Complete Barrett-Crane model and its causal structure

The causal structure is a quintessential element of continuum spacetime physics and needs to be properly encoded in theory Lorentzian quantum gravity. Established spin foam (and tensorial group field (TGFT)) models mostly work with relatively special classes triangulations (e.g. built from spacelike tetrahedra only), obscuring the explicit implementation local at microscopic level. We overcome this limitation construct full-fledged model for geometry building blocks which include spacelike, lightlike timelike tetrahedra. realize within context Barrett-Crane TGFT model. Following an characterization amplitudes via methods integral geometry, ensuing clear identification structure, we analyze model's respect its (space)time-orientation properties provide also more detailed comparison framework dynamical (CDT).