  As quotient spaces, Minkowski and de Sitter are fundamental,
non-gravitational spacetimes for the construction of physical theories. When
general relativity is constructed on a de Sitter spacetime, the usual
Riemannian structure is replaced by a more general structure called de
Sitter-Cartan geometry. In the contraction limit of an infinite cosmological
term, the de Sitter-Cartan spacetime reduces to a singular, flat, conformal
invariant four-dimensional cone spacetime, in which our ordinary notions of
time interval and space distance are absent. It is shown that such spacetime
satisfies all properties, including the Weyl curvature hypothesis, necessary to
play the role of the bridging spacetime connecting two aeons in Penrose's
conformal cyclic cosmology.
