Conformal Vacuum of dS$_4\times \mathbb R$ with Oppositely Oriented Boundaries

Abstract

We derive a dS$_4 \times \mathbb R$ quotient spacetime that is asymptotically dS$_4$, where the quotient makes its past boundary oppositely oriented relative to its future boundary. This introduces a lightlike singularity that severs the antipodes of the spacetime and simplifies its global vacuum to a trivial product on antipodal static patches. We show that this state is conformal to the vacuum of an infinite orientable cover of a non-orientable AdS$_3$ spacetime with an S$^2$ bundle. The vacuum's separability extends to its holographic dual, which is a product of Cardy states. We find that this candidate dS$_4$ vacuum state is perturbatively unstable within quantum gravity due to a vanishing Hagedorn temperature.

Figures (9)

• Figure 1: The Penrose diagram of dS$_d$ with static patches ('N' and 'S') at its antipodal worldsheets and top and bottom ('T' and 'B') time-dependent patches. While the north (N) and south (S) static patches are disjoint, they share their future and past cones behind a cosmological horizon indicated by the thick diagonal crossing lines. The thin filled (dashed) lines correspond to timelike (spacelike) geodesics in the static patches.
• Figure 2: ${\text{NO-AdS}_3}$ contains (a) a Mobius strip topology that can be seen in the two Poincaré patches. A timelike loop in its two (b) Poincaré patches returns to itself but with opposite orientation. This can be seen more clearly through its (c) orientable double cover since $\tau-x$ are exchanged after the equivalent loop.
• Figure 3: The two-dimensional boundary of the orientable double cover of ${\text{NO-AdS}_3}$ consists of two sections separated by fixed points along $t - \phi = \omega_- = 0, \pm \pi$ with $\partial_{\omega_-}$ flipped on the two sides. This can be described as two boundaries with opposite spacetime orientation, which are not causally connected because the timelike Killing vector vanishes on the fixed lines that separate them. Note that the situation is the same for its (non-orientable) single cover because the boundary is actually still orientable.
• Figure 4: (a) Two perspectives of the (orientable) infinite cover of ${\text{NO-AdS}_3}$ from unrolling global timelike $t$, which are rotated by $\frac{\pi}{2}$ with respect to each other. A Rindler patch and its antipode are colored red and blue, respectively. b) Antipodal Rindler patches can be fully contained within each Poincaré patch where their boundaries coincide with different strips. This infinite cover of ${\text{NO-AdS}_3}$ still contains the single cover's fixed lines (pink and green), unlike the universal cover of ${\text{AdS}_3}$, due to being an unrolling in the bulk $t$ direction compared to ${\text{NO-AdS}_3}$'s twist, which is along the bulk $\theta$ direction.
• Figure 5: (a) The orientable infinite cover of ${\text{NO-AdS}_3}$ with unwrapped $t$ with annotated boundary $\left\vert \omega \right\rangle$, $\left\vert \bar{\omega} \right\rangle$ and corresponding bulk $\left\vert \Omega \right\rangle$, $\left\vert \bar{\Omega} \right\rangle$ states, respectively. (b) The Penrose diagram of the orientable double cover of ${\text{NO-AdS}_3}$. The blue and red diamonds show the conformal projection of the two antipodal Rindler patches in Poincaré patch $P_1$ shown in Fig. \ref{['fig:Rindlerboundaries']} (note that these Rindler patches do not "fill out" the dropped dimensions at most points; one Rindler patch is "in front" of the other). $\left\vert \omega \right\rangle$ and $\left\vert \bar{\omega} \right\rangle$ are bulk states defined on antipodal Rindler patchs with boundary dual states $\left\vert \Omega \right\rangle$ and $\left\vert \bar{\Omega} \right\rangle$, respectively.