Evaporation of black holes in flat space entangled with an auxiliary universe

TL;DR

This work analyzes evaporating black holes in flat space by coupling a gravitating universe B to a non-gravitating auxiliary universe A in a thermofield-double state. The entanglement back-reaction in B lengthens the interior wormhole and reduces the horizon area, producing a Page-curve evolution for the entropy of A via the island formula: S(ρ_A) = min{S_no-island, S_island}, where S_island involves the dilaton boundary term Φ and CFT entropies. The authors compute the dilaton profile, quantum extremal surfaces, and CFT contributions, showing that at high entanglement temperature the island dominates and reproduces an evaporating-flat-space Page curve; they also study local operations (shock waves) in B, finding LOCC insertions decrease entropy while non-LOCC insertions can trigger multiple island/no-island transitions. The results clarify operational meaning of the island rule in flat space and reveal how entanglement monogamy and back-reaction govern the black hole’s evaporation and information flow across the two universes.

Abstract

We study a thermofield double type entangled state on two disjoint universes $A$ and $B$, where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy on the universe $A$ through the island formula, and argue that it naturally follows the Page curve of an evaporating black hole in flat space. We also study the effects of local operations in the gravitating universe with the black hole. We find that they accelerate the evaporation of the black hole, therefore disrupt the entanglement between two universes. Furthermore, we observe that depending on whether the operation can be regarded as an LOCC or not, the behavior of the entanglement entropy changes. In particular, when the operation is made neither in the entanglement wedge of the radiation system or that of the black hole, the transition between the island phase and the no-island phase can happen multiple times.

Figures (15)

• Figure 1: We consider a system with two disjoint asymptotically Minkowski spaces, A and B. In this figure, these universes are embedded in a larger Minkowki space.
• Figure 2: Left: The Penrose diagram of the black hole without the back-reaction. Right: The Penrose diagram of the black hole with the back-reaction of the source (\ref{['eq:stress']}). It develops a long wormhole region in its interior.
• Figure 3: The location of the island $C$ in the black hole with the back-reaction, denoted by the blue line.
• Figure 4: Left :Plot of the generalized entropy $S_{{\rm gen}} (x)$ as a function of the size of the island in the interior. Right: The resulting Page curve as a function of the entanglement temperature $T=1/\beta$. Here we set the parameters to be $\phi_0=1700,\; \Lambda=500,\; c=50$ in both figures and $\beta=1$ for left figure.
• Figure 5: The dilaton profile \ref{['eq:completedl']} in the presence of the shock wave. In the right wedge of the local operator, $x^{\pm} >x^{\pm}_{0}$, we have $\Phi= \Phi_{\beta, E}$ with \ref{['eq:phibetaE']}. On the left wedge, $x^{\pm} <x^{\pm}_{0}$, the dilaton profile coincides with $\Phi_{\beta,0}$, which is identical to \ref{['eq:dilbeta']} .