Entanglement entropy in holographic moving mirror and Page curve

TL;DR

The paper constructs a gravity dual for a 2D CFT with a moving mirror, modeling Hawking-like radiation and black-hole evaporation, and computes the time evolution of entanglement entropy. Using conformal maps, holographic AdS/BCFT with an end-of-the-world brane Q, and replica techniques, it demonstrates a Page-curve-like entropy evolution: linear growth from entangled-pair production and subsequent saturation due to island contributions, with explicit expressions in both free Dirac fermion and holographic CFTs. The brane-world interpretation links boundary entropy to an effective 2D gravitational sector and recasts the entropy evolution in terms of the island formula, offering a concrete strong-coupling realization of unitary black-hole radiation. The framework thus provides a tractable, holographic model to study unitarity and entropy dynamics in black hole evaporation, with avenues to incorporate singularities and gravitational radiation by adjusting the brane geometry.

Abstract

We calculate the time evolution of entanglement entropy in two dimensional conformal field theory with a moving mirror. For a setup modeling Hawking radiation, we obtain a linear growth of entanglement entropy and show that this can be interpreted as the production of entangled pairs. For a setup, which mimics black hole formation and evaporation, we find that the evolution follows the ideal Page curve. We perform these computations by constructing the gravity dual of the moving mirror model via holography. We also argue that our holographic setup provides a concrete model to derive the Page curve for black hole radiation in the strong coupling regime of gravity.

Figures (7)

• Figure 1: The moving mirror setup (left) and its conformal transformation into a static mirror (right). The Mirror trajectory is depicted by the thick curve. The shaded region shown in the right panel corresponds to an inside horizon region, which is missing in the left picture.
• Figure 2: The graphs in the left figure show the time evolution of entanglement entropy $S_A$. We choose the end point of $A$ to be $x_0=-t+\xi_0$, with $\xi_0=1$ (thick), $\xi_0=0.1$ (dashed) and $\xi_0=0.01$ (dotted). We set $\beta=1$, $\epsilon=0.1$ and $S_\text{bdy} = 0$. The right figure shows a quasi-particle picture of entanglement growth for the moving mirror. The black thick curve represents the mirror trajectory $x=Z(t)$. The purple dotted curve describes a spacelike curve defined by $v+p(u)=0$. The red line corresponds to the null line. The entangled pair production occurs on the purple dashed curve.
• Figure 3: Gravity dual of a moving mirror in the coordinates $(U,V,\eta)$ (left) and $(u,v,z)$ (right). We set ${\cal T}=0$. We also show the computation of holographic entanglement entropy.
• Figure 4: The time evolution of entanglement entropy for moving mirror \ref{['traj']} in free Dirac fermion CFT (left) and in holographic CFT (right). Here, we set the subsystem to be $A=[Z(t)+0.1,Z(t)+10]$ with $\beta=0.1$, $\epsilon=0.1$ and $S_\text{bdy} = 0$. In the right, the thick line and the dashed line show the disconnected and connected entanglement entropy, respectively.
• Figure 5: The profile of moving mirror, with the creation of entangled pairs and their reflection at the mirror, is depicted in the left picture. The right graph is the energy density $T_{uu}$ plotted as a function of $u$. We set $\beta=0.1$ and $u_0=5$.