De Sitter and Schwarzschild-De Sitter According to Schwarzschild and De Sitter

TL;DR

This paper explores the topological differences between de Sitter space with S^3 and RP^3 spatial sections, arguing that these differences are crucial for understanding horizon entropy in quantum cosmology.

Contribution

It challenges the common assumption of S^3 topology for de Sitter space, emphasizing the significance of RP^3 topology in quantum cosmological entropy calculations.

Findings

RP^3 and S^3 topologies lead to different horizon entropy perspectives

The topology choice impacts the interpretation of de Sitter spacetime in quantum cosmology

Classical cosmology does not favor S^3 over RP^3, but implications differ in quantum context

Abstract

When de Sitter first introduced his celebrated spacetime, he claimed, following Schwarzschild, that its spatial sections have the topology of the real projective space RP^3 (that is, the topology of the group manifold SO(3)) rather than, as is almost universally assumed today, that of the sphere S^3. (In modern language, Schwarzschild was disturbed by the non-local correlations enforced by S^3 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^3 to RP^3, 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(S^3)" and "dS(RP^3)" 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 RP^3 and S^3 actually leads to very different perspectives on this major question of quantum cosmology.