# Geometric derivation of Lorentz boosts

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.

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).