In two dimensions a very nice parametrization of the rotation group is obtained by the following line of arguments:

  1. The group of rotations is connected and compact. Therefore the exponential is surjective.
  2. For every not null vector $v$ there is a unique vector $\star v$ such that $(v,\star v)$ is a positively oriented orthogonal basis and $\|\star v\|=\|v\|$. Then $\star v$ has the simple interpretation of being the counter-clockwise rotation by $90^\circ$ of $v$. One immediately identifies $\star$ as the Hodge-star operator and proofs it is an element of the Lie algebra.
  3. By a simple calculation $e^{\varphi\star}v=\cos(\varphi)v+\sin(\varphi)\star v$ for every $\varphi\in\mathbb{R}$.

A similar path can be taken for the three-dimensional rotation group through the element of its Lie algebra $\varphi\cdot J$ defined by $\varphi\cdot Jv=\varphi\times v:=\star(\varphi\wedge v)$ for all vectors v. The idea is similarly that $(\varphi, \varphi\times v, \varphi\times(\varphi\times v))$ is an oriented basis. This leads to Rodrigues' formula.

I would like to know if anybody knows of such a simple characterization for the (proper orthochronous) Lorentz group? Since every element of the Lorentz group can be written using the product of a rotation and a boost, I am particularly interested in the boosts. I would like in particular a geometrical and basis-free description. I seem to be able to retain such descriptions much better. I imagine that the problem statement would be something like

Consider a unit timelike vector $t$ (indicating the 4-velocity of the observer) and a unit spatial vector $n$ (indicating the preferred direction in space). Consider a Lorentz transformation $L$ which leaves invariant $\text{span}\{t,n\}^\bot$. One should then obtain a relation like Rodrigues' formula (see wikipedia) using a geometrically inspired Lie algebra element (I don't understand the geometric content of the usual $K$'s). The parameters multiplying this Lie algebra element should then be clearly identifiable with rapidity or some related parameter.

  • $\begingroup$ I imagine further that $g(n,t)=0$. $\endgroup$ – Iván Mauricio Burbano Feb 28 '19 at 18:57

The following might be of interest.

Special Relativity in General Frames: From Particles to Astrophysics
By Éric Gourgoulhon
ISBN: 9783642372766

I transcribed his formula that is analogous to the Rodrigues formula. $\Lambda(\vec v)=-\Gamma(\vec u\cdot \vec v)\vec u+\frac{\Gamma}{c}[(\vec V\cdot \vec v)\vec u - (\vec u\cdot\vec v)\vec V] +\bot_u \vec v+ ({\Gamma-1})\displaystyle\frac{(\vec V\cdot\vec v)\vec V}{V^2}$

$\Lambda$ is a boost applied to a vector $\vec v$.
$\vec u$ is the [timelike] 4-velocity. $\Gamma$ is the time-dilation factor ($\cosh\psi$ where $\psi$ is the rapidity). $\vec V$ is the spacelike relative-velocity vector, whose magnitude is $\tanh\psi$. The quantity $\bot_u \vec v$ is the projection of $\vec v$ onto the $\vec u$-observer's rest space.

Here's a screenshot of page 197 from section 6.6 Properties of Lorentz Boosts (from Google Books).

p 197 from Gourgoulhon Special Relativity in General Frames: From Particles to Astrophysics


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