De Sitter and Schwarzschild-de Sitter according to Schwarzschild and de Sitter

When de Sitter first introduced his celebrated spacetime, he claimed, following Schwarzschild, that its spatial sections have the topology of the real projective space P³ (that is, the topology of the group manifold SO(3)) rather than, as is almost universally assumed today, that of the sphere S³. (In modern language, Schwarzschild was disturbed by the non-local correlations enforced by S³ geometry.) Thus, what we today call ``de Sitter space" would not have been accepted as such by de Sitter. There is no real basis within classical cosmology for preferring S<sup>3</sup> to P<sup>3</sup>, but the general feeling appears to be that the distinction is in any case of little importance. We wish to argue that, in the light of current concerns about the nature of de Sitter space, this is a mistake. In particular, we argue that the difference between ``dS(S³)" and ``dS(P³)" may be very important in attacking the problem of understanding horizon entropies. In the approach to de Sitter entropy via Schwarzschild-de Sitter spacetime, we find that the apparently trivial difference between P³ and S³ actually leads to very different perspectives on this major question of quantum cosmology.