# 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

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

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 $W$

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

## RotationEdit

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

$\bar{R} = R^\dagger.$

## BoostEdit

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 boostEdit

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 velocityEdit

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 velocityEdit

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 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

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

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

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