Entanglement entropy in quantum gravity and the Plateau problem

In a quantum gravity theory the entropy of entanglement $S$ between the fundamental degrees of freedom spatially divided by a surface is discussed. Classical gravity is considered as an emergent phenomenon and arguments are presented that (1) $S$ is a macroscopical quantity which can be determined without knowing a real microscopical content of the fundamental theory; (2) $S$ is given by the Bekenstein-Hawking formula in terms of the area of a codimension 2 hypesurface $\mathcal{B}$; (3) in static space-times $\mathcal{B}$ can be defined as a minimal hypersurface of a least volume separating the system in a constant-time slice. It is shown that properties of $S$ are in agreement with basic properties of the von Neumann entropy. Explicit variational formulae for $S$ in different physical examples are considered.

Figure 1

A constant-time slice of a system spatially divided by a hypersurface $\mathcal{B}$.

Figure 2

Construction of the space ${\mathcal{M}}_n$ for a 2D finite-temperature system set on an interval $\Sigma$. For $n=3$ the space ${\mathcal{M}}_n$ is obtained from 3 copies of the cylinders $\Sigma \times S^1$. The cylinders are cut and glued along the part ${\Sigma }_1$ of a “constant-time” slice.

Figure 3

In a quantum gravity one can specify the entangled regions of a system by dividing its spatial boundary $\mathcal{S}$ onto domains $D_1$ and $D_2$. The separating surface $\mathcal{B}$ is determined by a dynamical problem with the condition that $\mathcal{B}$ ends at the boundary $\mathcal{P}$ between $D_1$ and $D_2$.

Figure 4

Some examples of minimal surfaces corresponding to a single contour in a flat space.

Figure 5

The Carter-Penrose diagram for a Schwarzschild black hole. The black hole in cavity corresponds to a part of the diagram between the lines $r=r_0$. Shown at the diagram is one of the Cauchy hypersurfaces ${\Sigma }_t^L\bigcup _{}^{}{\Sigma }_t^R$.

Figure 6

Demonstrates the form of trajectories in the path-integral representation for the wave function $\Psi$ of the black hole and the density matrix $\widehat{\rho }$, Figs. 6a, 6b, respectively. Figure 6a shows half of the Gibbons-Hawking instanton; (b) shows the instanton with a cut.

Figure 7

Shown by dashed regions are different locations of states on the Einstein-Rosen bridge which are traced out. The boundary of these regions is $D_2$, the boundary of their completion is $D_1$. The cases (a), (b) correspond to the choice $D_1=S_1^R$, $D_2=S^L\bigcup _{}^{}{S}_2^R$, the case (c) is $D_1=S^L\bigcup _{}^{}{S}_2^R$, $D_2=S_1^R$, and the cases (d), (e) are $D_1=S_2^R$, $D_2=S^L\bigcup _{}^{}{S}_1^R$.

Figure 8

Separating surfaces in the presence of a black hole.

Figure 9

Figure 9a demonstrates a system with intersecting subsets (regions), 1 and 2. Boundaries of the regions are shown as lines $(A,D)$ and $(B,C)$. The domain of the intersection is dashed. The minimal surfaces are the internal lines connecting pairs of points, $A$ with $D$ and $B$ with $C$. Another set of minimal surfaces for the same configuration of the boundaries are the solid internal lines on Fig. 9b. They connect $A$ with $B$ and $C$ with $D$.