Black holes in dS$_3$

Abstract

In three-dimensional de Sitter space classical black holes do not exist, and the Schwarzschild-de Sitter solution instead describes a conical defect with a single cosmological horizon. We argue that the quantum backreaction of conformal fields can generate a black hole horizon, leading to a three-dimensional quantum de Sitter black hole. Its size can be as large as the cosmological horizon in a Nariai-type limit. We show explicitly how these solutions arise using braneworld holography, but also compare to a non-holographic, perturbative analysis of backreaction due to conformally coupled scalar fields in conical de Sitter space. We analyze the thermodynamics of this quantum black hole, revealing it behaves similarly to its classical four-dimensional counterpart, where the generalized entropy replaces the classical Bekenstein-Hawking entropy. We compute entropy deficits due to nucleating the three-dimensional black hole and revisit arguments for a possible matrix model description of dS spacetimes. Finally, we comment on the holographic dual description for dS spacetimes as seen from the braneworld perspective.

Figures (12)

• Figure 1: Bulk $\text{AdS}_{4}$ with a de Sitter$_3$ brane. The brane is represented as a (magenta) hyperboloid ($\sigma=\sigma_b$ in \ref{['eq:lineelementds3sec']}), and gravity is induced on it from integrating out the UV degrees of freedom of the $\text{CFT}_{3}$ excluded by the brane (dashed magenta region). One performs surgery by gluing two copies of the region $\sigma<\sigma_b$ along the two-sided brane. The brane is following an accelerating trajectory, and the bulk acceleration horizons give rise to cosmological horizons on the induced dS geometry. The red dashed line is the bifurcation surface of the Rindler-AdS horizons. When the brane is introduced, it becomes a compact surface of finite area \ref{['compacthor']}.
• Figure 2: Left: AdS$_4$ C-metric \ref{['eq:AdS4Ccoord']} with $\mu=0$, in $(r,x)$ coordinates in a slice at $t=0$ and constant $\phi$. Lines of constant $x$ are blue arcs; lines of constant $r$ are red arcs (full circles for $0 < r <\ell$). The thick blue circle $x=0$ is where we place the dS$_3$ brane; its interior is $0<x\leq1$, with $x=1$ the $\phi$ axis of rotation. The exterior region $x<0$ is excluded in the braneworld construction. The vertical red dashed line is the horizon at $r=R_3$. Its intersection with the brane yields a dS$_3$ cosmological horizon. The coordinates only cover half of the disk, with the other half being obtained through analytic continuation. Right: construction of black holes on a dS$_3$ braneworld when $\mu>0$. The dashed magenta region $x<0$ is excluded. The black hole horizon and the cosmological horizon are at constant $r$.
• Figure 3: Illustration of the bulk braneworld from the AdS$_4$ C-metric. The brane is placed at $x=0$ (magenta hyperboloid surface). The black lines denote $r=0$ and depict the worldlines of the accelerating black holes described by the C-metric. The AdS interior of the hyperboloid is kept, and the double-sided brane is glued to another copy of it.
• Figure 4: Penrose diagram of a static, neutral quantum black hole in dS$_3$.
• Figure 5: Plot of the horizon radii $r_{h}$ (blue) and $r_{c}$ (red) as a function of $\mu/\mu_{\text{N}}$. The black hole horizon becomes larger as $\mu$ grows, while the cosmological horizon shrinks.