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

$p\to p^{\prime }=Lp{L}^{†},$

where the spinor $L$ is subject to the condition of unimodularity

$L\overline{L}=1$

In $C{l}_3$, 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 $C{l}_3$), the Lorentz transformations is a rotation in the plane of the bivector

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

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

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

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

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

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

• Unimodular: $R\overline{R}=1$
• Unitary: $R{R}^{†}=1$

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

$\overline{R}=R^{†}.$

Boost

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

$R=e^{\frac{1}{2}\mathit{\eta }}$

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

$B=e^{\frac{1}{2}\eta {𝐞}_3},$

where the real scalar parameter $\eta$ is the rapidity.

The boost $B$ is seen to be:

• Unimodular: $B\overline{B}=1$
• Real: $B^{†}=B$

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

$L_{}=BR$

The boost factor can be extracted as

$B=\sqrt{L{L}^{†}}$

and the rotor is obtained from the even grades of $L$

$R=\frac{L+{\overline{L}}^{†}}{2⟨B{⟩}_S}$

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

$u=L{u}_{r_{}}{L}^{†},$

that can be written as

$u=L{L}^{†}=BR(BR{)}^{†}=BR{R}^{†}{B}^{†}=BB=B^2,$

so that

$B=\sqrt{u}=\frac{1+u}{\sqrt{2(1+⟨u{⟩}_S)}},$

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

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

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

$\mathit{\eta }=\eta \hat{\mathit{\eta }}$

the exponential can be expanded as

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

so that

${\gamma }_{}=\cosh \eta$

and

$\gamma \frac{𝐯}{c}=\hat{\mathit{\eta }}\sinh (\eta ),$

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

$\frac{𝐯}{c}=\hat{\mathit{\eta }}\eta$

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

$⟨u\overline{v}{⟩}_V\to ⟨u^{\prime }{\overline{v}}^{\prime }{⟩}_V$

applying the Lorentz transformation to the component paravectors

$⟨u^{\prime }{\overline{v}}^{\prime }{⟩}_V=⟨Lu{L}^{†}\overline{Lv{L}^{†}}{⟩}_V=⟨Lu{L}^{†}{\overline{L}}^{†}\overline{v}\overline{L}{⟩}_V=⟨Lu\overline{v}\overline{L}{⟩}_V=L⟨u\overline{v}{⟩}_V\overline{L},$

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

$F\to F^{{\prime }_{}}=LF\overline{L}$