Physics Using Geometric Algebra/Relativistic Classical Mechanics/Lorentz transformations

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


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



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




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