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AdS/CFT duality AdS/CFT duality

Double holography and black hole information

Double holography inserts a gravitating brane into AdS so that the same system admits three equivalent descriptions—a higher‑dimensional bulk with an end‑of‑the‑world brane, a brane‑world with induced gravity coupled to a bath, and a boundary/defect CFT. In this setup, standard RT/QES surfaces in the higher‑dimensional bulk automatically compute island contributions and reproduce the Page curve for Hawking radiation. The “mystery” of information recovery becomes a geometric phase transition of extremal surfaces.

1) From thermodynamics to entanglement

Section titled “1) From thermodynamics to entanglement”

Black holes behave as thermodynamic objects: Bekenstein and Hawking established

SBH=Area(H)4 G ℏ,T=κ2π,S_{\mathrm{BH}}=\frac{\mathrm{Area}(\mathcal{H})}{4\,G\,\hbar},\qquad T=\frac{\kappa}{2\pi},

but Hawking’s semiclassical calculation suggested that complete evaporation takes a pure state to a mixed state—the information paradox. A unitary evolution requires the Page curve, where the radiation entropy rises, peaks at the Page time, and then falls.

Holography (AdS/CFT) gives a geometric route to fine‑grained entropy. The Ryu–Takayanagi (RT) and Hubeny–Rangamani–Takayanagi (HRT) prescriptions relate boundary entanglement to bulk extremal surfaces, refined by FLM and the QES principle:

S(A)=min⁡X∼A[Area(X)4Gd+1+Sbulk(ΣX)],δSgen(X)=0.S(A)=\min_{\mathcal{X}\sim A}\left[ \frac{\mathrm{Area}(\mathcal{X})}{4 G_{d+1}} + S_{\text{bulk}}(\Sigma_{\mathcal{X}}) \right], \qquad \delta S_{\text{gen}}(\mathcal{X})=0.

In evaporating setups, the QES for the radiation region RR can jump at late times so that the entropy is computed using an island I\mathcal{I} inside the gravity region:

S(R)=min⁡I  ext[Area(∂I)4Gd+Sbulk(R∪I)].S(R)=\min_{\mathcal{I}}\;\mathrm{ext}\left[ \frac{\mathrm{Area}(\partial\mathcal{I})}{4 G_d} + S_{\text{bulk}}(R\cup \mathcal{I}) \right].

2) What is double holography?

Section titled “2) What is double holography?”

Double holography introduces a codimension‑one brane into AdSd+1_{d+1}. This produces three interlocking viewpoints:

A. Bulk perspective (AdSd+1_{d+1} with a brane)

Section titled “A. Bulk perspective (AdSd+1_{d+1}d+1​ with a brane)”
  • Work in (d+1)(d+1)-dimensional gravity with negative cosmological constant.
  • The brane is pure tension and its embedding is umbilic: Kab=λ habK_{ab}=\lambda\,h_{ab}, fixed by the Israel condition for a Z2\mathbb{Z}_2‑symmetric brane, Kab−K hab=−8πGd+1(Sab−1d−1S hab),Sab=−T hab ⇒ Kab=λhab, λ∝T.K_{ab}-K\,h_{ab} = -8\pi G_{d+1}\left(S_{ab}-\frac{1}{d-1}S\,h_{ab}\right), \quad S_{ab}=-T\,h_{ab}\ \Rightarrow\ K_{ab}=\lambda h_{ab},\ \lambda\propto T.
  • Holographic entropy for a boundary region RR is computed by a bulk extremal surface VV that may end on the brane. With an induced/“DGP” gravity term on the brane, the geometric functional is S(R)=min⁡ext[Area(V)4Gd+1+Area(V∩brane)4Gd+Sbulk(ΣV)].S(R)=\min_{\text{ext}}\left[ \frac{\mathrm{Area}(V)}{4G_{d+1}} + \frac{\mathrm{Area}(V\cap \text{brane})}{4G_{d}} + S_{\text{bulk}}(\Sigma_V) \right]. The brane segment V∩braneV\cap\text{brane} is the island boundary in the lower‑dimensional view.

B. Brane perspective (induced gravity on AdSd_d)

Section titled “B. Brane perspective (induced gravity on AdSd_dd​)”
  • Integrate out the extra dimension: gravity localizes on the brane (Randall–Sundrum/Karch–Randall). Parametrically, Gd∼Gd+1/LG_d \sim G_{d+1}/L (up to dimension‑dependent factors), with an AdSd_d geometry induced on the brane.
  • The same entropy is computed by the island/QES formula on the brane: S(R)=min⁡I  ext[Area(∂I)4Gd+Sbulk(R∪I)].S(R)=\min_{\mathcal{I}}\;\mathrm{ext}\left[ \frac{\mathrm{Area}(\partial\mathcal{I})}{4 G_{d}} + S_{\text{bulk}}(R\cup\mathcal{I}) \right].
  • At early times I=∅\mathcal{I}=\varnothing (Hawking’s result). After the Page time, a non‑empty island dominates, giving the unitary Page curve.

C. Boundary/defect CFT perspective (AdS/BCFT)

Section titled “C. Boundary/defect CFT perspective (AdS/BCFT)”
  • The boundary theory is a dd‑dimensional CFT coupled to a conformal defect where it meets the brane sector. This is the AdS/BCFT construction.
  • RT/HRT surfaces in AdSd+1_{d+1} that end on the brane compute BCFT entropies and mirror the island calculation.

These are three languages for the same physics. The extra bulk dimension turns “mysterious islands” into ordinary extremal surfaces undergoing a phase transition.

3) How islands emerge geometrically (Page curve in one picture)

Section titled “3) How islands emerge geometrically (Page curve in one picture)”

Consider a brane black hole coupled to a non‑gravitating bath (radiation measured at the asymptotic boundary). Two families of bulk extremal surfaces compete:

  1. Non‑crossing: VV hangs entirely outside the brane.
  2. Crossing: VV intersects the brane and encloses a piece of the brane interior.

As radiation time (or region size) grows, the crossing surface’s generalized entropy becomes smaller and wins. In the brane/QES language this means an island has appeared. The entropy then decreases and the Page curve is reproduced without adding ad‑hoc rules—just standard RT/HRT in one higher dimension.

Reconstruction. Because the crossing surface puts the island in the entanglement wedge of the radiation, interior operators can be represented on the radiation degrees of freedom (entanglement‑wedge reconstruction). This is precisely how information escapes Hawking’s paradox.

4) Remarks on the pure‑tension brane

Section titled “4) Remarks on the pure‑tension brane”
  • Variational principle & symmetry. The Israel/Neumann boundary condition with Sab=−ThabS_{ab}=-T h_{ab} leads to Kab∝habK_{ab}\propto h_{ab}, preserving AdSd_d symmetry on the brane and giving a clean, well‑posed gravitational problem.
  • Gravity localization. The warped AdS geometry binds a graviton mode near the brane, yielding an effective dd‑dimensional gravity sector.
  • RT/QES boundary condition. For an umbilic brane, extremal surfaces obey a simple endpoint condition (orthogonality in the simplest cases), allowing them to end on the brane and making the island contribution universal.
  • BCFT dictionary. The same condition implements the AdS/BCFT Neumann boundary condition, keeping the defect conformal so that the boundary picture remains simple.

5) Worked example

Section titled “5) Worked example”

Quantum BTZ on a brane. Embed a 3D BTZ black hole on an AdS3_3 brane inside AdS4_4. The 4D classical geometry encodes the 3D black hole with quantum backreaction from a large‑NN CFT. The generalized entropy (horizon area ++ exterior entanglement) satisfies the first law and the late‑time radiation entropy follows the Page curve. In the bulk picture, the extremal surface transition is manifest; on the brane, it is the appearance of an island behind the horizon.

  1. Choose the perspective (bulk geometry, brane QES, or BCFT).
  2. Specify the state/region (e.g. an eternal brane black hole in a thermal bath; the radiation interval RR).
  3. Find candidate extremal surfaces:
    • In the bulk: allow VV to end on the brane; include a brane area term.
    • On the brane: solve QES equations for ∂I\partial\mathcal{I}.
  4. Evaluate SgenS_{\text{gen}} and pick the minimal saddle.
  5. Check wedge inclusion to determine which bulk (or brane) regions are reconstructible from RR.

Timeline at a glance

Section titled “Timeline at a glance”

“History Doesn’t Repeat Itself, but It Often Rhymes” – Mark Twain

1973 — Black‑hole entropy.
Bekenstein proposes that a black hole carries entropy proportional to its horizon area, SBH=A/(4Gℏ)S_{\mathrm{BH}} = A / (4 G \hbar), inaugurating black‑hole thermodynamics. 1

1974–1975 — Hawking radiation.
Hawking shows that black holes radiate thermally with temperature T=κ/(2π)T = \kappa / (2\pi), implying evaporation. 2

1976 — The black‑hole information problem.
Hawking argues that complete evaporation produces a mixed state, signaling nonunitary evolution—“breakdown of predictability.” 3

1993 — Black‑hole complementarity & the Page curve.
Susskind–Thorlacius–Uglum articulate complementarity via the stretched‑horizon picture. 4
In the same year, Page analyzes typical entanglement of subsystems and sketches the Page curve for unitary evaporation. 5

1997–1998 — AdS/CFT correspondence.
Maldacena proposes gauge/gravity duality; Witten and Gubser–Klebanov–Polyakov formulate boundary correlators. 6 7 8

1999 — Randall–Sundrum braneworlds.
Warped AdS5_5 with branes localizes gravity and seeds later “brane + bath” models. 9 10

2000–2001 — Brane‑localized gravity & AdS C‑metrics.
Karch–Randall show gravity localized on an AdS4_4 brane in AdS5_5. 11
Emparan–Horowitz–Myers construct exact brane black holes using AdS C‑metrics—precursors to later double‑holographic black holes. 12 13

2006–2007 — Holographic entanglement entropy.
Ryu–Takayanagi (RT) relate boundary entanglement entropy to minimal areas; Hubeny–Rangamani–Takayanagi (HRT) generalize covariantly. 14 15

2007 — Hayden–Preskill decoding.
For old black holes, newly thrown‑in information can be recovered quickly from Hawking radiation after a scrambling delay. 16

2011 — AdS/BCFT.
Takayanagi (and Fujita–Takayanagi–Tonni) introduce holography with an end‑of‑the‑world brane obeying Kab=λ γabK_{ab} = \lambda\,\gamma_{ab}, providing a flexible platform for coupling CFTs to dynamical gravity on a brane. 17 18

2012 — AMPS firewall paradox.
Tension between purity of radiation, EFT outside the horizon, and “no drama” suggests firewalls unless entanglement structure is revised. 19

2013–2016 — Quantum corrections & entanglement wedge.
FLM give one‑loop (bulk‑entropy) corrections to RT. 20
Engelhardt–Wall propose quantum extremal surfaces (QES) extremizing generalized entropy. 21
JLMS equate boundary and bulk relative entropy; Dong–Harlow–Wall prove entanglement‑wedge reconstruction. 22 23

2019 — Entanglement islands arrive.
Penington shows a QES phase transition at the Page time, moving the wedge behind the horizon; Almheiri–Engelhardt–Marolf–Maxfield derive the island rule for radiation entropy. 24 25

2019–2020 — Replica wormholes.
Independently, Penington–Shenker–Stanford–Yang and Almheiri–Hartman–Maldacena–Shaghoulian–Tajdini show that new replica saddles (“replica wormholes”) justify the island formula and give the Page curve. 26 27

2020 — Double holography “made easy.”
Chen–Myers–Neuenfeld–Reyes–Sandor (Parts I & II) introduce tractable doubly holographic brane models where QES/islands are RT surfaces in one higher dimension; the Page curve follows from RT‑surface phase transitions crossing the brane. 28 29
Emparan et al. construct a quantum BTZ via a brane setup capturing strong CFT backreaction—another milestone for brane‑world black holes. 30

2021–2024 — Consolidation & string embeddings.
Reviews formalize the radiation‑entropy technology; stringy realizations and multi‑brane universes enrich double holography. 31 32 33

References

Section titled “References”

Footnotes

Section titled “Footnotes”

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  2. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199. https://doi.org/10.1007/BF02345020. ↩

  3. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460. https://doi.org/10.1103/PhysRevD.14.2460. ↩

  4. L. Susskind, L. Thorlacius, J. Uglum, The Stretched Horizon and Black Hole Complementarity, Phys. Rev. D 48 (1993) 3743. https://doi.org/10.1103/PhysRevD.48.3743. ↩

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  8. S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Gauge theory correlators from non‑critical string theory, Phys. Lett. B 428 (1998) 105; arXiv:hep-th/9802109. ↩

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  10. L. Randall, R. Sundrum, An Alternative to Compactification, Phys. Rev. Lett. 83 (1999) 4690. https://doi.org/10.1103/PhysRevLett.83.4690. ↩

  11. A. Karch, L. Randall, Locally Localized Gravity, JHEP 05 (2001) 008; arXiv:hep-th/0011156. ↩

  12. R. Emparan, G. T. Horowitz, R. C. Myers, Exact Description of Black Holes on Branes, arXiv:hep-th/9911043. ↩

  13. R. Emparan, G. T. Horowitz, R. C. Myers, Exact Description of Black Holes on Branes II, arXiv:hep-th/9912135. ↩

  14. S. Ryu, T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602; arXiv:hep-th/0603001. ↩

  15. V. E. Hubeny, M. Rangamani, T. Takayanagi, A Covariant Holographic Entanglement Entropy Proposal, JHEP 07 (2007) 062; arXiv:0705.0016. ↩

  16. P. Hayden, J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120; arXiv:0708.4025. ↩

  17. T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602; arXiv:1105.5165. ↩

  18. M. Fujita, T. Takayanagi, E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043; arXiv:1108.5152. ↩

  19. A. Almheiri, D. Marolf, J. Polchinski, J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062; arXiv:1207.3123. ↩

  20. T. Faulkner, A. Lewkowycz, J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074; arXiv:1307.2892. ↩

  21. N. Engelhardt, A. C. Wall, Quantum Extremal Surfaces, JHEP 01 (2015) 073; arXiv:1408.3203. ↩

  22. D. L. Jafferis, A. Lewkowycz, J. M. Maldacena, S. J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004; arXiv:1512.06431. ↩

  23. X. Dong, D. Harlow, A. C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge, Phys. Rev. Lett. 117 (2016) 021601; arXiv:1601.05416. ↩

  24. G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002; arXiv:1905.08255. ↩

  25. A. Almheiri, N. Engelhardt, D. Marolf, H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, arXiv:1905.08762. ↩

  26. G. Penington, S. H. Shenker, D. Stanford, Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205; arXiv:1911.11977. ↩

  27. A. Almheiri, T. Hartman, J. M. Maldacena, E. Shaghoulian, A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013; arXiv:1911.12333. ↩

  28. H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, J. Sandor, Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, arXiv:2006.04851. ↩

  29. H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, J. Sandor, Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, arXiv:2010.00018. ↩

  30. R. Emparan, M. Tomašević, R. Suzuki, K. Tanabe, Quantum BTZ Black Hole, arXiv:2007.15999. ↩

  31. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002; arXiv:2006.06872. ↩

  32. A. Karch, Double holography in string theory, JHEP 10 (2022) 012; arXiv:2206.11292. ↩

  33. R. C. Myers, et al., Double Holography of Entangled Universes, JHEP 07 (2024) 035; arXiv:2403.17483. ↩