Propagation of regularity and long time behavior of $3D$ massive relativistic transport equation I: Vlasov-Nordstr\"om system

In this paper, we introduce a new set of vector fields for the relativistic transport equation, which is applicable for general Vlasov-Wave type coupled systems. By combining the strength of Klainerman vector field method and Fourier method, we prove global regularity and scattering for the $3D$ massive relativistic Vlasov-Nordstr"om system for small initial data without any compact support assumption, which is widely used in the literature for the study of Vlasov solution.