Self-consistent spacetime metric for large-N black hole collapse and evaporation: the coupled four-dimensional semiclassical and five-dimensional holographic Einstein problem, from AdS regularization to the asymptotically flat limit

The open problem is to compute the explicit metric for gravitational collapse and black hole evaporation in the large-N limit, where the matter quantum effective action is supplied by AdS/CFT, so that the four-dimensional semiclassical Einstein equations couple to five-dimensional Einstein equations that must be solved together, probably numerically. This coupled problem was proposed as an open program in the open braneworld-holography literature — simulating the evaporation of a 4D black hole by Hawking radiation as a 5D classical process in the Randall-Sundrum braneworld (Tanaka 2003), and casting quantum-corrected 4D black holes as holograms of AdS braneworlds whose bulk solutions must include a radiation component (Emparan-Fabbri-Kaloper 2002). The question bundles three linked components.

Background

Black holes and the information paradox. A black hole formed by gravitational collapse emits thermal radiation (Hawking radiation) at a temperature inversely proportional to its mass, found by quantizing matter fields on a classical black hole background (Hawking 1975). If the radiation is exactly thermal, complete evaporation would take a pure initial quantum state to a mixed final state, contradicting the unitary evolution of quantum mechanics (Hawking 1976) — the information paradox. In his 2004 plenary talk at GR17, the 17th International Conference on General Relativity and Gravitation (held in Dublin), published in 2005 as both a proceedings chapter and a journal paper, Hawking conceded the famous bet and argued from the Euclidean path integral (quantum amplitudes written as integrals over positive-definite metrics obtained by analytically continuing physical Lorentzian spacetimes) that information is preserved: the sum over topologically trivial metrics gives unitary evolution, while the contributions of non-trivial, black-hole topologies decay at late times. The argument is naturally formulated in asymptotically anti-de Sitter space (AdS — the maximally symmetric solution of Einstein's equations with negative cosmological constant $\Lambda$), because the thermal partition function of an asymptotically flat black hole diverges owing to the infinite volume of flat space (an infrared problem), which Hawking proposed to circumvent by adding a small negative $\Lambda$ that is innocuous for a black hole much smaller than the AdS curvature radius. AdS black holes also have a rich thermal structure: below a critical temperature the canonical ensemble is dominated by thermal AdS, above it by stable 'giant' black holes, with an unstable 'small' branch resembling flat-space black holes (Hawking-Page 1983); boundary correlation functions decay in the black hole geometry yet remain of unitary size in thermal AdS (Maldacena 2001); and the AdS/CFT correspondence (Maldacena 1998) — the conjecture that quantum gravity in a $(d+1)$-dimensional AdS 'bulk' is equivalent to a $d$-dimensional conformal field theory (CFT) on its boundary — asserts exact unitarity of the boundary description.

The large-N effective problem. The quantitative question left open by this picture concerns the metric itself. In the semiclassical (leading-order in $\hbar$) approximation one solves the Einstein equations sourced by the expectation value of the quantum stress tensor,

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,\langle T_{\mu\nu}\rangle,$$

with $N$ matter fields so that $\langle T_{\mu\nu}\rangle$ scales like $N$ (or $N^2$ for a holographic gauge theory) while graviton loops are suppressed by $1/N$. When the matter sector is a holographic CFT, its quantum effective action — and hence $\langle T_{\mu\nu}\rangle$ — is given by the classical on-shell action of a five-dimensional AdS gravity problem whose boundary data is the four-dimensional metric itself. The four- and five-dimensional problems are therefore coupled: the 4D semiclassical equations above and the 5D Einstein equations must be solved together, the 4D metric serving simultaneously as the physical spacetime and as the boundary condition of the 5D problem. Equivalently (braneworld holography), one places a codimension-one brane (a thin hypersurface in the higher-dimensional space on which the matter fields are confined while gravity propagates in the whole space) in the AdS$_5$ bulk (the Randall-Sundrum setup): 4D gravity is induced on the brane, the bulk is dual to a cutoff CFT, and solving the classical 5D braneworld dynamics is equivalent to solving the 4D semiclassical equations with higher-curvature corrections. This coupled system was proposed as the arena in which black hole formation and evaporation could be followed dynamically, in general numerically (Tanaka 2003; Emparan-Fabbri-Kaloper 2002).

Regularization and the flat limit. Two technical prerequisites were flagged at the time. First, the gravitational action of an asymptotically AdS space diverges and must be regularized by boundary counterterms (holographic renormalization); beyond the volume and boundary-curvature counterterms, a third counterterm — logarithmic, tied to the conformal (Weyl) anomaly of the boundary theory in even boundary dimensions — resisted a covariant definition in early treatments. Second, the negative $\Lambda$ is only an infrared regulator: the ultimate target is a treatment of the evaporation of asymptotically flat black holes that is fully non-perturbative — not restricted to a fixed spacetime topology or a fixed background metric. Later work reshaped several of these questions: extended black horizons suffer the Gregory-Laflamme instability (the tendency of a black string to pinch off into localized black holes); the 5D horizon shapes dual to a 4D black hole interacting with the CFT are 'droplets' and 'funnels' (horizons anchored on the brane or stretching to the AdS boundary); and entanglement 'islands' — bulk regions whose quantum state is encoded in the radiation's entanglement entropy — reorganized the discussion of information recovery.

Problem Statement

The problem, proposed as an open program in the open braneworld-holography literature by Tanaka (2003) and Emparan-Fabbri-Kaloper (2002), is to carry out the explicit calculation of the metric for gravitational collapse and evaporation in the large-$N$ limit, using AdS/CFT to supply the quantum effective action of the matter fields. The large-$N$ limit here means the standard semiclassical approximation at large but finite $N$: $N$ matter-field degrees of freedom, so that the quantum stress tensor of matter scales with $N$ (like $N^2$ for a holographic gauge theory) while graviton loops are suppressed by $1/N$; at strictly $N\to\infty$ the evaporation of an isolated black hole survives only as a subleading $O(N^0)$ effect, so the evaporation setting must be chosen and justified as part of the solution (see the difficulties below). Concretely: solve the coupled four-dimensional and five-dimensional Einstein equations as a single self-consistent problem — the 4D semiclassical equations $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,\langle T_{\mu\nu}\rangle$ with the renormalized stress tensor of a large-$N$ holographic matter sector, and the 5D classical Einstein equations that compute that stress tensor with the 4D metric as boundary data — for the formation and subsequent evaporation of a black hole; this is expected to require numerical computation. Three linked components serve this single objective of obtaining the evaporation metric:

(1) The metric: an explicit (numerical or controlled analytic) solution of the coupled system at finite bulk dimension (e.g. $D=5$), exhibiting collapse and evaporation within the large-$N$ semiclassical framework defined above and in an evaporation setting that is chosen and justified as part of the solution (for instance an isolated black hole with the evaporation channel identified, or a leading-order bath-coupled configuration), and from which the mass loss, stress-energy distribution, and horizon dynamics can be read off. The target is an explicit metric defined by the following observables, rather than by the choice of setting: the collapse (formation) phase is exhibited and an explicit time evolution of the horizon (its position or area) and of the black-hole mass is then supplied over a stated interval, with a strictly nonzero mass-loss rate sourced by the large-$N$ matter sector and the corresponding emitted stress-energy flux read off from the same solution; a quasi-static or adiabatic treatment is admissible only for the evaporation phase after collapse and does not substitute for exhibiting the collapse (formation) phase. Because an isolated brane black hole has no $O(N^2)$ evaporation at strict leading order, leading-order evaporation occurs in a bath-coupled configuration, while in an isolated configuration only the explicit $O(N^0)$ emission and its backreaction are followed. A configuration whose mass-loss rate vanishes at every retained order — in particular a static or equilibrium metric — is not the object sought here.

(2) The regularization: a complete, covariantly well-defined counterterm regularization of the action for asymptotically anti-de Sitter space — including the third counterterm left non-covariantly defined in the early treatments — supplying the well-defined matter effective action and stress tensor that enter the 4D equations. The covariant obstruction itself has since been resolved by the holographic-renormalization program of 1999-2001; what remains of this component is its consistent dynamical implementation as a constituent of component (1): covariant counterterms, the anomaly-related logarithmic term, and the scheme ambiguities of holographic renormalization must be consistently realized in the time-dependent, inhomogeneous calculation that defines the renormalized stress tensor entering the 4D equations.

(3) The flat limit and the non-perturbative regime: control of the infrared problem of asymptotically flat space (for instance through a small negative cosmological constant, removed at the end of the calculation), and, ultimately, a fully non-perturbative treatment of the evaporation of asymptotically flat black holes, not limited to a fixed spacetime topology or a fixed semiclassical background.

The target is the metric and its construction: entropy-level statements about the unitarity of the radiation — for example Page curves, the plots of the entanglement entropy of the radiation against time that rise and then fall as unitary evolution requires — are not this object, and neither are static (equilibrium) configurations. Components (1) and (2), together with the infrared control of (3), are the problem addressed here, while the fully non-perturbative treatment of asymptotically flat evaporation named as the long-term target at the end of (3) is a longer-range goal.

References

1. Stephen Hawking, 'Black Holes and the Information Paradox', in General Relativity and Gravitation: Proceedings of the 17th International Conference on General Relativity and Gravitation (GR17), Dublin, 2004, World Scientific, 2005, pp. 56-62. DOI: 10.1142/9789812701688_0006, https://doi.org/10.1142/9789812701688_0006
2. S. W. Hawking, 'Information Loss in Black Holes', Phys. Rev. D 72, 084013 (2005). DOI: 10.1103/PhysRevD.72.084013, https://doi.org/10.1103/PhysRevD.72.084013. arXiv:hep-th/0507171, https://arxiv.org/abs/hep-th/0507171
3. S. W. Hawking, 'Particle Creation by Black Holes', Commun. Math. Phys. 43, 199-220 (1975). DOI: 10.1007/BF02345020, https://doi.org/10.1007/BF02345020
4. S. W. Hawking, 'Breakdown of Predictability in Gravitational Collapse', Phys. Rev. D 14, 2460-2473 (1976). DOI: 10.1103/PhysRevD.14.2460, https://doi.org/10.1103/PhysRevD.14.2460
5. S. W. Hawking and Don N. Page, 'Thermodynamics of Black Holes in Anti-de Sitter Space', Commun. Math. Phys. 87, 577-588 (1983). DOI: 10.1007/BF01208266, https://doi.org/10.1007/BF01208266
6. Juan M. Maldacena, 'The Large N Limit of Superconformal Field Theories and Supergravity', Adv. Theor. Math. Phys. 2, 231-252 (1998). DOI: 10.4310/ATMP.1998.V2.N2.A1, https://doi.org/10.4310/ATMP.1998.V2.N2.A1. arXiv:hep-th/9711200, https://arxiv.org/abs/hep-th/9711200
7. Juan Maldacena, 'Eternal Black Holes in AdS', JHEP 04 (2003) 021. DOI: 10.1088/1126-6708/2003/04/021, https://doi.org/10.1088/1126-6708/2003/04/021. arXiv:hep-th/0106112, https://arxiv.org/abs/hep-th/0106112
8. Vijay Balasubramanian and Per Kraus, 'A Stress Tensor for Anti-de Sitter Gravity', Commun. Math. Phys. 208, 413-428 (1999). DOI: 10.1007/s002200050764, https://doi.org/10.1007/s002200050764. arXiv:hep-th/9902121, https://arxiv.org/abs/hep-th/9902121
9. Roberto Emparan, Clifford V. Johnson, and Robert C. Myers, 'Surface Terms as Counterterms in the AdS/CFT Correspondence', Phys. Rev. D 60, 104001 (1999). DOI: 10.1103/PhysRevD.60.104001, https://doi.org/10.1103/PhysRevD.60.104001. arXiv:hep-th/9903238, https://arxiv.org/abs/hep-th/9903238
10. Sebastian de Haro, Kostas Skenderis, and Sergey N. Solodukhin, 'Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence', Commun. Math. Phys. 217, 595-622 (2001). DOI: 10.1007/s002200100381, https://doi.org/10.1007/s002200100381. arXiv:hep-th/0002230, https://arxiv.org/abs/hep-th/0002230
11. Lisa Randall and Raman Sundrum, 'An Alternative to Compactification', Phys. Rev. Lett. 83, 4690-4693 (1999). DOI: 10.1103/PhysRevLett.83.4690, https://doi.org/10.1103/PhysRevLett.83.4690. arXiv:hep-th/9906064, https://arxiv.org/abs/hep-th/9906064
12. Andreas Karch and Lisa Randall, 'Locally Localized Gravity', JHEP 05 (2001) 008. DOI: 10.1088/1126-6708/2001/05/008, https://doi.org/10.1088/1126-6708/2001/05/008. arXiv:hep-th/0011156, https://arxiv.org/abs/hep-th/0011156
13. Ruth Gregory and Raymond Laflamme, 'Black Strings and p-Branes are Unstable', Phys. Rev. Lett. 70, 2837-2840 (1993). DOI: 10.1103/PhysRevLett.70.2837, https://doi.org/10.1103/PhysRevLett.70.2837. arXiv:hep-th/9301052, https://arxiv.org/abs/hep-th/9301052
14. Takahiro Tanaka, 'Classical Black Hole Evaporation in Randall-Sundrum Infinite Braneworld', Prog. Theor. Phys. Suppl. 148, 307-316 (2003). DOI: 10.1143/PTPS.148.307, https://doi.org/10.1143/PTPS.148.307. arXiv:gr-qc/0203082, https://arxiv.org/abs/gr-qc/0203082
15. Roberto Emparan, Alessandro Fabbri, and Nemanja Kaloper, 'Quantum Black Holes as Holograms in AdS Braneworlds', JHEP 08 (2002) 043. DOI: 10.1088/1126-6708/2002/08/043, https://doi.org/10.1088/1126-6708/2002/08/043. arXiv:hep-th/0206155, https://arxiv.org/abs/hep-th/0206155
16. Pau Figueras, James Lucietti, and Toby Wiseman, 'Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua', Class. Quantum Grav. 28 (2011) 215018. DOI: 10.1088/0264-9381/28/21/215018, https://doi.org/10.1088/0264-9381/28/21/215018. arXiv:1104.4489, https://arxiv.org/abs/1104.4489
17. Pau Figueras and Toby Wiseman, 'Gravity and Large Black Holes in Randall-Sundrum II Braneworlds', Phys. Rev. Lett. 107 (2011) 081101. DOI: 10.1103/PhysRevLett.107.081101, https://doi.org/10.1103/PhysRevLett.107.081101. arXiv:1105.2558, https://arxiv.org/abs/1105.2558
18. Kazumi Kashiyama, Norihiro Tanahashi, Antonino Flachi, and Takahiro Tanaka, 'Quantum Back Reaction to asymptotically AdS Black Holes', JHEP 01 (2010) 099. DOI: 10.1007/JHEP01(2010)099, https://doi.org/10.1007/JHEP01(2010)099. arXiv:0910.5376, https://arxiv.org/abs/0910.5376
19. Nidal Haddad, 'Hawking Radiation from Small Black Holes at Strong Coupling and Large N', Class. Quantum Grav. 30 (2013) 195002. DOI: 10.1088/0264-9381/30/19/195002, https://doi.org/10.1088/0264-9381/30/19/195002. arXiv:1306.0086, https://arxiv.org/abs/1306.0086
20. Roberto Emparan, Raimon Luna, Ryotaku Suzuki, Marija Tomašević, and Benson Way, 'Holographic Duals of Evaporating Black Holes', JHEP 05 (2023) 182. DOI: 10.1007/JHEP05(2023)182, https://doi.org/10.1007/JHEP05(2023)182. arXiv:2301.02587, https://arxiv.org/abs/2301.02587
21. Geoff Penington, 'Entanglement Wedge Reconstruction and the Information Paradox', JHEP 09 (2020) 002. DOI: 10.1007/JHEP09(2020)002, https://doi.org/10.1007/JHEP09(2020)002. arXiv:1905.08255, https://arxiv.org/abs/1905.08255
22. Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield, 'The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole', JHEP 12 (2019) 063. DOI: 10.1007/JHEP12(2019)063, https://doi.org/10.1007/JHEP12(2019)063. arXiv:1905.08762, https://arxiv.org/abs/1905.08762
23. Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, 'Replica Wormholes and the Entropy of Hawking Radiation', JHEP 05 (2020) 013. DOI: 10.1007/JHEP05(2020)013, https://doi.org/10.1007/JHEP05(2020)013. arXiv:1911.12333, https://arxiv.org/abs/1911.12333
24. Koji Hashimoto, Norihiro Iizuka, and Yoshinori Matsuo, 'Islands in Schwarzschild Black Holes', JHEP 06 (2020) 085. DOI: 10.1007/JHEP06(2020)085, https://doi.org/10.1007/JHEP06(2020)085. arXiv:2004.05863, https://arxiv.org/abs/2004.05863
25. Adrian del Rio, 'The Backreaction Problem for Black Holes in Semiclassical Gravity', Gen. Rel. Grav. 57, 30 (2025). DOI: 10.1007/s10714-025-03352-x, https://doi.org/10.1007/s10714-025-03352-x
26. Niayesh Afshordi, Abhay Ashtekar, Enrico Barausse, Emanuele Berti, Richard Brito, Luca Buoninfante, Raúl Carballo-Rubio, Vitor Cardoso, Gregorio Carullo, Mihalis Dafermos, Mariafelicia De Laurentis, Adrian del Rio, Francesco Di Filippo, Astrid Eichhorn, Roberto Emparan, Ruth Gregory, Carlos A. R. Herdeiro, Jutta Kunz, Luis Lehner, Stefano Liberati, Samir D. Mathur, Samaya Nissanke, Paolo Pani, Alessia Platania, Frans Pretorius, Misao Sasaki, Paul Tiede, William Unruh, Matt Visser, and Robert M. Wald, 'Black Holes Inside and Out 2024: Visions for the Future of Black Hole Physics' (open-access volume containing the del Rio chapter). arXiv:2410.14414, https://arxiv.org/abs/2410.14414