Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane

TL;DR

This work extends the quantum extremal island framework to higher dimensions using a braneworld holographic model, where islands arise as standard RT-phase transitions of bulk surfaces crossing a Planck brane. The authors show that non-extremal black holes in any dimension exhibit islands that yield a unitary Page curve, computable by solving two ODEs, while extremal horizons in higher dimensions generally lack islands (as radiation ceases). In the two-dimensional limit, results reproduce known island behavior with calculable subleading corrections from the brane UV cutoff. The study also provides a detailed numerical analysis of island/no-island phases and Page curves for $d=3,4,5$, elucidating how brane tension, DGP coupling, and belt width shape entanglement dynamics. Overall, the braneworld approach offers analytic control and a clear geometric origin for islands without ensemble averaging, linking island physics to familiar RT-phase transitions in higher-dimensional AdS/CFT.

Abstract

We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in arXiv:2006.04851. The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with arXiv:1910.11077 at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time.

Figures (22)

• Figure 1: Illustration of doubly-holographic models: The top row illustrates (a time slice of) the three perspectives of the model in Almheiri:2019hni, while the bottom row displays the analogous descriptions of our construction in higher dimensions Chen:2020uac. In the latter, we are using the global conformal frame where the boundary CFT lives on $R\times S^{d-1}$ and the conformal defect appears on the equator of the ($d-1$)-sphere -- see discussion in section \ref{['sec:RS']} and Chen:2020uac. The bottom row reduces to the top upon setting $d=2$ and taking a $\mathbb{Z} _2$ quotient across the defect in the boundary or the brane in the bulk. The boundary, brane and bulk gravity perspectives correspond to panels a & d, b & e, and c & f, respectively.
• Figure 2: The choice of RT surfaces on a constant time slice in the presence of the brane (coloured green), showing the different ingredients involved in eq. \ref{['eq:island2']}.
• Figure 3: A timeslice of our Randall-Sundrum setup. In panel (a), we cut off the AdS$_{d+1}$ spacetime along an AdS$_d$ slice near the asymptotic boundary $\theta=0$, in the metric (\ref{['metric33']}). Two of these spaces are glued together in panel (b) and the brane is realized as the interface between the two geomeries.
• Figure 4: Our eternal black hole coupled to the CFT bath, as seen from the bulk perspective.
• Figure 5: The Euclidean path integral (orange region) prepares the Hawking-Hartle state. The black hole temperature $T=1/(2\pi R)$ is derived in section \ref{['sec:nonextremal']}.