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

Rotation[edit]

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

Boost[edit]

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 boost[edit]

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 velocity[edit]

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 velocity[edit]

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 biparavectors[edit]

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