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Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections

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Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections Instanton Tunneling for De Sitter Space with Real Projective Spatial Sections Yen Chin Ong∗ (1.) Center for Astronomy and Astrophysics, Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, Chinay. (2.) Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden (3.) Riemann Center for Geometry and Physics, Leibniz Universit¨atHannover, Appelstrasse 2, 30167 Hannover, Germany Dong-han Yeomz Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan The physics of tunneling from one spacetime to another is often understood in terms of instantons. For some instantons, it was recently shown in the literature that there are two complementary \interpretations" for their analytic continuations. Dubbed \something-to-something" and \nothing-to- something" interpretations, respectively, the former involves situation in which the initial and final hypersurfaces are connected by a Euclidean manifold, whereas the initial and final hypersurfaces in the latter case are not connected in such a way. We consider a de Sitter space with real projective 3 space RP spatial sections, as was originally understood by de Sitter himself. This original version of de Sitter space has several advantages over the usual de Sitter space with S3 spatial sections. In particular, the interpretation of the de Sitter entropy as entanglement entropy is much more natural. We discuss the subtleties involved in the tunneling of such a de Sitter space. I. DE SITTER SPACE ACCORDING TO metric tensor is a local quantity which is not sensitive to 3 DE SITTER: THE CASE FOR RP the global topology. This is the reason why Einstein's field equations | a system of differential equations | De Sitter space is a Lorentzian manifold of constant cannot uniquely fix the topology. Specifically, a 3-sphere 2 positive curvature, which is a solution to the Einstein's with sectional curvature 1=` admits a metric tensor of field equations with a positive cosmological constant Λ. It the form plays very prominent roles in theoretical cosmology, since g(S3) = `2 dχ2 + sin2 χ dθ2 + sin2 θ dφ2 ; (1) the observed accelerating expansion of the universe is very well explained by just such a Λ [1]. Furthermore, de where 0 6 χ, θ 6 π, and 0 6 φ < 2π. For a real projective Sitter space also provides the arena for early inflationary space, we have exactly the same metric expression: universe. Although in cosmology one often uses the flat 3 2 2 2 2 2 2 slicing of de Sitter space, such a slicing does not cover g(RP ) = ` dχ + sin χ dθ + sin θ dφ ; (2) the entire spacetime. Here we are more concerned with the actual, global, structure of de Sitter space. Globally, but now the coordinates on the 3-spheres are mapped in the topology of de Sitter space | as we have come to such a way that antipodal points are identified. Namely, arXiv:1612.07756v2 [gr-qc] 19 Apr 2017 3 know it | is R × S . However, a little known fact is χ 7! π − χ, θ 7! π − θ, and φ 7! π + φ. We will refer to that de Sitter himself did not have the same topology in this as the antipodal map. mind for \de Sitter space" [2]. Instead, he preferred to The topology of de Sitter space, as de Sitter understood 3 topologically identify the antipodal points on the S3. This it, is therefore R × RP , which, following [3], we shall 3 ∼ 3 3 3 yields the projective space P = S = 2, which, like S , denote by dS[RP ]. Similarly, the de Sitter space with R Z 3 3 is an orientable spin manifold, and thus is a well-behaved R × S topology will be denoted by dS[S ]. The reason 3 3 spatial geometry for a spacetime. This is not true for real de Sitter favoured dS[RP ] over dS[S ] is that, following projective spaces in all dimensions [3]. Schwarzschild's paper1 in 1900 [4], the fact that S3 has 3 3 RP admits the same metric as that of S , since the antipodal points means that the global geometry \forces a correlation" between these two otherwise faraway points. yPresent address ∗Electronic address: [email protected] 1 Note that this paper of Schwarzschild predated even Special zElectronic address: [email protected] Relativity. For an English translation of the paper, see [5]. 2 r=∞ r=∞ future infinity singularity future infinity cosmological horizon cosmological horizon r=ℓ event horizon A C A r=0 r=0 r=0 t=0 B t=0 B r=ℓ r=ℓ r=ℓ FIG. 2: Left: The Penrose diagram of a Schwarzschild de- Sitter black hole, SdS[S3]. Here the diagram extends indefi- nitely toward the right, as well as the left hand side. r=∞ r=∞ FIG. 1: Left: The Penrose diagram of dS[S3]; Right: the tensor reads 3 Penrose diagram of dS[RP ]. Each point of the double line 2 2 2 3 2M r dr corresponds to a RP on the equator of RP . Slanted lines g[SdS[S3]] = − 1 − − dt2 + are cosmological horizons. The dashed lines should be ignored r `2 2M r2 1 − − at this point; they will be discussed later. r `2 + r2(dθ2 + sin2 θ dφ2): (3) 3 Geometrically, any two coplanar geodesics in S would Here, the notation SdS[S3] only serves to remind us that 3 intersect twice, whereas in RP they would intersect only for M = 0, we recover a de Sitter space with S3 cross once. Since the intersection of geodesics is a matter of local section. The topology of this spacetime with a black hole 3 physics, it would seem strange that this would somehow is of course not R × S . Since there are two locations that force a correlation at the antipodal point, which is well correspond to r = 0 (namely the north and the south outside the current observable universe! It is for this poles of the S3), introducing a black hole at one pole 3 reason that de Sitter considered RP to be \the simpler apparently also implies the existence of another black case, and it is preferable to adopt this for the physical hole at the other pole. In fact, this then forces us to add world" [2]. more black holes ad infinitum, and the resulting Penrose 3 3 The Penrose diagram of both dS[S ] and dS[RP ] are diagram of Schwarzschild-de Sitter black hole extends shown in Fig.(1). In the familiar case of dS[S3], there are indefinitely in the horizontal direction, as in Fig.(2). This two causal patches that are out of causal contact. An scenario is not physical, since we do not expect that a local observer centered at the { arbitrary { pole r = 0 would gravitational collapse would produce another black hole not be able to see the antipode of the S3 (namely, the at the \other side" of the universe, much less infinitely 3 other r = 0) at the other end of the universe. In dS[RP ], many black holes. A possible way out of this conundrum due to the topological identification, there is no antipode. is to topologically identify the spacetime such that one Since the left and right hemispheres of the S3 are identified obtains what naively looks like the maximally extended by the antipodal map, only half of the Penrose diagram of version of an asymptotically flat Schwarzschild black hole dS[S3] remains. The equator of S3 is a S2, which, under (the Kruskal-Szekeres spacetime), as shown in Fig.(3). At 2 the antipodal map, becomes a RP . Thus, in the Penrose this point, one should ask: does this picture now represent 3 diagram of dS[RP ], which is on the right of Fig.(1), a physically acceptable geometry? In the asymptotically 2 each point on the double line corresponds to a RP . An flat case, the answer is yes | in the sense that in a real- observer, say, Alice, who started at r = 0 and travels istic gravitational collapse, the interior is replaced by the 2 toward the RP equator, would, upon traveling through infalling stellar matter so that the second asymptotically 2 RP , find herself now traveling back toward r = 0, since flat region on the other side of the Einstein-Rosen bridge she is “reflected” back by the topological identification. It is removed. However, this is not the case here. Due to the is, however, not exactly a reflection (since Alice is traveling topological identification, the second region that should 2 through { not within { RP ). Instead, she would find that be removed in a realistic gravitational collapse is, in fact, the universe seems to have been rotated upside down. See identical to the exterior spacetime. That is, to quote [3] for more discussions. Note that this “reflection" is not [3], \the `other' universe which we so casually consign imposed as a boundary condition by hand, instead it is an to non-existence is in fact our own." The conclusion is unavoidable consequence of the topological identification. that neither the picture in Fig.(2) nor the one in Fig.(3) 3 To our modern eyes, it does not seem that dS[RP ] can be considered physical. However, if we take de Sitter 3 3 with a nontrivial topology is \simpler" than dS[S ]. To space to be dS[RP ], then this problem does not arise, better appreciate what de Sitter meant, let us reproduce since introducing a black hole at \r = 0" does not entail the example in [3], which is to consider a Schwarzschild-de another black hole at the other pole, simply because there Sitter black hole.
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