American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2013;  3(3): 105-123

doi:10.5923/j.ajms.20130303.03

Properties of Different Types of Lorentz Transformations

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.

Article Outline

1. Introduction
    1.1. Special Lorentz Transformation
    1.2. Most General Lorentz Transformation
    1.3. Mixed Number Lorentz Transformation
    1.4. Geometric Product Lorentz Transformation
    1.5. Quaternion Lorentz Transformation
2. Reciprocal Property of Different Types of Lorentz Transformations
    2.1. Reciprocal Property of Special Lorentz Transformation
    2.2. Reciprocal Property of Most General Lorentz Transformation
    2.3. Reciprocal Property of Mixed Number Lorentz Transformation
    2.4. Reciprocal Property of Geometric Product Lorentz Transformation
    2.5. Reciprocal Property of Quaternion Lorentz Transformation
3. Associative Property of Different Types of Lorentz Transformations
    3.1. Associative Property of Special Lorentz Transformation
    3.2. Associative Property of Most General Lorentz Transformation
    3.3. Associative Property of Mixed Number Lorentz Transformation
    3.4. Associative Property of Geometric Product Lorentz Transformation
    3.5. Associative Property of Quaternion Lorentz Transformation
4. Isotropic Property of Lorentz Transformations
    4.1. Isotropic Property of Special Lorentz Transformation
    4.2. Isotropic Property of Most General Lorentz Transformation
    4.3. Isotropic Property of Mixed Number Lorentz Transformation
    4.4. Isotropic Property of Geometric Product Lorentz Transformation
    4.5. Isotropic Property of Quaternion Lorentz Transformation
5. Group Property of Lorentz Transformations
    5.1. Group Property of Special Lorentz Transformation
    5.2. Group Property of Most General Lorentz Transformation
    5.3. Group Property of Mixed Number Lorentz Transformation
    5.4. Group Property of Geometric Product Lorentz Transformation
    5.5. Group Property of Quaternion Product Lorentz Transformation
6. Conclusions

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 be
Written 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)
Similarly
If 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 , ,andwhere 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 andrelated 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
U
The 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 property
Table 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.

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