Neutrosophy is a theory developed by Florentin Smarandache in 1995 as a generalization of dialectics, which studies the origin, nature and properties of neutralities. This book applies neutrosophic method to the General Theory of Relativity, aiming to discover new effects hidden before.Studying Einstein's basic space-time, neutrosophic method displays new trajectories and particles never considered before. Such trajectories/particles are of two "mixed" kinds, which are (1) common for sub-light mass-bearing particles and massless photons (non-isotropic/isotropic trajectories) and (2) common for massless photons and super-light mass-bearing tachyons (isotropic/non-isotropic trajectories). As it is shown, such mass-bearing/light-like particles are accessible for observing, we can see them in different phenomena of nature.Solutions of Einstein's equations (evolution scenarios for the Universe) are collected in a table in this book. Each cell of the table is a cosmological model. Neutrosophic method, analyzing this cosmological table, displays new possible models of the evolution of the Universe.The foundation of Smarandache geometries is based on S-denying an axiom, i.e. in the same space an axiom is false in at least two different ways, or is false and also true. S-denying each of 4 signature conditions in Einstein's basic space-time we arrive to 4 kinds of expanded space-time for the General Theory of Relativity. The 4th expanded space-time type permits photon teleportation, well-known from the recent experiments, but which failed in the basic space-time, and also where virtual photons are permitted --- predicted by Quantum Electrodynamics instant-moving mediators between entangled regular particles.