Boosts in Lorentz-Minkowski space $\mathbb{L}^3$ (or $E_{1}^3$) (+ material) Can someone give me examples of boosts in $\mathbb{L}^3$? I understand that boosts are isometries that leave pointwise fixed a straight line $\mathcal{L}$. The only thing I can think of, until now, are rotations around said $\mathcal{L}$.
I'm using $\operatorname{diag}[1,1,-1]$, that is, $\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2 + y_1y_2 - z_1z_2$.
I'm using this text to study. Also, I'm having trouble finding more material on this. Everywhere I find anything about the subject, it is more related to physics than differential geometry, which makes it harder for me to understand. Can someone share something, please? Thanks in advance!
 A: http://www.dm.uniba.it/geloba2009/PresentazioniGeLoBa2009/Ortega.pdf
Try that...and try to find S. Haesen stuff
A: A boost is a special kind of  a Lorentz transformation. A Lorentz transformation (in dimension 3) is a linear transformation $\Lambda:\mathbb R^3\to\mathbb R^3$ that leaves invariant the quadratic form $x^2+y^2-z^2$. You can show that this condition implies $det(\Lambda)=\pm 1$. Let $v$ be an eigen vector of $\Lambda$ (every linear transformation on $\mathbb R^3$ has an eigen vector, because the characteristic polynomial has degree 3). Then $\langle v, v\rangle$ is either positive, negative or 0 (all three cases can occur). If it is positive then $v$ is called spacelike and you can show that the eigenvalue is $\pm 1$. If it is 1, and $det(\Lambda)=1$, $\Lambda$ is called a boost (in dimension $n$ a boost is required to have a pointwise fixed spacelike $n-2$ subspace). Take the plane (2-dimensional subspace)  $v^\perp \subset \mathbb R^3$ (the orthogonal complement of  $\mathbb Rv$ with respect to the bilinear form $\langle\cdot, \cdot\rangle$). Then $\Lambda$ leaves $v^\perp$ invariant and the  quadratic form  restricted to $v^\perp$ has signature $(1,1)$.   Let $v_1$ be a multiple of $v$ such that $\langle v_1,v_1\rangle=1.$ Complete it to an orthonormal basis of $\mathbb R^3$ by picking   $v_2, v_3\in v^\perp$ such that $\langle v_2,v_2\rangle=1,$ $\langle v_3,v_3\rangle=-1,$ and $\langle v_2,v_3\rangle=0.$ Then the matrix of $\Lambda$  with respect to the basis $v_1, v_2,v_3$ is
$$\begin{pmatrix}1 & 0 & 0 \\ 0 & \cosh(s) & \sinh(s) \\ 0 & \sinh(s) &\cosh(s) \end{pmatrix}$$
for some $s\in\mathbb R$. 
A: As rotations take place in planes, so too do boosts.  For any boost in a given plane, the set of null vectors in that plane are left invariant by the boost.  In general, there are two "cones" of null directions, and where the boost plane cuts these cones are the null vectors left invariant.
A: http://books.google.pt/books?id=1SKFQgAACAAJ&redir_esc=y
That book is very centered in Differential Geometry...
I hope it helps too...
