Flow-geometry microstates

Abstract

We construct geometric microstates for a class of two-dimensional flow geometries$-$spacetimes that interpolate from an asymptotic AdS$_2$ boundary to a dS$_2$ static patch in the interior$-$by inserting particles behind the horizon. We show that this mechanism produces dS microstates with an Einstein-Rosen bridge of infinite length behind the horizon. The state-counting of these microstates, including wormhole contributions, reproduces the Gibbons-Hawking entropy, $S_{\rm dS}=A^{\rm dS}_{\rm horizon}/4G$. Furthermore, we extend the microstate-counting method to the case of a finite-length Einstein-Rosen bridge. As a result, the Hilbert space of the dS horizon in the flow geometry can be spanned by states with a purely dS Einstein-Rosen bridge, containing no AdS portion on the time-symmetric slice. This provides a concrete realization of dS microstates within a controlled holographic framework.

Figures (22)

• Figure 1: (Left) Euclidean flow PETS geometry. By acting with an operator ${\cal O}(x)$, a heavy particle is created, which backreacts on the geometry. When the particle mass is sufficiently large, two disjoint dS regions are formed. (Right) Lorentzian flow PETS geometry. The particle (blue line) elongates the ER bridge. When the mass is large, an AdS region appears in the middle as in the figure; we call this the flow ER bridge. Conversely, if the mass is sufficiently small, there is no AdS region in the interior; we call this the dS ER bridge.
• Figure 2: Contour integration paths involved in the integrals for the resolvent $R(\lambda)$ in the complex $\lambda$-plane (\ref{['eq:Disc1']}). The poles of $R(\lambda)$ on the real axis correspond to the eigenvalues of the Gram matrix $\mathbf{G}$. The contour $\gamma_{0}$ which encircles the origin gives $\Omega - {\rm dim}(\mathcal{H}_{\mathbf{F}})$. The contour integral of $\gamma_{+}$, which encircles the positive region of the real axis, gives ${\rm dim}(\mathcal{H}_{\mathbf{F}})$. These contours can be deformed to the contour at infinity $\gamma_{\infty}$, and its contour integral gives $\Omega$.
• Figure 3: (Left) Euclidean flow PETS geometry. The worldline $\gamma_{1}$ with particle mass $m_{i}$ separates the geometry into left and right regions. The renormalized boundary lengths of the left and the right boundaries are, respectively, equal to $\widetilde{\beta}$. Therefore the total renormalized boundary length is $2\widetilde{\beta}$. $n_{L} (n_{R})$ represents the normal vector at $\gamma_{1}$ for the left (right) region. (Right) Right half of the geometry. Let $\beta_{i}$ be the full disc renormalized length, and let $\Delta \mathcal{T}(m_{i}, \beta_{i})$ be the time difference between the two endpoints of $\gamma_{1}$. They satisfy $\beta_{i} = \widetilde{\beta} + \Delta \mathcal{T}(m_{i}, \beta_{i})$.
• Figure 4: Qualitative behaviors of geodesics of various masses. The red circle represents the boundary between the AdS and dS regions. (Left) $m_{i}>\frac{\widetilde{\Phi}_{b} }{2 G_{2}\beta_{i}}$. When the mass is large, the rightmost geodesic has a small value of $\Delta \mathcal{T}$. As the mass decreases, the geodesics tend to wrap around the dS region and the value of $\Delta \mathcal{T}$ becomes large. When the mass is lower than a critical value, $\widetilde{\beta}$ is no longer positive. This case is depicted by the dashed curve. (Right) $m_{i}<\frac{\widetilde{\Phi}_{b} }{2 G_{2}\beta_{i}}$. In this mass range, every geodesic must pass through the 'north pole' and the 'south pole' of the dS region. When the mass is small, the geodesic is nearly a straight line, as in the figure, i.e.$\Delta \mathcal{T} \simeq \frac{\beta_{i}}{2}$. Increasing the mass also increases the value of $\Delta \mathcal{T}$. As in the previous case, when the mass exceeds another critical value, $\widetilde{\beta}$ is no longer positive. This case is depicted by the dashed curve.
• Figure 5: The behavior of $\beta_{i}(\widetilde{\beta}, m_{i})$. The blue curve corresponds to the $m_{i} > \frac{\widetilde{\Phi}_{b} }{2G_{2} \beta_{i}}$ case, and the red and pink curves correspond to the $m_{i} < \frac{\widetilde{\Phi}_{b} }{2G_{2} \beta_{i}}$ case. For both figures, the horizontal axes and the vertical axis of the right figure are divided by $2G_{2}\widetilde{\Phi}_{b}$. (Left) Relationship between $\beta_{i}$ and $\widetilde{\beta}$, normalized by $1/m_{i}$. The red curve terminates around $(\beta_{i}/\widetilde{\beta}, \frac{2G_{2}}{\widetilde{\Phi}_{b}}\widetilde{\beta} m_{i}) \simeq (3.5023, 0.1677)$. (Right) Relationship between $\widetilde{\beta}$ and $E$, normalized by $\beta_{i}$. There are no real Euclidean saddles in the energy range given by Eq. \ref{['eq:EnergyRange']}.