LORENTZ INVARIANCE AND SPIN PROPERTIES OF REFLECTION SYMMETRIC SUM OF VECTORS
Non-Associativity of Lorentz Transformation and Associative Reflection Symmetric 
Transformation 
Mushfiq Ahmad 
Department of Physics, Rajshahi University, Rajshahi, Bangladesh. 
E-mail: mushfiqahmad@ru.ac.bd 
M. Shah Alam 
Department of Physics, Shah Jalal University of Science and Technology, Sylhet, 
Bangladesh.  
E-mail: salam@sust.edu  
Abstract 
Each of the two moving observers observes the relative velocity of the other. The two 
velocities should be equal and opposite. We have shown that this relativistic requirement 
is not fulfilled by Lorentz transformation. We have also shown that the reason is that 
Lorentz transformation is not associative. Reciprocal symmetric transformation is 
associative and fulfills relativistic requirements. 
1. Introduction 
Two observers are moving with velocities U and V. Their relativity velocity according to 
one observer should be W. According to the other observer it should be –W. Arrows give 
the positive directions of the vectors. Negative vectors have the opposite directions. 
According to triangular law of addition of vectors, we must have 
andUVW +−= ˆ)( VUW +−=− ˆ)( . Therefore,   
U}V{ +−− ˆ)( VU +−= ˆ)(                                                          (1.1) 
If  is non-commutative +̂
U}V{ +−− ˆ)( )(ˆ UV −+≠                                                          (1.2) 
2. Lorentz-Einstein Transformation  
mailto:mushfiqahmad@
http://ru.ac.bd:20000/ym/Compose?To=salam@sust.edu
If a body is traveling with velocity U and an observer is traveling with velocity V, the 
relative velocity, , according to Lorentz-Einstein transformation is given by.  EL−W
−−−+−
=+−=−
1)/(11)/(1
           (2.1) 
We have used the symbol  to represent Lorentz-Einstein addition of velocities. If the 
object and the observer swap positions so that U is the velocity of the observer and V is 
the velocity of the object, the relative velocity,
, is given by 
)(U/11V)(U/1
V(-U)W
−−−+−
            (2.2) 
In general they are not equal and opposite. 
ELEL −− −≠ WW
                                                   (2.3) 
An exception is when U and V are parallel. In that case they are equal.  
(2.1) and (2.2) show that  
ELEl −− ≠−=−+ WWUV
                                         (2.4) 
3. Non-Associativity of Lorentz Transformation 
Consider 3 vectors U, V and W using definition (2.1) of+
, we can calculate 
 and 
 and show that in general WV)(U ++ W)(VU ++
≠++ WV)(U
W)(VU ++
                                                   (3.1) 
We can by pass the cumbersome calculation and prove (3.1) by using the following 
theorem. 
4. Theorem: 
If operation  is associative +̂
 )(ˆ)()ˆ( UVVU −+−=+−                                                    (4.1a) 
And if operation  is also non-commutative +̂
)(ˆ)()ˆ( VUVU −+−≠+−                                                    (4.1b) 
(4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for 
associativity and non-commutativity.  
Proof: 
Consider 
)(ˆ)(ˆ}ˆ{ UVVU −+−++                                              (4.2) 
Using associativity and (4.2) becomes 0)(ˆ =−+ VV
=−+−++ )(ˆ)(ˆ}ˆ{ UVVU 0)(ˆ)}(ˆ{ˆ =−+−++ UVVU                       (4.3) 
Therefore, 
0)(ˆ)(ˆˆ =−+−++ }UV{V}{U                                          (4.5) 
Comparing (4.5) with 
0ˆˆˆ =+−++ V)(UV)(U                                              (4.6) 
We find that 
)(ˆ)()ˆ( UVVU −+−=+−                                            (4.7) 
If  is also non-commutative +̂
U}{V +− ˆ )(ˆ)( UV −+−≠                                            (4.8) 
5. Non-Associativity of Lorentz Transformation 
(2.4) contradicts (4.7) and (4.8). Therefore, Lorentz transformation is not associative. 
6. Associativity of Reciprocal Symmetric Transformation 
Relative velocity according to reciprocal transformation is defined as 
VxUUV
U(-V)W
ˆ                                   (6.1) 
A direct calculation will show that for any U, V and W 
=++ WV)(U ˆˆ W)(VU ++ ˆˆ                                          (6.2) 
(6.2) can also be seen from the fact that (6.1) is built from Pauli quaternion1 and Pauli 
quaternion is associative.  
Definition (6.1) fulfills requirement (4.1) 
)(ˆ)()ˆ( VUUV −+−=+−                                            (6.3) 
7. Comparison between Lorentz and Reciprocal Symmetric Transformations 
Lorentz transformation and reciprocal symmetric transformations have the same 
magnitude. 
|)||ˆ| VUUV +=+
                                              (7.1) 
Reciprocal symmetric transformation has a complex x-product term. This x-product term 
ensures (6.3). In Lorentz transformation (2.1) the product term comes as a dot-product, 
which does not change sing when order is changed. We have already seen2  the complex 
part explicitly shows the rotation hidden in Lorentz transformation. We have also seen3 
that the complex part explicitly shows the origin of spin. 
8. Conclusion 
We have shown that Lorentz Transformation is not associative and that Reciprocal 
Symmetric transformation is associative. Reciprocal Symmetric transformation fulfills 
requirement of Lorentz invariance4. It is complex. This complex part gives rise to spin5. 
                                                                                                                                                                                 
1 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of 
Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 
2 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of 
Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 
3 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 
4 Mushfiq AhmadReciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical 
Concepts. http://www.arxiv.org/abs/math-ph/0611024. 
5 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
	Abstract
	(1.1)
	(1.2)
	If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity,�, according to Lorentz-Einstein transformation is given by.
	(2.1)
	We have used the symbol � to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity,�, is given by
	(2.2)
	(2.3)
	(2.4)
	(3.1)
	And if operation � is also non-commutative
	(4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity.
	Proof:
	Consider
	(4.2)
	(4.3)
	(4.5)
	(6.2)
	(6.3)
	(7.1)

  Each of the two moving observers observes the relative velocity of the other.
The two velocities should be equal and opposite. We have shown that this
relativistic requirement is not fulfilled by Lorentz transformation. We have
also shown that the reason is that Lorentz transformation is not associative.
Reciprocal symmetric transformation is associative and fulfills relativistic
requirements.

Introduction 
Two observers are moving with velocities U and V. Their relativity velocity according to 
one observer should be W. According to the other observer it should be –W. Arrows give 
the positive directions of the vectors. Negative vectors have the opposite directions. 
According to triangular law of addition of vectors, we must have 
andUVW +−= ˆ)( VUW +−=− ˆ)( . Therefore,   
U}V{ +−− ˆ)( VU +−= ˆ)(                                                          (1.1) 
If  is non-commutative +̂
U}V{ +−− ˆ)( )(ˆ UV −+≠                                                          (1.2) 
2. Lorentz-Einstein Transformation  
mailto:mushfiqahmad@
http://ru.ac.bd:20000/ym/Compose?To=salam@sust.edu
If a body is traveling with velocity U and an observer is traveling with velocity V, the 
relative velocity, , according to Lorentz-Einstein transformation is given by.  EL−W
−−−+−
=+−=−
1)/(11)/(1
           (2.1) 
We have used the symbol  to represent Lorentz-Einstein addition of velocities. If the 
object and the observer swap positions so that U is the velocity of the observer and V is 
the velocity of the object, the relative velocity,
, is given by 
)(U/11V)(U/1
V(-U)W
−−−+−
            (2.2) 
In general they are not equal and opposite. 
ELEL −− −≠ WW
                                                   (2.3) 
An exception is when U and V are parallel. In that case they are equal.  
(2.1) and (2.2) show that  
ELEl −− ≠−=−+ WWUV
                                         (2.4) 
3. Non-Associativity of Lorentz Transformation 
Consider 3 vectors U, V and W using definition (2.1) of+
, we can calculate 
 and 
 and show that in general WV)(U ++ W)(VU ++
≠++ WV)(U
W)(VU ++
                                                   (3.1) 
We can by pass the cumbersome calculation and prove (3.1) by using the following 
theorem. 
4. Theorem: 
If operation  is associative +̂
 )(ˆ)()ˆ( UVVU −+−=+−                                                    (4.1a) 
And if operation  is also non-commutative +̂
)(ˆ)()ˆ( VUVU −+−≠+−                                                    (4.1b) 
(4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for 
associativity and non-commutativity.  
Proof: 
Consider 
)(ˆ)(ˆ}ˆ{ UVVU −+−++                                              (4.2) 
Using associativity and (4.2) becomes 0)(ˆ =−+ VV
=−+−++ )(ˆ)(ˆ}ˆ{ UVVU 0)(ˆ)}(ˆ{ˆ =−+−++ UVVU                       (4.3) 
Therefore, 
0)(ˆ)(ˆˆ =−+−++ }UV{V}{U                                          (4.5) 
Comparing (4.5) with 
0ˆˆˆ =+−++ V)(UV)(U                                              (4.6) 
We find that 
)(ˆ)()ˆ( UVVU −+−=+−                                            (4.7) 
If  is also non-commutative +̂
U}{V +− ˆ )(ˆ)( UV −+−≠                                            (4.8) 
5. Non-Associativity of Lorentz Transformation 
(2.4) contradicts (4.7) and (4.8). Therefore, Lorentz transformation is not associative. 
6. Associativity of Reciprocal Symmetric Transformation 
Relative velocity according to reciprocal transformation is defined as 
VxUUV
U(-V)W
ˆ                                   (6.1) 
A direct calculation will show that for any U, V and W 
=++ WV)(U ˆˆ W)(VU ++ ˆˆ                                          (6.2) 
(6.2) can also be seen from the fact that (6.1) is built from Pauli quaternion1 and Pauli 
quaternion is associative.  
Definition (6.1) fulfills requirement (4.1) 
)(ˆ)()ˆ( VUUV −+−=+−                                            (6.3) 
7. Comparison between Lorentz and Reciprocal Symmetric Transformations 
Lorentz transformation and reciprocal symmetric transformations have the same 
magnitude. 
|)||ˆ| VUUV +=+
                                              (7.1) 
Reciprocal symmetric transformation has a complex x-product term. This x-product term 
ensures (6.3). In Lorentz transformation (2.1) the product term comes as a dot-product, 
which does not change sing when order is changed. We have already seen2  the complex 
part explicitly shows the rotation hidden in Lorentz transformation. We have also seen3 
that the complex part explicitly shows the origin of spin. 
8. Conclusion 
We have shown that Lorentz Transformation is not associative and that Reciprocal 
Symmetric transformation is associative. Reciprocal Symmetric transformation fulfills 
requirement of Lorentz invariance4. It is complex. This complex part gives rise to spin5. 
                                                                                                                                                                                 
1 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of 
Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 
2 Mushfiq Ahmad, M. Shah Alam, M.O.G. Talukder. Comparison between Spin and Rotation Properties Of 
Lorentz Einstein and Reflection Symmetric Transformations.http://www.arxiv.org/abs/math-ph/0701067 
3 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 
4 Mushfiq AhmadReciprocal Symmetry and Equivalence between Relativistic and Quantum Mechanical 
Concepts. http://www.arxiv.org/abs/math-ph/0611024. 
5 Mushfiq Ahmad. Reciprocal Symmetry and the Origin of Spin. http://www.arxiv.org/abs/math-ph/0702043 
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
http://www.arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1
	Abstract
	(1.1)
	(1.2)
	If a body is traveling with velocity U and an observer is traveling with velocity V, the relative velocity,�, according to Lorentz-Einstein transformation is given by.
	(2.1)
	We have used the symbol � to represent Lorentz-Einstein addition of velocities. If the object and the observer swap positions so that U is the velocity of the observer and V is the velocity of the object, the relative velocity,�, is given by
	(2.2)
	(2.3)
	(2.4)
	(3.1)
	And if operation � is also non-commutative
	(4.1a) is a necessary condition for associativity. (4.1b) is a necessary condition for associativity and non-commutativity.
	Proof:
	Consider
	(4.2)
	(4.3)
	(4.5)
	(6.2)
	(6.3)
	(7.1)
