  The classical evolution of the universe can be seen as a parametrised
worldline of the minisuperspace, with the time variable $t$ the parameter that
parametrises the worldline. The time reversal symmetry of the field equations
implies that for any positive oriented solution there can be a symmetric
negative oriented one that, in terms of the same time variable, represent an
expanding and a contracting universe, respectively. However, the choice of the
time variable induced by the correct value of the Schr\"odinger equation in the
two universes makes that their physical time variables could be reversely
related. In that case, the two universes would be both expanding universes from
the point of view of their internal inhabitants, who identify matter with the
particles that move in their spacetimes and antimatter with the particles that
move in the time reversely symmetric universe. If the assumptions considered
are consistent with a realistic scenario of our universe, the creation of a
universe-antiuniverse pair might explain two main and related problems in
cosmology: the time asymmetry and the primordial matter-antimatter asymmetry of
our universe.
