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 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 $s\in\mathbb R$.