Thermal Phases and AdS Black Holes

AdS black holes organize thermal physics through a hierarchy of statements: boundary data select an ensemble, Euclidean fillings supply candidate saddles, renormalized actions compare their free energies, fluctuation operators test local stability, and finite-N effects limit the classical phase picture. This chapter keeps that hierarchy explicit while deriving the standard transition first exhibited by Hawking and Page 1983 and the black-brane results.

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.

Enter this chapter

Start every thermal claim by recording:

• the boundary spatial manifold and Euclidean thermal circle;
• fixed temperature, angular velocities, chemical potentials, or conserved charges;
• the allowed bulk topologies and matter boundary conditions;
• the complete renormalized action and entropy functional;
• the large-N, derivative, loop, and spatial-volume limits;
• the distinction between equilibrium dominance, local stability, decay, and exact finite-N behavior.

Two routes then become available. The ensemble-first route fixes sources and charges before comparing actions, Hessians, and Legendre transforms. The geometry-first route follows neutral, charged, rotating, two-sided, and higher-derivative solutions before asking which phase statement each supports.

Chapter map

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.

One calculation, five different conclusions

For a saddle $g_*$ in a fixed ensemble,

$$Z\simeq\sum_* e^{-I_E[g_*]}, \qquad F_*=\frac{I_E[g_*]}{\beta}.$$

This compact formula supports several logically distinct results:

Calculation	Question answered	Extra requirement	Does not establish
Compare $I_E$	Which included saddle dominates?	matched boundary data and contour membership	completeness of saddles
Differentiate $I_E$	What are $E$, $S$, and charges?	fixed-source convention and boundary terms	microscopic state count
Test the Hessian	Is the saddle locally thermodynamically stable?	correct ensemble variables	Lorentzian stability in every channel
Solve perturbations	Does a mode grow or bifurcate?	physical horizon and boundary conditions	nonlinear endpoint or dominance
Sum competing sectors	How is a large-N crossing rounded?	finite-N weights and volume limit	exact spectrum or recurrence pattern

Keeping the columns separate prevents a classical horizon from being used as evidence for claims it was never designed to answer.

Canonical examples and checks

The spherical Einstein solution gives

$$T(r_h)=\frac{d r_h^2+(d-2)L^2}{4\pi L^2r_h},$$

with a Hawking–Page crossing at $r_h=L$. The planar solution gives $T=dr_h/(4\pi L^2)$ and $\varepsilon=(d-1)p$. These are different boundary topologies: the spherical problem has an intrinsic scale $L$ and competing thermal-AdS saddle, whereas the planar conformal problem has neither without an additional deformation or compactification.

Every extension should reproduce an independent identity. Charged and rotating saddles satisfy the first law with the correct potentials. Higher-derivative saddles match Wald entropy to Euclidean derivatives. A proposed new phase has a normalizable zero mode at onset and lower potential only after nonlinear continuation. A finite-N treatment retains both competing contributions instead of taking their minimum prematurely.

Scientific boundaries

Thermal and Nonequilibrium QFT owns general ensembles, phases, KMS structure, response, and nucleation. QFT in Curved Spacetime owns semiclassical horizon thermodynamics and entropy laws. This chapter owns their realization by asymptotically AdS saddles and the boundary phase dictionary.

The separation matters most at the frontier. Classical saddle dominance does not determine an exact discrete spectrum. Wald entropy is not a microscopic degeneracy. A Euclidean bridge is not a proof of exact interior factorization. A Page-like thermodynamic curve is not an evaporation S-matrix. Those questions require the later wormhole, microstate, and black-hole-information chapters.

Review the chapter

1. Why must proper thermal-circle lengths be matched before subtracting two Euclidean actions?
2. Which boundary term distinguishes fixed charge from fixed potential?
3. Why is there no spherical Hawking–Page temperature for an undeformed planar CFT?
4. What additional work turns a charged-scalar zero mode into a new thermodynamic phase?
5. How can a negative-heat-capacity black hole appear microcanonically but not canonically?
6. Why is the finite-N crossover width near a two-saddle crossing not the same as a tunneling rate?

A complete response invokes identical induced data for question 1, the Maxwell Legendre term for question 2, absence of an intrinsic planar scale and competing saddle for question 3, nonlinear backreaction and a same-ensemble free-energy comparison for question 4, constrained versus fluctuating energy for question 5, and equilibrium mixing versus a bounce barrier for question 6.

Where to go next

Thermal and Real-Time Holography turns these equilibrium backgrounds into causal correlators, quasinormal modes, chaos diagnostics, and late-time questions. Holographic Matter and Transport adds density, symmetry breaking, and response. Wormholes, Gravitational Path Integrals, and Ensembles asks which saddles and contours define the gravitational sum itself.

Chapter-scale structure and validity checks

The chapter-scale structure map locates this page’s result inside the full reasoning chain. Follow the solid arrows through the declared inputs and checks; the dashed final arrow marks the point where an additional inference would be required.

A holographic phase is an ensemble-specific saddle comparison; classical dominance does not determine exact finite-N spectra or recurrences. The diagram is an original schematic, is not to scale, and uses the dashed final arrow to mark the claim boundary.

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The companion validity map turns three common overclaims into explicit failure tests. Read each row from its declared object to the diagnostic, then compare the licensed conclusion with the dashed “not” endpoint.

A holographic phase is an ensemble-specific saddle comparison; classical dominance does not determine exact finite-N spectra or recurrences. Each row pairs a diagnostic with the strongest supported conclusion and an explicitly unsupported promotion. The diagram is an original schematic and is not to scale.

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Claim-domain comparison

The table below gives a screen-reader-friendly comparison of three representative claims. It keeps the required declaration, approximation status, evidence timing, counterevidence, falsifier, failure condition, and licensed conclusion in one reading order.

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.	renormalized free-energy crossing	a sharp finite-N transition	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.	first law and Hessian stability	global stability in every 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.	Wald and Euclidean checks	microscopic state counting	corrected saddle entropy

Download the structured table data (JSON).

References

• Gibbons, G. W., and Stephen W. Hawking. “Action Integrals and Partition Functions in Quantum Gravity.” Physical Review D 15, 2752–2756 (1977). DOI.
• Hawking, Stephen W., and Don N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87, 577–588 (1983). DOI.
• Wald, Robert M. “Black Hole Entropy Is the Noether Charge.” Physical Review D 48, R3427–R3431 (1993). DOI; arXiv:gr-qc/9307038.
• 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.