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 IRP 3 (that is, the topology of the group manifold SO(3)) rather than, as is almost universally assumed today, that of the sphere S. (In modern language, Schwarzschild was disturbed by the non-local correlations enforced by S 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 to IRP , 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)” and “dS(IRP )” 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 IRP 3 and S actually leads to very different perspectives on this major question of quantum cosmology. 1. The Superfluous Antipodes Understanding de Sitter spacetime — the finiteness of its entropy, its embedding in string theory — is one of the key problems of current physics. Obviously, therefore, it is important to be confident that we know exactly what “de Sitter spacetime” actually is. From its first appearance [1], however, it has been clear that there is a fundamental ambiguity in the very definition of “de Sitter spacetime”, arising from the well-known fact that the Einstein equations do not completely fix the topology of spacetime. (de Sitter’s paper can be found at http://adsabs.harvard.edu/ads abstracts.html.) This kind of topological ambiguity has recently attracted a great deal of attention from an observational point of view (see for example [2][3]), but far less attention has been paid to its theoretical implications. The purpose of this work is to draw attention to the fact that these theoretical implications have a bearing on issues of great current interest. Today we are accustomed to the idea that the topology of de Sitter space is that of IR×S3, where S is the three-sphere. (Sometimes it is convenient to use other slicings, such as that by flat three-dimensional spaces, but these do not cover the entire spacetime and so they tell us nothing about its global structure.) However, de Sitter himself explicitly rejected this interpretation: for him, the topology of “de Sitter spacetime” was IR×IRP . Here IRP 3 is the real projective space, obtained from S by identifying all points with their antipodes. (The distinction between the two is precisely that between the group manifolds of SO(3) and SU(2); for it is easy to see that SU(2) has the topology of S, and it is well known that it covers SO(3) twice, so the group manifold of SO(3) is IRP .) The metric of IRP 3 is exactly that of S — that is, if the sectional curvature is 1/L, it is g(IRP , 1/L) = g(S, 1/L) = L[dχ⊗ dχ+ sin(χ)[dθ ⊗ dθ + sin(θ)dφ⊗ dφ)]], (1) but now the angles χ, θ, and φ (which on S range respectively from zero to π, zero to π, and zero to 2π) are subject to identifications according to the antipodal map א3 on S, defined by χ → π − χ θ → π − θ φ → π + φ. (2) Thus IRP 3 is the quotient S/ZZ2 = S /{1,א3}. Comparing the two, de Sitter states that IRP 3 “is really the simpler case, and it is preferable to adopt this for the physical world.” (He also reports a letter from Einstein to the effect that the latter agreed with him — though it appears that Einstein later [4] changed his opinion, on extremely tenuous aesthetic grounds.) de Sitter was apparently influenced by a much earlier (1900) paper of Schwarzschild [5] (translated as [6]). Usually this paper is cited as one of the first attempts to discuss, from an observational point of view, the possibility that the spatial sections of the Universe may not have the geometry or topology of ordinary Euclidean space. Schwarzschild’s paper is strange to modern eyes, however, in that, when he considers positively curved space, he only discusses IRP , which he calls “the simplest of the spaces with spherical trigonometry.” In fact, he explicitly rejects S as a physically acceptable model for spatial geometry, on the grounds