The geodesics on a relativistic manifold are given by the well-known equation involving the Christoffel coefficients. This equation can be solved numerically step by step and transposed into a quantization. We study here the effect of this quantization on the Schwarzschild spacetime, more precisely in the Kruskal-Szekeres map.

The Schwarzschild solution of Einstein’s field equation comes in two variants, one for a single spherical mass (black hole), and one for the universe as a whole (in which dark energy prevails over matter). The cosmic variant represents a special form of expanding De-Sitter space (and is a good match with cosmic reality). In the cosmic variant, the Schwarzschild horizon of a black hole is replaced by the cosmic event horizon. The world-lines of escaping galaxies and also of photons is given by a function r(t) in which no square root of a negative number appears, so that the functionvalue is a real number even when it comes to r larger than the distance to the horizon. It turns out that no singularity exists beyond the cosmic event horizon. A surprising symmetry is thereby revealed (with the cosmic event-horizon functioning as a symmetry line). This symmetry comprises timereversals along world-lines, and the possibility of crossing the cosmic event horizon in both directions. As a consequence thereof, loop-shaped world lines of objects are within reach (similar to those in Gödel’s universe), even in the vicinity of planet Earth. As Gödel had correctly pointed out, loop-shaped world lines are proof of the static model of time, according to which present, past and future events occurring at the same spatial location are equally part of physical reality, not different from events occurring at the same time at different places. Kruskal-Szekeres charts do not obstruct any of these results. These charts have so far been used in an incorrect manner. When using them correctly, no singularity exists beyond event horizons, neither in the back-hole case, nor in the cosmic case. Doubts with regard to the existence of singularities, recently expressed by R. P. Kerr (2023), are thus proved to be justified.