Relativistic Doppler’s Effect
The Doppler’s effect[15] is an apparent change in frequency of a wave that results from the motion of source or observer or both. The relativistic Doppler’s effect[16] is the change in frequency and wavelength of light, caused by the relative motion of the source and observer such as in the regular Doppler’s effect, when taking into account effects of special theory of relativity. The relativistic Doppler’s effect is different from the true (non – relativistic) Doppler’s effect as the equations include the time dilation effect of the special relativity. They described the total difference in the frequencies and possess the Lorentz symmetry.We can analyse the Doppler’s effect[17] of light by considering a light source as a clock that ticks 
times per second and emits a wave of light with each tick. In the case of observer receding from the light source, the observer travels the distance 
away from the source between the ticks, which mean that the light wave from a given tick takes 
longer to reach him than the previous one. Hence the total time between the arrivals of successive waves is 
and the observed frequency is 
In the case of observer approaching the light source, the observed frequency is | (28) |
3.1. Relativistic Doppler’s Effect of Special Lorentz Transformation
Consider two frames of references 
and 
where 
is moving along common X-axis with velocity V with respect to 
frame. Let the transmitter and receiver be situated at origins 
and 
of frames 
and 
respectively. Let two light signals or pulses be transmitted at time 
and
, T being the true period of light pulses. Let 
be the interval between the receptions of the pulses by the receiver in the frame
. Since the observer and receiver is at rest in frame
, 
is the proper time interval 
between these pulses. Since the observer continues to be at 
all the time, the distance 
covered by him in the frame 
during the reception of the two pulses is zero.Using the inverse Lorentz transformation we can write, | (29) |
 | (30) |
Using the equations (29) and (30) we can write[13] | (31) |
If 
and 
be the actual and observed frequencies of light pulses, respectively .This equation (31) is the formula of the relativistic Doppler’s effect of Special Lorentz transformation for light.  | Figure 3. The frame  is moving along common X-axis with velocity V with respect to  frame |
3.2. Relativistic Doppler’s Effect of most General Lorentz Transformation
Consider two frames of references 
and 
the velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Let the transmitter and receiver be situated at origins 
and 
of frames 
and 
respectively. Let two light signals or pulses be transmitted at time 
and
, T being the true period of light pulses. Let 
be the interval between the receptions of the pulses by the receiver in the frame
. Since the observer and receiver is at rest in frame
, 
is the proper time interval 
between these pulses. Since the observer continues to be at 
all the time, the distance 
covered by him in the frame 
during the reception of the two pulses is zero. | Figure 4. The frame  is moving along any arbitrary direction with velocity  with respect to  frame |
Using the space part and time part of inverse most general Lorentz transformation[2] any one can get the formula of relativistic Doppler’s effect[18] in the most general Lorentz transformation which is | (32) |
This formula of the relativistic Doppler’s effect coincides with the formula of Special Lorentz transformation
3.3. Relativistic Doppler’s Effect of Mixed Number Lorentz Transformation
Consider the same situation described in Figure 4. Using the similar way of most general Lorentz transformation[2] we can find the formula of Relativistic Doppler’s Effect of mixed number Lorentz transformation[18] which is given by  | (33) |
3.4. Relativistic Doppler’s Effect of Quaternion Lorentz Transformation
Consider two frames of references 
and 
as described in Figure 4. If 
and 
be the actual and observed frequencies of light pulses, respectively, then the formula for the relativistic Doppler’s Effect of quaternion Lorentz transformation[14] can be found as  | (34) |
3.5. Relativistic Doppler’s Effect of Geometric Product Lorentz Transformation
Consider two frames of references 
and
. The velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Let the transmitter and receiver be situated at origins 
and 
of frames 
and 
respectively. Let two light signals or pulses be transmitted at time 
and
, T being the true period of light pulses. Let 
be the interval between the receptions of the pulses by the receiver in the frame
. Since the observer and receiver is at rest in frame
, 
is the proper time interval 
between these pulses. Since the observer continues to be at 
all the time, the distance 
covered by him in the frame 
during the reception of the two pulses is zero.Using the space part of the inverse geometric product Lorentz transformation of equation (13) | (35) |
Since 
and 
then equation (35) can be written as
or, | (36) |
This equation (36) shows the second pulse has to travel this extra distance 
then the first pulse in the frame 
to be able to reach at origin 
in the moving frame
.Using the time part of the geometric product Lorentz transformation of equation (13) | (37) |
Since 
and 
then equation (37) can be written as  | (38) |
This relation includes both the actual time period T of the pulses and the time taken 
by the second pulse to cover the extra distance 
in the frame
, i.e. Using 
the equation (39) can be written as  | (40) |
Using (36), (38) and (40) we can write  | (41) |
Using equation (23), the equation (41) can be written as  | (42) |
If 
and 
be the actual and observed frequencies of light pulses, respectively, we have 
and 
then equation (42) gives  | (43) |
this equation (43) is the formula of the relativistic Doppler’s effect of geometric product Lorentz transformation for light.
4. Reflection of Light by a moving Mirror
4.1. Reflection of light by a moving mirror in Special Lorentz Transformation
Consider two frames of references 
and 
where 
is moving along common X-axis with velocity V with respect to 
frame. Consider a mirror M to be fixed in the 
plane of system
. Consider the mirror is perfectly reflecting, moving along the direction of its normal relative to
. Let a ray of light in
be incident at angle 
at
in system
.As mirror M is stationary in system 
ordinary laws of reflection hold good and so the angle of reflection will be 
and the reflected ray lies in 
plane. Let the angles of incidence and reflection are measured in system S be
, we get[2] | (44) |
Using 
we get from (44) | (45) |
This equation (45) is the law of reflection of light by a moving mirror[2] in the case of special Lorentz transformation
4.2. Reflection of Light by a moving Mirror in Most General Lorentz Transformation
Consider two frames of references 
and
. The velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has two components Vx, and Vy . Consider a mirror M to be fixed in the 
plane of system
. Consider the mirror is perfectly reflecting, moving along the direction of its normal relative to
. Let a ray of light in
be incident at angle 
at
in system
.As mirror M is stationary in system 
ordinary laws of reflection hold good and so the angle of reflection will be 
and the reflection ray lies in 
plane .Let the angles of incidence and reflection are measured in system S be
, we get[3] | Figure 5. The reflection of light by a moving mirror where M is fixed in the  plane of system |
Let the incident ray in system 
and 
be represented by 
and 
whereand  | (47) |
As phase is a Lorenz invariant quantity, using equations (46) and (47) we must have  | (48) |
Using 
we get from equation (48)  | (49) |
Using equations (3), (4) and (49) and considering the velocity V has two components 
and
, we can get the formula[19] of the reflection of light ray by a moving mirror in most general Lorentz transformation 
 | Figure 6. The frame  is moving along any arbitrary direction with velocity  with respect to  frame where a mirror M is fixed in the  plane of system |
4.3. Reflection of Light by a moving Mirror in mixed number Lorentz Transformation
Consider two frames of reference 
and 
. The velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Consider same situation as described in the Figure 6.Using equation (7) and (49) we have considering the velocity V has two components 
and 
 | (50) |
Equating the coefficient of t of equation (50) we get, | (51) |
Equating the coefficient of x of equation (50) we get, | (52) |
Equating the coefficient of y of equation (50) we get, | (53) |
Now from equation (52) and (53) we get, | (54) |
Using equation (8) and (49) we have considering the velocity V has two components 
and 
 | (55) |
Equating the coefficient of 
of equation (55) we get, | (56) |
Equating the coefficient of 
of equation (55) we get, | (57) |
Equating the coefficient of 
of equation (55) we get, | (58) |
Now from equation (57) and (58) we get, | (59) |
Similarly, for reflected ray the relation between angle of reflection in system 
and 
can be obtained as  | (60) |
From equation (59) and (60), we get,  | (61) |
This equation (61) is the law of reflection of light ray by a moving mirror in mixed number Lorentz transformation.
4.4. Reflection of Light by a moving Mirror in Quaternion Lorentz Transformation
Consider two frames of reference 
and 
. The velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Consider same situation as described in the Figure 6.Using equation (11) and (49) we have considering the velocity V has two components 
and 
we can get by the similar process of mixed number Lorentz transformation, the same formula for the reflection of light ray by a moving mirror in quaternion Lorentz transformation | (62) |
4.5. Reflection of Light by a moving Mirror in Geometric Product Lorentz Transformation
Consider two frames of reference 
and 
. The velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Consider same situation as described in the Figure 6.Using equation (15) and (49), we have considering the velocity V has two components 
and 
then after a little bit lengthy calculation according to the similar process of mixed number Lorentz transformation we can get the same formula for the reflection of light ray by a moving mirror in geometric product Lorentz transformation | (63) |
5. Comparison of the Study
5.1. Comparison of Relativistic Aberrations of Special, Most General, Mixed Number, Quaternion and Geometric Product Lorenz Transformations
5.2. Comparison of Relativistic Doppler’s Effect of Special, Most General, Mixed Number, Quaternion and Geometric Product Lorenz Transformations
5.3. Comparison of the Formulae of the Reflection of Light by a Moving Mirror in Special, Most General, Mixed Number, Quaternion and Geometric Product Lorenz Transformations
6. Conclusions
The Relativistic aberration, Doppler’s effect and the reflection of light ray by a moving mirror formulae of special, most general, mixed number, quaternion and geometric product Lorenz transformations have been derived clearly. We have found that the formulae for Relativistic aberration and for the reflection of light ray by a moving mirror of mixed number, quaternion and geometric product Lorenz transformations are simpler than the formula of most general Lorenz transformation .
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Abstract
Reference
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