Nuts and Bolts for Creating Space

Abstract

We discuss the way in which field theory quantities assemble the spatial geometry of three-dimensional anti-de Sitter space (AdS3). The field theory ingredients are the entanglement entropies of boundary intervals. A point in AdS3 corresponds to a collection of boundary intervals, which is selected by a variational principle we discuss. Coordinates in AdS3 are integration constants of the resulting equation of motion. We propose a distance function for this collection of points, which obeys the triangle inequality as a consequence of the strong subadditivity of entropy. Our construction correctly reproduces the static slice of AdS3 and the Ryu-Takayanagi relation between geodesics and entanglement entropies. We discuss how these results extend to quotients of AdS3 -- the conical defect and the BTZ geometries. In these cases, the set of entanglement entropies must be supplemented by other field theory quantities, which can carry the information about lengths of non-minimal geodesics.

Figures (15)

• Figure 1: The notation introduced in Sec. \ref{['rev']}. We consider the set of spacelike geodesics tangent to a given curve $R = R(\tilde{\theta})$, where $\tilde{\theta}$ is the bulk angular coordinate. The geodesics are centered at $\theta(\tilde{\theta})$ and have width $\alpha(\tilde{\theta})$. The variable $\theta$ is reserved for the boundary coordinate.
• Figure 2: A graphical representation of eq. (\ref{['differentials']}). The black, continuous geodesics are tangent to the bulk curve; their lengths are $S(I_k)$. The purple, dotted geodesics subtend $I_k \cap I_{k+1}$. We illustrate the effect of the limit $N \to \infty$ by displaying finite combinations of geodesics at $N = 20$ and $N = 80$ for a circle. We also display the $N = 80$ finite sum for the curve shown in Fig. \ref{['notation']}.
• Figure 3: The differential entropy proof of the triangle inequality (\ref{['triineq']}). Terms on the left hand side are drawn in continuous brown while the right hand side is in dashed red. See text below for more explanation.
• Figure 4: The functions $\alpha_{A,B}(\theta)$ of two points $A$ and $B$, which approach the boundary at fixed angular coordinates $\tilde{\theta} = \theta_{A,B}$. The pointwise minimum $\gamma(\theta) = \min\{\alpha_A(\theta), \alpha_B(\theta)\}$ approaches the causal diamond (shaded) of the interval $(\theta_A, \theta_B)$ and of its complement.
• Figure 5: Upper left: $\alpha(\theta)$ (green and blue), whose derivative is discontinuous at a point $(\theta_k, \alpha_k)$ (red). Upper right: the geodesics defined by $\alpha(\theta)$ and their (color coded) joint outer envelope. The two bulk open curves have the same tangent geodesic at their endpoints (red), because $\alpha(\theta)$ is continuous. The thickened part of this geodesic stretches between the endpoints of the open curves; its length is $f(\alpha_k, \theta_k, \tilde{\theta}_{\rm blue})-f(\alpha_k, \theta_k, \tilde{\theta}_{\rm green})$, viz. eq. (\ref{['kinkinaexpl']}). The differential entropy of $\alpha(\theta)$ in the upper left panel is the length of this outer envelope. Lower panels: the individual bulk open curves and their tangent geodesics.

Theorems & Definitions (1)

• Definition 1