Thermal Phases and AdS Black Holes

Thermal AdS and AdS black holes are competing gravitational realizations of boundary thermal states. Comparing them requires common boundary data and an ensemble; free-energy dominance, local stability, and lifetime are different conclusions.

Enter this chapter

Begin with Euclidean Saddles, Thermal States, and Hawking–Page Transitions to compare Euclidean fillings with the same thermal boundary.

Helpful background. Thermodynamic Limits, Phases, and Ensemble Equivalence supplies the statistical meaning of a phase. Black-Hole Thermodynamics at the QFT Interface supplies the horizon laws. States, Geometries, and Radial Quantization and Renormalized One-Point Functions and the Variational Problem supply the holographic state and charge maps.

Topics

1. Euclidean Saddles, Thermal States, and Hawking–Page Transitions — Matches the thermal boundary circle and derives the spherical transition temperature.
2. Euclidean Actions, Boundary Terms, and Free-Energy Comparisons — Gives the variationally complete action protocol.
3. AdS Black Branes and Holographic Thermodynamics — Derives temperature, entropy density, energy, pressure, and the conformal equation of state.
4. Rotating and Charged AdS Black Holes — Adds potentials, charges, angular momenta, extremality, and ensemble-dependent stability.
5. Two-Sided Black Holes and Thermofield-Double States — Distinguishes exact purification from its leading smooth-bridge saddle.
6. Imported Wald Entropy in Holographic Higher-Derivative Thermodynamics — Evaluates the Noether-charge correction and checks it against the first law.
7. Confinement and Deconfinement Dictionaries — Combines free-energy scaling, center symmetry, loops, worldsheets, and topology.
8. Black-Hole Instabilities and New Phases — Separates thermodynamic, Euclidean, Lorentzian, and global stability.
9. Ensemble Inequivalence and Microcanonical Saddles — Explains negative heat capacity and constrained gravitational saddles.
10. Finite-N Smoothing, Tunneling, and Metastability — Replaces the sharp saddle crossing by the appropriate crossover and decay scales.

How the ideas fit together

Fixing the boundary thermal circle defines the canonical comparison. Renormalized Euclidean actions give the free-energy difference between saddles; for a spherical boundary this yields the familiar thermal-AdS/black-hole transition and its confinement interpretation in suitable dual theories Witten 1998, §§ 2.3–2.4.

A locally stable black hole need not dominate that ensemble. Charge, rotation, and microcanonical constraints also change the comparison. Finite-N corrections round the classical saddle picture, while tunneling and instability calculations determine lifetimes through information absent from a free-energy crossing alone.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim-domain comparison

Representative claim domains and validity boundaries for Thermal Phases and AdS Black Holes

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
Hawking-Page transition	Declare boundary topology and canonical ensemble; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: ensemble and boundary data → Euclidean gravitational saddles → action, entropy, and charges → stability and finite-N checks → thermal phase claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “renormalized free-energy crossing” check is counterevidence to the promoted claim.	In the same boundary geometry and ensemble, the claimed free-energy crossing is absent or another admitted saddle has lower potential in the proposed dominance region.	Inferring a sharp finite-N transition from leading saddle dominance.	dominant semiclassical saddle
charged or rotating saddle	Declare chemical potentials and regularity; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: ensemble and boundary data → Euclidean gravitational saddles → action, entropy, and charges → stability and finite-N checks → thermal phase claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “first law and Hessian stability” check is counterevidence to the promoted claim.	With charge conventions fixed, the proposed thermodynamics violates the first law; a negative allowed Hessian direction separately defeats a local-stability claim.	Inferring global stability in every ensemble from thermodynamics in one ensemble.	thermodynamics in a stated ensemble
higher-derivative entropy	Declare covariant action and boundary terms; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: ensemble and boundary data → Euclidean gravitational saddles → action, entropy, and charges → stability and finite-N checks → thermal phase claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Wald and Euclidean checks” check is counterevidence to the promoted claim.	For the same covariant action and stationary horizon, the stated Wald entropy disagrees with the Euclidean thermodynamic derivative beyond the retained-order error.	Identifying a corrected saddle entropy with microscopic state counting.	corrected saddle entropy

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Where to go next

Continue to Lorentzian, Nonequilibrium, and Chaotic Holography to derive retarded response from the bulk boundary-value problem.

References

• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2, 505–532 (1998). DOI; arXiv:hep-th/9803131.