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 },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19435ab26685b2f28e99ef03ccb186f06d1f451e)
where the spinor
is subject to the condition of unimodularity
![{\displaystyle L{\bar {L}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b503208afaed3df3213bcd0536b242cd0c6fca86)
In
, the spinor
can be written as the
exponential of a biparavector
![{\displaystyle L_{{}_{}}=e^{W}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6301dbc050698964d70630617d199ed210aaa607)
If the biparavector
contains only a bivector (complex vector in
), the Lorentz transformations is a rotation in the plane of the bivector
![{\displaystyle R=e^{-i{\frac {1}{2}}{\boldsymbol {\theta }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8490823e5f6e68d3c50d51214a52a2514d0d1746)
for example, the following expression represents a rotor that applies a rotation angle
around the direction
according to the right hand rule
![{\displaystyle R=e^{-{\frac {\theta }{2}}\mathbf {e} _{12}}=e^{-i{\frac {\theta }{2}}\mathbf {e} _{3}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19a376a0498338972508eb52526f7b09f8aff7e)
applying this rotor to the unit vector along
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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84729a2c4cb7883402c86c821fd6fb4e8d8fb1e8)
The rotor
has two fundamental properties. It is said to be unimodular and
unitary, such that
- Unimodular:
![{\displaystyle R{\bar {R}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc965c71aa8747a5895ac039a6823b06127cda86)
- Unitary:
![{\displaystyle RR^{\dagger }=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f7dc82861d56ac39a575200e8b1533ca846842)
In the case of rotors, the bar conjugation and the reversion have the same effect
![{\displaystyle {\bar {R}}=R^{\dagger }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a690399e83098f44dc094fc073e80623e0820033)
If the biparavector
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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e24ef036a337d966d02c7658d08dca7b95d892)
for example, the following expression represents a boost along the
direction
![{\displaystyle B=e^{{\frac {1}{2}}\eta \,\mathbf {e} _{3}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30c6413cf9e1f1956a5ae6ba7cf0ec55a2ffc630)
where the real scalar parameter
is the rapidity.
The boost
is seen to be:
- Unimodular:
![{\displaystyle B{\bar {B}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed632f9361cacabb9eac159796c666e10385b75)
- Real:
![{\displaystyle B^{\dagger }=B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc3a7fbe7759d7806ae3e3ebce23051095f61a2)
In general, the spinor of the proper Lorentz transformation can be written
as the product of a boost and a rotor
![{\displaystyle L_{{}_{}}=BR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee52428aa0583096507116797ccbe9608ad337b)
The boost factor can be extracted as
![{\displaystyle B={\sqrt {LL^{\dagger }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c41fa12f185acc34e0ddb21afdef680ff89606)
and the rotor is obtained from the even grades of
![{\displaystyle R={\frac {L+{\bar {L}}^{\dagger }}{2\langle B\rangle _{S}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb3c391749776a0058fe314e08836838373b5e6)
The proper velocity of a particle at rest is equal to one
![{\displaystyle u_{r_{}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e080a6a81e555ca4f08d6ed2dec29a73641a73e)
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 },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a516df24f0282f0449553ff4bdbcda0b53ecb94)
that can be written as
![{\displaystyle u=LL^{\dagger }=BR(BR)^{\dagger }=BRR^{\dagger }B^{\dagger }=BB=B^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f857c10961fcc41681893db5ffae971fba9815d)
so that
![{\displaystyle B={\sqrt {u}}={\frac {1+u}{\sqrt {2(1+\langle u\rangle _{S})}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8878d436676ff21028435c9af3ebcb0eee8f8842)
where the explicit formula of the square root for a unit length paravector was used.
The proper velocity is the square of the boost
![{\displaystyle u=B^{2^{}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a135cfc9bf2526b7d6c2c6ae12519c3d539eac80)
so that
![{\displaystyle \gamma (1+{\frac {\mathbf {v} }{c}})=e^{\boldsymbol {\eta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4a5ae387c13240467839de7d2e8a5050af839b)
rewriting the rapidity in terms of the product of its magnitude and respective
unit vector
![{\displaystyle {\boldsymbol {\eta }}=\eta {\hat {\boldsymbol {\eta }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9387209604a207444578ca53576c93bdb5d8e5d5)
the exponential can be expanded as
![{\displaystyle \gamma +\gamma {\frac {\mathbf {v} }{c}}=\cosh(\eta )+{\hat {\boldsymbol {\eta }}}\sinh(\eta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e52728d1f5a3bf52df334e40ea871d9057c110e1)
so that
![{\displaystyle \gamma _{{}_{}}=\cosh {\eta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce61d76083cefe2c5ca7a74386fa37d2705779b9)
and
![{\displaystyle \gamma {\frac {\mathbf {v} }{c}}={\hat {\boldsymbol {\eta }}}\sinh(\eta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a053770bc099f30d7af58f7ab7950fd282f8861)
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3598b4f95c833044e241d9ea205e260767f2429b)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd28d680c960bea22bdf4b2def0f0c74723eb934)
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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f15010eba072276908b2841da978c09000d3ea)
so that if
is a biparavector, the Lorentz transformations is given
by
![{\displaystyle F\rightarrow F^{\prime _{}}=LF{\bar {L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/217184c851236fcde74a269900fe34c448ec7ffc)