<p>A time-domain elastic scattering problem is considered in three dimensions. In problem setting, a rigid obstacle is immersed in an unbounded domain filled with homogeneous and isotropic elastic medium. In order to analyze the well-posedness of the target problem, we reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval by using a compressed coordinate transformation. The Galerkin method is adopted to prove the uniqueness results, and the energy method is used to prove the stability of the scattering problem. In addition, we derive a priori estimate with explicit time dependence.</p>