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

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

where the spinor ${\displaystyle L}$ is subject to the condition of unimodularity

${\displaystyle L{\bar {L}}=1}$

In ${\displaystyle Cl_{3}}$, the spinor ${\displaystyle L}$ can be written as the exponential of a biparavector ${\displaystyle W}$

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

RotationEdit

If the biparavector ${\displaystyle W}$  contains only a bivector (complex vector in ${\displaystyle Cl_{3}}$ ), the Lorentz transformations is a rotation in the plane of the bivector

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

for example, the following expression represents a rotor that applies a rotation angle ${\displaystyle \theta }$  around the direction ${\displaystyle \mathbf {e} _{3}}$  according to the right hand rule

${\displaystyle R=e^{-{\frac {\theta }{2}}\mathbf {e} _{12}}=e^{-i{\frac {\theta }{2}}\mathbf {e} _{3}},}$

applying this rotor to the unit vector along ${\displaystyle \mathbf {e} _{1}}$  gives the expected result

${\displaystyle \mathbf {e} _{1}\rightarrow e^{-i{\frac {\theta }{2}}\mathbf {e} _{3}}\mathbf {e} _{1}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}=\mathbf {e} _{1}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}=\mathbf {e} _{1}e^{i\theta \mathbf {e} _{3}}=\mathbf {e} _{1}(\cos(\theta )+i\mathbf {e} _{3}\sin(\theta ))=\mathbf {e} _{1}\cos(\theta )+\mathbf {e} _{2}\sin(\theta )}$

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

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

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

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

BoostEdit

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

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

for example, the following expression represents a boost along the ${\displaystyle \mathbf {e} _{3}}$  direction

${\displaystyle B=e^{{\frac {1}{2}}\eta \,\mathbf {e} _{3}},}$

where the real scalar parameter ${\displaystyle \eta }$  is the rapidity.

The boost ${\displaystyle B}$  is seen to be:

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

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

${\displaystyle L_{{}_{}}=BR}$

The boost factor can be extracted as

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

and the rotor is obtained from the even grades of ${\displaystyle L}$

${\displaystyle R={\frac {L+{\bar {L}}^{\dagger }}{2\langle B\rangle _{S}}}}$

Boost in terms of the required proper velocityEdit

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

${\displaystyle u_{r_{}}=1}$

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

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

that can be written as

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

so that

${\displaystyle B={\sqrt {u}}={\frac {1+u}{\sqrt {2(1+\langle u\rangle _{S})}}},}$

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

${\displaystyle u=B^{2^{}},}$

so that

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

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

${\displaystyle {\boldsymbol {\eta }}=\eta {\hat {\boldsymbol {\eta }}}}$

the exponential can be expanded as

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

so that

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

and

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

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

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

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

${\displaystyle \langle u{\bar {v}}\rangle _{V}\rightarrow \langle u^{\prime }{\bar {v}}^{\prime }\rangle _{V}}$

applying the Lorentz transformation to the component paravectors

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

so that if ${\displaystyle F}$  is a biparavector, the Lorentz transformations is given by

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