In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory. In this spirit, I will present a correspondence between vector fields on the hyperbolic plane and surfaces in the so-called half-pipe space, which is the dual of Minkowski space. Using this construction, I investigate the problem of extending vector fields on the circle to the hyperbolic plane. Following this, I will focus on two types of extensions: infinitesimal earthquakes and harmonic Lagrangian vector fields. Time permitting, I will also discuss how the properties of these vector fields may be expressed in terms of the properties of the corresponding surfaces in half-pipe space, as well as their asymptotic boundaries.