  We investigate the two-dimensional behavior of gravity coupled to a dynamical
unit timelike vector field, i.e. "Einstein-aether theory". The classical
solutions of this theory in two dimensions depend on one coupling constant.
When this coupling is positive the only solutions are (i) flat spacetime with
constant aether, (ii) de Sitter or anti-de Sitter spacetimes with a uniformly
accelerated unit vector invariant under a two-dimensional subgroup of SO(2,1)
generated by a boost and a null rotation, and (iii) a non-constant curvature
spacetime that has no Killing symmetries and contains singularities. In this
case the sign of the curvature is determined by whether the coupling is less or
greater than one. When instead the coupling is negative only solutions (i) and
(iii) are present. This classical study of the behavior of Einstein-aether
theory in 1+1 dimensions may provide a starting point for further
investigations into semiclassical and fully quantum toy models of quantum
gravity with a dynamical preferred frame.
