Physics Using Geometric Algebra/Relativistic Classical Mechanics/Lorentz transformations

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A Lorentz transformation is a linear transformation that maintains the length of paravectors. Lorentz transformations include rotations and boosts as proper Lorentz transformations and reflections and non-orthochronous transformations among the improper Lorentz transformations.

A proper Lorentz transformation can be written in spinorial form as

where the spinor is subject to the condition of unimodularity

In , the spinor can be written as the exponential of a biparavector

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RotationEdit

If the biparavector   contains only a bivector (complex vector in  ), the Lorentz transformations is a rotation in the plane of the bivector

 

for example, the following expression represents a rotor that applies a rotation angle   around the direction   according to the right hand rule

 

applying this rotor to the unit vector along   gives the expected result

 

The rotor   has two fundamental properties. It is said to be unimodular and unitary, such that

  • Unimodular:  
  • Unitary:  

In the case of rotors, the bar conjugation and the reversion have the same effect

 

BoostEdit

If the biparavector   contains only a real vector, the Lorentz transformation is a boost along the direction of the respective vector

 

for example, the following expression represents a boost along the   direction

 

where the real scalar parameter   is the rapidity.

The boost   is seen to be:

  • Unimodular:  
  • Real:  

The Lorentz transformation as a composition of a rotation and a boostEdit

In general, the spinor of the proper Lorentz transformation can be written as the product of a boost and a rotor

 

The boost factor can be extracted as

 

and the rotor is obtained from the even grades of  

 

Boost in terms of the required proper velocityEdit

The proper velocity of a particle at rest is equal to one

 

Any proper velocity, at least instantaneously, can be obtained from an active Lorentz transformation from the particle at rest, such that

 

that can be written as

 

so that

 

where the explicit formula of the square root for a unit length paravector was used.

Rapidity and velocityEdit

The proper velocity is the square of the boost

 

so that

 

rewriting the rapidity in terms of the product of its magnitude and respective unit vector

 

the exponential can be expanded as

 

so that

 

and

 

where we see that in the non-relativistic limit the rapidity becomes the velocity divided by the speed of light

 

Lorentz transformation applied to biparavectorsEdit

The Lorentz transformation applied to biparavector has a different form from the Lorentz transformation applied to paravectors. Considering a general biparavector written in terms of paravectors

 

applying the Lorentz transformation to the component paravectors

 

so that if   is a biparavector, the Lorentz transformations is given by