A gas turbine engine has a turbine section and an air moving section upstream of the turbine section. A seal is between a static surface and a rotating surface in at least one of the turbine section and air moving section. The seal includes a seal shoe which is biased at least in part by a spring force, with the spring force created by cuts formed to provide a gap to allow arms associated with the seal to provide the spring force. The movement of the shoe is along a direction having at least a component parallel to an axis of rotation of the gas turbine engine. In addition, an axially moving non-contact seal is claimed.