Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness

By using stationary-to-Randers correspondence (SRC, see Caponio et al. in Rev Mat Iberoamericana 27:919–952, 2011), a characterization of light and time- convexity of the boundary of a region of a standard stationary (n + 1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n + 1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of hypersurfaces S n−1 (r ) × R, where S n−1 (r ) is a sphere of large radius in a spacelike section of an end, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e., the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.