  We study the neutral K\"ahler metric on the space of time-like lines in
Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the
tangent bundle to the hyperbolic plane. We find all of the infinitesimal
isometries of this metric, as well as the geodesics, and interpret them in
terms of the Lorentzian metric on ${\Bbb{E}}^3_1$. In addition, we give a new
characterisation of Weingarten surfaces in Euclidean ${\Bbb{E}}^3$ and
Lorentzian ${\Bbb{E}}^3_1$ as the vanishing of the scalar curvature of the
associated normal congruence in the space of oriented lines. Finally, we relate
our construction to the classical Weierstrass representation of minimal and
maximal surfaces in ${\Bbb{E}}^3$ and ${\Bbb{E}}^3_1$.
