Asymptotically flat black hole spacetimes with multiple injections

In quantum gravity, Hawking radiation presents several fundamental problems. One of the problems is the black hole (BH) information paradox, in which the entanglement entropy (EE), which quantifies quantum entanglement, exceeds its upper bound. In the absence of the paradox, EE follows the Page curve. Recent progress has been made in resolving this paradox using the island formula, a method for computing EE that successfully reproduces the expected Page curve. In this approach, a portion of the black hole interior is treated as part of the radiation region.

Meanwhile, an alternative scenario has been proposed where multiple collapsing shells prevent the formation of a well-defined event horizon [1]. In this case, radiation is emitted throughout the collapse process, shifting dynamically the Schwarzschild radius inward, and a surface structure is formed just outside. This leads to a distinction between the conventional event horizon and the surface, introducing an intermediate region between the Schwarzschild radius and the surface. Interestingly, this model also suggests that part of the black hole interior effectively belongs to the radiation region, drawing a possible parallel to the island formula.

In this talk, we explore spacetimes with multiple energy injections in asymptotically flat two-dimensional black hole backgrounds and analyze the entanglement entropy in such scenarios. Since considering backreaction in gravitational collapse in two dimensions is difficult, we instead construct a spacetime solution with multiple energy injections and analyze EE within this background. The main focus of this talk is to derive the spacetime and examine its properties. Additionally, we perform EE calculations in parallel with previous studies [2], which consider the case of α single injection, and confirm that the behavior of EE depends on the interval between energy injections.