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

$p\rightarrow p^{\prime }=LpL^{\dagger },$

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

$L_{{}_{}}=BR$

The boost factor can be extracted as

$B={\sqrt {LL^{\dagger }}}$

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

$u=Lu_{r_{}}L^{\dagger },$

that can be written as

$u=LL^{\dagger }=BR(BR)^{\dagger }=BRR^{\dagger }B^{\dagger }=BB=B^{2},$

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

$\langle u^{\prime }{\bar {v}}^{\prime }\rangle _{V}=\langle LuL^{\dagger }\,\,{\overline {LvL^{\dagger }}}\rangle _{V}=\langle LuL^{\dagger }\,{\bar {L}}^{\dagger }{\bar {v}}{\bar {L}}\rangle _{V}=\langle Lu{\bar {v}}{\bar {L}}\rangle _{V}=L\langle u{\bar {v}}\rangle _{V}{\bar {L}},$

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

$F\rightarrow F^{\prime _{}}=LF{\bar {L}}$