Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity

Abstract

We study the semi-classical thermodynamics of two-dimensional de Sitter space ($\text{dS}_{2}$) in Jackiw-Teitelboim (JT) gravity coupled to conformal matter. We extend the quasi-local formalism of Brown and York to $\text{dS}_{2}$, where a timelike boundary is introduced in the static patch to uniquely define conserved charges, including quasi-local energy. The boundary divides the static patch into two systems, a cosmological system and a black hole system, the former being unstable under thermal fluctuations while the latter is stable. A semi-classical quasi-local first law is derived, where the Gibbons--Hawking entropy is replaced by the generalized entropy. In the microcanonical ensemble the generalized entropy is stationary. Further, we show the on-shell Euclidean microcanonical action of a causal diamond in semi-classical JT gravity equals minus the generalized entropy of the diamond, hence extremization of the entropy follows from minimizing the action. Thus, we provide a first principles derivation of the island rule for $U(1)$ symmetric $\text{dS}_{2}$ backgrounds, without invoking the replica trick. We discuss the implications of our findings for static patch de Sitter holography.

Figures (13)

• Figure 1: Penrose diagram of de Sitter space. The left and right (green) regions describe the static patch of de Sitter space. The dotted curves (red) represent anchor curves which we use to define quasi-local thermodynamics. The boundaries of the (blue) bulk spatial surface anchored between the two (stretched) cosmological horizons are extremal surfaces whose area is proposed to compute the entanglement entropy.
• Figure 2: Two-dimensional de Sitter space in the half reduction model. The left and right (green) triangles represent the two static patches. In the half reduction model, the dilaton $\phi\geq0$, where it formally diverges at past and future infinity $\mathcal{I}^{\pm}$ and vanishes at the poles.
• Figure 3: Two-dimensional de Sitter space in the full reduction model. The left and right green causal diamonds are the static patches, while both the blue shaded and unshaded white regions are referred to as hyperbolic patches. The near-Nariai black hole geometry is imprinted in the two-dimensional geometry via the "black hole" interiors, the white regions, with past and future singularities residing inside, where the dilaton takes arbitrarily large negative values. The left and right edges are identified.
• Figure 4: Introducing a Brown-York timelike boundary $B$ (red) at radius $r=r_{B}$ in dS$_2$. We define quasi-local charges with respect to this boundary, a surface with a fixed Tolman temperature. In the full reduction model $B$ rests somewhere between $r=-L$ and $r=L$ in the static patch. The shaded blue region refers to black hole system, while the shaded magenta region describes the cosmological system. The constant-$t$ slice $\Sigma$ has boundary $\partial\Sigma=\mathcal{S}\cup\mathcal{H}$ with $\mathcal{S}$ being the intersection of $\Sigma$ and $B$, and $\mathcal{H}$ is the bifurcation point of the Killing horizon located at $r_{\text{h}}$ or $r_{\text{c}}$ for the black hole or cosmological system, respectively.
• Figure 5: Plot of $E$ (left) and $C_{\phi_{B}}$ (right) as a function of radius $r_{B}$ for both cosmological (violet) and black hole (blue) systems. We have set $\phi_{r}=L=G_{2}=1$.