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


p\rightarrow p^\prime = L p L^\dagger,

where the spinor  L is subject to the condition of unimodularity

 L \bar{L} = 1

In Cl_3 , the spinor  L can be written as the exponential of a biparavector  W


L_{{ }_{ }} = e^{W}

Rotation[edit]

If the biparavector W contains only a bivector (complex vector in Cl_3 ), the Lorentz transformations is a rotation in the plane of the bivector


R  = e^{ -i \frac{1}{2} \boldsymbol{\theta} }

for example, the following expression represents a rotor that applies a rotation angle  \theta around the direction \mathbf{e}_3 according to the right hand rule


R  = e^{-\frac{\theta}{2} \mathbf{e}_{12}}=e^{ -i \frac{\theta}{2} \mathbf{e}_3 },

applying this rotor to the unit vector along \mathbf{e}_1 gives the expected result


\mathbf{e}_1 \rightarrow  e^{ -i \frac{\theta}{2} \mathbf{e}_3 } \mathbf{e}_1
e^{ i \frac{\theta}{2} \mathbf{e}_3 } = \mathbf{e}_1 e^{ i \frac{\theta}{2} \mathbf{e}_3 }e^{ i \frac{\theta}{2} \mathbf{e}_3 } = \mathbf{e}_1 e^{ i \theta \mathbf{e}_3 } =
\mathbf{e}_1 ( \cos(\theta) + i \mathbf{e}_3 \sin(\theta) ) =
\mathbf{e}_1 \cos(\theta) + \mathbf{e}_2 \sin(\theta)

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

  • Unimodular: R \bar{R} = 1
  • Unitary: RR^\dagger = 1

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

 \bar{R} = R^\dagger.

Boost[edit]

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


R  = e^{ \frac{1}{2}\boldsymbol{\eta} }

for example, the following expression represents a boost along the \mathbf{e}_3 direction


B  = e^{  \frac{1}{2} \eta \, \mathbf{e}_3 },

where the real scalar parameter \eta is the rapidity.

The boost B is seen to be:

  • Unimodular: B \bar{B} = 1
  • Real: B^\dagger = B

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


L_{{}_{ }}  = B R

The boost factor can be extracted as


B = \sqrt{L L^\dagger}

and the rotor is obtained from the even grades of L


R = \frac{L + \bar{L}^\dagger}{2 \langle B \rangle_S}

Boost in terms of the required proper velocity[edit]

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

u_{r_{ }} = 1

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


u = L u_{r_{}} L^\dagger,

that can be written as


u = L L^\dagger = BR (BR)^\dagger = B R R^\dagger B^\dagger = BB = B^2,

so that


B = \sqrt{u} = \frac{1+u}{\sqrt{2(1+\langle u \rangle_S)}},

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


u = B^{2^{ }},

so that


\gamma(1+\frac{\mathbf{v}}{c}) = e^{\boldsymbol{\eta}},

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

 \boldsymbol{\eta} = \eta \hat{\boldsymbol{\eta}}

the exponential can be expanded as


\gamma +  \gamma\frac{\mathbf{v}}{c} = \cosh(\eta) + \hat{\boldsymbol{\eta}}\sinh(\eta),

so that

 \gamma_{{ }_{ }} = \cosh{\eta}

and

 \gamma\frac{\mathbf{v}}{c}=\hat{\boldsymbol{\eta}}\sinh(\eta),

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

 \frac{\mathbf{v}}{c} = \hat{\boldsymbol{\eta}} \eta

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


\langle u \bar{v}  \rangle_V  \rightarrow \langle u^\prime \bar{v}^\prime \rangle_V

applying the Lorentz transformation to the component paravectors


\langle u^\prime \bar{v}^\prime  \rangle_V = 
\langle L u L^\dagger \,\, \overline{ L v L^\dagger} \rangle_V=
 \langle L u L^\dagger\, \bar{L}^\dagger\bar{v} \bar{L} \rangle_V =
  \langle L u \bar{v} \bar{L} \rangle_V = 
 L\langle u \bar{v}  \rangle_V\bar{L},

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


F \rightarrow F^{\prime_{ }} = L F \bar{L}