Atikur Rahman Baizid1, 2, Md. Shah Alam2
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
It is obvious that Lorentz transformation is the starting point of Relativistic mechanics. There are different types of Lorentz transformations such as Special, Most general, Mixed number, Geometric product, and Quaternion Lorentz transformations. To study relativistic mechanics, we must need to know the properties of different types of Lorentz transformations. In this paper we have studied reciprocal property, associative property, isotropic property and group property of the above Lorentz transformations.
Keywords:
Special Lorentz Transformation, Most General Lorentz Transformation, Mixed Number Lorentz Transformation, Geometric Product Lorentz Transformation, Reciprocal Property, Associative Property , Isotropic Property, Group Property
Cite this paper: Atikur Rahman Baizid, Md. Shah Alam, Properties of Different Types of Lorentz Transformations, American Journal of Mathematics and Statistics, Vol. 3 No. 3, 2013, pp. 105-123. doi: 10.5923/j.ajms.20130303.03.
1. Introduction
In most treatments on special relativity, the line of motion is aligned with the x-axis. This is a natural choice because in such a situation the 
coordinates are invariant under the Lorentz transformations. However; it is of interest to study the case when the line of motion does not coincide with any of the coordinate axes. Practical instances of such a situation are an airplane during landing or take off. The ground at the airfield has a natural coordinate system with the x-axis parallel to the ground, whereas the airplane ascends or descends at an angle with ground. For this reason we need to study the properties of different types Lorentz transformations where the line of action is along x-axis as well as along any arbitrary line. We have studied the Reciprocal property, Associative property, Isotropic property, Group property of different types of Lorentz transformations.
1.1. Special Lorentz Transformation
Consider two inertial frames of Reference 
and 
where the frame 
is at rest and the frame 
is moving along X-axis with velocity V with respect to 
frame. The space and time coordinates of 
and 
are (x, y, z, t) and (x′, y′, z′, t′) respectively. The relation between the coordinates of 
and 
is called Special Lorentz transformation which can be written as [1] | (1) |
And  | (2) |
1.2. Most General Lorentz Transformation
When the velocity 
of 
with respect to 
is not along X-axis i.e. the velocity 
has three components Vx, Vy and Vz. Then the relation between the coordinates of 
and 
is called Most general Lorentz transformation which can be written as [2] | (3) |
and | (4) |
Where
,
in unit of C
1.3. Mixed Number Lorentz Transformation
In the case of Most General Lorentz transformation, the velocity 
of S' with respect to 
is not along X-axis; i.e., the velocity 
has three components, Vx, Vy, and Vz. Let in this case Z and Z' be the space parts in 
and S' frames, respectively. Then using the mixed product 
, Mixed number Lorentz transformations [3 – 6] can be written as | (5) |
And  | (6) |
1.4. Geometric Product Lorentz Transformation
In this case the velocity 
of S' with respect to 
also has three components, Vx, Vy, and Vz as the Most general Lorentz transformation. Let in this case 
and 
be the space parts in 
and S' frames respectively. Then using geometric product of two vectors 
the geometric product Lorentz transformation [7, 8] can be written as  | (7) |
And | (8) |
1.5. Quaternion 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 S' frames respectively. Then using quaternion product 
the Quaternion Lorentz transformation [9-12] can be written as  | (9) |
And  | (10) |
2. Reciprocal Property of Different Types of Lorentz Transformations
2.1. Reciprocal Property of Special Lorentz Transformation
The velocity addition formula for special Lorentz transformation [13] can beWritten as | (11) |
If we replace U by P where 
then 
will be change to 
where 
Reciprocal property demands that if 
then 
Now, | (12) |
Consequently, special Lorentz transformation satisfies the reciprocal property.
2.2. Reciprocal Property of Most General Lorentz Transformation
From the transformation equations of addition of velocities of most general Lorentz transformation [14] we have | (13) |
If we replace 
by 
where 
then 
will be change to 
where 
Reciprocal property demands that if 
then 
Now,
Can be written as
So,  | (14) |
Consequently, the most general Lorentz transformation does not satisfy the reciprocal property.
2.3. Reciprocal Property of Mixed Number Lorentz Transformation
From the transformation equations of addition of velocities of mixed number Lorentz transformation [14] we have | (15) |
If we replace 
by 
where 
then 
will be change to 
where 
Reciprocal property demands that if 
then
Now,
can be written as
 | (16) |
SimilarlyIf we replace 
by 
where 
then 
will be change to 
where 
Consequently, Mixed number Lorentz transformation satisfies the reciprocal property.
2.4. Reciprocal Property of Geometric Product Lorentz Transformation
From the transformation equations of addition of velocities of geometric product Lorentz transformation [14] we have, | (17) |
If we replace 
by 
where 
then 
will be change to 
where
Reciprocal property demands that if 
then 
Now, 
Can be written as | (18) |
Consequently, geometric product Lorentz transformation does not satisfy the reciprocal property.
2.5. Reciprocal Property of Quaternion Lorentz Transformation
From the transformation equations of addition of velocities of Quaternion Lorentz transformation [14] we have | (19) |
If we replace 
by 
where 
then 
will be change to 
where 
Reciprocal property demands that if 
then 
Now, 
Can be written as | (20) |
Consequently, the Quaternion Lorentz transformation does not satisfy the reciprocal property.
3. Associative Property of Different Types of Lorentz Transformations
Consider three inertial frames of reference S , 
,
and
where the frame S is at rest and the frame 
is moving with velocity 
with respect to S, 
is moving with velocity 
with respect to
, 
is moving with velocity 
respect to 
then associative property says that 
[Fig-1]. We are going to discuss the associative property of different Lorentz transformations in unit of c. Let 
are the symbols of the Lorentz sum of Special, Most general, Mixed number, geometric product, and Quaternion product Lorentz transformations respectively.
3.1. Associative Property of Special Lorentz Transformation
The velocity addition formula for special Lorentz transformation [13] can be written as | (21) |
Now , if 
moves with velocity 
with respect to S , then according to the velocity addition formula for special Lorentz transformation [13] can be written as | (22) |
Again, Let 
moves with velocity V with respect to 
and 
moves with velocity W with respect to 
then according to the velocity addition formula for the Special Lorentz transformation [13] the resultant velocity of V and W can be written as  | (23) |
 | Figure 1. Associative Property of Lorentz transformations |
Finally, let 
moves with velocity U with respect to S and 
moves with velocity 
with respect to 
then the resultant velocity [13] of U and 
can be written as | (24) |
Hence, | (25) |
Consequently, Special Lorentz transformation satisfies the Associative property.
3.2. Associative Property of Most General Lorentz Transformation
The velocity addition formula for most general Lorentz transformation [14] can be written as  | (26) |
Let us consider 
moves with velocity 
with respect to S and 
moves with velocity 
respect to 
then according to the velocity addition formula for most general Lorentz transformation [14] can be written as
Substituting the value of 
we have | (27) |
Again, Let 
moves with velocity 
with respect to 
and 
moves with velocity 
with respect to 
then according to the velocity addition formula for most general Lorentz transformation [14] the resultant velocity of 
and 
can be written as  | (28) |
Finally, let 
moves with velocity 
with respect to S and 
moves with velocity 
with respect to 
then the resultant velocity [14] of 
and 
can be written as | (29) |
Substituting the value of 
the above expression can be written as | (30) |
Hence, | (31) |
Consequently, the Most general Lorentz transformation does not satisfy the Associative property.
3.3. Associative Property of Mixed Number Lorentz Transformation
The velocity addition formula for mixed Number Lorentz transformation [14] can be written as  | (32) |
Now, if 
moves with velocity 
with respect to S and 
moves with velocity 
respect to 
then according to the velocity addition formula for mixed number Lorentz transformation [14] can be written as | (33) |
Again, Let 
moves with velocity 
with respect to 
and 
moves with velocity 
with respect to 
then according to the velocity addition formula for mixed number Lorentz transformation [14] the resultant velocity of 
and 
can be written as  | (34) |
Finally, let 
moves with velocity 
with respect to S and 
moves with velocity 
with respect to 
then using (34) the resultant velocity [14] of 
and 
can be written as | (35) |
Can be written as
Hence, | (36) |
Consequently, mixed number Lorentz transformation satisfies the Associative property.
3.4. Associative Property of Geometric Product Lorentz Transformation
The velocity addition formula for geometric product Lorentz transformation [14] can be written as  | (37) |
Now, if 
moves with velocity 
with respect to S and 
moves with velocity 
respect to 
then according to the velocity addition formula for geometric product Lorentz transformation [14] can be written as | (38) |
Again, Let 
moves with velocity 
with respect to 
and 
moves with velocity 
with respect to 
then according to the velocity addition formula for the Geometric product Lorentz transformation [14] the resultant velocity of 
and 
can be written as  | (39) |
Finally, let 
moves with velocity 
with respect to S and 
moves with velocity 
with respect to 
then using (39) the resultant velocity [14] of 
and 
can be written as
can be written as | (40) |
Hence, | (41) |
Consequently, geometric product Lorentz transformation does not satisfy the Associative property.
3.5. Associative Property of Quaternion Lorentz Transformation
The velocity addition formula for Quaternion Lorentz transformation [14] can be written as  | (42) |
Now, if 
moves with velocity 
with respect to S and 
moves with velocity 
respect to 
then according to the velocity addition formula for the Quaternion Lorentz transformation [14] can be written as | (43) |
Again, Let 
moves with velocity 
with respect to 
and 
moves with velocity 
with respect to 
then according to the velocity addition formula for the Quaternion Lorentz transformation [14] the resultant velocity of 
and 
can be written as  | (44) |
Finally, let 
moves with velocity 
with respect to S and 
moves with velocity 
with respect to 
then using (44) the resultant velocity [14] of 
and 
can be written as | (45) |
Can be written as
Hence, | (46) |
Consequently, Quaternion Lorentz transformation does not satisfy the Associative property.It can be easily shown that the above Lorentz Transformations satisfy the associative property if Vy=0 and Vz =0, reducing these to Special Lorentz Transformation
4. Isotropic Property of Lorentz Transformations
Consider three inertial frames of Reference 
, 
and 
where 
moves with velocity 
with respect to
and 
moves with velocity 
with respect to
. If 
be the velocity of 
with respect to
. | Figure 2. Isotropic property of Lorentz Transformations |
If a, b and c are the three angles of the triangle
, then according to Lorentz sum we can write | (47) |
 | (48) |
 | (49) |
and their product
Now the isotropic property demands that if
then 
4.1. Isotropic Property of Special Lorentz Transformation
For special Lorentz transformation, there will be no option to make a triangle. So, isotropic property is not applicable for special Lorentz transformation.
4.2. Isotropic Property of Most General Lorentz Transformation
The velocity addition formula for most general Lorentz transformation [14] can be written as | (50) |
Using equation (47) and (50) we can write (taking c = 1)
 | (51) |
Putting A = B = C in equation (51) we get 
 | (52) |
Using equation (48) and (50) we can write (taking c = 1)
 | (53) |
Putting A = B = C in equation (53) we get 
 | (54) |
Using equation (49) and (50) we can write (taking c = 1) 
 | (55) |
Putting A = B = C in equation (55) we get 
 | (56) |
Hence, from equations (52), (54) and (56) we have 
Hence, most general Lorentz transformation satisfies the isotropic property
4.3. Isotropic Property of Mixed Number Lorentz Transformation
The velocity addition formula for mixed number Lorentz transformation [14] can be written as  | (57) |
Using equation (47) and (57) we can write (taking c = 1) | (58) |
Putting A = B = C in equation (22) we get 
 | (59) |
Using equation (48) and (57) we can write (taking c = 1) 
 | (60) |
Putting A = B = C in equation (60) we get 
 | (61) |
Using equation (49) and (57) we can write (taking
)
 | (62) |
Putting A = B = C in equation (26) we get 
 | (63) |
Hence, from equations (59), (61) and (63) we have 
Hence, Mixed number Lorentz transformation satisfies the isotropic property.
4.4. Isotropic Property of Geometric Product Lorentz Transformation
The velocity addition formula for Geometric product Lorentz transformation [14] can be written as  | (64) |
Using equation (47) and (64) we can write (taking c = 1) | (65) |
Putting A = B = C in equation (65) we get
 | (66) |
Using equation (48) and (64) we can write (taking c = 1)  | (67) |
Putting A = B = C in equation (67) we get
 | (68) |
Using equation (49) and (64) we can write (taking c = 1) | (69) |
Putting A = B = C in equation (69) we get 
 | (70) |
Hence, from equations (66), (68) and (70) we have 
Hence, geometric product Lorentz Transformation satisfies the isotropic property
4.5. Isotropic Property of Quaternion Lorentz Transformation
The velocity addition formula for Quaternion Lorentz transformation [14] can be written as  | (71) |
Using equation (47) and (71) we can write (taking c = 1)
 | (72) |
Putting A = B = C in equation (72) we get 
 | (73) |
Using equation (48) and (71) we can write (taking c = 1)  | (74) |
Putting A = B = C in equation (74) we get 
 | (75) |
Using equation (49) and (71) we can write (taking c = 1)  | (76) |
Putting A = B = C in equation (76) we get 
 | (77) |
Hence, from equations (73), (75) and (77) we have 
Hence, Quaternion Lorentz transformation satisfies the isotropic property.
5. Group Property of Lorentz Transformations
The result of two Lorentz transformations is itself a Lorentz transformation [13]
5.1. Group Property of Special Lorentz Transformation
Consider three inertial frames of reference 
, 
and 
where 
has relative velocity 
with respect to 
along positive x-axis and 
has relative velocity 
with respect to 
along the same direction as shown in fig.1. According to special Lorentz transformation we get | (78) |
 | Figure 3. Special Lorentz Transformations |
Similarly if the origins in 
and 
coincide at 
the space and time co-ordinates in 
and
related as  | (79) |
Suppose, 
be the resultant velocity of 
and 
then from the velocity addition formula [14] we get | (80) |
is the velocity of system 
relative to
. To prove this result we have to show that | (81) |
Now, 
Can be written as
 | (82) |
Now using (15) 
Can be written as
So,  | (83) |
Again, from equation (82)
Can be written as 
Thus the result of two Lorentz transformations is itself a Lorentz transformation. Hence Special Lorentz transformations form a group.
5.2. Group Property of Most General Lorentz Transformation
Consider three inertial frames of Reference 
, 
and 
where 
moves with velocity
with respect to
and 
moves with velocity 
with respect to
. If 
be the velocity of 
with respect to 
 | Figure 4. Most General Lorentz Transformation |
So the relation between the co-ordinates 
in 
and 
in 
can be expressed by most general Lorentz transformation by | (84) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by most general Lorentz transformation by | (85) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by most general Lorentz transformation by | (86) |
From the velocity addition formula for Most General Lorentz transformation [14] we can write
Now we have from (85)
Where 
Now we have to show 
We know 
Can be written as
Hence, the time part of the Most General Lorentz transformation does not satisfy the group property.Again from equation (85)
Which can be written as 
Where the operator 
in general is different from the unit operator. 
is the velocity of the system 
relative to 
and 
is the velocity of the system 
relative to
.We get 
and 
Now  | (87) |
Hence we have 
UThe rotation operator thus represents an infinitesimal rotation around the direction of the vector 
[3]Hence, most general Lorentz transformation does not satisfy the group property without rotation
5.3. Group Property of Mixed Number Lorentz Transformation
Figure-2 describes that three inertial frames of reference 
, 
and 
where 
moves with velocity 
with respect to
and 
moves with velocity 
with respect to
.If 
be the velocity of 
with respect to 
then the relation between the co-ordinates 
in 
and 
in 
can be expressed by mixed number Lorentz transformation by | (88) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by Mixed Number Lorentz transformation by  | (89) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by mixed Number Lorentz transformation by  | (90) |
The velocity addition formula for mixed Number Lorentz transformation [14] can be written as | (91) |
Now from equation (89) we can write  | (92) |
Again from equation (91) we can write 
Where 
Now we have to show that 
Hence, 
Therefore, mixed Number Lorentz transformation satisfies the group property
5.4. Group Property of Geometric Product Lorentz Transformation
Figure-2 describes that three inertial frames of Reference 
, 
and 
where 
moves with velocity 
with respect to
and 
moves with velocity 
with respect to
.If 
be the velocity of 
with respect to 
then the relation between the co-ordinates 
in 
and 
in 
can be expressed by geometric product Lorentz transformation by  | (93) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by geometric product Lorentz transformation by  | (94) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by geometric Product Lorentz transformation by  | (95) |
The velocity addition formula for geometric product Lorentz transformation [14] can be written as  | (96) |
Now from equation (94) we can write 
Again from equation (94) we can write 
Where
Now we have to show that 
Therefore, Geometric Product Lorentz transformation does not satisfy the isotropic property
5.5. Group Property of Quaternion Product Lorentz Transformation
Figure-2 describes that three inertial frames of Reference 
, 
and 
where 
moves with velocity 
with respect to
and 
moves with velocity 
with respect to
.If 
be the velocity of 
with respect to 
then the relation between the co-ordinates 
in 
and 
in 
can be expressed by Quaternion Product Lorentz transformation by  | (97) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by Quaternion Product Lorentz transformation by  | (98) |
Similarly, the relation between the co-ordinates 
in 
and 
in 
can be expressed by Quaternion Product Lorentz transformation by  | (99) |
The velocity addition formula for Quaternion Product Lorentz transformation [14] can be written as  | (100) |
Now from equation (98) we can write 
Or, 
Where 
Again from equation (98) we can write 
Can be written as
Where 
Now we have to show that 
Therefore, Quaternion Product Lorentz transformation does not satisfy the group propertyTable 1. Comparison of the properties of different Lorentz transformations
 |
| |
|
6. Conclusions
We have discussed different properties of different Lorentz transformations and obtained that special and mixed number Lorentz transformations satisfy the reciprocal property. Most general, geometric product and Quaternion Lorentz transformations do not satisfy the reciprocal property. Special and mixed number Lorentz transformations satisfy Associative property but the most general, geometric product and Quaternion Lorentz transformations do not satisfy the Associative property. Isotropic property is not applicable for special Lorentz transformation. Most general, mixed number, geometric product and Quaternion Lorentz transformations satisfy the isotropic property. Special and mixed number Lorentz transformations satisfy the group property. Most general, geometric product and Quaternion Lorentz transformations do not satisfy the group property.
References
| [1] | Resnick, Robert. Introduction to special relativity, Wiley Eastern Limited New age international limited - 1994 |
| [2] | Moller C. The Theory of Relativity Oxford University Press-1972 |
| [3] | Alam, M.S. 2000. Study of Mixed Number, Proc. Pakistan Acad. of Sci. 37(1):119-122. |
| [4] | Alam, M.S. 2001. Mixed product of vectors, Journal of Theoretics, 3(4). http://www.journaloftheoretics.com/ |
| [5] | Alam, M.S. 2003. Comparative study of mixed product and quaternion product, Indian J. Physics A 77: 47-49. |
| [6] | Alam, M.S. 2001. Mixed product of vectors, Journal of Theoretics, 3(4). http://www.journaloftheoretics.com/ |
| [7] | Datta, B.K., De Sabbata V. and Ronchetti L. 1998. Quantization of gravity in real space time, Il Nuovo Cimento, 113B. |
| [8] | Datta, B.K., Datta R. 1998. Einstein field equations in spinor formalism, Foundations of Physics letters, 11, 1. |
| [9] | Kyrala, A. 1967. Theoretical Physics, W.B. Saunders Company. Philadelphia & London, Toppan Company Limited. Tokyo, Japan. |
| [10] | http://mathworld.wolfram.com/Quaternion.html |
| [11] | http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quaternions1.html |
| [12] | Alam, M.S. 2003. Different types of product of vectors, NewsBull.Cal.Math.Soc. 26(1,2 & 3) 21-24. |
| [13] | Satya Prokash. Relativistic Mechanics, Pragati Prakashan-2000, India. |
| [14] | Md. Shah Alam and Khurshida Begum. Different Types of Lorentz Transformations, Jahangirnagar Physics studies-2009. |
Abstract
Reference
Full-Text PDF
Full-text HTML