  This article proposes a boundary element method for the dynamic contact
between a linearly elastic body and a rigid obstacle. The Signorini contact
problem is formulated as a variational inequality for the Poincar\'{e}-Steklov
operator for the elastodynamic equations on the boundary, which is solved in a
mixed formulation using boundary elements in the time domain. We obtain an a
priori estimate for the resulting Galerkin approximations. Numerical
experiments confirm the stability and convergence of the proposed method for
the contact problem in flat and curved two-dimensional geometries, as well as
for moving obstacles.
