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} $$

Rotation
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
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
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
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
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
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} $$