Holographic Black Holes

• Holographic black holes are gravitational systems defined by encoding high-dimensional bulk information on lower-dimensional screens, enforcing the Bekenstein–Hawking entropy bound.
• They provide a framework to study black hole thermodynamics and dualities via AdS/CFT, linking pressure, volume, and enthalpy through holographic heat engines.
• The approach addresses quantum complexity and the information paradox using horizon-to-bulk correspondences, quantum extremal surfaces, and island prescriptions.

A holographic black hole is a gravitational system in which the bulk spacetime geometry, thermodynamics, quantum information properties, and microscopic structure are fundamentally controlled by holographic principles, most notably the encoding of high-dimensional bulk data on lower-dimensional screens such as the event horizon or asymptotic boundaries. The term encompasses both the original 't Hooft–Susskind area-law holography, the landscape of AdS/CFT dual black holes, generalized horizon-to-bulk correspondences, and recent developments in black hole thermodynamics, complexity, evaporation, and quantum microstate structure. The paradigmatic feature is the saturation of the holographic entropy bound, with black hole states and their dual field theory representations providing an arena for testing the interplay between gravity, thermodynamics, and quantum information in strongly coupled regimes.

1. Black Hole Entropy and the Holographic Bound

The foundational signature of holographic black holes is the Bekenstein–Hawking entropy formula: $S_{BH} = \frac{A_{\rm hor}}{4G_N}$ where $A_{\rm hor}$ is the area of the event horizon and $G_N$ is Newton’s constant (units $\hbar=c=k_B=1$). This area law suggests that all microstates of the black hole are encoded on the horizon, rather than distributed throughout its spatial volume. Extended frameworks, such as spontaneously induced gravity or higher-curvature gravity with appropriate phase transitions at $r = 2GM$, show that the volume inside any interior sphere can vanish ($V(r) \to 0$), while the area remains finite, enforcing a local saturation of the 't Hooft–Susskind bound: $S(r) = \frac{A(r)}{4G}$ for any $0 < r < 2GM$, with all information stored on interior shells (“onion-layer” holography) (Davidson et al., 2010, Davidson et al., 2010).

2. Holographic Duality and Bulk–Boundary Correspondence

The AdS/CFT correspondence generically equates gravity in an asymptotically AdS spacetime with a large-$N$ quantum field theory on its boundary. For black holes, the dual boundary state is typically a thermal (or near-thermal) density matrix, where the temperature, chemical potentials (for rotation, charge), and thermodynamic quantities are controlled by the bulk horizon data. Key distinctions with pure black-hole holography include:

• The physical entropy of the boundary theory is generally not simply given by the area of any asymptotic boundary surface, except when that surface coincides with the event horizon.
• Quantum information paradoxes (e.g., the fate of Hawking radiation) are addressed by specific gravitational microstate constructions (e.g., fuzzballs), not by the general properties of the boundary CFT (Mathur, 2012).

Horizon-to-bulk holography can be formalized in “near-horizon geometry” (NHG) frameworks, where the local two-dimensional horizon screen, equipped with intrinsic metric and rotation one-form, fully determines the bulk geometry in a neighborhood via constraint equations. For extremal black holes, all local data near the horizon are encoded holographically in the screen data (Lewandowski et al., 2017).

3. Extended Thermodynamics and Holographic Heat Engines

Holographic black holes admit a generalized thermodynamic interpretation (“black hole chemistry”) with cosmological constant promoted to thermodynamic pressure: $P = -\frac{\Lambda}{8\pi}$ and conjugate volume $V = (\partial M/\partial P)_{S,J,\ldots}$. In the AdS/CFT duality, varying $P$ is mapped to changing the central charge $N^2$ or the boundary CFT volume. Holographic heat engines are constructed as closed cycles in the $P-V$ plane, with the black hole acting as the working fluid. Their efficiency is determined via black hole enthalpy (identified with ADM mass) and the first law of black hole thermodynamics. Exact results are available for both rectangular and circular cycles. For black holes with vanishing specific heat at constant volume ($C_V=0$), a universal efficiency bound independent of spacetime dimension holds: $$\eta \leq \eta_0 = \frac{2\pi}{\pi + 4} \simeq 0.8798 < \eta_{\rm Carnot}$$ which is saturated for extremal cycles (Hennigar et al., 2017). The efficiency encodes geometric/topological data and higher-curvature corrections, reflecting the dual CFT’s angular momentum sectors, genus, and finite-$N$ corrections.

4. Holographic Complexity of Black Holes

Quantum computational complexity in holographic black holes is captured by bulk geometric functionals anchored to the boundary time slice, notably the “complexity equals action” (CA) and “complexity equals volume” (CV) prescriptions:

• CA: complexity $\mathcal{C} = I_{\rm WdW}/\pi$, with the action $I_{\rm WdW}$ evaluated on the Wheeler–DeWitt patch. At late times, the growth rate is controlled by the difference of internal energies (or generalized potentials) between inner/outer horizons:

$$\lim_{t\rightarrow\infty} \frac{d\mathcal{C}}{dt} = \frac{1}{\pi}\left[ (M - \Omega_+J)_+ - (M - \Omega_-J)_- \right]$$

For charged/rotating holes, this generalizes to include electric and chiral anomaly potentials (Jiang et al., 2020, Mounim et al., 2021).

• CV: complexity $$\mathcal{C}_{\mathcal{V}} = \mathrm{Vol}(\mathcal{B})/(G_N R)$$. For large/hyperbolic black holes, the complexity of formation and growth rate can be directly proportional to thermodynamic volume or entropy, and in some cases, the form

$\mathcal{C}_{\mathcal{V}} \sim S \log(1/T)$

emerges, especially near extremality (Barbon et al., 2015, Wang et al., 2023).

The complexity growth obeys the generalized Lloyd bound, $d\mathcal{C}/dt \leq 2M/\pi$, in Einstein gravity, but can be violated in cases with higher-derivative corrections or causally nontrivial horizon structure (e.g., hyperbolic EMD black holes, stringy corrections) (Fan et al., 2019, Wang et al., 2023, Balushi et al., 2020).

Quantum corrections, as in quantum BTZ black holes, can dramatically change the complexity growth behavior, especially in the action prescription, where the presence of an interior singularity can give non-smooth classical limits, while volume complexity remains continuous (Emparan et al., 2021, Chen et al., 2023).

5. Black Hole Microstates, Entropy Packing, and Hollowgraphy

Holographic black holes can realize microstate structure via two mechanisms:

• String-theoretic fuzzballs: The black hole is replaced by a family of horizonless, smooth microstate geometries; the area law arises from the coarse-grained envelope of these geometries. Hawking’s calculation fails as the horizon is replaced by microstate structure; radiation is non-thermal and purifies the exterior density matrix. High-energy infall can be approximately duplicated by auxiliary constructions (“approximate complementarity”) (Mathur, 2012).
• Hollowgraphy: In extensions of GR driven by phase transitions at the would-be horizon, the black hole interior obtains vanishing (or arbitrarily small) spatial volume, while the area remains finite. All mass and entropy is distributed on thin shells at fixed radii, resulting in a local (onion-layer) saturation of the Bekenstein–Hawking bound; the Komar mass and material energy increase monotonically to the horizon, and the interior Newton constant runs to zero (Davidson et al., 2010, Davidson et al., 2010). This geometric mechanism enforces holography without recourse to string dualities.

6. Quantum Evaporation, Islands, and the Page Curve

Holographic models of evaporating AdS black holes (e.g., in braneworld setups) utilize a classical bulk black hole horizon intersecting a brane. During evaporation, the horizon slides off the brane, shrinking the induced (apparent) horizon. The evolution can be tracked precisely in large-$D$ effective theory, including the rate of mass loss and the geometric profile (Emparan et al., 2023).

Information recovery and the Page curve are captured by the quantum extremal surface (QES) or “island” prescription for radiation entropy: $$S_{\rm rad}(t) = \operatorname{min}_{I} \operatorname{ext}_{\partial I} \left[ \frac{\operatorname{Area}(\partial I)}{4G_{\rm eff}} + S_{\rm bulk}(I \cup \mathcal{R}) \right]$$ Islands, when favored, ensure that the entropy of radiation decreases in tandem with the shrinking black hole area, yielding a unitary Page curve. The dynamical endpoint can be complete evaporation or an equilibrium funnel, depending on initial conditions and horizon instabilities.

7. Holography in Rotating, Charged, and Nontrivial Topologies

Generalizations to rotating, charged, and hyperbolic black holes introduce additional structures:

• Rotating Kerr–AdS and Kerr–Newman–AdS black holes: Extended thermodynamics, inclusion of angular momentum as a chemical potential, and dual CFT interpretations with varying central charge and phase transitions (e.g., first- and second-order transitions in the canonical and grand-canonical boundary ensembles). Complexities are sensitive to both thermodynamic volume and internal horizon data (Hennigar et al., 2017, Gong et al., 2023).
• Hyperbolic and toroidal black holes: Hyperbolic black holes exhibit first-order complexity phase transitions, log-divergent ground-state complexity at low temperatures, and frequent Lloyd bound violations (Barbon et al., 2015, Wang et al., 2023). Small toroidal AdS black holes provide clean probes of boundary IR physics, with unique emission and stability properties (McInnes, 2022).
• Hairy/dilaton black holes: The inclusion of scalars/hair can accelerate complexity growth and produces nontrivial thermodynamic and complexity phase structures (Wang et al., 2023).

Table: Key Physical Quantities and Their Holographic Interpretation

Quantity	Gravitational Side	Dual Field Theory Interpretation
Entropy $S$	Horizon area, $A/4G_N$, Komar mass scaling	Central charge, number of degrees of freedom
Temperature $T$	Surface gravity, periodicity of Euclidean time	Thermal state temperature
Pressure $P$	Negative cosmological constant, $-\Lambda/8\pi$	Central charge scaling, CFT volume
Volume $V$	Thermodynamic volume (conjugate to $P$)	Field theory volume or number of species
Complexity $\mathcal{C}$	WdW action or maximal volume functional in the bulk	Circuit complexity/preparation cost
Work $W$	Area enclosed by cycle in $P-V$	Energy extraction, process in theory space
Island region	Bulk subregion including parts of black hole interior	States purifying Hawking radiation

• Hennigar, Kubiznak, Mann. "Holographic heat engines: general considerations and rotating black holes" (Hennigar et al., 2017)
• Davidson, Gurwich. "Holographic Entropy Packing inside a Black Hole" (Davidson et al., 2010); "Hollowgraphy Driven Holography" (Davidson et al., 2010)
• Mathur. "Black holes and holography" (Mathur, 2012)
• Barbon, Martin-Garcia. "Holographic Complexity Of Cold Hyperbolic Black Holes" (Barbon et al., 2015)
• Wang, Ren. "Holographic complexity of hyperbolic black holes" (Wang et al., 2023)
• Fan, Guo. "Holographic complexity and thermodynamics of AdS black holes" (Fan et al., 2019)
• Emparan, Frassino, Yu. "Holographic Complexity of Rotating Quantum Black Holes" (Chen et al., 2023)
• Carmi, Chapman, Myers et al. "Holographic complexity of rotating black holes" (Balushi et al., 2020)
• Mounim, Mück. "Reparameterization Dependence and Holographic Complexity of Black Holes" (Mounim et al., 2021)
• Marolf, Maxfield, Peach. "Holographic black hole cosmologies" (Sahu et al., 2024)
• Emparan, Grumiller, Mandlik. "Holographic duals of evaporating black holes" (Emparan et al., 2023)
• McInnes. "The Special Role of Toroidal Black Holes in Holography" (McInnes, 2022)
• Wu, Liu. "Holographic Complexity in a Charged Supersymmetric Black Holes" (Jiang et al., 2020)
• "Holographic Q-Picture of Black Holes in Five Dimensional Minimal Supergravity" (Chen et al., 2010)
• "A Holographic, Hydrodynamic Model of a Schwarzschild Black Hole" (MacKay, 29 Apr 2025)

These works collectively establish the central pillars, methodologies, and phenomenology of holographic black holes. The theme unifies geometric entropy, thermodynamics, quantum information, and boundary field theory in a coupled framework where lower-dimensional data holographically encode all bulk properties, both classically and at the quantum level.