Euclidean Saddles and Gravitational Thermodynamics

Euclidean gravity turns stationary black-hole thermodynamics into a sequence of geometric tests: remove the horizon cone, complete and renormalize the action, differentiate the replica angle, compare saddles with identical boundary sources, and inspect the quadratic spectrum. Each test answers a different question. Smoothness fixes temperature; an on-shell action fixes a potential; a Hessian tests local stability; an action difference tests global dominance; and a contour determines whether a real or complex saddle contributes at all.

Helpful background. Quantum Fields in Curved Spacetime supplies thermal path integrals; Einstein Equations supplies the stationary saddles; Black-Hole Geometry and Mechanics supplies horizons and conserved charges; and Gravitational Actions and Boundary Terms supplies the Euclidean variational principle.

The Euclidean saddle logic

For fixed boundary inverse temperature and chemical potentials, the leading semiclassical partition function is schematically

$$Z(\beta,\Omega,\Phi) \simeq\sum_\sigma n_\sigma \exp\!\left[-I_{E,\mathrm{ren}}(g_\sigma)\right].$$

The notation hides four obligations. Each $g_\sigma$ must solve the equations with the same induced boundary geometry and ensemble data. Its horizon must be smooth, or any conical defect must be deliberately included as an off-shell or replica construction. The bulk action, Gibbons–Hawking–York term, matter terms, and subtraction or counterterms must be evaluated in one scheme. Finally, $n_\sigma$ is determined by the integration cycle, not by the saddle equation.

For a conventional real stationary saddle,

$$I_{E,\mathrm{ren}} =\beta(E-\Omega J-\Phi Q)-S.$$

This relation connects action and thermodynamics but does not collapse all stability questions into one. Gibbons and Hawking established the action/partition-function calculation in Gibbons and Hawking 1977, pp. 2752–2756.

Follow the six calculations in order

1. Euclidean Regularity and Thermal Periodicity expands the nonextremal horizon into a polar plane and obtains $\beta=2\pi/\kappa$. It keeps the horizon generator normalized to the same infinity or cavity clock used to define energy.
2. On-Shell Action and Boundary Terms assembles the bulk, Gibbons–Hawking–York, reference or counterterm, and ensemble Legendre contributions. A Schwarzschild calculation recovers energy, area entropy, and negative canonical heat capacity.
3. Entropy from Conical Defects varies the replica opening angle. Localized curvature produces $A/(4G)$, while matter and higher-curvature terms combine into renormalized generalized entropy.
4. Hawking–Page and Cavity Phase Transitions computes action crossings in two distinct canonical systems: Schwarzschild–AdS with a conformal boundary and asymptotically flat Schwarzschild with a finite reflecting wall.
5. Negative Modes and Thermodynamic Stability separates gauge and conformal directions from the physical transverse-traceless spectrum. It pairs Prestidge’s Schwarzschild–AdS threshold with a reproducible, explicitly proxy-level radial eigensolver.
6. Complex Saddles and Contour Questions explains why an on-shell complex solution contributes only when its Lefschetz thimble has nonzero coefficient in a physically defined cycle. Its comparison of results available at the stated date describes the still-unresolved full gravitational contour.

The order is cumulative. Regularity identifies which geometry represents the boundary temperature; the completed action makes saddle comparison finite; conical variation identifies entropy; phase competition motivates the quadratic and contour questions.

Keep five decisions separate

Question	Diagnostic	What it does not decide
Is the horizon smooth?	Polar regularity and thermal identification	Whether the saddle dominates
Is the variational problem defined?	Complete action with fixed sources	Whether the functional integral converges
Is the saddle locally stable?	Response Hessian and allowed negative modes	Whether another stable saddle has lower free energy
Which phase dominates?	Renormalized action difference at identical boundary data	Whether transition kinetics are fast
Does a complex saddle contribute?	Intersection number with the original contour	Whether its action is merely small

The distinctions prevent common errors. Local positive heat capacity does not imply global dominance: the large Schwarzschild–AdS branch is metastable between $T_{\min}$ and $T_{\mathrm{HP}}$. Global dominance does not prove Lorentzian stability. A negative mode is ensemble dependent, because changing which boundary variable is fixed changes the allowed fluctuation space. A complex metric that passes an allowability test can still have zero intersection number.

Prestidge’s computation gives a controlled example of the local distinction Prestidge 2000, §§2–5, PDF. Witten’s allowability proposal and discussion of the missing global cycle give the contour boundary Witten 2022, §§3–5, PDF.

Regimes and normalization checks

The chapter works at leading semiclassical saddle order unless a one-loop fluctuation is explicitly discussed. Curvature radii must exceed the effective cutoff, and the action includes renormalized local couplings. One-loop determinants are the first omitted correction in phase comparisons; near a zero mode or Stokes wall, an isolated Gaussian saddle expansion fails and must be replaced by a uniform treatment.

Three quick checks catch many errors. First, rescaling the horizon generator must rescale $\kappa$, boundary time, energy, and chemical potentials consistently. Second, every action comparison must match the induced boundary metric, including the proper Euclidean circle. Third, differentiating a replica action must use $(\alpha\partial_\alpha-1)I_E$ so the extensive smooth-copy contribution cancels. Solodukhin gives the conical and renormalization details Solodukhin 2011, §§2.6–2.9 and §§3.11–3.12, PDF.

Review the chapter

For four-dimensional Schwarzschild–AdS,

$$T(r_+)=\frac{1+3r_+^2/L^2}{4\pi r_+}, \qquad F(r_+)=\frac{r_+}{4G}\left(1-\frac{r_+^2}{L^2}\right).$$

Find the local-stability turning point and the global Hawking–Page crossing. Describe the temperature interval in which the large black hole is locally stable but globally subdominant.

Solution

Differentiating $T$ gives

$$\frac{dT}{dr_+} =\frac{1}{4\pi}\left(-\frac{1}{r_+^2}+\frac{3}{L^2}\right),$$

so the turning point is $r_+=L/\sqrt3$ and $T_{\min}=\sqrt3/(2\pi L)$. The large branch above that radius has positive heat capacity.

The nontrivial free-energy zero is $r_+=L$, where $T_{\mathrm{HP}}=1/(\pi L)$. Therefore, for

$$\frac{\sqrt3}{2\pi L}\lt T\lt\frac{1}{\pi L},$$

the large black hole is a local canonical minimum but thermal AdS has lower free energy. This is metastability: the Hessian and the global action difference answer distinct questions.

References

• Gibbons, G. W., and S. W. Hawking (1977), “Action Integrals and Partition Functions in Quantum Gravity,” Physical Review D 15, 2752–2756. DOI.
• Prestidge, T. (2000), “Dynamic and Thermodynamic Stability and Negative Modes in Schwarzschild–Anti-de Sitter,” Physical Review D 61, 084002. Open PDF.
• Solodukhin, S. N. (2011), “Entanglement Entropy of Black Holes,” Living Reviews in Relativity 14, 8. Open PDF.
• Witten, E. (2022), “A Note on Complex Spacetime Metrics,” arXiv:2111.06514v2. Open PDF.